/*!
* This file is part of GPBoost a C++ library for combining
*	boosting with Gaussian process and mixed effects models
*
* Copyright (c) 2020 - 2026 Fabio Sigrist, Tim Gyger, and Pascal Kuendig. All rights reserved.
*
* Licensed under the Apache License Version 2.0. See LICENSE file in the project root for license information.
*/
#ifndef GPB_RE_MODEL_TEMPLATE_H_
#define GPB_RE_MODEL_TEMPLATE_H_

#define _USE_MATH_DEFINES // for M_PI
#include <cmath>
#include <GPBoost/type_defs.h>
#include <GPBoost/re_comp.h>
#include <GPBoost/sparse_matrix_utils.h>
#include <GPBoost/Vecchia_utils.h>
#include <GPBoost/GP_utils.h>
#include <GPBoost/likelihoods.h>
#include <GPBoost/utils.h>
#include <GPBoost/optim_utils.h>
#include <LBFGSpp/BFGSMat.h>
//#include <Eigen/src/misc/lapack.h>

#include <memory>
#include <mutex>
#include <vector>
#include <algorithm>    // std::shuffle
#include <chrono>  // only for debugging
#include <thread> // only for debugging

#ifndef M_PI
#define M_PI      3.1415926535897932384626433832795029
#endif

#include <LightGBM/utils/log.h>
using LightGBM::Log;
using LightGBM::LogLevelRE;
#include <LightGBM/utils/common.h>
#include <LightGBM/meta.h>
using LightGBM::label_t;

#ifdef USE_CUDA_GP
#include <cuda_runtime.h>
#endif

namespace GPBoost {

	/*!
	* \brief Template class used in the wrapper class REModel
	* The template parameters <T_mat, T_chol> can be <den_mat_t, chol_den_mat_t>, <sp_mat_t, chol_sp_mat_t>, <sp_mat_rm_t, chol_sp_mat_rm_t>
	*	depending on whether dense or sparse linear matrix algebra is used
	*/
	template<typename T_mat, typename T_chol>
	class REModelTemplate {
	public:
		/*! \brief Null costructor */
		REModelTemplate();

		/*!
		* \brief Costructor
		* \param num_data Number of data points
		* \param cluster_ids_data IDs / labels indicating independent realizations of random effects / Gaussian processes (same values = same process realization)
		* \param re_group_data Labels of group levels for the grouped random effects in column-major format (i.e. first the levels for the first effect, then for the second, etc.). Every group label needs to end with the null character '\0'
		* \param num_re_group Number of grouped (intercept) random effects
		* \param re_group_rand_coef_data Covariate data for grouped random coefficients
		* \param ind_effect_group_rand_coef Indices that relate every random coefficients to a "base" intercept grouped random effect. Counting start at 1.
		* \param num_re_group_rand_coef Number of grouped random coefficient
		* \param drop_intercept_group_rand_effect Indicates whether intercept random effects are dropped (only for random coefficients). If drop_intercept_group_rand_effect[k] > 0, the intercept random effect number k is dropped. Only random effects with random slopes can be dropped.
		* \param num_gp Number of (intercept) Gaussian processes
		* \param gp_coords_data Coordinates (features) for Gaussian process
		* \param dim_gp_coords Dimension of the coordinates (=number of features) for Gaussian process
		* \param gp_rand_coef_data Covariate data for Gaussian process random coefficients
		* \param num_gp_rand_coef Number of Gaussian process random coefficients
		* \param cov_fct Type of covariance function for Gaussian process (GP)
		* \param cov_fct_shape Shape parameter of covariance function (=smoothness parameter for Matern and Wendland covariance. This parameter is irrelevant for some covariance functions such as the exponential or Gaussian
		* \param gp_approx Type of GP-approximation for handling large data
		* \param cov_fct_taper_range Range parameter of the Wendland covariance function and Wendland correlation taper function. We follow the notation of Bevilacqua et al. (2019, AOS)
		* \param cov_fct_taper_shape Shape parameter of the Wendland covariance function and Wendland correlation taper function. We follow the notation of Bevilacqua et al. (2019, AOS)
		* \param num_neighbors The number of neighbors used in the Vecchia approximation
		* \param vecchia_ordering Ordering used in the Vecchia approximation. "none" = no ordering, "random" = random ordering
		* \param num_ind_points Number of inducing points / knots for, e.g., a predictive process approximation
		* \param cover_tree_radius Radius (= "spatial resolution") for the cover tree algorithm
		* \param ind_points_selection Method for choosing inducing points
		* \param likelihood Likelihood function for the observed response variable
		* \param likelihood_additional_param Additional parameter for the likelihood which cannot be estimated (e.g., degrees of freedom for likelihood = "t")
		* \param matrix_inversion_method Method which is used for matrix inversion
		* \param seed Seed used for model creation (e.g., random ordering in Vecchia approximation)
		* \param num_parallel_threads Number of parallel threads for OMP
		* \param GPU_use If TRUE, GPU acceleration will be used if supported.
		* \param has_weights True, if sample weights should be used
		* \param weights Sample weights
		* \param likelihood_learning_rate Likelihood learning rate for generalized Bayesian inference (only non-Gaussian likelihoods)
		*/
		REModelTemplate(data_size_t num_data,
			const data_size_t* cluster_ids_data,
			const char* re_group_data,
			data_size_t num_re_group,
			const double* re_group_rand_coef_data,
			const data_size_t* ind_effect_group_rand_coef,
			data_size_t num_re_group_rand_coef,
			const int* drop_intercept_group_rand_effect,
			data_size_t num_gp,
			const double* gp_coords_data,
			int dim_gp_coords,
			const double* gp_rand_coef_data,
			data_size_t num_gp_rand_coef,
			const char* cov_fct,
			double cov_fct_shape,
			const char* gp_approx,
			double cov_fct_taper_range,
			double cov_fct_taper_shape,
			int num_neighbors,
			const char* vecchia_ordering,
			int num_ind_points,
			double cover_tree_radius,
			const char* ind_points_selection,
			const char* likelihood,
			double likelihood_additional_param,
			const char* matrix_inversion_method,
			int seed,
			int num_parallel_threads,
			bool GPU_use,
			bool has_weights,
			const double* weights,
			double likelihood_learning_rate) {
			// Check if cuda device is available
			bool can_use_cuda = false;
			if (GPU_use) {
#ifdef USE_CUDA_GP
				int device_count = 0;
				cudaError_t err = cudaGetDeviceCount(&device_count);

				if (err == cudaSuccess && device_count > 0) {
					can_use_cuda = true;
					Log::REInfo("CUDA device detected (%i device(s)). GPU acceleration will be used where supported.",device_count);
				}
				else {
					Log::REWarning("CUDA requested but no compatible device found. Falling back to CPU.");
				}
#else
				Log::REWarning("CUDA requested but this build was compiled without CUDA support. Falling back to CPU.");
#endif
			}
			GPU_use_ = can_use_cuda;
			if (num_parallel_threads > 0) {
				Eigen::setNbThreads(num_parallel_threads);
				omp_set_num_threads(num_parallel_threads);
			}
			CHECK(num_data > 0);
			num_data_ = num_data;
			//Initialize RNG
			CHECK(seed >= 0);
			rng_ = RNG_t(seed);
			seed_rng_ = seed;
			//Set up likelihood
			string_t likelihood_strg;
			if (likelihood == nullptr) {
				likelihood_strg = "gaussian";
			}
			else {
				likelihood_strg = Likelihood<T_mat, T_chol>::ParseLikelihoodAlias(std::string(likelihood));
			}
			gauss_likelihood_ = likelihood_strg == "gaussian";
			if (has_weights) {
				if (gauss_likelihood_) {
					Log::REFatal("'weights' are currently not supported for likelihood = 'gaussian' ");
				}
			}
			likelihood_additional_param_ = likelihood_additional_param;
			//Set up matrix inversion method
			if (matrix_inversion_method != nullptr) {
				matrix_inversion_method_user_provided_ = std::string(matrix_inversion_method);
			}
			std::string cov_fct_strg = (cov_fct != nullptr) ? std::string(cov_fct) : "";
			//Set up GP approximation
			if (gp_approx == nullptr) {
				gp_approx_ = "none";
			}
			else {
				gp_approx_ = std::string(gp_approx);
				if (gp_approx_ == "full_scale_tapering_pred_var_stochastic_stable" ||
					gp_approx_ == "full_scale_tapering_pred_var_exact_stable" ||
					gp_approx_ == "full_scale_tapering_pred_var_exact") {
					if (gp_approx_ == "full_scale_tapering_pred_var_stochastic_stable") {
						calc_pred_cov_var_FSA_cholesky_ = "stochastic_stable";
					}
					else if (gp_approx_ == "full_scale_tapering_pred_var_exact_stable") {
						calc_pred_cov_var_FSA_cholesky_ = "exact_stable";
					}
					else if (gp_approx_ == "full_scale_tapering_pred_var_exact") {
						calc_pred_cov_var_FSA_cholesky_ = "exact";
					}
					gp_approx_ = "full_scale_tapering";
				}
				if (gp_approx_ == "full_scale_tapering" && !gauss_likelihood_) {
					Log::REFatal("Approximation '%s' is currently not supported for non-Gaussian likelihoods ", gp_approx_.c_str());
				}
				if (gp_approx_ == "full_scale_vecchia_correlation_based" ||
					gp_approx_ == "vif_correlation_based" || gp_approx_ == "VIF_correlation_based" || 
					gp_approx_ == "full_scale_vecchia_correlation" || gp_approx_ == "vif_correlation" || gp_approx_ == "VIF_correlation") {
					gp_approx_ = "full_scale_vecchia";
					vecchia_neighbor_selection_ = "residual_correlation";
				}
				if (gp_approx_ == "vif" || gp_approx_ == "VIF") {
					gp_approx_ = "full_scale_vecchia";
				}
				vecchia_latent_approx_gaussian_ = false;
				if (gp_approx_ == "vecchia_latent") {
					gp_approx_ = "vecchia";
					vecchia_latent_approx_gaussian_ = true;
					gauss_likelihood_ = false;
				}
				if (gp_approx_ == "vecchia_correlation_based" || gp_approx_ == "vecchia_correlation") {
					gp_approx_ = "vecchia";
					vecchia_neighbor_selection_ = "correlation";
				}
				if (cov_fct_strg == "space_time_gneiting") {
					if (gp_approx_ == "vecchia") {
						vecchia_neighbor_selection_ = "correlation";//default is correlation-based neighbor search
					}
					else if (gp_approx_ == "vecchia_not_correlation" || gp_approx_ == "vecchia_not_correlation_based" ||
						gp_approx_ == "vecchia_euclidean" || gp_approx_ == "vecchia_euclidean_based") {
						gp_approx_ = "vecchia";
						vecchia_neighbor_selection_ = "nearest";
					}
				}
			}
			if (SUPPORTED_GP_APPROX_.find(gp_approx_) == SUPPORTED_GP_APPROX_.end()) {
				Log::REFatal("GP approximation '%s' is currently not supported ", gp_approx_.c_str());
			}
			//Set up cluster IDs
			SetUpClusterIds(num_data_, cluster_ids_data, num_data_per_cluster_, data_indices_per_cluster_, unique_clusters_, num_clusters_);
			num_comps_total_ = 0;
			//Do some checks for grouped RE components and set meta data (number of components etc.)
			std::vector<std::vector<re_group_t>> re_group_levels;//Matrix with group levels for the grouped random effects (re_group_levels[j] contains the levels for RE number j)
			if (num_re_group > 0) {
				num_re_group_ = num_re_group;
				num_group_variables_ = num_re_group_;
				drop_intercept_group_rand_effect_ = std::vector<bool>(num_re_group_);
				for (int j = 0; j < num_re_group_; ++j) {
					drop_intercept_group_rand_effect_[j] = false;
				}
				CHECK(re_group_data != nullptr);
				if (num_re_group_rand_coef > 0) {
					num_re_group_rand_coef_ = num_re_group_rand_coef;
					CHECK(re_group_rand_coef_data != nullptr);
					CHECK(ind_effect_group_rand_coef != nullptr);
					for (int j = 0; j < num_re_group_rand_coef_; ++j) {
						CHECK(0 < ind_effect_group_rand_coef[j] && ind_effect_group_rand_coef[j] <= num_re_group_);
					}
					ind_effect_group_rand_coef_ = std::vector<int>(ind_effect_group_rand_coef, ind_effect_group_rand_coef + num_re_group_rand_coef_);
					if (drop_intercept_group_rand_effect != nullptr) {
						drop_intercept_group_rand_effect_ = std::vector<bool>(num_re_group_);
						for (int j = 0; j < num_re_group_; ++j) {
							drop_intercept_group_rand_effect_[j] = drop_intercept_group_rand_effect[j] > 0;
						}
						for (int j = 0; j < num_re_group_; ++j) { // check that all dropped intercept random effects have at least on random slope
							if (drop_intercept_group_rand_effect_[j] &&
								std::find(ind_effect_group_rand_coef_.begin(), ind_effect_group_rand_coef_.end(), j) != ind_effect_group_rand_coef_.end()) {
								Log::REFatal("Cannot drop intercept random effect number %d as this random effect has no corresponding random coefficients", j);
							}
						}
					}
				}
				num_re_group_total_ = num_re_group_ + num_re_group_rand_coef_;
				num_comps_total_ += num_re_group_total_;
				// Convert characters in 'const char* re_group_data' to matrix (num_group_variables_ x num_data_) with strings of group labels
				re_group_levels = std::vector<std::vector<re_group_t>>(num_group_variables_, std::vector<re_group_t>(num_data_));
				if (num_group_variables_ > 0) {
					ConvertCharToStringGroupLevels(num_data_, num_group_variables_, re_group_data, re_group_levels);
				}
			}//end num_re_group > 0
			//Do some checks for GP components and set meta data (number of components etc.)
			if (num_gp > 0) {
				if (num_gp > 1) {
					Log::REFatal("num_gp can only be either 0 or 1 in the current implementation");
				}
				num_gp_ = num_gp;
				ind_intercept_gp_ = num_comps_total_;
				CHECK(dim_gp_coords > 0);
				CHECK(gp_coords_data != nullptr);
				CHECK(cov_fct != nullptr);
				dim_gp_coords_ = dim_gp_coords;
				if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
					if (num_neighbors > 0) {
						num_neighbors_ = num_neighbors;
					}
					else {
						if (gp_approx_ == "vecchia") {
							num_neighbors_ = 20;
						}
						else {
							num_neighbors_ = 30;// gp_approx_ == "full_scale_vecchia"
						}
					}
					num_neighbors_pred_ = 2 * num_neighbors_;
					if (vecchia_ordering == nullptr) {
						vecchia_ordering_ = "none";
					}
					else {
						vecchia_ordering_ = std::string(vecchia_ordering);
						if (SUPPORTED_VECCHIA_ORDERING_.find(vecchia_ordering_) == SUPPORTED_VECCHIA_ORDERING_.end()) {
							Log::REFatal("Ordering of type '%s' is not supported for the Veccia approximation ", vecchia_ordering_.c_str());
						}
					}
					if (ind_points_selection == nullptr) {
						ind_points_selection_ = "kmeans++";
					}
					else {
						ind_points_selection_ = std::string(ind_points_selection);
						if (SUPPORTED_METHOD_INDUCING_POINTS_.find(ind_points_selection_) == SUPPORTED_METHOD_INDUCING_POINTS_.end()) {
							Log::REFatal("Method '%s' is not supported for choosing the inducing points ", ind_points_selection_.c_str());
						}
					}
				}//end if gp_approx_ == "vecchia"
				if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
					if (num_ind_points > 0) {
						num_ind_points_ = num_ind_points;
					}
					else {
						if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
							num_ind_points_ = 500;
						}
						else {
							num_ind_points_ = 200;// gp_approx_ == "full_scale_vecchia"
						}
					}
					CHECK(cover_tree_radius > 0);
					cover_tree_radius_ = cover_tree_radius;
					ind_points_selection_ = std::string(ind_points_selection);
					if (SUPPORTED_METHOD_INDUCING_POINTS_.find(ind_points_selection_) == SUPPORTED_METHOD_INDUCING_POINTS_.end()) {
						Log::REFatal("Method '%s' is not supported for choosing the inducing points ", ind_points_selection_.c_str());
					}
				}
				if (num_gp_rand_coef > 0) {//Random slopes
					CHECK(gp_rand_coef_data != nullptr);
					num_gp_rand_coef_ = num_gp_rand_coef;
				}
				num_gp_total_ = num_gp_ + num_gp_rand_coef_;
				num_comps_total_ += num_gp_total_;
			}//end num_gp > 0
			DetermineSpecialCasesModelsEstimationPrediction(cov_fct_strg);
			for (const auto& cluster_i : unique_clusters_) {
				if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
					std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_ip_cluster_i;
					std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cross_cov_cluster_i;
					std::vector<std::shared_ptr<RECompGP<T_mat>>> re_comps_resid_cluster_i;
					if (gp_approx_ == "full_scale_vecchia") {
						if (vecchia_ordering_ == "random" || vecchia_ordering_ == "time_random_space") {
							std::shuffle(data_indices_per_cluster_[cluster_i].begin(), data_indices_per_cluster_[cluster_i].end(), rng_);
						}
					}
					CreateREComponentsFITC_FSA(num_data_, data_indices_per_cluster_, cluster_i, gp_coords_data,
						cov_fct_strg, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape,
						re_comps_ip_cluster_i, re_comps_cross_cov_cluster_i, re_comps_resid_cluster_i, false);
					re_comps_ip_[cluster_i][0] = re_comps_ip_cluster_i;
					re_comps_cross_cov_[cluster_i][0] = re_comps_cross_cov_cluster_i;
					re_comps_resid_[cluster_i][0] = re_comps_resid_cluster_i;
				}
				if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
					std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cluster_i;
					std::vector<std::vector<int>> nearest_neighbors_cluster_i;
					std::vector<den_mat_t> dist_obs_neighbors_cluster_i;
					std::vector<den_mat_t> dist_between_neighbors_cluster_i;
					std::vector<Triplet_t> entries_init_B_cluster_i;
					std::vector<std::vector<den_mat_t>> z_outer_z_obs_neighbors_cluster_i;
					if (gp_approx_ == "vecchia" && num_data_per_cluster_[cluster_i] > 300000 && dim_gp_coords_ > 30 && vecchia_neighbor_selection_ != "correlation" && !GPU_use_) {
						Log::REInfo("Because many high-dimensional GP coordinates are used, we recommend setting gp_approx = 'vecchia_correlation_based'."
							"This option exploits a cover-tree algorithm to accelerate the Vecchia neighbor search.");
					}
					CreateREComponentsVecchia(num_data_, dim_gp_coords_, data_indices_per_cluster_, cluster_i,
						num_data_per_cluster_, gp_coords_data, gp_rand_coef_data,
						re_comps_cluster_i, nearest_neighbors_cluster_i, dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i,
						entries_init_B_cluster_i, z_outer_z_obs_neighbors_cluster_i, Vecchia_calculations_on_RE_scale_, has_duplicates_coords_,
						vecchia_ordering_, num_neighbors_, vecchia_neighbor_selection_, true, rng_, num_gp_rand_coef_, num_gp_total_, num_comps_total_, gauss_likelihood_,
						cov_fct_strg, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape, gp_approx_ == "tapering", save_distances_isotropic_cov_fct_Vecchia_,
						gp_approx_, nearest_neighbors_determined_, GPU_use_, grouped_RE_and_vecchia_GP_);
					only_one_GP_calculations_on_RE_scale_ = Vecchia_calculations_on_RE_scale_;
					nearest_neighbors_[cluster_i][0] = nearest_neighbors_cluster_i;
					dist_obs_neighbors_[cluster_i][0] = dist_obs_neighbors_cluster_i;
					dist_between_neighbors_[cluster_i][0] = dist_between_neighbors_cluster_i;
					entries_init_B_[cluster_i][0] = entries_init_B_cluster_i;
					z_outer_z_obs_neighbors_[cluster_i][0] = z_outer_z_obs_neighbors_cluster_i;
					re_comps_vecchia_[cluster_i][0] = re_comps_cluster_i;
				}//end gp_approx_ == "vecchia"
				if (gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" ||
					grouped_RE_and_vecchia_GP_) {
					data_size_t num_gp_temp = num_gp_;
					if (grouped_RE_and_vecchia_GP_) num_gp_ = 0;//hack that GP components are not created in 'CreateREComponents' below if grouped_RE_and_vecchia_GP_
					std::vector<std::shared_ptr<RECompBase<T_mat>>> re_comps_cluster_i;
					bool calculateZZt = !use_woodbury_identity_ && !grouped_RE_and_vecchia_GP_;
					CreateREComponents(num_data_, data_indices_per_cluster_, cluster_i,
						re_group_levels, num_data_per_cluster_, re_group_rand_coef_data,
						gp_coords_data, gp_rand_coef_data, calculateZZt,
						cov_fct_strg, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape, re_comps_cluster_i);
					re_comps_[cluster_i][0] = re_comps_cluster_i;
					if (grouped_RE_and_vecchia_GP_) num_gp_ = num_gp_temp;
				}
			}//end loop over clusters
			if (has_weights) {
				has_weights_ = true;
				if (GPBoost::HasNegativeValues<double>(weights, num_data_)) {
					Log::REFatal(" Found negative values in 'weights' ");
				}
				double sum_weights = 0.;
#pragma omp parallel for schedule(static) reduction(+:sum_weights)
				for (data_size_t i = 0; i < num_data_; ++i) {
					sum_weights += weights[i];
				}
				if (GPBoost::IsZero(sum_weights)) {
					Log::REFatal("The total sum of the 'weights' is zero ");
				}
				if (likelihood_strg != "binomial_logit" && likelihood_strg != "binomial_probit" && likelihood_strg != "beta_binomial") {
					if (std::abs(sum_weights - num_data_) > 0.001 * num_data_) {
						Log::REInfo("The total sum of the weights (%g) does not equal the number of data points (%d). This is not necessarily an issue ",
							sum_weights, num_data_);
					}
				}
				for (const auto& cluster_i : unique_clusters_) {
					weights_[cluster_i] = vec_t(num_data_per_cluster_[cluster_i]);
					for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
						weights_[cluster_i][j] = weights[data_indices_per_cluster_[cluster_i][j]];
					}
				}
				// old version where weights are normalized
//				double corr_fact = num_data_ / sum_weights;
//				for (const auto& cluster_i : unique_clusters_) {
//					weights_[cluster_i] = vec_t(num_data_per_cluster_[cluster_i]);
//					for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
//						weights_[cluster_i][j] = weights[data_indices_per_cluster_[cluster_i][j]] * corr_fact;
//					}
//				}
			}
			else {
				for (const auto& cluster_i : unique_clusters_) {
					weights_[cluster_i] = vec_t();
				}
			}//end has_weights
			CHECK(likelihood_learning_rate > 0.);
			likelihood_learning_rate_ = likelihood_learning_rate;
			//Create matrices Z and ZtZ if Woodbury identity is used (used only if there are only grouped REs and no GPs)
			if (use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_ || grouped_RE_and_vecchia_GP_) {
				InitializeMatricesForUseWoodburyIdentity();
			}
			InitializeIdentityMatricesForGaussianData();// applies only if gauss_likelihood_ && gp_approx_ != "vecchia" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" && gp_approx_ != "full_scale_vecchia"
			if (num_re_group_total_ == 1 && num_comps_total_ == 1) {
				if (num_clusters_ == 1 && re_comps_[unique_clusters_[0]][0][0]->GetNumUniqueREs() == 1) {
					iid_model_ = true;
				}
			}
			InitializeLikelihoods(likelihood_strg);
			SetDefaultMatrixInversionMethod();//needs to be run after 'InitializeLikelihoods' since defaults can depend on likelihood
			DetermineCovarianceParameterIndicesNumCovPars();
			InitializeDefaultSettings();
			CheckCompatibilitySpecialOptions();
			SetPropertiesLikelihood();
			string_t cov_fct_name = CovFunctionName();
			if (cov_fct_name == "hurst" || cov_fct_name == "hurst_ard") {//Info message for Hurst covariance
				string_t range_msg = cov_fct_name == "hurst" ? "" : " and the range parameter for the first coordinate is 1 (due to identifiability with the marginal variance)";
				string_t stdz_msg = cov_fct_name == "hurst" ? "" : "and standardize ";
				string_t msg = "For the '" + cov_fct_name + "' covariance, the GP is anchored at 0 (i.e., b(0) = 0 a.s.)" + range_msg + 
					". In general, it is thus recommened to mean-center " + stdz_msg + " the coordinates first (in case you have not done this already) ";
				Log::REInfo(msg.c_str());
			}
		}//end REModelTemplate

		/*! \brief Destructor */
		~REModelTemplate() {
		}

		/*! \brief Disable copy */
		REModelTemplate& operator=(const REModelTemplate&) = delete;

		/*! \brief Disable copy */
		REModelTemplate(const REModelTemplate&) = delete;

		/*!
		* \brief Returns true if the likelihood is Gaussian
		*/
		bool IsGaussLikelihood() const {
			return(gauss_likelihood_);
		}

		/*!
		* \brief Returns the type of likelihood
		*/
		string_t GetLikelihood() {
			return(likelihood_[unique_clusters_[0]]->GetLikelihood());
		}

		bool LikelihoodSupported(const string_t& likelihood) {
			return(likelihood_[unique_clusters_[0]]->LikelihoodSupported(likelihood));
		}

		/*!
		* \brief Transform from the latent to the response variable scale (often this is the inverse link function)
		*			This is only used by the 'ConvertOutput()' function in regression_objective.hpp
		*/
		double TransformToReponseScale(const double value) {
			return(likelihood_[unique_clusters_[0]]->TransformToReponseScale(value));
		}

		/*!
		* \brief Returns the type of covariance function
		*/
		string_t CovFunctionName() {
			std::string cov_fct = "";
			if (num_gp_total_ > 0) {
				if (gp_approx_ == "none" || gp_approx_ == "tapering") {
					std::shared_ptr<RECompGP<T_mat>> re_comp_gp_clus0 = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[unique_clusters_[0]][0][ind_intercept_gp_]);
					cov_fct = re_comp_gp_clus0->CovFunctionName();
				}
				else if (gp_approx_ == "vecchia") {
					std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp_clus0 = re_comps_vecchia_[unique_clusters_[0]][0][0];
					cov_fct = re_comp_gp_clus0->CovFunctionName();
				}
				else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
					std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp_clus0 = re_comps_ip_[unique_clusters_[0]][0][0];
					cov_fct = re_comp_gp_clus0->CovFunctionName();
				}
				else {
					Log::REFatal("CovFunctionName not implemented for gp_approx = '%s' ", gp_approx_.c_str());
				}
			}
			return(cov_fct);
		}//end CovFunctionName

		/*!
		* \brief Returns the number of CG steps when the CG method was last run
		*/
		int GetNumCGSteps() {
			bool cg_steps_recorded = ((num_re_group_total_ > 1 && num_re_group_total_ == num_comps_total_) || grouped_RE_and_vecchia_GP_)
				&& matrix_inversion_method_ == "iterative";
			if (!cg_steps_recorded) {
				Log::REWarning("GetNumCGSteps: this function is currently only implemented when having multiple grouped random effects and iterative methods are used ");
			}
			if (num_clusters_ > 1) {
				Log::REWarning("GetNumCGSteps: this function is not properly implemented when having multiple 'clusters' ('cluster_ids'). A value is returned for only one cluster ");
			}
			if (gauss_likelihood_) {
				return(num_cg_steps_last_);
			}
			else {
				return(likelihood_[unique_clusters_[0]]->GetNumCGSteps());
			}
		}

		/*!
		* \brief Returns the number of CG steps when the CG method was last run for the SLQ method
		*/
		int GetNumCGStepsTridiag() {
			bool cg_steps_recorded = ((num_re_group_total_ > 1 && num_re_group_total_ == num_comps_total_) || grouped_RE_and_vecchia_GP_)
				&& matrix_inversion_method_ == "iterative";
			if (!cg_steps_recorded) {
				Log::REWarning("GetNumCGStepsTridiag: this function is currently only implemented when having multiple grouped random effects and iterative methods are used ");
			}
			if (num_clusters_ > 1) {
				Log::REWarning("GetNumCGStepsTridiag: this function is not properly implemented when having multiple 'clusters' ('cluster_ids'). A value is returned for only one cluster ");
			}
			if (gauss_likelihood_) {
				return(num_cg_steps_tridiag_last_);
			}
			else {
				return(likelihood_[unique_clusters_[0]]->GetNumCGStepsTridiag());
			}
		}

		/*!
		* \brief Returns the number of mode finding steps from the last mode finding in a Laplace approximation
		*/
		int GetNumModeFindingSteps() {
			if (gauss_likelihood_) {
				Log::REWarning("Mode is not found for a 'gaussian' likelihood ");
				return(0);
			}
			else {
				if (num_clusters_ > 1) {
					Log::REWarning("GetNumModeFindingSteps: this function is not properly implemented when having multiple 'clusters' ('cluster_ids'). A value is returned for only one cluster ");
				}
				return(likelihood_[unique_clusters_[0]]->GetNumModeFindingSteps());				
			}
		}

		/*!
		* \brief Set / change the type of likelihood
		* \param likelihood Likelihood name
		*/
		void SetLikelihood(const string_t& likelihood) {
			bool gauss_likelihood_before = gauss_likelihood_;
			bool only_one_grouped_RE_calculations_on_RE_scale_before = only_one_grouped_RE_calculations_on_RE_scale_;
			bool only_one_GP_calculations_on_RE_scale_before = only_one_GP_calculations_on_RE_scale_;
			bool only_grouped_REs_use_woodbury_identity_before = use_woodbury_identity_;
			gauss_likelihood_ = (Likelihood<T_mat, T_chol>::ParseLikelihoodAlias(likelihood) == "gaussian") && !vecchia_latent_approx_gaussian_;
			std::string cov_fct = CovFunctionName();
			DetermineSpecialCasesModelsEstimationPrediction(cov_fct);
			CheckCompatibilitySpecialOptions();
			//Make adaptions in re_comps_ for special options when switching between Gaussian and non-Gaussian likelihoods
			if (gauss_likelihood_before && !gauss_likelihood_) {
				if ((gp_approx_ == "vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia") && has_duplicates_coords_) {
					Log::REFatal("Cannot change the likelihood to '%s' from 'gaussian' when gp_approx = '%s' and having duplicate coordinates ", likelihood.c_str(), gp_approx_.c_str());
				}
				if (only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_) {
					CHECK(gp_approx_ != "full_scale_tapering");
					for (const auto& cluster_i : unique_clusters_) {
						for (int igp = 0; igp < num_sets_re_; ++igp) {
							if (gp_approx_ == "vecchia") {
								re_comps_vecchia_[cluster_i][igp][0]->DropZ();
							}
							else {
								re_comps_[cluster_i][igp][0]->DropZ();
							}
						}
					}
				}
				if (estimate_cov_par_index_has_been_set_) {
					estimate_cov_par_index_.erase(estimate_cov_par_index_.begin());//drop nugget effect
				}
			}//end gauss_likelihood_before && !gauss_likelihood_
			else if (!gauss_likelihood_before && gauss_likelihood_) {
				if (only_one_GP_calculations_on_RE_scale_before && (gp_approx_ == "vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia")) {
					Log::REFatal("Cannot change the likelihood to 'gaussian' when gp_approx = '%s' and having duplicate coordinates ", gp_approx_.c_str());
				}
				if (only_one_GP_calculations_on_RE_scale_before || only_one_grouped_RE_calculations_on_RE_scale_before) {
					CHECK(gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" && gp_approx_ != "full_scale_vecchia");
					for (const auto& cluster_i : unique_clusters_) {
						for (int igp = 0; igp < num_sets_re_; ++igp) {
							if (gp_approx_ == "vecchia") {
								re_comps_vecchia_[cluster_i][igp][0]->AddZ();
							}
							else {
								re_comps_[cluster_i][igp][0]->AddZ();
							}
						}
					}
				}
				if (estimate_cov_par_index_has_been_set_) {
					estimate_cov_par_index_.insert(estimate_cov_par_index_.begin(), 1);
				}
			}//end !gauss_likelihood_before && gauss_likelihood_
			//Matrices used when use_woodbury_identity_==true 
			if ((use_woodbury_identity_ && !only_grouped_REs_use_woodbury_identity_before) ||
				(use_woodbury_identity_ && only_one_grouped_RE_calculations_on_RE_scale_before && !only_one_grouped_RE_calculations_on_RE_scale_)) {
				InitializeMatricesForUseWoodburyIdentity();
			}
			else if (!use_woodbury_identity_ && !grouped_RE_and_vecchia_GP_) {
				//Delete not required matrices
				Zt_ = std::map<data_size_t, sp_mat_t>();
				P_Zt_ = std::map<data_size_t, sp_mat_t>();
				ZtZ_ = std::map<data_size_t, sp_mat_t>();
				cum_num_rand_eff_ = std::map<data_size_t, std::vector<data_size_t>>();
				Zj_square_sum_ = std::map<data_size_t, std::vector<double>>();
				ZtZj_ = std::map<data_size_t, std::vector<sp_mat_t>>();
				P_ZtZj_ = std::map<data_size_t, std::vector<sp_mat_t>>();
			}
			//Identity matrices for Gaussian data
			if (!gauss_likelihood_before && gauss_likelihood_) {
				InitializeIdentityMatricesForGaussianData();
			}
			else if (gauss_likelihood_before && !gauss_likelihood_) {
				//Delete not required matrices
				Id_ = std::map<data_size_t, T_mat>();
				P_Id_ = std::map<data_size_t, T_mat>();
			}
			InitializeLikelihoods(likelihood);
			SetDefaultMatrixInversionMethod();
			DetermineCovarianceParameterIndicesNumCovPars();
			InitializeDefaultSettings();
			CheckPreconditionerType();
			SetPropertiesLikelihood();
		}//end SetLikelihood

		/*!
		* \brief Calculate test negative log-likelihood using adaptive GH quadrature
		* \param y_test Test response variable
		* \param pred_mean Predictive mean of latent random effects
		* \param pred_var Predictive variances of latent random effects
		* \param num_data Number of data points
		*/
		double TestNegLogLikelihoodAdaptiveGHQuadrature(const label_t* y_test,
			const double* pred_mean,
			const double* pred_var,
			const data_size_t num_data) {
			return(likelihood_[unique_clusters_[0]]->TestNegLogLikelihoodAdaptiveGHQuadrature(y_test, pred_mean, pred_var, num_data));
		}

		LBFGSpp::BFGSMat<double>& GetMBFGS() {
			return(m_bfgs_);
		}

		/*!
		* \brief Set configuration parameters for the optimizer
		* \param lr Learning rate for covariance parameters. If lr<= 0, internal default values are used (0.1 for "gradient_descent" and 1. for "fisher_scoring")
		* \param acc_rate_cov Acceleration rate for covariance parameters for Nesterov acceleration (only relevant if nesterov_schedule_version == 0).
		* \param max_iter Maximal number of iterations
		* \param delta_rel_conv Convergence tolerance. The algorithm stops if the relative change in eiher the log-likelihood or the parameters is below this value
		* \param use_nesterov_acc Indicates whether Nesterov acceleration is used in the gradient descent for finding the covariance parameters (only used for "gradient_descent")e
		* \param nesterov_schedule_version Which version of Nesterov schedule should be used (only relevant if use_nesterov_acc)
		* \param optimizer_cov Optimizer for covariance parameters
		* \param momentum_offset Number of iterations for which no mometum is applied in the beginning (only relevant if use_nesterov_acc)
		* \param convergence_criterion The convergence criterion used for terminating the optimization algorithm. Options: "relative_change_in_log_likelihood" or "relative_change_in_parameters"
		* \param lr_coef Learning rate for fixed-effect linear coefficients
		* \param acc_rate_coef Acceleration rate for coefficients for Nesterov acceleration (only relevant if nesterov_schedule_version == 0)
		* \param optimizer_coef Optimizer for linear regression coefficients
		* \param cg_max_num_it Maximal number of iterations for conjugate gradient algorithm
		* \param cg_max_num_it_tridiag Maximal number of iterations for conjugate gradient algorithm when being run as Lanczos algorithm for tridiagonalization
		* \param cg_delta_conv Tolerance level for L2 norm of residuals for checking convergence in conjugate gradient algorithm when being used for parameter estimation
		* \param num_rand_vec_trace Number of random vectors (e.g. Rademacher) for stochastic approximation of the trace of a matrix
		* \param reuse_rand_vec_trace If true, random vectors (e.g. Rademacher) for stochastic approximation of the trace of a matrix are sampled only once at the beginning and then reused in later trace approximations, otherwise they are sampled everytime a trace is calculated
		* \param cg_preconditioner_type Type of preconditioner used for the conjugate gradient algorithm
		* \param seed_rand_vec_trace Seed number to generate random vectors (e.g. Rademacher) for stochastic approximation of the trace of a matrix
		* \param fitc_piv_chol_preconditioner_rank Rank of the FITC and pivoted Cholesky preconditioners of the conjugate gradient algorithm
		* \param estimate_aux_pars If true, any additional parameters for non-Gaussian likelihoods are also estimated (e.g., shape parameter of gamma likelihood)
		* \param estimate_cov_par_index If estimate_cov_par_index[0] >= 0, some covariance parameters might not be estimated, estimate_cov_par_index[i] is then bool and indicates which ones are estimated
		* \param m_lbfgs Number of corrections to approximate the inverse Hessian matrix for the lbfgs optimizer
		* \param delta_conv_mode_finding Used for checking convergence in mode finding algorithm for non-Gaussian likelihoods
		*/
		void SetOptimConfig(double lr,
			double acc_rate_cov,
			int max_iter,
			double delta_rel_conv,
			bool use_nesterov_acc,
			int nesterov_schedule_version,
			const char* optimizer,
			int momentum_offset,
			const char* convergence_criterion,
			double lr_coef,
			double acc_rate_coef,
			const char* optimizer_coef,
			int cg_max_num_it,
			int cg_max_num_it_tridiag,
			double cg_delta_conv,
			int num_rand_vec_trace,
			bool reuse_rand_vec_trace,
			const char* cg_preconditioner_type,
			int seed_rand_vec_trace,
			int fitc_piv_chol_preconditioner_rank,
			bool estimate_aux_pars,
			const int* estimate_cov_par_index,
			int m_lbfgs,
			double delta_conv_mode_finding) {
			lr_cov_init_ = lr;
			lr_cov_after_first_iteration_ = lr;
			lr_cov_after_first_optim_boosting_iteration_ = lr;
			acc_rate_cov_ = acc_rate_cov;
			max_iter_ = max_iter;
			delta_rel_conv_init_ = delta_rel_conv;
			use_nesterov_acc_ = use_nesterov_acc;
			nesterov_schedule_version_ = nesterov_schedule_version;
			if (optimizer != nullptr) {
				if (std::string(optimizer) != "") {
					optimizer_cov_pars_ = std::string(optimizer);
					optimizer_cov_pars_has_been_set_ = true;
					if (optimizer_cov_pars_ == "gradient_descent_constant_change" ||
						optimizer_cov_pars_ == "newton_constant_change" ||
						optimizer_cov_pars_ == "fisher_scoring_constant_change") {
						learning_rate_constant_first_order_change_ = true;
					}
					else {
						learning_rate_constant_first_order_change_ = false;
					}
					if (optimizer_cov_pars_ == "gradient_descent_constant_change" ||
						optimizer_cov_pars_ == "gradient_descent_increase_lr" ||
						optimizer_cov_pars_ == "gradient_descent_reset_lr") {
						optimizer_cov_pars_ = "gradient_descent";
					}
					if (optimizer_cov_pars_ == "newt_constant_change") {
						optimizer_cov_pars_ = "newton";
					}
					if (optimizer_cov_pars_ == "fisher_scoring_constant_change") {
						optimizer_cov_pars_ = "fisher_scoring";
					}
					if (optimizer_cov_pars_ == "gradient_descent_increase_lr") {
						increase_learning_rate_again_ = true;
					}
					else {
						increase_learning_rate_again_ = false;
					}
					if (optimizer_cov_pars_ == "gradient_descent_reset_lr") {
						reset_learning_rate_every_iteration_ = true;
					}
					else {
						reset_learning_rate_every_iteration_ = false;
					}
				}
			}
			momentum_offset_ = momentum_offset;
			if (convergence_criterion != nullptr) {
				convergence_criterion_ = std::string(convergence_criterion);
				if (SUPPORTED_CONV_CRIT_.find(convergence_criterion_) == SUPPORTED_CONV_CRIT_.end()) {
					Log::REFatal("Convergence criterion '%s' is not supported.", convergence_criterion_.c_str());
				}
			}
			lr_coef_init_ = lr_coef;
			lr_coef_after_first_iteration_ = lr_coef;
			lr_coef_after_first_optim_boosting_iteration_ = lr_coef;
			acc_rate_coef_ = acc_rate_coef;
			if (optimizer_coef != nullptr) {
				if (std::string(optimizer_coef) != "") {
					optimizer_coef_ = std::string(optimizer_coef);
					coef_optimizer_has_been_set_ = true;
				}
			}
			num_rand_vec_trace_ = num_rand_vec_trace;
			seed_rand_vec_trace_ = seed_rand_vec_trace;
			reuse_rand_vec_trace_ = reuse_rand_vec_trace;
			// Conjugate gradient algorithm related parameters
			if (matrix_inversion_method_ == "iterative") {
				cg_max_num_it_ = cg_max_num_it;
				cg_max_num_it_tridiag_ = cg_max_num_it_tridiag;
				cg_delta_conv_ = cg_delta_conv;
				if (cg_preconditioner_type != nullptr) {
					if (cg_preconditioner_type_ != std::string(cg_preconditioner_type) &&
						model_has_been_estimated_) {
						Log::REFatal("Cannot change 'cg_preconditioner_type' after a model has been fitted ");
					}
					cg_preconditioner_type_ = ParsePreconditionerAlias(std::string(cg_preconditioner_type));
					CheckPreconditionerType();
					cg_preconditioner_type_has_been_set_ = true;
				}
				if (fitc_piv_chol_preconditioner_rank > 0) {
					fitc_piv_chol_preconditioner_rank_ = fitc_piv_chol_preconditioner_rank;
					fitc_piv_chol_preconditioner_rank_has_been_set_ = true;
				}
				else {
					if (cg_preconditioner_type_ == "fitc") {
						fitc_piv_chol_preconditioner_rank_ = default_fitc_preconditioner_rank_;
					}
					else if (cg_preconditioner_type_ == "pivoted_cholesky") {
						fitc_piv_chol_preconditioner_rank_ = default_piv_chol_preconditioner_rank_;
					}
				}
			}

			estimate_aux_pars_ = estimate_aux_pars;
			if (lr > 0) {
				lr_aux_pars_init_ = lr;
				lr_aux_pars_after_first_iteration_ = lr;
				lr_aux_pars_after_first_optim_boosting_iteration_ = lr;
			}
			set_optim_config_has_been_called_ = true;
			if (estimate_cov_par_index[0] >= 0.) {
				estimate_cov_par_index_ = std::vector<int>(num_cov_par_);
				for (int ipar = 0; ipar < num_cov_par_; ++ipar) {
					estimate_cov_par_index_[ipar] = estimate_cov_par_index[ipar];
				}
				estimate_cov_par_index_has_been_set_ = true;
			}
			if (m_lbfgs > 0) {
				m_lbfgs_ = m_lbfgs;
			}
			if (delta_conv_mode_finding > 0.) {
				delta_conv_mode_finding_ = delta_conv_mode_finding;
			}
			SetPropertiesLikelihood();
		}//end SetOptimConfig

		/*!
		* \brief Find covariance parameters and linear regression coefficients (if there are any) that minimize the (approximate) negative log-ligelihood
		*		 Note: You should pre-allocate memory for optim_cov_pars and optim_coef. Their length equal the number of covariance parameters and the number of regression coefficients
		*           If calc_std_dev, you also need to pre-allocate memory for std_dev_cov_par and std_dev_coef of the same length for the standard deviations
		* \param y_data Response variable data
		* \param covariate_data Covariate data (=independent variables, features). Set to nullptr if there is no covariate data
		* \param num_covariates Number of covariates
		* \param[out] optim_cov_pars Optimal covariance parameters
		* \param[out] optim_coef Optimal regression coefficients
		* \param[out] num_it Number of iterations
		* \param init_cov_pars Initial values for covariance parameters of RE components
		* \param init_coef Initial values for the regression coefficients (can be nullptr)
		* \param fixed_effects Additional fixed effects that are added to the linear predictor (= offset) (can be nullptr)
		* \param learn_covariance_parameters If true, covariance parameters are estimated
		* \param called_in_GPBoost_algorithm If true, this function is called in the GPBoost algorithm, otherwise for the estimation of a GLMM
		* \param reuse_learning_rates_from_previous_call If true, the learning rates for the covariance and potential auxiliary parameters are kept at the values from a previous call and
		*			not re-initialized (can only be set to true if called_in_GPBoost_algorithm is true). This option is only used for "gradient_descent"
		* \param only_intercept_for_GPBoost_algo True if the covariates contain only an intercept and this function is called from the GPBoost algorithm for finding an initial score
		* \param find_learning_rate_for_GPBoost_algo True if this function is called from the GPBoost algorithm for finding an optimal learning rate
		*/
		void OptimLinRegrCoefCovPar(const double* y_data,
			const double* covariate_data,
			int num_covariates,
			double* optim_cov_pars,
			double* optim_coef,
			int& num_it,
			double* init_cov_pars,
			double* init_coef,
			const double* fixed_effects,
			bool learn_covariance_parameters,
			bool called_in_GPBoost_algorithm,
			bool reuse_learning_rates_from_previous_call,
			bool only_intercept_for_GPBoost_algo,
			bool find_learning_rate_for_GPBoost_algo) {
			if (iid_model_) {
				if (gauss_likelihood_) {
					estimate_cov_par_index_[1] = 0;
					init_cov_pars[1] = 1e-20 / init_cov_pars[0];
				}
				else {
					estimate_cov_par_index_[0] = 0;
					init_cov_pars[0] = 1e-20;
				}
				estimate_cov_par_index_has_been_set_ = true;
			}
			if (NumAuxPars() == 0) {
				estimate_aux_pars_ = false;
			}
			if (covariate_data == nullptr) {
				has_covariates_ = false;
			}
			else {
				has_covariates_ = true;
			}
			bool reuse_m_bfgs_from_previous_call = reuse_learning_rates_from_previous_call && called_in_GPBoost_algorithm && learn_covariance_parameters &&
				cov_pars_have_been_estimated_once_ && cov_pars_have_been_estimated_during_last_call_;
			OptimParamsSetInitialValues();
			InitializeOptimSettings(reuse_learning_rates_from_previous_call);
			// Some checks
			if (SUPPORTED_OPTIM_COV_PAR_.find(optimizer_cov_pars_) == SUPPORTED_OPTIM_COV_PAR_.end()) {
				Log::REFatal("Optimizer option '%s' is not supported for covariance parameters ", optimizer_cov_pars_.c_str());
			}
			if (optimizer_cov_pars_ == "fisher_scoring" && !gauss_likelihood_) {
				Log::REFatal("Optimizer option '%s' is not supported for covariance parameters for non-Gaussian likelihoods ", optimizer_cov_pars_.c_str());
			}
			if (optimizer_cov_pars_ == "fisher_scoring" && estimate_aux_pars_) {
				Log::REFatal("Optimizer option '%s' is not supported when estimating additional auxiliary parameters for non-Gaussian likelihoods ", optimizer_cov_pars_.c_str());
			}
			if (has_covariates_) {
				if (gauss_likelihood_) {
					if (SUPPORTED_OPTIM_COEF_GAUSS_.find(optimizer_coef_) == SUPPORTED_OPTIM_COEF_GAUSS_.end()) {
						Log::REFatal("Optimizer option '%s' is not supported for linear regression coefficients.", optimizer_coef_.c_str());
					}
				}
				else {
					if (SUPPORTED_OPTIM_COEF_NONGAUSS_.find(optimizer_coef_) == SUPPORTED_OPTIM_COEF_NONGAUSS_.end()) {
						Log::REFatal("Optimizer option '%s' is not supported for linear regression coefficients for non-Gaussian likelihoods ", optimizer_coef_.c_str());
					}
				}
				// check whether optimizer_cov_pars_ and optimizer_coef_ are compatible
				if (optimizer_coef_ != optimizer_cov_pars_ &&
					((OPTIM_EXTERNAL_.find(optimizer_cov_pars_) != OPTIM_EXTERNAL_.end() &&
						!(optimizer_coef_ == "wls" && OPTIM_EXTERNAL_SUPPORT_WLS_.find(optimizer_cov_pars_) != OPTIM_EXTERNAL_SUPPORT_WLS_.end())) ||
						OPTIM_EXTERNAL_.find(optimizer_coef_) != OPTIM_EXTERNAL_.end())) { // optimizers are not equal and one of them is an external optimizer (except when optimizer_coef_ == "wls" and optimizer_cov_pars_ supports this)
					if (optimizer_cov_pars_has_been_set_ && coef_optimizer_has_been_set_) {
						Log::REFatal("Cannot use optimizer_cov = '%s' when optimizer_coef = '%s' ",
							optimizer_cov_pars_.c_str(), optimizer_coef_.c_str());
					}
					else if (optimizer_cov_pars_has_been_set_ && !coef_optimizer_has_been_set_) {
						if ((gauss_likelihood_ && SUPPORTED_OPTIM_COEF_GAUSS_.find(optimizer_cov_pars_) == SUPPORTED_OPTIM_COEF_GAUSS_.end()) ||
							(!gauss_likelihood_ && SUPPORTED_OPTIM_COEF_NONGAUSS_.find(optimizer_coef_) == SUPPORTED_OPTIM_COEF_NONGAUSS_.end())) {
							Log::REFatal("Cannot use optimizer_cov = '%s' when optimizer_coef = '%s' ",
								optimizer_cov_pars_.c_str(), optimizer_coef_.c_str());
						}
						else {
							Log::REDebug("'%s' is also used for estimating regression coefficients (optimizer_coef = '%s' is ignored) ",
								optimizer_cov_pars_.c_str(), optimizer_coef_.c_str());
							optimizer_coef_ = optimizer_cov_pars_;
						}
					}
					else if (!optimizer_cov_pars_has_been_set_ && coef_optimizer_has_been_set_) {
						if (SUPPORTED_OPTIM_COV_PAR_.find(optimizer_cov_pars_) == SUPPORTED_OPTIM_COV_PAR_.end()) {
							Log::REFatal("Cannot use optimizer_cov = '%s' when optimizer_coef = '%s' ",
								optimizer_cov_pars_.c_str(), optimizer_coef_.c_str());
						}
						else {
							Log::REWarning("'%s' is also used for estimating covariance parameters (optimizer_cov = '%s' is ignored) ",
								optimizer_coef_.c_str(), optimizer_cov_pars_.c_str());
							optimizer_cov_pars_ = optimizer_coef_;
						}
					}
				}
			}//end has_covariates_
			if (optimizer_cov_pars_ == "fisher_scoring" || optimizer_cov_pars_ == "newton" || optimizer_cov_pars_ == "nelder_mead") {
				bool has_cov_par_not_estimated = std::any_of(estimate_cov_par_index_.begin(), estimate_cov_par_index_.end(), [](int x) { return x <= 0; });
				if (has_cov_par_not_estimated) {
					Log::REFatal("Holding fix some covariance parameters (via 'estimate_cov_par_index') when using optimizer_cov = '%s' as optimizer is currently not supported ", optimizer_cov_pars_.c_str());
				}
			}
			// Profiling out sigma (=use closed-form expression for error / nugget variance) is often better (the paremeters usually live on different scales and the nugget needs a smaller learning rate than the others...)
			profile_out_error_variance_ = gauss_likelihood_ &&
				(optimizer_cov_pars_ == "gradient_descent" || optimizer_cov_pars_ == "nelder_mead" || optimizer_cov_pars_ == "adam" ||
					optimizer_cov_pars_ == "lbfgs" || optimizer_cov_pars_ == "lbfgs_linesearch_nocedal_wright");
			bool gradient_contains_error_var = gauss_likelihood_ && !profile_out_error_variance_;//If true, the error variance parameter (=nugget effect) is also included in the gradient, otherwise not
			if (optimizer_cov_pars_ == "lbfgs_not_profile_out_nugget") {
				optimizer_cov_pars_ = "lbfgs";
			}
			// Check response variable data
			if (y_data != nullptr) {
				if (LightGBM::Common::HasNAOrInf(y_data, num_data_)) {
					Log::REFatal("NaN or Inf in response variable / label ");
				}
			}
			// Initialization of variables
			bool use_nesterov_acc = use_nesterov_acc_;
			bool use_nesterov_acc_coef = use_nesterov_acc_;
			//Nesterov acceleration is only used for gradient descent and not for other methods
			if (optimizer_cov_pars_ != "gradient_descent") {
				use_nesterov_acc = false;
			}
			if (optimizer_coef_ != "gradient_descent") {
				use_nesterov_acc_coef = false;
			}
			bool terminate_optim = false;
			learning_rate_decreased_first_time_ = false;
			learning_rate_increased_after_descrease_ = false;
			learning_rate_decreased_after_increase_ = false;
			num_ll_evaluations_ = 0;
			num_iter_ = 0;
			num_it = max_iter_;
			has_intercept_ = false; //If true, the covariates contain an intercept column (only relevant if there are covariates)
			intercept_col_ = -1;
			// Check whether one of the columns contains only 1's (-> has_intercept_)
			if (has_covariates_) {
				num_covariates_ = num_covariates;
				X_ = Eigen::Map<const den_mat_t>(covariate_data, num_data_, num_covariates_);
				for (int icol = 0; icol < num_covariates_; ++icol) {
					bool var_is_constant = true;
#pragma omp parallel for schedule(static)
					for (data_size_t i = 1; i < num_data_; ++i) {
						if (var_is_constant) {
							if (!(TwoNumbersAreEqual<double>(X_.coeff(i, icol), X_.coeff(0, icol)))) {
#pragma omp critical
								{
									var_is_constant = false;
								}
							}
						}
					}
					if (var_is_constant) {
						has_intercept_ = true;
						intercept_col_ = icol;
						break;
					}
				}
				if (num_covariates_ > 1) {
					Eigen::ColPivHouseholderQR<den_mat_t> qr_decomp(X_);
					int rank = (int)qr_decomp.rank();
					// If X_ was a sparse matrix, use the following code:
					//Eigen::SparseQR<sp_mat_t, Eigen::COLAMDOrdering<int>> qr_decomp;
					//qr_decomp.compute(X_);
					if (rank < num_covariates_) {
						Log::REWarning("The linear regression covariate data matrix (fixed effect) is rank deficient. "
							"This is not necessarily a problem when using gradient descent. "
							"If this is not desired, consider dropping some columns / covariates ");
					}
				}
			}//end if has_covariates_
			if (only_intercept_for_GPBoost_algo) {
				CHECK(!find_learning_rate_for_GPBoost_algo);
				CHECK(has_intercept_);
			}
			if (find_learning_rate_for_GPBoost_algo) {
				CHECK(!only_intercept_for_GPBoost_algo);
			}
			if (only_intercept_for_GPBoost_algo || find_learning_rate_for_GPBoost_algo) {
				CHECK(has_covariates_);
				CHECK(num_covariates_ == 1);
				CHECK(!learn_covariance_parameters);
			}
			if (reuse_learning_rates_from_previous_call) {
				CHECK(called_in_GPBoost_algorithm);
				CHECK(learn_covariance_parameters || find_learning_rate_for_GPBoost_algo);
			}
			if (called_in_GPBoost_algorithm && find_learning_rate_for_GPBoost_algo && gauss_likelihood_) {
				// Use explicit formula for optimal learning rate for Gaussian likelihood
				SetY(covariate_data);
				CalcYAux(1., false);//y_aux = Psi^-1 * f^t, where f^t (= covariate_data) is the new base learner
				double numer = 0., denom = 0.;
				for (const auto& cluster_i : unique_clusters_) {
					vec_t y_vec_cluster_i(num_data_per_cluster_[cluster_i]);
					for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
						y_vec_cluster_i[j] = y_vec_[data_indices_per_cluster_[cluster_i][j]];
					}
					numer = y_vec_cluster_i.dot(y_aux_[cluster_i]);
					denom = y_[cluster_i].dot(y_aux_[cluster_i]);
				}
				optim_coef[0] = numer / denom;
				Log::REDebug(" ");
				Log::REDebug("GPModel: optimal learning rate = %g ", -optim_coef[0]);
				return;
			}
			// Assume that this function is only called for initialization of the GPBoost algorithm
			//	when (i) there is only an intercept (and not other covariates) and (ii) the covariance parameters are not learned
			const double* fixed_effects_ptr = fixed_effects;
			if (fixed_effects != nullptr && !called_in_GPBoost_algorithm) {//save offset / fixed_effects for prediction
				has_fixed_effects_ = true;
				fixed_effects_ = Eigen::Map<const vec_t>(fixed_effects, num_data_ * num_sets_re_);
			}
			// Initialization of covariance parameters
			int num_cov_par_estimate = num_cov_par_;
			if (estimate_aux_pars_) {
				num_cov_par_estimate += NumAuxPars();
			}
			vec_t cov_aux_pars = vec_t(num_cov_par_estimate);
			for (int i = 0; i < num_cov_par_; ++i) {
				cov_aux_pars[i] = init_cov_pars[i];
			}
			if (gauss_likelihood_) {
				sigma2_ = cov_aux_pars[0];
			}
			cov_pars_set_first_time_ = cov_aux_pars.segment(0, num_cov_par_);
			optimization_running_currently_ = true;
			// Set response variabla data (if needed). Note: for the GPBoost algorithm this is set a prior by calling SetY. For Gaussian data with covariates, this is set later repeatedly.
			if ((!has_covariates_ || !gauss_likelihood_) && y_data != nullptr) {
				SetY(y_data);
			}
			if (!has_covariates_ || !gauss_likelihood_) {
				CHECK(y_has_been_set_);//response variable data needs to have been set at this point for non-Gaussian likelihoods and for Gaussian data without covariates
			}
			if (gauss_likelihood_ && !find_learning_rate_for_GPBoost_algo) {
				// If find_learning_rate_for_GPBoost_algo, 'y_data' does not need to be provided but y_vec_ from the last call (when optimizing the covariance parameters) can be reused 
				CHECK(y_data != nullptr);
				// Copy of response data (used only for Gaussian data and if there are also linear covariates since then y_ is modified during the optimization algorithm and this contains the original data)
				y_vec_ = Eigen::Map<const vec_t>(y_data, num_data_);
			}
			// Initialization of linear regression coefficients related variables
			vec_t beta_lag1, beta_init, beta_after_grad_aux, beta_after_grad_aux_lag1, beta_before_lr_cov_small, beta_before_lr_aux_pars_small, fixed_effects_vec;
			scale_covariates_ = false;
			if (has_covariates_) {
				scale_covariates_ = (optimizer_coef_ == "gradient_descent" || (optimizer_cov_pars_ == "bfgs_optim_lib" && !gauss_likelihood_) ||
					((optimizer_cov_pars_ == "lbfgs" || optimizer_cov_pars_ == "lbfgs_linesearch_nocedal_wright") && optimizer_coef_ != "wls")) &&
					!(has_intercept_ && num_covariates_ == 1);//if there is only an intercept, we don't need to scale the covariates
				// Scale covariates (in order that the gradient is less sample-size dependent)
				if (scale_covariates_) {
					loc_transf_ = vec_t(num_covariates_);
					scale_transf_ = vec_t(num_covariates_);
					vec_t col_i_centered;
					for (int icol = 0; icol < num_covariates_; ++icol) {
						if (!has_intercept_ || icol != intercept_col_) {
							loc_transf_[icol] = X_.col(icol).mean();
							col_i_centered = X_.col(icol);
							col_i_centered.array() -= loc_transf_[icol];
							scale_transf_[icol] = std::sqrt(col_i_centered.array().square().sum() / num_data_);
							X_.col(icol) = col_i_centered / scale_transf_[icol];
						}
					}
					if (has_intercept_) {
						loc_transf_[intercept_col_] = 0.;
						scale_transf_[intercept_col_] = 1.;
					}
				}
				beta_ = vec_t(num_covariates_ * num_sets_re_);
				if (init_coef == nullptr) {
					beta_.setZero();
				}
				else {
					beta_ = Eigen::Map<const vec_t>(init_coef, num_covariates_ * num_sets_re_);
				}
				if (init_coef == nullptr || only_intercept_for_GPBoost_algo) {
					if (has_intercept_) {
						double tot_var_mean_re = GetTotalVarComps(cov_aux_pars.segment(0, num_cov_par_), 0);
						if (num_sets_re_ > 1) {
							CHECK(num_sets_re_ == 2); // check whether this makes sense if other models with num_sets_re_> 1 are implemented in the future
						}
						for (int igp = 0; igp < num_sets_re_; ++igp) {
							if (y_data == nullptr) {
								vec_t y_temp(num_data_);
								GetY(y_temp.data());
								beta_[num_covariates * igp + intercept_col_] = likelihood_[unique_clusters_[0]]->FindInitialIntercept(y_temp.data(), num_data_, tot_var_mean_re, fixed_effects, igp);
							}
							else {
								beta_[num_covariates * igp + intercept_col_] = likelihood_[unique_clusters_[0]]->FindInitialIntercept(y_data, num_data_, tot_var_mean_re, fixed_effects, igp);
							}
						}
					}
				}
				else if (scale_covariates_) {
					// transform initial coefficients
					TransformCoef(beta_, beta_);
				}
				beta_after_grad_aux = beta_;
				beta_after_grad_aux_lag1 = beta_;
				beta_init = beta_;
				UpdateFixedEffects(beta_, fixed_effects, fixed_effects_vec);
				if (!gauss_likelihood_) {
					fixed_effects_ptr = fixed_effects_vec.data();
				}
				// Determine constants C_mu and C_sigma2 used for checking whether step sizes for linear regression coefficients are clearly too large
				if (y_data == nullptr) {
					vec_t y_temp(num_data_);
					GetY(y_temp.data());
					likelihood_[unique_clusters_[0]]->FindConstantsCapTooLargeLearningRateCoef(y_temp.data(), num_data_, fixed_effects, C_mu_, C_sigma2_);
				}
				else {
					likelihood_[unique_clusters_[0]]->FindConstantsCapTooLargeLearningRateCoef(y_data, num_data_, fixed_effects, C_mu_, C_sigma2_);
				}
			}//end if has_covariates_
			if (!has_intercept_ && !called_in_GPBoost_algorithm && !find_learning_rate_for_GPBoost_algo && fixed_effects == nullptr) {
				CHECK(y_data != nullptr);
				string_t cov_function_name = CovFunctionName();
				if (cov_function_name == "hurst" || cov_function_name == "hurst_ard") {
					Log::REWarning("There is no intercept (= a column of 1's) included in the covariates 'X'. For the '%s' covariance, it is recommended to include an intercept since the GP is anchored at 0 (i.e., b(0) = 0 a.s.) ", cov_function_name.c_str());
				}
				else {
					double tot_var = GetTotalVarComps(cov_aux_pars.segment(0, num_cov_par_), 0);
					if (likelihood_[unique_clusters_[0]]->ShouldHaveIntercept(y_data, num_data_, tot_var, fixed_effects)) {
						string_t msg = "There is no intercept (= a column of 1's) included in the covariates 'X'. Consider including one ";
						if (has_covariates_) {
							Log::REWarning(msg.c_str());
						}
						else {
							Log::REDebug(msg.c_str());
						}
					}
				}
			}//end intercept message
			if (!has_intercept_ && !find_learning_rate_for_GPBoost_algo) {
				Log::REDebug("The covariate data contains no column of ones, i.e., no intercept is included ");
			}
			if (!has_covariates_ && fixed_effects != nullptr && gauss_likelihood_) {
				vec_t resid = y_vec_;
#pragma omp parallel for schedule(static)
				for (data_size_t i = 0; i < num_data_; ++i) {
					resid[i] -= fixed_effects[i];
				}
				SetY(resid.data());
			}
			if (estimate_aux_pars_) {
				// Find initial values for additional likelihood parameters (aux_pars) if they have not been given
				if (!(likelihood_[unique_clusters_[0]]->AuxParsHaveBeenSet())) {//If initial values have been provided, these were set in re_model previously
					const double* aux_pars;
					if (y_data == nullptr) {
						vec_t y_aux_temp(num_data_);
						GetY(y_aux_temp.data());
						aux_pars = likelihood_[unique_clusters_[0]]->FindInitialAuxPars(y_aux_temp.data(), fixed_effects_ptr, num_data_);
						y_aux_temp.resize(0);
					}
					else {
						aux_pars = likelihood_[unique_clusters_[0]]->FindInitialAuxPars(y_data, fixed_effects_ptr, num_data_);
					}
					SetAuxPars(aux_pars);
				}
				for (int i = 0; i < NumAuxPars(); ++i) {
					cov_aux_pars[num_cov_par_ + i] = GetAuxPars()[i];
				}
			}//end estimate_aux_pars_
			bool profile_out_coef = optimizer_coef_ == "wls" && gauss_likelihood_ && has_covariates_;
			SetLag1ProfiledOutVariables(profile_out_error_variance_, profile_out_coef);
			// Initialize auxiliary variables for e.g. Nesterov acceleration
			vec_t cov_aux_pars_lag1 = vec_t(num_cov_par_estimate);
			vec_t cov_aux_pars_init = cov_aux_pars;
			vec_t cov_pars_after_grad_aux = cov_aux_pars, cov_aux_pars_after_grad_aux_lag1 = cov_aux_pars;//auxiliary variables used only if use_nesterov_acc == true
			vec_t cov_aux_pars_before_lr_coef_small, aux_pars_before_lr_cov_small, cov_pars_before_lr_aux_pars_small;//auxiliary variables
			// Print out initial information
			if (called_in_GPBoost_algorithm) {
				Log::REDebug(" ");
			}
			if (called_in_GPBoost_algorithm && only_intercept_for_GPBoost_algo) {
				Log::REDebug("GPModel: start finding initial intercept ... ");
			}
			if (called_in_GPBoost_algorithm && find_learning_rate_for_GPBoost_algo) {
				Log::REDebug("GPModel: start finding optimal learning rate ... ");
			}
			Log::REDebug("GPModel: initial parameters: ");
			PrintTraceParameters(cov_aux_pars.segment(0, num_cov_par_), beta_, cov_aux_pars.data() + num_cov_par_, true);

			if (max_iter_ > 0) {
			// Initialize optimizer:
			// - factorize the covariance matrix (Gaussian data) or calculate the posterior mode of the random effects for use in the Laplace approximation (non-Gaussian likelihoods)
			// - calculate initial value of objective function
			// - Note: initial values of aux_pars (additional parameters of likelihood) are set in likelihoods.h
			if (ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(true)) {
				SetCovParsComps(cov_aux_pars.segment(0, num_cov_par_));
				RedetermineNearestNeighborsVecchiaInducingPointsFITC(true);//called if gp_approx_ == "vecchia" or  gp_approx_ == "full_scale_vecchia" and neighbors are selected based on correlations and not distances or gp_approx_ == "fitc" with ard kernel
			}
			bool na_or_inf_occurred = false;
			if ((optimizer_cov_pars_ != "lbfgs" && optimizer_cov_pars_ != "lbfgs_linesearch_nocedal_wright") ||
					(gauss_likelihood_ && (profile_out_coef || profile_out_error_variance_))) {
				//Calculate initial log-likelihood whenever not lbfgs or also when maxit = 0 or some variables are later profiled out
				CalcCovFactorOrModeAndNegLL(cov_aux_pars.segment(0, num_cov_par_), fixed_effects_ptr);
				// TODO: for likelihood evaluation we don't need y_aux = Psi^-1 * y but only Psi^-0.5 * y. So, if has_covariates_==true, we might skip this step here and save some time
				string_t ll_str;
				if (gauss_likelihood_) {
					ll_str = "negative log-likelihood";
				}
				else {
					ll_str = "approximate negative marginal log-likelihood";
				}
				string_t init_coef_str = "";
				if (has_covariates_) {
					init_coef_str = " and 'init_coef'";
				}
				string_t problem_str = "none";
				if (std::isnan(neg_log_likelihood_)) {
					problem_str = "NaN";
				}
				else if (std::isinf(neg_log_likelihood_)) {
					problem_str = "Inf";
				}
				if (problem_str != "none") {
					Log::REFatal((problem_str + " occurred in initial " + ll_str + ". "
						"Possible solutions: try other initial values ('init_cov_pars'" + init_coef_str + ") "
						"or other tuning parameters in case you apply the GPBoost algorithm (e.g., learning_rate)").c_str());
				}
				Log::REDebug("Initial %s: %g", ll_str.c_str(), neg_log_likelihood_);
			}
			if (OPTIM_EXTERNAL_.find(optimizer_cov_pars_) != OPTIM_EXTERNAL_.end()) {
					OptimExternal<T_mat, T_chol>(this, cov_aux_pars, beta_, fixed_effects, max_iter_,
						delta_rel_conv_, convergence_criterion_, num_it, learn_covariance_parameters,
						optimizer_cov_pars_, profile_out_error_variance_, profile_out_coef,
						neg_log_likelihood_, num_cov_par_, NumAuxPars(), GetAuxPars(), has_covariates_, lr_cov_init_, reuse_m_bfgs_from_previous_call,
						m_lbfgs_);
					// Check for NA or Inf
					if (optimizer_cov_pars_ == "bfgs_optim_lib" || optimizer_cov_pars_ == "lbfgs" || optimizer_cov_pars_ == "lbfgs_linesearch_nocedal_wright") {
						if (learn_covariance_parameters) {
							for (int i = 0; i < (int)cov_aux_pars.size(); ++i) {
								if (std::isnan(cov_aux_pars[i]) || std::isinf(cov_aux_pars[i])) {
									na_or_inf_occurred = true;
								}
							}
						}
						if (has_covariates_ && !na_or_inf_occurred) {
							for (int i = 0; i < (int)beta_.size(); ++i) {
								if (std::isnan(beta_[i]) || std::isinf(beta_[i])) {
									na_or_inf_occurred = true;
								}
							}
						}
					} // end check for NA or Inf
			} // end use of external optimizer
			else {
				// Start optimization with internal optimizers such as "gradient_descent" or "fisher_scoring"
				bool lr_cov_is_small = false, lr_aux_pars_is_small = false, lr_coef_is_small = false;
				for (num_iter_ = 0; num_iter_ < max_iter_; ++num_iter_) {
					if (reset_learning_rate_every_iteration_) {
						InitializeOptimSettings(reuse_learning_rates_from_previous_call);//reset learning rates to their initial values
					}
					neg_log_likelihood_lag1_ = neg_log_likelihood_;
					cov_aux_pars_lag1 = cov_aux_pars;
					// Update linear regression coefficients using gradient descent or generalized least squares (the latter option only for Gaussian data)
					if (has_covariates_) {
						beta_lag1 = beta_;
						if (optimizer_coef_ == "gradient_descent") {// one step of gradient descent
							vec_t grad_beta;
							// Calculate gradient for linear regression coefficients
							vec_t unused_dummy;
							CalcGradPars(cov_aux_pars, cov_aux_pars[0], false, true, unused_dummy, grad_beta, false, false, fixed_effects_ptr, false);
							AvoidTooLargeLearningRateCoef(beta_, grad_beta);
							CalcDirDerivArmijoAndLearningRateConstChangeCoef(grad_beta, beta_, beta_after_grad_aux, use_nesterov_acc_coef);
							if (called_in_GPBoost_algorithm && reuse_learning_rates_from_previous_call &&
								coef_have_been_estimated_once_ && optimizer_coef_ == "gradient_descent") {//potentially increase learning rates again in GPBoost algorithm
								PotentiallyIncreaseLearningRateCoefForGPBoostAlgorithm();
							}//end called_in_GPBoost_algorithm / potentially increase learning rates again
							// Update linear regression coefficients, do learning rate backtracking, and recalculate mode for Laplace approx. (only for non-Gaussian likelihoods)
							UpdateLinCoef(beta_, grad_beta, cov_aux_pars[0], use_nesterov_acc_coef, num_iter_, beta_after_grad_aux, beta_after_grad_aux_lag1,
								acc_rate_coef_, nesterov_schedule_version_, momentum_offset_, fixed_effects, fixed_effects_vec);
							if (num_iter_ == 0) {
								lr_coef_after_first_iteration_ = lr_coef_;
								lr_is_small_threshold_coef_ = lr_coef_ / 1e4;
								if (called_in_GPBoost_algorithm && reuse_learning_rates_from_previous_call &&
									!coef_have_been_estimated_once_ && optimizer_coef_ == "gradient_descent") {
									lr_coef_after_first_optim_boosting_iteration_ = lr_coef_;
								}
							}
							fixed_effects_ptr = fixed_effects_vec.data();
							// In case lr_coef_ is very small, we monitor whether cov_aux_pars continues to change. If it does, we will reset lr_coef_ to its initial value
							if (lr_coef_ < lr_is_small_threshold_coef_ && learn_covariance_parameters && !lr_coef_is_small) {
								if ((beta_ - beta_lag1).norm() < LR_IS_SMALL_REL_CHANGE_IN_PARS_THRESHOLD_ * beta_lag1.norm()) {//also require that relative change in parameters is small
									lr_coef_is_small = true;
									cov_aux_pars_before_lr_coef_small = cov_aux_pars;
								}
							}
						}
						else if (optimizer_coef_ == "wls") {// coordinate descent using generalized least squares (only for Gaussian data)
							ProfileOutCoef(fixed_effects, fixed_effects_vec);
							EvalNegLogLikelihoodOnlyUpdateFixedEffects(cov_aux_pars[0], neg_log_likelihood_after_lin_coef_update_);
						}
						// Reset lr_cov_ to its initial values in case beta changes substantially after lr_cov_ is very small
						bool mode_hast_just_been_recalculated = false;
						if (lr_cov_is_small && learn_covariance_parameters) {
							if ((beta_ - beta_before_lr_cov_small).norm() > MIN_REL_CHANGE_IN_OTHER_PARS_FOR_RESETTING_LR_ * beta_before_lr_cov_small.norm()) {
								lr_cov_ = lr_cov_after_first_iteration_;
								lr_cov_is_small = false;
								RecalculateModeLaplaceApprox(fixed_effects_ptr);
								mode_hast_just_been_recalculated = true;
							}
						}
						// Reset lr_aux_pars_ to its initial values in case beta changes substantially after lr_aux_pars_ is very small
						if (lr_aux_pars_is_small && estimate_aux_pars_ && learn_covariance_parameters) {
							if ((beta_ - beta_before_lr_cov_small).norm() > MIN_REL_CHANGE_IN_OTHER_PARS_FOR_RESETTING_LR_ * beta_before_lr_cov_small.norm()) {
								lr_aux_pars_ = lr_aux_pars_after_first_iteration_;
								lr_aux_pars_is_small = false;
								if (!mode_hast_just_been_recalculated) {
									RecalculateModeLaplaceApprox(fixed_effects_ptr);
								}
							}
						}
					}//end has_covariates_
					else {
						neg_log_likelihood_after_lin_coef_update_ = neg_log_likelihood_lag1_;
					}// end update regression coefficients
					// Update covariance parameters using one step of gradient descent or Fisher scoring
					if (learn_covariance_parameters) {
						// Calculate gradient or natural gradient = FI^-1 * grad (for Fisher scoring)
						vec_t grad, neg_step_dir; // gradient and negative step direction. E.g., neg_step_dir = grad for gradient descent and neg_step_dir = FI^-1 * grad for Fisher scoring (="natural" gradient)
						den_mat_t approx_Hessian;
						if (profile_out_error_variance_) {
							cov_aux_pars[0] = ProfileOutSigma2();
							//EvalNegLogLikelihoodOnlyUpdateNuggetVariance(cov_aux_pars[0], neg_log_likelihood_after_lin_coef_update_);//todo: enable this and change tests
						}
						vec_t unused_dummy;
						if (optimizer_cov_pars_ == "gradient_descent" || optimizer_cov_pars_ == "newton") {
							CalcGradPars(cov_aux_pars.segment(0, num_cov_par_), 1., true, false, grad, unused_dummy, gradient_contains_error_var, false, fixed_effects_ptr, false);
							if (optimizer_cov_pars_ == "gradient_descent") {
								neg_step_dir = grad;
							}
							else if (optimizer_cov_pars_ == "newton") {
								CalcHessianCovParAuxPars(cov_aux_pars, gradient_contains_error_var, fixed_effects_ptr, approx_Hessian);
								neg_step_dir = approx_Hessian.llt().solve(grad);
							}
						}
						else if (optimizer_cov_pars_ == "fisher_scoring") {
							CHECK(gauss_likelihood_);
							// We don't profile out sigma2 since this seems better for Fisher scoring (less iterations)	
							CalcGradPars(cov_aux_pars.segment(0, num_cov_par_), 1., true, false, grad, unused_dummy, gradient_contains_error_var, true, fixed_effects_ptr, false);
							CalcFisherInformation(cov_aux_pars.segment(0, num_cov_par_), approx_Hessian, true, gradient_contains_error_var, true);
							neg_step_dir = approx_Hessian.llt().solve(grad);
						}
						if (optimizer_cov_pars_ == "gradient_descent") {
							AvoidTooLargeLearningRatesCovAuxPars(neg_step_dir);// Avoid too large learning rates for covariance parameters and aux_pars (for fisher_scoring and newton, this is done non-permanently in 'UpdateCovAuxPars')
						}
						CalcDirDerivArmijoAndLearningRateConstChangeCovAuxPars(grad, neg_step_dir, cov_aux_pars, cov_pars_after_grad_aux, use_nesterov_acc);
						if (called_in_GPBoost_algorithm && reuse_learning_rates_from_previous_call &&
							cov_pars_have_been_estimated_once_ && optimizer_cov_pars_ == "gradient_descent") {//potentially increase learning rates again in GPBoost algorithm
							PotentiallyIncreaseLearningRatesForGPBoostAlgorithm();
						}//end called_in_GPBoost_algorithm / potentially increase learning rates again
						// Update covariance and additional likelihood parameters, do learning rate backtracking, factorize covariance matrix, and calculate new value of objective function
						UpdateCovAuxPars(cov_aux_pars, neg_step_dir, use_nesterov_acc, num_iter_,
							cov_pars_after_grad_aux, cov_aux_pars_after_grad_aux_lag1, acc_rate_cov_, nesterov_schedule_version_, momentum_offset_, fixed_effects_ptr);
						if (num_iter_ == 0) {
							lr_cov_after_first_iteration_ = lr_cov_;
							lr_is_small_threshold_cov_ = lr_cov_ / 1e4;
							if (estimate_aux_pars_) {
								lr_aux_pars_after_first_iteration_ = lr_aux_pars_;
								lr_is_small_threshold_aux_ = lr_aux_pars_ / 1e4;
							}
							if (called_in_GPBoost_algorithm && reuse_learning_rates_from_previous_call &&
								!cov_pars_have_been_estimated_once_ && optimizer_cov_pars_ == "gradient_descent") {
								lr_cov_after_first_optim_boosting_iteration_ = lr_cov_;
								if (estimate_aux_pars_) {
									lr_aux_pars_after_first_optim_boosting_iteration_ = lr_aux_pars_;
								}
							}
						}
						// In case lr_cov_ is very small, we monitor whether the other parameters (beta, aux_pars) continue to change. If yes, we will reset lr_cov_ to its initial value
						if (lr_cov_ < lr_is_small_threshold_cov_ && (has_covariates_ || estimate_aux_pars_) && !lr_cov_is_small) {
							if ((cov_aux_pars.segment(0, num_cov_par_) - cov_aux_pars_lag1.segment(0, num_cov_par_)).norm() <
								LR_IS_SMALL_REL_CHANGE_IN_PARS_THRESHOLD_ * cov_aux_pars_lag1.segment(0, num_cov_par_).norm()) {//also require that relative change in parameters is small
								lr_cov_is_small = true;
								if (has_covariates_) {
									beta_before_lr_cov_small = beta_;
								}
								if (estimate_aux_pars_) {
									aux_pars_before_lr_cov_small = cov_aux_pars.segment(num_cov_par_, NumAuxPars());
								}
							}
						}
						// In case lr_aux_pars_ is very small, we monitor whether the other parameters (beta, covariance parameters) continue to change. If yes, we will reset lr_aux_pars_ to its initial value
						if (estimate_aux_pars_) {
							if (lr_aux_pars_ < lr_is_small_threshold_aux_ && !lr_cov_is_small) {
								if ((cov_aux_pars.segment(num_cov_par_, NumAuxPars()) - cov_aux_pars_lag1.segment(num_cov_par_, NumAuxPars())).norm() <
									LR_IS_SMALL_REL_CHANGE_IN_PARS_THRESHOLD_ * cov_aux_pars_lag1.segment(num_cov_par_, NumAuxPars()).norm()) {//also require that relative change in parameters is small
									lr_aux_pars_is_small = true;
									if (has_covariates_) {
										beta_before_lr_aux_pars_small = beta_;
									}
									cov_pars_before_lr_aux_pars_small = cov_aux_pars.segment(0, num_cov_par_);
								}
							}
						}
						// Reset lr_coef_ to its initial value in case cov_aux_pars changes substantially after lr_coef_ is very small
						bool mode_hast_just_been_recalculated = false;
						if (lr_coef_is_small && has_covariates_) {
							if ((cov_aux_pars - cov_aux_pars_before_lr_coef_small).norm() > MIN_REL_CHANGE_IN_OTHER_PARS_FOR_RESETTING_LR_ * cov_aux_pars_before_lr_coef_small.norm()) {
								lr_coef_ = lr_coef_after_first_iteration_;
								lr_coef_is_small = false;
								RecalculateModeLaplaceApprox(fixed_effects_ptr);
								mode_hast_just_been_recalculated = true;
							}
						}
						// Reset lr_aux_pars_ to its initial values in case covariance paremeters change substantially after lr_aux_pars_ is very small
						if (lr_aux_pars_is_small && estimate_aux_pars_) {
							if ((cov_aux_pars.segment(0, num_cov_par_) - cov_pars_before_lr_aux_pars_small).norm() > MIN_REL_CHANGE_IN_OTHER_PARS_FOR_RESETTING_LR_ * cov_pars_before_lr_aux_pars_small.norm()) {
								lr_aux_pars_ = lr_aux_pars_after_first_iteration_;
								lr_aux_pars_is_small = false;
								if (!mode_hast_just_been_recalculated) {
									RecalculateModeLaplaceApprox(fixed_effects_ptr);
									mode_hast_just_been_recalculated = true;
								}
							}
						}
						// Reset lr_cov_ to its initial values in case aux_pars changes substantially after lr_cov_ is very small
						if (lr_cov_is_small && estimate_aux_pars_) {
							if ((cov_aux_pars.segment(num_cov_par_, NumAuxPars()) - aux_pars_before_lr_cov_small).norm() > MIN_REL_CHANGE_IN_OTHER_PARS_FOR_RESETTING_LR_ * aux_pars_before_lr_cov_small.norm()) {
								lr_cov_ = lr_cov_after_first_iteration_;
								lr_cov_is_small = false;
								if (!mode_hast_just_been_recalculated) {
									RecalculateModeLaplaceApprox(fixed_effects_ptr);
								}
							}
						}
					}//end if (learn_covariance_parameters)
					else {
						neg_log_likelihood_ = neg_log_likelihood_after_lin_coef_update_;
					}// end update covariance parameters
					// Check for NA or Inf
					if (std::isnan(neg_log_likelihood_) || std::isinf(neg_log_likelihood_)) {
						na_or_inf_occurred = true;
						terminate_optim = true;
					}
					else {
						if (learn_covariance_parameters) {
							for (int i = 0; i < (int)cov_aux_pars.size(); ++i) {
								if (std::isnan(cov_aux_pars[i]) || std::isinf(cov_aux_pars[i])) {
									na_or_inf_occurred = true;
									terminate_optim = true;
								}
							}
						}
					}
					if (!na_or_inf_occurred) {
						// Check convergence
						terminate_optim = CheckOptimizerHasConverged(cov_aux_pars, cov_aux_pars_lag1, beta_lag1);
						if (learn_covariance_parameters && ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(terminate_optim)) {
							RedetermineNearestNeighborsVecchiaInducingPointsFITC(terminate_optim);//called only in certain iterations if gp_approx_ == "vecchia" and neighbors are selected based on correlations and not distances
							if (convergence_criterion_ == "relative_change_in_log_likelihood") {
								//recalculate old and new objective function when neighbors have been redetermined
								if (has_covariates_) {
									UpdateFixedEffects(beta_lag1, fixed_effects, fixed_effects_vec);
									fixed_effects_ptr = fixed_effects_vec.data();
								}
								if (estimate_aux_pars_) {
									SetAuxPars(cov_aux_pars_lag1.data() + num_cov_par_);
								}
								CalcCovFactorOrModeAndNegLL(cov_aux_pars_lag1.segment(0, num_cov_par_), fixed_effects_ptr);//recalculate old log-likelihood
								neg_log_likelihood_lag1_ = neg_log_likelihood_;
								if (has_covariates_) {
									UpdateFixedEffects(beta_, fixed_effects, fixed_effects_vec);
									fixed_effects_ptr = fixed_effects_vec.data();
								}
								if (estimate_aux_pars_) {
									SetAuxPars(cov_aux_pars.data() + num_cov_par_);
								}
								CalcCovFactorOrModeAndNegLL(cov_aux_pars.segment(0, num_cov_par_), fixed_effects_ptr);//recalculate new log-likelihood
								terminate_optim = CheckOptimizerHasConverged(cov_aux_pars, cov_aux_pars_lag1, beta_lag1);
							}
						}
						// Trace output for convergence monitoring
						if ((num_iter_ < 10 || ((num_iter_ + 1) % 10 == 0 && (num_iter_ + 1) < 100) || ((num_iter_ + 1) % 100 == 0 && (num_iter_ + 1) < 1000) ||
							((num_iter_ + 1) % 1000 == 0 && (num_iter_ + 1) < 10000) || ((num_iter_ + 1) % 10000 == 0)) && (num_iter_ != (max_iter_ - 1)) && !terminate_optim) {
							Log::REDebug("GPModel: parameters after optimization iteration number %d: ", num_iter_ + 1);
							PrintTraceParameters(cov_aux_pars.segment(0, num_cov_par_), beta_, cov_aux_pars.data() + num_cov_par_, learn_covariance_parameters);
							if (gauss_likelihood_) {
								Log::REDebug("Negative log-likelihood: %g", neg_log_likelihood_);
							}
							else {
								Log::REDebug("Approximate negative marginal log-likelihood: %g", neg_log_likelihood_);
							}
						} // end trace output
					}// end not na_or_inf_occurred
					// Check whether to terminate
					if (terminate_optim) {
						num_it = num_iter_ + 1;
						break;
					}
					//increase learning rates again
					else if (increase_learning_rate_again_ && optimizer_cov_pars_ == "gradient_descent" && (num_iter_ + 1) >= 10 &&
						learning_rate_decreased_first_time_ && !learning_rate_decreased_after_increase_ && !na_or_inf_occurred) {
						if ((neg_log_likelihood_lag1_ - neg_log_likelihood_) < INCREASE_LR_CHANGE_LL_THRESHOLD_ * std::max(std::abs(neg_log_likelihood_lag1_), 1.)) {
							if (has_covariates_) {
								lr_coef_ /= LR_SHRINKAGE_FACTOR_;
							}
							if (learn_covariance_parameters) {
								lr_cov_ /= LR_SHRINKAGE_FACTOR_;
							}
							if (estimate_aux_pars_) {
								lr_aux_pars_ /= LR_SHRINKAGE_FACTOR_;
							}
							learning_rate_increased_after_descrease_ = true;
							Log::REDebug("GPModel covariance parameter estimation: The learning rates have been increased in iteration number %d ", num_iter_ + 1);
						}
					}
				}//end for loop for optimization
			}//end internal optimizers such as "gradient_descent" or "fisher_scoring"
			// redo optimization with "nelder_mead" in case NA or Inf occurred
			if (na_or_inf_occurred && optimizer_cov_pars_ != "nelder_mead") {
				string_t optimizers = "";
				for (auto elem : SUPPORTED_OPTIM_COV_PAR_) {
					if (gauss_likelihood_ || elem != "fisher_scoring") {
						optimizers += " '" + elem + "'";
					}
				}
				Log::REWarning("NaN or Inf occurred in covariance parameter optimization using '%s'. "
					"The optimization will be started a second time using 'nelder_mead'. "
					"If you want to avoid this, try directly using a different optimizer. "
					"If you have used 'gradient_descent', you can also consider using a smaller learning rate ", optimizer_cov_pars_.c_str());
				cov_aux_pars = cov_aux_pars_init;
				if (has_covariates_) {
					beta_ = beta_init;
				}
				if (!gauss_likelihood_) { // reset the initial modes to 0
					for (const auto& cluster_i : unique_clusters_) {
						likelihood_[cluster_i]->InitializeModeAvec();
					}
				}
				delta_rel_conv_ = delta_rel_conv_init_;
				OptimExternal<T_mat, T_chol>(this, cov_aux_pars, beta_, fixed_effects, max_iter_,
					delta_rel_conv_, convergence_criterion_, num_it,
					learn_covariance_parameters, "nelder_mead", profile_out_error_variance_, false,
					neg_log_likelihood_, num_cov_par_, NumAuxPars(), GetAuxPars(), has_covariates_, lr_cov_init_, reuse_m_bfgs_from_previous_call,
					m_lbfgs_);
			}
			if (num_it == max_iter_) {
				Log::REDebug("GPModel: no convergence after the maximal number of iterations "
					"(%d, nb. likelihood evaluations = %d) ", max_iter_, num_ll_evaluations_);
			}
			else {
				Log::REDebug("GPModel: parameter estimation finished after %d iteration "
					"(nb. likelihood evaluations = %d) ", num_it, num_ll_evaluations_);
			}
			PrintTraceParameters(cov_aux_pars.segment(0, num_cov_par_), beta_, cov_aux_pars.data() + num_cov_par_, learn_covariance_parameters);
			if (gauss_likelihood_) {
				Log::REDebug("Negative log-likelihood: %g", neg_log_likelihood_);
			}
			else {
				Log::REDebug("Approximate negative marginal log-likelihood: %g", neg_log_likelihood_);
			}
			CovarianceParameterRangeWarning(cov_aux_pars.segment(0, num_cov_par_));
			vec_t cov_pars_var_const_maybe;
			MaybeKeepVarianceConstant(cov_aux_pars.segment(0, num_cov_par_), cov_pars_var_const_maybe);
			for (int i = 0; i < num_cov_par_; ++i) {
				cov_aux_pars[i] = cov_pars_var_const_maybe[i];
				optim_cov_pars[i] = cov_aux_pars[i];
			}
			if (estimate_aux_pars_) {
				SetAuxPars(cov_aux_pars.data() + num_cov_par_);
			}
			if (has_covariates_) {
				if (scale_covariates_) {
					// transform coefficients back to original scale
					TransformBackCoef(beta_, beta_);
						}
				for (int i = 0; i < num_covariates_ * num_sets_re_; ++i) {
					optim_coef[i] = beta_[i];
				}
			}
			model_has_been_estimated_ = true;
			if (learn_covariance_parameters) {
				cov_pars_have_been_estimated_once_ = true;
				cov_pars_have_been_estimated_during_last_call_ = true;
			}
			else {
				cov_pars_have_been_estimated_during_last_call_ = false;
			}
			if (has_covariates_ && !only_intercept_for_GPBoost_algo) {
				coef_have_been_estimated_once_ = true;
			}
			}//end max_iter_ > 0
			if (has_covariates_) {
				if (called_in_GPBoost_algorithm) {
					has_covariates_ = false;
					// When this function is called in the GPBoost algorithm for finding an intial intercept or a learning rate,
					//	we set has_covariates_ to false in order to avoid potential problems when making predictions with the GPBoostOOS algorithm,
					//	since in the second phase of the GPBoostOOS algorithm covariance parameters are not estimated (and thus has_covariates_ is not set to false)
					//	but this function is called for initialization of the GPBoost algorithm.
				}
				if (scale_covariates_) {
					//transform covariates back
					for (int icol = 0; icol < num_covariates_; ++icol) {
						if (!has_intercept_ || icol != intercept_col_) {
							X_.col(icol).array() *= scale_transf_[icol];
							X_.col(icol).array() += loc_transf_[icol];
			}
					}
					if (has_intercept_) {
						X_.col(intercept_col_).array() = 1.;
					}
				}//end scale_covariates_
			}//end has_covariates_
			optimization_running_currently_ = false;
		}//end OptimLinRegrCoefCovPar

		bool CanCalculateStandardErrorsCovPars() const {
			return (gauss_likelihood_ && gp_approx_ != "full_scale_vecchia");
		}

		/*!
		* \brief Calculate standard errors of covariance parameters (for Gaussian likelihood only)
		* \para optim_cov_pars Optimal covariance parameters
		* \param[out] std_dev_cov_par Standard deviations for the covariance parameters (can be nullptr, used only if calc_std_dev)
		*/
		void CalculateStandardErrorsCovPars(const double* optim_cov_pars,
			double* std_dev_cov_par) {
			vec_t std_dev_cov(num_cov_par_);
			vec_t cov_pars(num_cov_par_);
			for (int i = 0; i < num_cov_par_; ++i) {
				cov_pars[i] = optim_cov_pars[i];
			}
			if (gauss_likelihood_) {
				if (gp_approx_ == "full_scale_vecchia") {
					Log::REFatal("Calculation of standard errors of covariance parameters for the approximation '%s' is currently not supported ", gp_approx_.c_str());
				}
				CalcStdDevCovPar(cov_pars, std_dev_cov);//TODO: maybe another call to CalcCovFactor can be avoided in CalcStdDevCovPar (need to take care of cov_pars[0])
			}
			else {
				std_dev_cov.setZero();// Calculation of standard deviations for covariance parameters is not supported for non-Gaussian likelihoods
				Log::REFatal("Calculation of standard errors of covariance parameters for non-Gaussian likelihoods is currently not supported ");
			}
			for (int i = 0; i < num_cov_par_; ++i) {
				std_dev_cov_par[i] = std_dev_cov[i];
			}
		}//end CalculateStandardErrorsCovPars

		/*!
		* \brief Calculate standard errors of regression coefficients
		* \para optim_cov_pars Optimal covariance parameters
		* \param[out] std_dev_coef Standard deviations for the coefficients (can be nullptr, used only if calc_std_dev and if covariate_data is not nullptr)
		*/
		void CalculateStandardErrorsCoefs(const double* optim_cov_pars,
			double* std_dev_coef) {
			vec_t cov_pars(num_cov_par_);
			for (int i = 0; i < num_cov_par_; ++i) {
				cov_pars[i] = optim_cov_pars[i];
			}
			CHECK(has_covariates_);
			vec_t std_dev_beta(num_covariates_);
			if (gauss_likelihood_) {
				CalcStdDevCoef(cov_pars, X_, std_dev_beta);
			}
			else {
				const double* fixed_effects_ptr = nullptr;
				if (has_fixed_effects_) {
					fixed_effects_ptr = fixed_effects_.data();
				}
				CalcStdDevCoefNonGaussian(num_covariates_, beta_, cov_pars, fixed_effects_ptr, std_dev_beta);
			}
			for (int i = 0; i < num_covariates_; ++i) {
				std_dev_coef[i] = std_dev_beta[i];
			}
		}//end CalculateStandardErrorsCoefs

		bool CheckOptimizerHasConverged(const vec_t& cov_aux_pars,
			const vec_t& cov_aux_pars_lag1,
			const vec_t& beta_lag1) const {
			if (convergence_criterion_ == "relative_change_in_parameters") {
				if (has_covariates_) {
					if (((beta_ - beta_lag1).norm() <= delta_rel_conv_ * beta_lag1.norm()) && ((cov_aux_pars - cov_aux_pars_lag1).norm() < delta_rel_conv_ * cov_aux_pars_lag1.norm())) {
						return true;
					}
				}
				else {
					if ((cov_aux_pars - cov_aux_pars_lag1).norm() <= delta_rel_conv_ * cov_aux_pars_lag1.norm()) {
						return true;
					}
				}
			}
			else if (convergence_criterion_ == "relative_change_in_log_likelihood") {
				if ((neg_log_likelihood_lag1_ - neg_log_likelihood_) <= delta_rel_conv_ * std::max(std::abs(neg_log_likelihood_lag1_), 1.)) {
					return true;
				}
			} // end check convergence
			return false;
		}//CheckOptimizerHasConverged

		/*!
		* \brief Calculate gradient wrt the covariance and auxiliary parameters and regression coefficients
		*	Call 'CalcCovFactorOrModeAndNegLL' first since this function assumes that the covariance matrix has been factorized (by 'CalcCovFactor') and
		*		that y_aux or y_tilde/y_tilde2 (if use_woodbury_identity_) have been calculated (by 'CalcYAux' or 'CalcYtilde')
		*	The gradient wrt covariance and auxiliary parameters is calculated on the log-scale
		* \param cov_pars_in Covariance parameters
		* \param marg_var Marginal variance parameters sigma^2 (only used for Gaussian data)
		* \param calc_cov_aux_par_grad If true, the gradient wrt covariance and auxiliary parameters is calculated
		* \param calc_beta_grad If true, the gradient wrt the regression coefficients is calculated
		* \param[out] grad_cov_aux_par Gradient wrt the covariance parameters and any additional parameters for the likelihood for non-Gaussian likelihoods
		* \param[out] grad_beta Gradient for linear regression coefficients
		* \param include_error_var If true, the gradient with respect to the error variance parameter (=nugget effect) is also calculated, otherwise not (set this to true if the nugget effect is not calculated by using the closed-form solution)
		* \param save_psi_inv_for_FI If true, the inverse covariance matrix Psi^-1 is saved for reuse later (e.g. when calculating the Fisher information in Fisher scoring). This option is ignored if the Vecchia approximation is used.
		*		 For iterative methods for grouped random effects, if true, P^(-1) z_i is saved for later reuse when calculating the Fisher information.
		* \param fixed_effects Fixed effects component of location parameter (used only for non-Gaussian likelihoods)
		* \param call_for_std_dev_coef If true, the function is called for calculating standard deviations of linear regression coefficients
		*/
		void CalcGradPars(const vec_t& cov_pars_in,
			double marg_var,
			bool calc_cov_aux_par_grad,
			bool calc_beta_grad,
			vec_t& grad_cov_aux_par,
			vec_t& grad_beta,
			bool include_error_var,
			bool save_psi_inv_for_FI,
			const double* fixed_effects,
			bool call_for_std_dev_coef) {
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			if ((gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") && calc_cov_aux_par_grad) {
				CalcGradientVecchia(true, 1., false);
			}
			if (gauss_likelihood_) {//Gaussian likelihood
				if (calc_cov_aux_par_grad) {
					grad_cov_aux_par = include_error_var ? vec_t::Zero(num_cov_par_) : vec_t::Zero(num_cov_par_ - 1);
					int first_cov_par = include_error_var ? 1 : 0;
					for (const auto& cluster_i : unique_clusters_) {
						if (gp_approx_ == "vecchia") {//Vechia approximation
							vec_t u(num_data_per_cluster_[cluster_i]);
							vec_t uk(num_data_per_cluster_[cluster_i]);
							if (include_error_var) {
								u = B_[cluster_i][0] * y_[cluster_i];
								if (estimate_cov_par_index_[0] > 0) {
									grad_cov_aux_par[0] += -1. * ((double)(u.transpose() * D_inv_[cluster_i][0] * u)) / cov_pars[0] / 2. + num_data_per_cluster_[cluster_i] / 2.;
								}
								u = D_inv_[cluster_i][0] * u;
							}
							else {
								u = D_inv_[cluster_i][0] * B_[cluster_i][0] * y_[cluster_i];//TODO: this is already calculated in CalcYAux -> save it there and re-use here?
							}
							for (int j = 0; j < num_comps_total_; ++j) {
								int num_par_comp = re_comps_vecchia_[cluster_i][0][j]->num_cov_par_;
								for (int ipar = 0; ipar < num_par_comp; ++ipar) {
									if (estimate_cov_par_index_[ind_par_[j] + ipar] > 0) {
										uk = B_grad_[cluster_i][0][num_par_comp * j + ipar] * y_[cluster_i];
										grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += ((uk.dot(u) - 0.5 * u.dot(D_grad_[cluster_i][0][num_par_comp * j + ipar] * u)) / cov_pars[0] +
											0.5 * (D_inv_[cluster_i][0].diagonal()).dot(D_grad_[cluster_i][0][num_par_comp * j + ipar].diagonal()));
									}
								}
							}
						}//end gp_approx_ == "vecchia"
						else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
							CalcGradPars_FITC_FSA_GaussLikelihood_Cluster_i(cov_pars[0], grad_cov_aux_par, include_error_var, first_cov_par, cluster_i);
						}// end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
						else if (use_woodbury_identity_) {
							CalcGradPars_Only_Grouped_REs_Woodbury_GaussLikelihood_Cluster_i(cov_pars, grad_cov_aux_par, include_error_var, save_psi_inv_for_FI, first_cov_par, cluster_i);
						}//end use_woodbury_identity_
						else {//not use_woodbury_identity_
							T_mat psi_inv;
							CalcPsiInv(psi_inv, cluster_i, !save_psi_inv_for_FI);
							if (save_psi_inv_for_FI) {//save for latter use when calculating the Fisher information
								psi_inv_[cluster_i] = psi_inv;
							}
							if (include_error_var) {
								if (estimate_cov_par_index_[0] > 0) {
									grad_cov_aux_par[0] += -1. * ((double)(y_[cluster_i].transpose() * y_aux_[cluster_i])) / cov_pars[0] / 2. + num_data_per_cluster_[cluster_i] / 2.;
								}
							}
							for (int j = 0; j < num_comps_total_; ++j) {
								for (int ipar = 0; ipar < re_comps_[cluster_i][0][j]->num_cov_par_; ++ipar) {
									if (estimate_cov_par_index_[ind_par_[j] + ipar] > 0) {
										std::shared_ptr<T_mat> gradPsi = re_comps_[cluster_i][0][j]->GetZSigmaZtGrad(ipar, true, 1.);
										grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += -1. * ((double)(y_aux_[cluster_i].transpose() * (*gradPsi) * y_aux_[cluster_i])) / cov_pars[0] / 2. +
											((double)(((*gradPsi).cwiseProduct(psi_inv)).sum())) / 2.;
									}
								}
							}
						}//end not use_woodbury_identity_
					}// end loop over clusters
				}//end grad_cov_aux_par
				if (calc_beta_grad) {
					if (use_woodbury_identity_ && matrix_inversion_method_ != "iterative") {// calculate y_aux = Psi^-1*y (in most cases, this has been already calculated when calling 'CalcCovFactorOrModeAndNegLL' before this)
						CalcYAux(1., false);
					}
					vec_t y_aux(num_data_);
					GetYAux(y_aux);
					grad_beta = (-1. / marg_var) * (X_.transpose()) * y_aux;
				}//end calc_beta_grad
			}//end gauss_likelihood_
			else {//not gauss_likelihood_
				vec_t grad_cov_aux_cluster_i, grad_F;
				const double* fixed_effects_cluster_i_ptr = nullptr;
				double* grad_cov_clus_i_ptr = nullptr;
				double* grad_aux_clus_i_ptr = nullptr;
				vec_t fixed_effects_cluster_i;
				if(calc_cov_aux_par_grad) {
					CHECK(!include_error_var);
					int length_cov_grad = num_cov_par_;
					if (estimate_aux_pars_) {
						length_cov_grad += NumAuxPars();
					}
					grad_cov_aux_par = vec_t::Zero(length_cov_grad);
					grad_cov_aux_cluster_i = vec_t::Zero(length_cov_grad);
					grad_cov_clus_i_ptr = grad_cov_aux_cluster_i.data();
					if (estimate_aux_pars_) {
						grad_aux_clus_i_ptr = grad_cov_aux_cluster_i.data() + num_cov_par_;
					}
				}
				if (calc_beta_grad) {
					grad_F = vec_t(num_data_ * num_sets_re_);
				}
				bool calc_grad_aux_par = calc_cov_aux_par_grad && estimate_aux_pars_;
				for (const auto& cluster_i : unique_clusters_) {
					vec_t grad_F_cluster_i;
					if (calc_beta_grad) {
						grad_F_cluster_i = vec_t(num_data_per_cluster_[cluster_i] * num_sets_re_);
					}
					//map fixed effects to clusters (if needed)
					if (num_clusters_ == 1 && (gp_approx_ != "vecchia" || vecchia_ordering_ == "none")) {//only one cluster / independent realization and order of data does not matter
						fixed_effects_cluster_i_ptr = fixed_effects;
					}
					else if (fixed_effects != nullptr) {//more than one cluster and order of samples matters
						fixed_effects_cluster_i = vec_t(num_data_per_cluster_[cluster_i] * num_sets_re_);
						for (int igp = 0; igp < num_sets_re_; ++igp) {
#pragma omp parallel for schedule(static)
							for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
								fixed_effects_cluster_i[j + num_data_per_cluster_[cluster_i] * igp] = fixed_effects[data_indices_per_cluster_[cluster_i][j] + num_data_ * igp];
							}
						}
						fixed_effects_cluster_i_ptr = fixed_effects_cluster_i.data();
					}
					if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
						likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxVecchia(y_[cluster_i].data(), y_int_[cluster_i].data(),
							fixed_effects_cluster_i_ptr, B_[cluster_i], D_inv_[cluster_i], B_grad_[cluster_i], D_grad_[cluster_i],
							calc_cov_aux_par_grad, calc_beta_grad, calc_grad_aux_par,
							grad_cov_clus_i_ptr, grad_F_cluster_i,
							grad_aux_clus_i_ptr, false, num_comps_total_, call_for_std_dev_coef, re_comps_ip_preconditioner_[cluster_i][0],
							re_comps_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0],
							cluster_i, this, estimate_cov_par_index_, GPU_use_);
					}
					else if (gp_approx_ == "fitc") {
						likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxFITC(y_[cluster_i].data(), y_int_[cluster_i].data(),
							fixed_effects_cluster_i_ptr, re_comps_ip_[cluster_i][0][0]->GetZSigmaZt(), chol_fact_sigma_ip_[cluster_i][0],
							re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr(), fitc_resid_diag_[cluster_i], re_comps_ip_[cluster_i][0], re_comps_cross_cov_[cluster_i][0],
							calc_cov_aux_par_grad, calc_beta_grad, calc_grad_aux_par,
							grad_cov_clus_i_ptr, grad_F_cluster_i, grad_aux_clus_i_ptr, false, call_for_std_dev_coef, estimate_cov_par_index_, GPU_use_);
					}
					else if (gp_approx_ == "full_scale_vecchia") {
						likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxFSVA(y_[cluster_i].data(), y_int_[cluster_i].data(),
							fixed_effects_cluster_i_ptr, chol_fact_sigma_ip_[cluster_i][0],
							chol_fact_sigma_woodbury_[cluster_i], chol_ip_cross_cov_[cluster_i][0], sigma_woodbury_[cluster_i], re_comps_ip_[cluster_i][0], re_comps_cross_cov_[cluster_i][0],
							B_[cluster_i][0], D_inv_[cluster_i][0], B_T_D_inv_B_cross_cov_[cluster_i][0], D_inv_B_cross_cov_[cluster_i][0],
							sigma_ip_inv_cross_cov_T_[cluster_i][0], B_grad_[cluster_i][0], D_grad_[cluster_i][0],
							calc_cov_aux_par_grad, calc_beta_grad, calc_grad_aux_par, grad_cov_clus_i_ptr, grad_F_cluster_i, grad_aux_clus_i_ptr,
							false, call_for_std_dev_coef, re_comps_ip_preconditioner_[cluster_i][0], re_comps_cross_cov_preconditioner_[cluster_i][0],
							chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0], estimate_cov_par_index_, GPU_use_);
					}
					else if ((use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) || grouped_RE_and_vecchia_GP_) {
						likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxGroupedRE(y_[cluster_i].data(), y_int_[cluster_i].data(),
							fixed_effects_cluster_i_ptr, SigmaI_[cluster_i], grouped_RE_and_vecchia_GP_, B_[cluster_i][0], D_inv_[cluster_i][0], B_grad_[cluster_i][0], D_grad_[cluster_i][0], 
							cum_num_rand_eff_[cluster_i], calc_cov_aux_par_grad, calc_beta_grad, calc_grad_aux_par,
							grad_cov_clus_i_ptr, grad_F_cluster_i, grad_aux_clus_i_ptr, false, call_for_std_dev_coef, estimate_cov_par_index_);
					}
					else if (only_one_grouped_RE_calculations_on_RE_scale_) {
						likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxOnlyOneGroupedRECalculationsOnREScale(y_[cluster_i].data(), y_int_[cluster_i].data(),
							fixed_effects_cluster_i_ptr, re_comps_[cluster_i][0][0]->cov_pars_[0],
							calc_cov_aux_par_grad, calc_beta_grad, calc_grad_aux_par,
							grad_cov_clus_i_ptr, grad_F_cluster_i, grad_aux_clus_i_ptr, false, call_for_std_dev_coef, estimate_cov_par_index_);
					}
					else {
						likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxStable(y_[cluster_i].data(), y_int_[cluster_i].data(),
							fixed_effects_cluster_i_ptr, ZSigmaZt_[cluster_i], re_comps_[cluster_i][0],
							calc_cov_aux_par_grad, calc_beta_grad, calc_grad_aux_par,
							grad_cov_clus_i_ptr, grad_F_cluster_i, grad_aux_clus_i_ptr, false, call_for_std_dev_coef, estimate_cov_par_index_);
					}
					if(calc_cov_aux_par_grad) {
						grad_cov_aux_par += grad_cov_aux_cluster_i;
					}
					if (calc_beta_grad) {
						if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {//only one cluster / independent realization and order of data does not matter
#pragma omp parallel for schedule(static)//write on output
							for (int j = 0; j < num_data_ * num_sets_re_; ++j) {
								grad_F[j] = grad_F_cluster_i[j];
							}
						}
						else {//more than one cluster and order of samples matters
							for (int igp = 0; igp < num_sets_re_; ++igp) {
#pragma omp parallel for schedule(static)
								for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
									grad_F[data_indices_per_cluster_[cluster_i][j] + igp * num_data_] = grad_F_cluster_i[j + igp * num_data_per_cluster_[cluster_i]];
								}
							}
						} // end more than one cluster
					}
				}// end loop over clusters
				if (calc_beta_grad) {
					grad_beta = vec_t(num_sets_re_ * num_covariates_);
					for (int igp = 0; igp < num_sets_re_; ++igp) {
						grad_beta.segment(igp * num_covariates_, num_covariates_) = (X_.transpose()) * (grad_F.segment(igp * num_data_, num_data_));
					}
				}
			}//end not gauss_likelihood_
			// Check for NAs and Inf
			if (calc_cov_aux_par_grad) {
				for (int i = 0; i < (int)grad_cov_aux_par.size(); ++i) {
					if (std::isnan(grad_cov_aux_par[i])) {
						grad_cov_aux_par[i] = 0.;
						Log::REWarning("NaN occured in gradient wrt covariance / auxiliary parameter number %d (counting starts at 1, total nb. par. = %d). This is replaced with 0 ", i + 1, grad_cov_aux_par.size());
					}
					else if (std::isinf(grad_cov_aux_par[i])) {
						grad_cov_aux_par[i] = 0.;
						Log::REWarning("Inf occured in gradient wrt covariance / auxiliary parameter number %d (counting starts at 1, total nb. par. = %d). This is replaced with 0 ", i + 1, grad_cov_aux_par.size());
					}
				}
			}
			if (calc_beta_grad) {
				for (int i = 0; i < (int)grad_beta.size(); ++i) {
					if (std::isnan(grad_beta[i])) {
						grad_beta[i] = 0.;
						Log::REWarning("NaN occured in gradient wrt regression coefficient number %d (counting starts at 1, total nb. par. = %d). This is replaced with 0 ", i + 1, grad_beta.size());
					}
					else if (std::isinf(grad_beta[i])) {
						grad_beta[i] = 0.;
						Log::REWarning("Inf occured in gradient wrt regression coefficient number %d (counting starts at 1, total nb. par. = %d). This is replaced with 0 ", i + 1, grad_beta.size());
					}
				}
			}
			// For debugging
			//if (calc_cov_aux_par_grad) {
			//	for (int i = 0; i < (int)grad_cov_aux_par.size(); ++i) {
			//		Log::REDebug("grad_cov_aux_par[%d]: %g", i, grad_cov_aux_par[i]);
			//	}
			//}
			//if (calc_beta_grad) {
			//	for (int i = 0; i < (int)grad_beta.size(); ++i) {
			//		Log::REDebug("grad_beta[%d]: %g", i, grad_beta[i]);
			//	}
			//}
		}//end CalcGradPars

		/*!
		* \brief Calculate gradient wrt the covariance and auxiliary parameters for the FITC and FSA approximations for Gaussian likelihoods
		*	The gradient wrt covariance and auxiliary parameters is calculated on the log-scale
		*/
		void CalcGradPars_FITC_FSA_GaussLikelihood_Cluster_i(double error_var,
			vec_t& grad_cov_aux_par,
			bool include_error_var,
			int first_cov_par,
			data_size_t cluster_i) {
			CHECK(gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia");
			CHECK(gauss_likelihood_);
			if (include_error_var) {
				if (estimate_cov_par_index_[0] > 0) {
					grad_cov_aux_par[0] += -1. * ((double)(y_[cluster_i].transpose() * y_aux_[cluster_i])) / error_var / 2. + num_data_per_cluster_[cluster_i] / 2.;
				}
			}
			for (int j = 0; j < num_comps_total_; ++j) {
				int num_par_comp = re_comps_ip_[cluster_i][0][j]->num_cov_par_;
				// sigma_cross_cov
				const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][j]->GetSigmaPtr();
				// sigma_ip^-1 * sigma_cross_cov * sigma^-1 * y
				vec_t sigma_ip_inv_cross_cov_y_aux = chol_fact_sigma_ip_[cluster_i][0].solve((*cross_cov).transpose() * y_aux_[cluster_i]);
				// Initialize Matrices
				den_mat_t sigma_resid_inv_cross_cov_T;
				std::shared_ptr<T_mat> sigma_resid;
				den_mat_t rand_vec_probe_P_inv;
				if (matrix_inversion_method_ == "cholesky" && gp_approx_ == "full_scale_tapering") {
					// sigma_resid^-1 * t(cross_cov)
					sigma_resid_inv_cross_cov_T = chol_fact_resid_[cluster_i].solve((*cross_cov));
					sigma_resid = re_comps_resid_[cluster_i][0][j]->GetZSigmaZt();
					// sigma_resid^-1 with sparsity pattern of sigma_resid
					T_mat Identity(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
					Identity.setIdentity();
					T_mat chol_fact_resid_inv;
					TriangularSolveGivenCholesky<T_chol, T_mat, T_mat, T_mat>(chol_fact_resid_[cluster_i], Identity, chol_fact_resid_inv, false);
					CalcLtLGivenSparsityPattern<T_mat>(chol_fact_resid_inv, (*sigma_resid), true);
					chol_fact_resid_inv.resize(0, 0);
				}
				else if (matrix_inversion_method_ == "iterative") {
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia") {
						Log::REFatal("'iterative' methods are not implemented for gp_approx = '%s'. Use 'cholesky' ", gp_approx_.c_str());
					}
					// P^-1 * sample vectors
					if (cg_preconditioner_type_ == "fitc") {
						const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][j]->GetSigmaPtr();
						den_mat_t diag_sigma_resid_inv_Z = diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * rand_vec_probe_[cluster_i];
						//rand_vec_probe_P_inv = diag_sigma_resid_inv_Z - (diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * ((*cross_cov_preconditioner) * chol_fact_woodbury_preconditioner_[cluster_i].solve((*cross_cov_preconditioner).transpose() * diag_sigma_resid_inv_Z)));
						den_mat_t rand_vec_probe_P_inv_interim1;
						GPBoost::matmul((*cross_cov_preconditioner).transpose(), diag_sigma_resid_inv_Z, rand_vec_probe_P_inv_interim1, GPU_use_);
						den_mat_t rand_vec_probe_P_inv_interim2;
						GPBoost::solve_linear_sys(chol_fact_woodbury_preconditioner_[cluster_i], rand_vec_probe_P_inv_interim1, rand_vec_probe_P_inv_interim2, GPU_use_);
						den_mat_t rand_vec_probe_P_inv_interim3;
						GPBoost::matmul((*cross_cov_preconditioner), rand_vec_probe_P_inv_interim2, rand_vec_probe_P_inv_interim3, GPU_use_);
						rand_vec_probe_P_inv = diag_sigma_resid_inv_Z - (diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * rand_vec_probe_P_inv_interim3);
					}
					else {
						rand_vec_probe_P_inv = rand_vec_probe_[cluster_i];
					}
				}
				vec_t vecchia_y, cross_cov_vecchia_y, woodbury_vecchia_y;
				if (gp_approx_ == "full_scale_vecchia") {
					vecchia_y = D_inv_rm_[cluster_i][0] * (B_rm_[cluster_i][0] * y_[cluster_i]);
					cross_cov_vecchia_y = B_T_D_inv_B_cross_cov_[cluster_i][0].transpose() * y_[cluster_i];
					woodbury_vecchia_y = chol_fact_sigma_woodbury_[cluster_i].solve(cross_cov_vecchia_y);
				}
				for (int ipar = 0; ipar < num_par_comp; ++ipar) {
					if (estimate_cov_par_index_[ind_par_[j] + ipar] > 0) {
						// Derivative of Components
						std::shared_ptr<den_mat_t> cross_cov_grad = re_comps_cross_cov_[cluster_i][0][j]->GetZSigmaZtGrad(ipar, true, 0.);
						den_mat_t sigma_ip_stable_grad = *(re_comps_ip_[cluster_i][0][j]->GetZSigmaZtGrad(ipar, true, 0.));
						den_mat_t cross_cov_grad_t = (*cross_cov_grad).transpose();
						// Trace of sigma_ip^-1 * sigma_ip_grad
						if (matrix_inversion_method_ == "cholesky") {
							den_mat_t sigma_ip_inv_sigma_ip_stable_grad = chol_fact_sigma_ip_[cluster_i][0].solve(sigma_ip_stable_grad);
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] -= 0.5 * sigma_ip_inv_sigma_ip_stable_grad.trace();
						}
						grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += ((0.5 * sigma_ip_inv_cross_cov_y_aux.dot((sigma_ip_stable_grad)*sigma_ip_inv_cross_cov_y_aux)
							- ((cross_cov_grad_t)*y_aux_[cluster_i]).dot(sigma_ip_inv_cross_cov_y_aux)) / error_var);
						// sigma_woodbury_grad
						den_mat_t sigma_woodbury_grad;
						den_mat_t cross_cov_grad_sigma_resid_inv_cross_cov_T;
						if (gp_approx_ == "full_scale_vecchia") {
							// row-major
							sp_mat_rm_t B_grad_rm = sp_mat_rm_t(B_grad_[cluster_i][0][num_par_comp * j + ipar]);
							// B_grad * cross_cov
							den_mat_t cross_cov_B_grad(num_data_per_cluster_[cluster_i], num_ind_points_);
#pragma omp parallel for schedule(static)   
							for (int i = 0; i < num_ind_points_; ++i) {
								cross_cov_B_grad.col(i) = B_grad_rm * (*cross_cov).col(i);
							}
							// row-major
							sp_mat_rm_t D_grad_rm = sp_mat_rm_t(D_grad_[cluster_i][0][num_par_comp * j + ipar]);
							// D_grad * D^-1 * B * t(cross_cov)
							den_mat_t D_grad_sigma_resid_inv_cross_cov_T(num_data_per_cluster_[cluster_i], num_ind_points_);
#pragma omp parallel for schedule(static)   
							for (int i = 0; i < num_ind_points_; ++i) {
								D_grad_sigma_resid_inv_cross_cov_T.col(i) = D_grad_rm * D_inv_B_cross_cov_[cluster_i][0].col(i);
							}
							// cross_crov_grad *  sigma_resid^-1 * t(cross_cov)
							//cross_cov_grad_sigma_resid_inv_cross_cov_T = (*cross_cov_grad).transpose() * B_T_D_inv_B_cross_cov_[cluster_i][0];
							GPBoost::matmul(cross_cov_grad_t, B_T_D_inv_B_cross_cov_[cluster_i][0], cross_cov_grad_sigma_resid_inv_cross_cov_T, GPU_use_);
							// cross_cov * B_grad * D^-1 * B * t(cross_cov)
							//den_mat_t cross_cov_B_grad_cross_cov_T = cross_cov_B_grad.transpose() * D_inv_B_cross_cov_[cluster_i][0];
							den_mat_t cross_cov_B_grad_cross_cov_T;
							GPBoost::matmul(cross_cov_B_grad.transpose(), D_inv_B_cross_cov_[cluster_i][0], cross_cov_B_grad_cross_cov_T, GPU_use_);
							// cross_crov_grad *  sigma_resid^-1 * t(cross_cov) + cross_crov *  sigma_resid^-1 * t(cross_cov_grad) - cross_cov * sigma_resid^-1 * sigma_resid_grad * sigma_resid^-1 * t(cross_cov)
							den_mat_t cross_cov_D_grad_cross_cov_T;
							GPBoost::matmul(D_inv_B_cross_cov_[cluster_i][0].transpose(), D_grad_sigma_resid_inv_cross_cov_T, cross_cov_D_grad_cross_cov_T, GPU_use_);
							sigma_woodbury_grad = cross_cov_grad_sigma_resid_inv_cross_cov_T + cross_cov_grad_sigma_resid_inv_cross_cov_T.transpose() +
								cross_cov_B_grad_cross_cov_T.transpose() + cross_cov_B_grad_cross_cov_T - cross_cov_D_grad_cross_cov_T;
							// t(y) * (vecchia^-1)_grad * y
							vec_t vecchia_grad_y = (B_grad_rm.transpose() * vecchia_y) - (B_t_D_inv_rm_[cluster_i][0] * (D_grad_rm * vecchia_y)) +
								(B_t_D_inv_rm_[cluster_i][0] * (B_grad_rm * y_[cluster_i]));
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += 0.5 * (D_inv_rm_[cluster_i][0].diagonal()).dot(D_grad_rm.diagonal());
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += (0.5 * y_[cluster_i].dot(vecchia_grad_y) - ((*cross_cov).transpose() * vecchia_grad_y).dot(woodbury_vecchia_y) +
								woodbury_vecchia_y.dot(cross_cov_B_grad_cross_cov_T * woodbury_vecchia_y) -
								0.5 * (D_inv_B_cross_cov_[cluster_i][0] * woodbury_vecchia_y).dot((D_grad_sigma_resid_inv_cross_cov_T * woodbury_vecchia_y))) / error_var;
						}
						else if (gp_approx_ == "full_scale_tapering") {
							// Initialize Residual Process
							re_comps_resid_[cluster_i][0][j]->CalcSigma();
							std::shared_ptr<T_mat> sigma_resid_grad = re_comps_resid_[cluster_i][0][j]->GetZSigmaZtGrad(ipar, true, 1.);
							// sigma_ip^-1 * sigma_cross_cov
							//den_mat_t sigma_ip_inv_sigma_cross_cov = chol_fact_sigma_ip_[cluster_i][0].solve((*cross_cov).transpose());
							den_mat_t sigma_ip_inv_sigma_cross_cov;
							GPBoost::solve_linear_sys(chol_fact_sigma_ip_[cluster_i][0], (*cross_cov).transpose(), sigma_ip_inv_sigma_cross_cov, GPU_use_);
							// Subtract gradient of predictive process covariance
							den_mat_t sigma_ip_stable_grad_sigma_ip_inv_sigma_cross_cov;// = sigma_ip_stable_grad * sigma_ip_inv_sigma_cross_cov;
							GPBoost::matmul(sigma_ip_stable_grad, sigma_ip_inv_sigma_cross_cov, sigma_ip_stable_grad_sigma_ip_inv_sigma_cross_cov, GPU_use_);
							SubtractProdFromMat<T_mat>(*sigma_resid_grad, -sigma_ip_inv_sigma_cross_cov, sigma_ip_stable_grad_sigma_ip_inv_sigma_cross_cov, true);
							SubtractProdFromMat<T_mat>(*sigma_resid_grad, cross_cov_grad_t, sigma_ip_inv_sigma_cross_cov, false);
							SubtractProdFromMat<T_mat>(*sigma_resid_grad, sigma_ip_inv_sigma_cross_cov, cross_cov_grad_t, false);
							// Apply taper
							re_comps_resid_[cluster_i][0][j]->ApplyTaper(*(re_comps_resid_[cluster_i][0][j]->dist_), *sigma_resid_grad);
							if (matrix_inversion_method_ == "cholesky") {
								// cross_crov_grad *  sigma_resid^-1 * t(cross_cov)
								cross_cov_grad_sigma_resid_inv_cross_cov_T = (cross_cov_grad_t)*sigma_resid_inv_cross_cov_T;
								// cross_crov_grad *  sigma_resid^-1 * t(cross_cov) + cross_crov *  sigma_resid^-1 * t(cross_cov_grad) - cross_cov * sigma_resid^-1 * sigma_resid_grad * sigma_resid^-1 * t(cross_cov)
								sigma_woodbury_grad = cross_cov_grad_sigma_resid_inv_cross_cov_T + cross_cov_grad_sigma_resid_inv_cross_cov_T.transpose() -
									sigma_resid_inv_cross_cov_T.transpose() * ((*sigma_resid_grad) * sigma_resid_inv_cross_cov_T);
								grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += 0.5 * ((double)(((*sigma_resid).cwiseProduct(*sigma_resid_grad)).sum()));
							}
							else if (matrix_inversion_method_ == "iterative") {// Conjugate Gradient
								// Derivative of Woodbury preconditioner matrix (Cm + Cmn * diag(Cs)^-1 * Cnm) or (Lambda + t(EVects of Cm) * Cmn * diag(Cs)^-1 * Cnm * EVects of Cm)
								den_mat_t sigma_woodbury_preconditioner_grad;
								vec_t diagonal_approx_grad_preconditioner;
								den_mat_t sigma_ip_stable_grad_preconditioner;
								std::shared_ptr<den_mat_t> cross_cov_grad_preconditioner;
								if (cg_preconditioner_type_ == "fitc") {
									const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][j]->GetSigmaPtr();
									sigma_ip_stable_grad_preconditioner = sigma_ip_stable_grad;
									cross_cov_grad_preconditioner = cross_cov_grad;
									diagonal_approx_grad_preconditioner = (*sigma_resid_grad).diagonal();
									den_mat_t sigma_cross_cov_diag_sigma_resid_inv = (*cross_cov_preconditioner).transpose() * diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal();
									den_mat_t D_cross_cov = (diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * (*cross_cov_preconditioner));
									//den_mat_t cross_cov_grad_d_inv_cross_cov = (*cross_cov_grad).transpose() * (diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * (*cross_cov_preconditioner));
									den_mat_t cross_cov_grad_d_inv_cross_cov;
									GPBoost::matmul((*cross_cov_grad).transpose(), D_cross_cov, cross_cov_grad_d_inv_cross_cov, GPU_use_);
									den_mat_t D_grad_cross_cov = (diagonal_approx_grad_preconditioner.asDiagonal()) * sigma_cross_cov_diag_sigma_resid_inv.transpose();
									den_mat_t cross_cov_grad_d_inv_grad_cross_cov;
									GPBoost::matmul(sigma_cross_cov_diag_sigma_resid_inv, D_grad_cross_cov, cross_cov_grad_d_inv_grad_cross_cov, GPU_use_);
									sigma_woodbury_preconditioner_grad = sigma_ip_stable_grad_preconditioner + cross_cov_grad_d_inv_cross_cov + cross_cov_grad_d_inv_cross_cov.transpose() - sigma_cross_cov_diag_sigma_resid_inv * ((diagonal_approx_grad_preconditioner.asDiagonal()) * sigma_cross_cov_diag_sigma_resid_inv.transpose());
								}
								den_mat_t sigma_ip_inv_sigma_ip_stable_grad;
								sigma_ip_inv_sigma_ip_stable_grad = chol_fact_sigma_ip_[cluster_i][0].solve(sigma_ip_stable_grad);
								// (Derivative of Sigma) * P^-1 * sample vectors
								den_mat_t sigma_resid_grad_Z(num_data_per_cluster_[cluster_i], rand_vec_probe_P_inv.cols());
								sigma_resid_grad_Z.setZero();
#pragma omp parallel for schedule(static)   
								for (int i = 0; i < rand_vec_probe_P_inv.cols(); ++i) {
									sigma_resid_grad_Z.col(i) += (*sigma_resid_grad) * rand_vec_probe_P_inv.col(i); //parallelization in for loop is much faster
								}
								den_mat_t sigma_ip_inv_sigma_cross_cov_Z = sigma_ip_inv_sigma_cross_cov * rand_vec_probe_P_inv;
								den_mat_t sigma_grad_Z = sigma_resid_grad_Z + (*cross_cov_grad) * sigma_ip_inv_sigma_cross_cov_Z +
									sigma_ip_inv_sigma_cross_cov.transpose() * ((*cross_cov_grad).transpose() * rand_vec_probe_P_inv) -
									sigma_ip_inv_sigma_cross_cov.transpose() * (sigma_ip_stable_grad * sigma_ip_inv_sigma_cross_cov_Z);
								// Stochastic Trace
								vec_t sample_Sigma = (solution_for_trace_[cluster_i].cwiseProduct(sigma_grad_Z)).colwise().sum();
								double stochastic_tr = sample_Sigma.mean();
								// Variance Reduction 
								if (cg_preconditioner_type_ == "fitc") {
									const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][j]->GetSigmaPtr();
									den_mat_t sigma_cross_cov_diag_sigma_resid_inv = (*cross_cov_preconditioner).transpose() * diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal();
									den_mat_t cross_cov_grad_d_inv_cross_cov = (*cross_cov_grad_preconditioner).transpose() * (diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * (*cross_cov_preconditioner));
									// (Derivative of P) * P^-1 * sample vectors
									den_mat_t P_G_Z = sigma_grad_Z + (diagonal_approx_grad_preconditioner.asDiagonal()) * rand_vec_probe_P_inv - sigma_resid_grad_Z;
									// Stochastic Trace (Preconditioner)
									vec_t sample_P = (rand_vec_probe_P_inv.cwiseProduct(P_G_Z)).colwise().sum();
									double Tr_P_stoch = sample_P.mean();
									// Exact Trace (Preconditioner)
									double Tr_P = 0;
									Tr_P -= sigma_ip_inv_sigma_ip_stable_grad.trace();
									Tr_P += diagonal_approx_inv_preconditioner_[cluster_i].dot(diagonal_approx_grad_preconditioner);
									den_mat_t sigma_woodbury_inv_sigma_woodbury_stable_grad = chol_fact_woodbury_preconditioner_[cluster_i].solve(sigma_woodbury_preconditioner_grad);
									Tr_P += sigma_woodbury_inv_sigma_woodbury_stable_grad.trace();
									// Calculate optimal c
									double c_opt;
									CalcOptimalC(sample_Sigma, sample_P, stochastic_tr, Tr_P, c_opt);
									// Reduce variance
									stochastic_tr += c_opt * (Tr_P - Tr_P_stoch);
								}
								grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += 0.5 * stochastic_tr;
							}
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] -= 0.5 * y_aux_[cluster_i].dot((*sigma_resid_grad) * y_aux_[cluster_i]) / error_var;
						}//end gp_approx_ == "full_scale_tapering"
						else { // fitc
							// Derivative of diagonal part
							vec_t FITC_Diag_grad = vec_t::Zero(num_data_per_cluster_[cluster_i]);
							if (re_comps_cross_cov_[cluster_i][0][0]->VarianceOnDiagonal()) {
								FITC_Diag_grad.array() += sigma_ip_stable_grad.coeffRef(0, 0);
							}
							else {
								CHECK(num_data_per_cluster_[cluster_i] == re_comps_cross_cov_[cluster_i][0][j]->GetNumUniqueREs());
#pragma omp parallel for schedule(static)
								for (int ii = 0; ii < re_comps_cross_cov_[cluster_i][0][j]->GetNumUniqueREs(); ++ii) {
									FITC_Diag_grad[ii] += re_comps_cross_cov_[cluster_i][0][j]->GetZSigmaZtGradDiagonal_ii(ii, ipar, true, 0.);//uses coords of all data points to calculate Sigma[ii,ii]
								}
							}
							//den_mat_t sigma_ip_inv_sigma_cross_cov = chol_fact_sigma_ip_[cluster_i][j].solve((*cross_cov).transpose());
							den_mat_t sigma_ip_inv_sigma_cross_cov;
							GPBoost::solve_linear_sys(chol_fact_sigma_ip_[cluster_i][j], (*cross_cov).transpose(), sigma_ip_inv_sigma_cross_cov, GPU_use_);
							//den_mat_t sigma_ip_grad_inv_sigma_cross_cov = sigma_ip_stable_grad * sigma_ip_inv_sigma_cross_cov;
							den_mat_t sigma_ip_grad_inv_sigma_cross_cov;
							GPBoost::matmul(sigma_ip_stable_grad, sigma_ip_inv_sigma_cross_cov, sigma_ip_grad_inv_sigma_cross_cov, GPU_use_);
#pragma omp parallel for schedule(static)
							for (int ii = 0; ii < num_data_per_cluster_[cluster_i]; ++ii) {
								FITC_Diag_grad[ii] -= 2 * sigma_ip_inv_sigma_cross_cov.col(ii).dot((*cross_cov_grad).transpose().col(ii))
									- sigma_ip_inv_sigma_cross_cov.col(ii).dot(sigma_ip_grad_inv_sigma_cross_cov.col(ii));
							}
							sigma_ip_inv_sigma_cross_cov.resize(0, 0);
							sigma_ip_grad_inv_sigma_cross_cov.resize(0, 0);
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] -= 0.5 * y_aux_[cluster_i].dot(FITC_Diag_grad.asDiagonal() * y_aux_[cluster_i]) / error_var;
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += 0.5 * FITC_Diag_grad.dot(fitc_resid_diag_[cluster_i].cwiseInverse());
							// Derivative of Woodbury Matrix
							vec_t fitc_resid_diag_I = fitc_resid_diag_[cluster_i].cwiseInverse();
							//cross_cov_grad_sigma_resid_inv_cross_cov_T = (*cross_cov).transpose() * fitc_resid_diag_I.asDiagonal() * (*cross_cov_grad);
							den_mat_t D_cross_cov_grad = fitc_resid_diag_I.asDiagonal() * (*cross_cov_grad);
							GPBoost::matmul((*cross_cov).transpose(), D_cross_cov_grad, cross_cov_grad_sigma_resid_inv_cross_cov_T, GPU_use_);
							sigma_woodbury_grad = cross_cov_grad_sigma_resid_inv_cross_cov_T + cross_cov_grad_sigma_resid_inv_cross_cov_T.transpose();
							fitc_resid_diag_I.array() *= fitc_resid_diag_I.array();
							fitc_resid_diag_I.array() *= FITC_Diag_grad.array();
							den_mat_t D_cross_cov = fitc_resid_diag_I.asDiagonal() * (*cross_cov);
							den_mat_t cross_cov_D_cross_cov;
							GPBoost::matmul((*cross_cov).transpose(), D_cross_cov, cross_cov_D_cross_cov, GPU_use_);
							sigma_woodbury_grad -= cross_cov_D_cross_cov;
						}
						// sigma_woodbury^-1 * sigma_woodbury_grad
						if (matrix_inversion_method_ == "cholesky") {
							sigma_woodbury_grad += (sigma_ip_stable_grad);
							den_mat_t sigma_woodbury_inv_sigma_woodbury_grad = chol_fact_sigma_woodbury_[cluster_i].solve(sigma_woodbury_grad);
							grad_cov_aux_par[first_cov_par + ind_par_[j] - 1 + ipar] += 0.5 * ((sigma_woodbury_inv_sigma_woodbury_grad.trace()));
						}
					}//end estimate_cov_par_index_[ind_par_[j] + ipar] > 0
				}//end loop over ipar
			}//end loop over comps	
		}//end CalcGradPars_FITC_FSA_GaussLikelihood_Cluster_i

		/*!
		* \brief Calculate gradient wrt the covariance and auxiliary parameters when having only grouped REs for Gaussian likelihoods
		*	The gradient wrt covariance and auxiliary parameters is calculated on the log-scale
		*/
		void CalcGradPars_Only_Grouped_REs_Woodbury_GaussLikelihood_Cluster_i(const vec_t& cov_pars,
			vec_t& grad_cov_aux_par,
			bool include_error_var,
			bool save_psi_inv_for_FI,
			int first_cov_par,
			data_size_t cluster_i) {
			CHECK(use_woodbury_identity_);
			CHECK(gauss_likelihood_);
			if (include_error_var) {
				double yTPsiInvy;
				CalcYTPsiIInvY(yTPsiInvy, false, cluster_i, true, true);
				if (estimate_cov_par_index_[0] > 0) {
					grad_cov_aux_par[0] += -1. * yTPsiInvy / cov_pars[0] / 2. + num_data_per_cluster_[cluster_i] / 2.;
				}
			}
			if (matrix_inversion_method_ == "cholesky") {
				std::vector<T_mat> LInvZtZj_cluster_i;
				if (save_psi_inv_for_FI) {
					LInvZtZj_[cluster_i].clear();
					LInvZtZj_cluster_i = std::vector<T_mat>(num_comps_total_);
				}
				for (int j = 0; j < num_comps_total_; ++j) {
					if (estimate_cov_par_index_[ind_par_[j]] > 0) {
						vec_t y_tilde_j, y_tilde2_j;
						if (linear_kernel_use_woodbury_identity_) {
							CHECK(num_comps_total_ == 1);
							y_tilde_j = Zt_[cluster_i] * y_[cluster_i];
							y_tilde2_j = Zt_[cluster_i] * y_tilde2_[cluster_i];
						}
						else {
							sp_mat_t* Z_j = re_comps_[cluster_i][0][j]->GetZ();
							y_tilde_j = (*Z_j).transpose() * y_[cluster_i];
							y_tilde2_j = (*Z_j).transpose() * y_tilde2_[cluster_i];
						}
						double yTPsiIGradPsiPsiIy = y_tilde_j.transpose() * y_tilde_j - 2. * (double)(y_tilde_j.transpose() * y_tilde2_j) + y_tilde2_j.transpose() * y_tilde2_j;
						yTPsiIGradPsiPsiIy *= cov_pars[j + 1];
						T_mat LInvZtZj;
						if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ == ZtZj_ and L_inv are diagonal  
							LInvZtZj = ZtZ_[cluster_i];
							LInvZtZj.diagonal().array() /= sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array();
						}
						else {
							// Note: the following is often the bottleneck (= slower than Cholesky dec.) when there are multiple REs and the number of random effects is large
							if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
								TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), P_ZtZj_[cluster_i][j], LInvZtZj, false);
							}
							else {
								TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), ZtZj_[cluster_i][j], LInvZtZj, false);
							}
						}
						if (save_psi_inv_for_FI) {//save for latter use when calculating the Fisher information
							LInvZtZj_cluster_i[j] = LInvZtZj;
						}
						double trace_PsiInvGradPsi = Zj_square_sum_[cluster_i][j] - LInvZtZj.squaredNorm();
						trace_PsiInvGradPsi *= cov_pars[j + 1];
						grad_cov_aux_par[first_cov_par + j] += -1. * yTPsiIGradPsiPsiIy / cov_pars[0] / 2. + trace_PsiInvGradPsi / 2.;
					}// end estimate_cov_par_index_[ind_par_[j]] > 0
				}//end loop over comps
				if (save_psi_inv_for_FI) {
					LInvZtZj_[cluster_i] = LInvZtZj_cluster_i;
				}
			}//end cholesky
			else if (matrix_inversion_method_ == "iterative") {
				//Calculate P^(-1)z_i
				den_mat_t PI_RV(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_), L_inv_RV, DI_L_plus_D_t_PI_RV;
				if (cg_preconditioner_type_ == "incomplete_cholesky") {
					L_inv_RV.resize(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						L_inv_RV.col(i) = (L_SigmaI_plus_ZtZ_rm_[cluster_i].template triangularView<Eigen::Lower>()).solve(rand_vec_probe_P_[cluster_i].col(i));
					}
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						PI_RV.col(i) = (L_SigmaI_plus_ZtZ_rm_[cluster_i].transpose().template triangularView<Eigen::Upper>()).solve(L_inv_RV.col(i));
					}
				}//end "incomplete_cholesky"
				else if (cg_preconditioner_type_ == "ssor") {
					DI_L_plus_D_t_PI_RV.resize(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
					L_inv_RV.resize(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						L_inv_RV.col(i) = (P_SSOR_L_D_sqrt_inv_rm_[cluster_i].template triangularView<Eigen::Lower>()).solve(rand_vec_probe_P_[cluster_i].col(i));
					}
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						PI_RV.col(i) = (P_SSOR_L_D_sqrt_inv_rm_[cluster_i].transpose().template triangularView<Eigen::Upper>()).solve(L_inv_RV.col(i));
					}
					//For variance reduction
					den_mat_t L_plus_D_t_PI_RV(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						L_plus_D_t_PI_RV.col(i) = (SigmaI_plus_ZtZ_rm_[cluster_i].template triangularView<Eigen::Upper>()) * PI_RV.col(i);
					}
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						DI_L_plus_D_t_PI_RV.col(i) = P_SSOR_D_inv_[cluster_i].asDiagonal() * L_plus_D_t_PI_RV.col(i); //alternative D^-0.5 * P_SSOR_L_D_sqrt_inv_rm_ m*t vs nnz
					}
				}//end "ssor"
				else {
					Log::REFatal("Preconditioner type '%s' is not supported for calculating gradients ", cg_preconditioner_type_.c_str());
				}
				for (int j = 0; j < num_comps_total_; ++j) {
					if (estimate_cov_par_index_[ind_par_[j]] > 0) {
						vec_t y_tilde_j, y_tilde2_j;
						if (linear_kernel_use_woodbury_identity_) {
							y_tilde_j = Zt_[cluster_i] * y_[cluster_i];
							y_tilde2_j = Zt_[cluster_i] * y_tilde2_[cluster_i];
						}
						else {
							sp_mat_t* Z_j = re_comps_[cluster_i][0][j]->GetZ();
							y_tilde_j = (*Z_j).transpose() * y_[cluster_i];
							y_tilde2_j = (*Z_j).transpose() * y_tilde2_[cluster_i];
						}
						double yTPsiIGradPsiPsiIy = y_tilde_j.transpose() * y_tilde_j - 2. * (double)(y_tilde_j.transpose() * y_tilde2_j) + y_tilde2_j.transpose() * y_tilde2_j;
						yTPsiIGradPsiPsiIy *= cov_pars[j + 1];
						//-dSigma^(-1)/dtheta_j
						vec_t neg_SigmaI_deriv = vec_t::Zero(cum_num_rand_eff_[cluster_i][num_comps_total_]);
#pragma omp parallel for schedule(static)
						for (int i = cum_num_rand_eff_[cluster_i][j]; i < cum_num_rand_eff_[cluster_i][j + 1]; ++i) {
							neg_SigmaI_deriv[i] = 1. / cov_pars[j + 1];
						}
						//Stochastic trace: tr((Sigma^(-1) + Z^T Z)^(-1) dSigma^(-1)/dtheta_j)
						vec_t zt_SigmaI_plus_ZtZ_inv_SigmaI_deriv_PI_z = -1. * ((solution_for_trace_[cluster_i].cwiseProduct(neg_SigmaI_deriv.asDiagonal() * PI_RV)).colwise().sum()).transpose();
						double trace_PsiInvGradPsi = zt_SigmaI_plus_ZtZ_inv_SigmaI_deriv_PI_z.mean();
						if (cg_preconditioner_type_ == "ssor") {
							//Variance reduction
							//deterministic tr(D^(-1) dSigma^(-1)/dtheta_j)
							double tr_D_inv_SigmaI_deriv = -1. * (P_SSOR_D_inv_[cluster_i].cwiseProduct(neg_SigmaI_deriv)).sum();//cum_num_rand_eff_[cluster_i][j] - cum_num_rand_eff_[cluster_i][j + 1]; //-n_j
							//stochastic tr(P^(-1) dP/dtheta_j)
							den_mat_t neg_SigmaI_deriv_DI_L_plus_D_t_PI_RV = neg_SigmaI_deriv.asDiagonal() * DI_L_plus_D_t_PI_RV;
							vec_t zt_PI_P_deriv_PI_z = -2. * ((PI_RV.cwiseProduct(neg_SigmaI_deriv_DI_L_plus_D_t_PI_RV)).colwise().sum()).transpose();
							zt_PI_P_deriv_PI_z += ((DI_L_plus_D_t_PI_RV.cwiseProduct(neg_SigmaI_deriv_DI_L_plus_D_t_PI_RV)).colwise().sum()).transpose();
							double tr_PI_P_deriv = zt_PI_P_deriv_PI_z.mean();
							//optimal c
							double c_opt;
							CalcOptimalC(zt_SigmaI_plus_ZtZ_inv_SigmaI_deriv_PI_z, zt_PI_P_deriv_PI_z, trace_PsiInvGradPsi, tr_PI_P_deriv, c_opt);
							trace_PsiInvGradPsi += c_opt * (tr_D_inv_SigmaI_deriv - tr_PI_P_deriv);
						}
						trace_PsiInvGradPsi += cum_num_rand_eff_[cluster_i][j + 1] - cum_num_rand_eff_[cluster_i][j]; //tr(Sigma)^(-1) dSigma/dtheta_j)
						grad_cov_aux_par[first_cov_par + j] += -1. * yTPsiIGradPsiPsiIy / cov_pars[0] / 2. + trace_PsiInvGradPsi / 2.;
					}// end estimate_cov_par_index_[ind_par_[j]] > 0
				}//end loop over comps
				if (save_psi_inv_for_FI) { //save for latter use when calculating the Fisher information
					PI_RV_[cluster_i] = PI_RV;
				}
			}//end iterative
			else {
				Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
			}
		}//end CalcGradPars_Only_Grouped_REs_Woodbury_GaussLikelihood_Cluster_i

		/*!
		* \brief Reset mode to previous value. Used when, e.g., NA or Inf occurred
		*/
		void ResetLaplaceApproxModeToPreviousValue() {
			CHECK(!gauss_likelihood_);
			for (const auto& cluster_i : unique_clusters_) {
				likelihood_[cluster_i]->ResetModeToPreviousValue();
			}
		}

		/*!
		* \brief Profile out sigma2 (=use closed-form expression for error / nugget variance)
		* \return sigma2_
		*/
		double ProfileOutSigma2() {
			if (estimate_cov_par_index_[0] > 0) {
				sigma2_ = yTPsiInvy_ / num_data_;
			}
			return sigma2_;
		}

		/*!
		* \brief Get sigma2
		* \return sigma2_
		*/
		double Sigma2() {
			return sigma2_;
		}

		/*!
		* \brief Profile out the linear regression coefficients (=use closed-form WLS expression)
		* \param fixed_effects Externally provided fixed effects component of location parameter (only used for non-Gaussian likelihoods)
		* \param[out] fixed_effects_vec Vector of fixed effects (used only for non-Gaussian likelihoods)
		*/
		void ProfileOutCoef(const double* fixed_effects,
			vec_t& fixed_effects_vec) {
			CHECK(gauss_likelihood_);
			CHECK(has_covariates_);
			if (fixed_effects != nullptr) {
				vec_t resid = y_vec_;
#pragma omp parallel for schedule(static)
				for (data_size_t i = 0; i < num_data_; ++i) {
					resid[i] -= fixed_effects[i];
				}
				SetY(resid.data());
			}
			else {
				SetY(y_vec_.data());
			}
			CalcYAux(1., false);
			UpdateCoefGLS();
			UpdateFixedEffects(beta_, fixed_effects, fixed_effects_vec);// Set y_ to resid = y - X*beta for updating covariance parameters
		}

		/*!
		* \brief Get beta_
		* \param[out] beta
		*/
		void GetBeta(vec_t& beta) {
			beta = beta_;
		}

		/*!
		* \brief Return value of neg_log_likelihood_
		* \return neg_log_likelihood_
		*/
		double GetNegLogLikelihood() {
			return neg_log_likelihood_;
		}

		/*!
		* \brief Return num_cov_par_
		* \return num_cov_par_
		*/
		int GetNumCovPar() {
			return num_cov_par_;
		}

		/*!
		* \brief Return num_covariates_
		* \return num_covariates_
		*/
		int GetNumCoef() {
			return num_covariates_ * num_sets_re_;
		}

		/*!
		* \brief Return num_sets_re_
		* \return num_sets_re_
		*/
		int GetNumSetsRE() {
			return num_sets_re_;
		}

		/*!
		* \brief Return has_covariates_
		* \return has_covariates_
		*/
		bool HasCovariates() {
			return has_covariates_;
		}

		/*!
		* \brief Return estimate_aux_pars_
		*/
		bool EstimateAuxPars() const {
			return estimate_aux_pars_;
		}

		/*!
		* \brief Return true if aux_pars have been set or estimated
		*/
		bool AuxParsHaveBeenSetOrEstimated() {
			return likelihood_[unique_clusters_[0]]->AuxParsHaveBeenSet();
		}

		/*!
		* \brief Number of auxiliary parameters in likelihoods.h
		*/
		int NumAuxPars() {
			return likelihood_[unique_clusters_[0]]->NumAuxPars();
		}

		/*!
		* \brief Returns a pointer to aux_pars_
		*/
		const double* GetAuxPars() {
			return likelihood_[unique_clusters_[0]]->GetAuxPars();
		}

		/*!
		* \brief Set aux_pars_
		* \param aux_pars New values for aux_pars_
		*/
		void SetAuxPars(const double* aux_pars) {
			for (const auto& cluster_i : unique_clusters_) {
				likelihood_[cluster_i]->SetAuxPars(aux_pars);
			}
		}

		void GetNamesAuxPars(string_t& name) {
			likelihood_[unique_clusters_[0]]->GetNamesAuxPars(name);
		}

		/*!
		* \brief Set num_iter_. Used by external optimizers from OptimLib
		* \param num_iter New values for num_iter_
		*/
		void SetNumIter(int num_iter) {
			num_iter_ = num_iter;
		}

		/*!
		* \brief Write the current values of profiled-out variables (if there are any such as nugget effects, regression coefficients) to their lag1 variables. Used only by external optimizers
		* \param profile_out_marginal_variance If true, the error variance sigma is profiled out (= use closed-form expression for error / nugget variance)
		* \param profile_out_regression_coef If true, the linear regression coefficients are profiled out (= use closed-form WLS expression)
		*/
		void SetLag1ProfiledOutVariables(bool profile_out_marginal_variance,
			bool profile_out_regression_coef) {
			if (profile_out_marginal_variance) {
				sigma2_lag1_ = sigma2_;
			}
			if (profile_out_regression_coef) {
				beta_lag1_ = beta_;
			}
		}//end SetLag1ProfiledOutVariables

		/*!
		* \brief Reset the profiled-out variables (if there are any such as nugget effects, regression coefficients) to their lag1 variables
		* \param profile_out_marginal_variance If true, the error variance sigma is profiled out (= use closed-form expression for error / nugget variance)
		* \param profile_out_regression_coef If true, the linear regression coefficients are profiled out (= use closed-form WLS expression)
		*/
		void ResetProfiledOutVariablesToLag1(bool profile_out_marginal_variance,
			bool profile_out_regression_coef) {
			if (profile_out_marginal_variance) {
				sigma2_ = sigma2_lag1_;
			}
			if (profile_out_regression_coef) {
				beta_ = beta_lag1_;
			}
		}//end ResetProfiledOutVariablesToLag1

		/*!
		* \brief Factorize the covariance matrix (Gaussian data) or
		*	calculate the posterior mode of the random effects for use in the Laplace approximation (non-Gaussian likelihoods)
		*	And calculate the negative log-likelihood (Gaussian data) or the negative approx. marginal log-likelihood (non-Gaussian likelihoods)
		* \param cov_pars_in Covariance parameters
		* \param fixed_effects Fixed effects component of location parameter (only used for non-Gaussian likelihoods, othetwise, this is already set before in y_)
		*/
		void CalcCovFactorOrModeAndNegLL(const vec_t& cov_pars_in,
			const double* fixed_effects) {
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			SetCovParsComps(cov_pars);
			CalcCovFactor(true, 1.);
			if (gauss_likelihood_) {
				if (use_woodbury_identity_ && matrix_inversion_method_ != "iterative") {
					CalcYtilde(true);//y_tilde = L^-1 * Z^T * y and y_tilde2 = Z * L^-T * L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z)
				}
				else {
					CalcYAux(1., true);//y_aux = Psi^-1 * y
				}
				EvalNegLogLikelihoodGauss(nullptr, cov_pars.data(), nullptr, neg_log_likelihood_, true, true, true, false);
			}//end gauss_likelihood_
			else {//not gauss_likelihood_
				neg_log_likelihood_ = -CalcModePostRandEffCalcMLL(fixed_effects, true);//calculate mode and approximate marginal likelihood
			}//end not gauss_likelihood_
		}//end CalcCovFactorOrModeAndNegLL

		/*!
		* \brief Update fixed effects with new linear regression coefficients
		* \param beta Linear regression coefficients
		* \param fixed_effects Externally provided fixed effects component of location parameter (only used for non-Gaussian likelihoods)
		* \param[out] fixed_effects_vec Vector of fixed effects (used only for non-Gaussian likelihoods)
		*/
		void UpdateFixedEffects(const vec_t& beta,
			const double* fixed_effects,
			vec_t& fixed_effects_vec) {
			if (gauss_likelihood_) {
				vec_t resid = y_vec_ - (X_ * beta);
				if (fixed_effects != nullptr) {//add external fixed effects to linear predictor
#pragma omp parallel for schedule(static)
					for (data_size_t i = 0; i < num_data_; ++i) {
						resid[i] -= fixed_effects[i];
					}
				}
				SetY(resid.data());
			}
			else {
				fixed_effects_vec = vec_t(num_data_ * num_sets_re_);
				for (int igp = 0; igp < num_sets_re_; ++igp) {
					fixed_effects_vec.segment(igp * num_data_, num_data_) = X_ * (beta.segment(num_covariates_ * igp, num_covariates_));
				}
				if (fixed_effects != nullptr) {//add external fixed effects to linear predictor
#pragma omp parallel for schedule(static)
					for (int i = 0; i < num_data_ * num_sets_re_; ++i) {
						fixed_effects_vec[i] += fixed_effects[i];
					}
				}
			}
		}//end UpdateFixedEffects

		/*!
		* \brief Calculate the value of the negative log-likelihood for a "gaussian" likelihood
		* \param y_data Response variable data
		* \param cov_pars Values for covariance parameters of RE components
		* \param fixed_effects Externally provided fixed effects component of location parameter
		* \param[out] negll Negative log-likelihood
		* \param CalcCovFactor_already_done If true, it is assumed that the covariance matrix has already been factorized
		* \param CalcYAux_already_done If true, it is assumed that y_aux_=Psi^-1y_ has already been calculated (only relevant if not use_woodbury_identity_)
		* \param CalcYtilde_already_done If true, it is assumed that y_tilde = L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z), has already been calculated (only relevant for use_woodbury_identity_)
		* \param redetermine_neighbors_vecchia If true, 'RedetermineNearestNeighborsVecchiaInducingPointsFITC' is called if applicable
		*/
		void EvalNegLogLikelihoodGauss(const double* y_data,
			const double* cov_pars,
			const double* fixed_effects,
			double& negll,
			bool CalcCovFactor_already_done,
			bool CalcYAux_already_done,
			bool CalcYtilde_already_done,
			bool redetermine_neighbors_vecchia) {
			CHECK(gauss_likelihood_);
			CHECK(!(CalcYAux_already_done && !CalcCovFactor_already_done));// CalcYAux_already_done && !CalcCovFactor_already_done makes no sense
			if (fixed_effects != nullptr) {
				if (y_data == nullptr) {
					Log::REFatal("EvalNegLogLikelihoodGauss: 'y_data' cannot nullptr when 'fixed_effects' is provided ");
				}
				vec_t y_minus_lp(num_data_);
#pragma omp parallel for schedule(static)
				for (data_size_t i = 0; i < num_data_; ++i) {
					y_minus_lp[i] = y_data[i] - fixed_effects[i];
				}
				SetY(y_minus_lp.data());
			}
			else {
				if (y_data != nullptr) {
					SetY(y_data);
				}
			}
			if (!CalcCovFactor_already_done) {
				const vec_t cov_pars_vec = Eigen::Map<const vec_t>(cov_pars, num_cov_par_);
				SetCovParsComps(cov_pars_vec);
				if (redetermine_neighbors_vecchia) {
					if (ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(true)) {
						RedetermineNearestNeighborsVecchiaInducingPointsFITC(true);//called if gp_approx_ == "vecchia" or  gp_approx_ == "full_scale_vecchia" and neighbors are selected based on correlations and not distances or gp_approx_ == "fitc" with ard kernel
					}
				}
				CalcCovFactor(true, 1.);//Create covariance matrix and factorize it
			}
			//Calculate quadratic form y^T Psi^-1 y
			CalcYTPsiIInvY(yTPsiInvy_, true, 1, CalcYAux_already_done, CalcYtilde_already_done);
			//Calculate log determinant
			if (matrix_inversion_method_ == "iterative") {
				if (saved_rand_vec_.size() == 0) {
					for (const auto& cluster_i : unique_clusters_) {
						saved_rand_vec_[cluster_i] = false;
					}
				}
			}
			log_det_Psi_ = 0;
			for (const auto& cluster_i : unique_clusters_) {
				if (gp_approx_ == "vecchia") {
					log_det_Psi_ -= D_inv_[cluster_i][0].diagonal().array().log().sum();
				}
				else {
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						if (matrix_inversion_method_ == "cholesky") {//Cholesky
							log_det_Psi_ -= 2. * (((den_mat_t)chol_fact_sigma_ip_[cluster_i][0].matrixL()).diagonal().array().log().sum());
							log_det_Psi_ += 2. * (((den_mat_t)chol_fact_sigma_woodbury_[cluster_i].matrixL()).diagonal().array().log().sum());
							////alternative way for calculating determinants with Woodbury (does not solve numerical stability issue, 05.06.2024)
							//log_det_Psi_ += 2. * (((den_mat_t)chol_fact_sigma_woodbury_stable_[cluster_i].matrixL()).diagonal().array().log().sum());
							if (gp_approx_ == "full_scale_tapering") {
								log_det_Psi_ += 2. * (((T_mat)chol_fact_resid_[cluster_i].matrixL()).diagonal().array().log().sum());
							}
							else if (gp_approx_ == "full_scale_vecchia") {
								log_det_Psi_ -= D_inv_rm_[cluster_i][0].diagonal().array().log().sum();
							}
							else {
								log_det_Psi_ += fitc_resid_diag_[cluster_i].array().log().sum();
							}
						}//end Cholesky
						else if (matrix_inversion_method_ == "iterative") {//Conjugate Gradient
							// Sample probe vectors
							if (!saved_rand_vec_[cluster_i]) {
								rand_vec_probe_[cluster_i].resize(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
								GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vec_probe_[cluster_i]);
								// Sample probe vectors from N(0,P)
								if (cg_preconditioner_type_ == "fitc") {
									rand_vec_probe_low_rank_[cluster_i].resize(num_ind_points_, num_rand_vec_trace_);
									GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vec_probe_low_rank_[cluster_i]);
									rand_vec_probe_P_[cluster_i] = rand_vec_probe_[cluster_i];
								}
								if (reuse_rand_vec_trace_) {//Use same random vectors for each iteration && cluster_i == end(unique_cluster) Tim
									saved_rand_vec_[cluster_i] = true;
								}
							}
							if (cg_preconditioner_type_ == "fitc") {
								den_mat_t chol_ip_cross_cov_Z = chol_ip_cross_cov_preconditioner_[cluster_i][0].transpose() * rand_vec_probe_low_rank_[cluster_i];
								rand_vec_probe_[cluster_i] = chol_ip_cross_cov_Z + diagonal_approx_preconditioner_[cluster_i].cwiseSqrt().asDiagonal() * rand_vec_probe_P_[cluster_i];
							}
							const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr();
							std::shared_ptr<T_mat> sigma_resid = re_comps_resid_[cluster_i][0][0]->GetZSigmaZt();
							// Initialize Solution Sigma^-1 (u_1,...,u_t) 
							solution_for_trace_[cluster_i].resize(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
							solution_for_trace_[cluster_i].setZero();
							// Initialize tridiagonal Matrices
							int cg_max_num_it_tridiag = cg_max_num_it_tridiag_;
							if (first_update_) {
								cg_max_num_it_tridiag = (int)round(cg_max_num_it_tridiag_ / 3);
							}
							std::vector<vec_t> Tdiags_(num_rand_vec_trace_, vec_t(cg_max_num_it_tridiag));
							std::vector<vec_t> Tsubdiags_(num_rand_vec_trace_, vec_t(cg_max_num_it_tridiag - 1));
							// Conjuagte Gradient with Lanczos
							if (cg_preconditioner_type_ == "fitc") {
								const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
								CGTridiagFSA<T_mat>(*sigma_resid, *cross_cov_preconditioner, chol_ip_cross_cov_[cluster_i][0], rand_vec_probe_[cluster_i],
									Tdiags_, Tsubdiags_, solution_for_trace_[cluster_i], NaN_found, num_data_per_cluster_[cluster_i],
									num_rand_vec_trace_, cg_max_num_it_tridiag, cg_delta_conv_, cg_preconditioner_type_,
									chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
							}
							else {
								CGTridiagFSA<T_mat>(*sigma_resid, *cross_cov, chol_ip_cross_cov_[cluster_i][0], rand_vec_probe_[cluster_i],
									Tdiags_, Tsubdiags_, solution_for_trace_[cluster_i], NaN_found, num_data_per_cluster_[cluster_i],
									num_rand_vec_trace_, cg_max_num_it_tridiag, cg_delta_conv_, cg_preconditioner_type_,
									chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
							}
							if (NaN_found) {
								Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
							}
							// LogDet Estimation
							LogDetStochTridiag(Tdiags_, Tsubdiags_, log_det_Psi_, num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
							// Correction for Preconditioner (necessary if using preconditioner)
							if (cg_preconditioner_type_ == "fitc") {
								log_det_Psi_ -= 2. * (((den_mat_t)chol_fact_sigma_ip_preconditioner_[cluster_i][0].matrixL()).diagonal().array().log().sum());
								log_det_Psi_ += 2. * (((den_mat_t)chol_fact_woodbury_preconditioner_[cluster_i].matrixL()).diagonal().array().log().sum());
								log_det_Psi_ -= diagonal_approx_inv_preconditioner_[cluster_i].array().log().sum();
							}
						}//end Conjugate Gradient
						else {
							Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
						}
					}//end gp_approx_ == "fitc" or "full_scale_tapering"
					else {
						if (use_woodbury_identity_) {
							if (num_re_group_total_ == 1 && num_comps_total_ == 1) {
								log_det_Psi_ += (2. * sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().log().sum());
							}
							else {
								if (matrix_inversion_method_ == "cholesky") {
									log_det_Psi_ += (2. * chol_facts_[cluster_i].CholFactMatrix().diagonal().array().log().sum());
								}//end cholesky
								else if (matrix_inversion_method_ == "iterative") {
									// Sample probe vectors
									if (!saved_rand_vec_[cluster_i]) {
										rand_vec_probe_[cluster_i].resize(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_); //N(0,I)
										GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vec_probe_[cluster_i]);
										if (reuse_rand_vec_trace_) {
											saved_rand_vec_[cluster_i] = true;
										}
										rand_vec_probe_P_[cluster_i].resize(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_); //N(0,P)
									}
									CHECK(rand_vec_probe_[cluster_i].cols() == num_rand_vec_trace_);
									CHECK(rand_vec_probe_P_[cluster_i].cols() == num_rand_vec_trace_);
									CHECK(rand_vec_probe_[cluster_i].rows() == cum_num_rand_eff_[cluster_i][num_comps_total_]);
									CHECK(rand_vec_probe_P_[cluster_i].rows() == cum_num_rand_eff_[cluster_i][num_comps_total_]);
									// Get probe vectors from N(0,P)
									if (cg_preconditioner_type_ == "incomplete_cholesky") {
										//P = L L^T: u_i = L r_i, where r_i ~ N(0,I)
#pragma omp parallel for schedule(static)   
										for (int i = 0; i < num_rand_vec_trace_; ++i) {
											rand_vec_probe_P_[cluster_i].col(i) = L_SigmaI_plus_ZtZ_rm_[cluster_i] * rand_vec_probe_[cluster_i].col(i);
										}
									}
									else if (cg_preconditioner_type_ == "ssor") {
										//P = L D^-1 L^T: u_i = L D^-0.5 r_i, where r_i ~ N(0,I)
#pragma omp parallel for schedule(static)
										for (int i = 0; i < num_rand_vec_trace_; ++i) {
											rand_vec_probe_P_[cluster_i].col(i) = P_SSOR_L_D_sqrt_inv_rm_[cluster_i] * rand_vec_probe_[cluster_i].col(i);
										}
									}
									else if (cg_preconditioner_type_ == "diagonal") {
										//P = diag(Sigma^-1 + Z^T Z): ui = diag(Sigma^-1 + Z^T Z)^0.5 r_i, where r_i ~ N(0,I)
#pragma omp parallel for schedule(static)   
										for (int i = 0; i < num_rand_vec_trace_; ++i) {
											rand_vec_probe_P_[cluster_i].col(i) = SigmaI_plus_ZtZ_inv_diag_[cluster_i].cwiseInverse().cwiseSqrt().asDiagonal() * rand_vec_probe_[cluster_i].col(i);
										}
									}
									else if (cg_preconditioner_type_ == "none") {
										rand_vec_probe_P_[cluster_i] = rand_vec_probe_[cluster_i];
									}
									else {
										Log::REFatal("Preconditioner type '%s' is not supported.", cg_preconditioner_type_.c_str());
									}
									// Initialize Solution Sigma^-1 (u_1,...,u_t) 
									solution_for_trace_[cluster_i].resize(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
									solution_for_trace_[cluster_i].setZero();
									// Initialize tridiagonal Matrices
									int cg_max_num_it_tridiag = cg_max_num_it_tridiag_;
									if (first_update_) {
										cg_max_num_it_tridiag = (int)round(cg_max_num_it_tridiag_ / 3);
									}
									std::vector<vec_t> Tdiags_(num_rand_vec_trace_, vec_t(cg_max_num_it_tridiag));
									std::vector<vec_t> Tsubdiags_(num_rand_vec_trace_, vec_t(cg_max_num_it_tridiag - 1));
									// Conjuagte Gradient with Lanczos
									CGTridiagRandomEffects(SigmaI_plus_ZtZ_rm_[cluster_i], rand_vec_probe_P_[cluster_i],
										Tdiags_, Tsubdiags_, solution_for_trace_[cluster_i], NaN_found, cum_num_rand_eff_[cluster_i][num_comps_total_],
										num_rand_vec_trace_, cg_max_num_it_tridiag, cg_delta_conv_, cg_preconditioner_type_,
										L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i], SigmaI_plus_ZtZ_inv_diag_[cluster_i], num_cg_steps_tridiag_last_);
									if (NaN_found) {
										Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
									}
									// LogDet Estimation
									LogDetStochTridiag(Tdiags_, Tsubdiags_, log_det_Psi_, cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
									// Correction for preconditioner
									if (cg_preconditioner_type_ == "incomplete_cholesky") {
										//log|P| = log|L| + log|L^T|
										log_det_Psi_ += 2 * (L_SigmaI_plus_ZtZ_rm_[cluster_i].diagonal().array().log().sum());
									}
									else if (cg_preconditioner_type_ == "ssor") {
										//log|P| = log|L| + log|D^-1| + log|L^T|
										log_det_Psi_ += 2 * (P_SSOR_L_D_sqrt_inv_rm_[cluster_i].diagonal().array().log().sum());
									}
									else if (cg_preconditioner_type_ == "diagonal") {
										//log|P| = - log|diag(Sigma^-1 + Z^T Z)^(-1)|
										log_det_Psi_ -= SigmaI_plus_ZtZ_inv_diag_[cluster_i].array().log().sum();
									}
									else if (cg_preconditioner_type_ != "none") {
										Log::REFatal("Preconditioner type '%s' is not supported.", cg_preconditioner_type_.c_str());
									}
								}//end iterative
								else {
									Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
								}
							}
							for (int j = 0; j < num_comps_total_; ++j) {
								int num_rand_eff = cum_num_rand_eff_[cluster_i][j + 1] - cum_num_rand_eff_[cluster_i][j];
								log_det_Psi_ += (num_rand_eff * std::log(re_comps_[cluster_i][0][j]->cov_pars_[0]));
							}
						}
						else {
							log_det_Psi_ += (2. * chol_facts_[cluster_i].CholFactMatrix().diagonal().array().log().sum());
						}
					}
				}//end not "vecchia"
			}//end loop cluster_i
			negll = yTPsiInvy_ / 2. / cov_pars[0] + log_det_Psi_ / 2. + num_data_ / 2. * (std::log(cov_pars[0]) + std::log(2 * M_PI));
		}//end EvalNegLogLikelihoodGauss

		/*!
		* \brief Calculate the value of the negative log-likelihood when yTPsiInvy_ and log_det_Psi_ is already known
		* \param sigma2 Nugget / error term variance
		* \param[out] negll Negative log-likelihood
		*/
		void EvalNegLogLikelihoodOnlyUpdateNuggetVariance(const double sigma2,
			double& negll) {
			negll = yTPsiInvy_ / 2. / sigma2 + log_det_Psi_ / 2. + num_data_ / 2. * (std::log(sigma2) + std::log(2 * M_PI));
		}//end EvalNegLogLikelihoodOnlyUpdateNuggetVariance

		/*!
		* \brief Calculate the value of the negative log-likelihood when only the fixed effects part has changed and the covariance matrix has not changed
		*	Note: It is assuzmed that y_ has been set before by calling 'SetY' with the residuals = y - fixed_effcts
		* \param sigma2 Nugget / error term variance
		* \param[out] negll Negative log-likelihood
		*/
		void EvalNegLogLikelihoodOnlyUpdateFixedEffects(const double sigma2,
			double& negll) {
			// Calculate y_aux = Psi^-1 * y (if not use_woodbury_identity_) or y_tilde and y_tilde2 (if use_woodbury_identity_) for covariance parameter update (only for Gaussian data)
			if (use_woodbury_identity_ && matrix_inversion_method_ != "iterative") {
				CalcYtilde(true);//y_tilde = L^-1 * Z^T * y and y_tilde2 = Z * L^-T * L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z)
			}
			else {
				CalcYAux(1., true);//y_aux = Psi^-1 * y
			}
			//Calculate quadratic form y^T Psi^-1 y
			CalcYTPsiIInvY(yTPsiInvy_, true, 1, true, true);
			negll = yTPsiInvy_ / 2. / sigma2 + log_det_Psi_ / 2. + num_data_ / 2. * (std::log(sigma2) + std::log(2 * M_PI));
		}

		/*!
		* \brief Calculate the value of the approximate negative marginal log-likelihood obtained when using a Laplace approximation
		* \param y_data Response variable data
		* \param cov_pars Values for covariance parameters of RE components
		* \param[out] negll Approximate negative marginal log-likelihood
		* \param fixed_effects Fixed effects component of location parameter
		* \param InitializeModeCovMat If true, posterior mode is initialized to 0 and the covariance matrix is calculated. Otherwise, existing values are used
		* \param CalcModePostRandEff_already_done If true, it is assumed that the posterior mode of the random effects has already been calculated
		* \param redetermine_neighbors_vecchia If true, 'RedetermineNearestNeighborsVecchiaInducingPointsFITC' is called if applicable
		*/
		void EvalLaplaceApproxNegLogLikelihood(const double* y_data,
			const double* cov_pars,
			double& negll,
			const double* fixed_effects,
			bool InitializeModeCovMat,
			bool CalcModePostRandEff_already_done,
			bool redetermine_neighbors_vecchia) {
			CHECK(!gauss_likelihood_);
			if (y_data != nullptr) {
				SetY(y_data);
			}
			else {
				if (!CalcModePostRandEff_already_done) {
					CHECK(y_has_been_set_);
				}
			}
			if (InitializeModeCovMat) {
				CHECK(cov_pars != nullptr);
			}
			if (CalcModePostRandEff_already_done) {
				negll = neg_log_likelihood_;//Whenever the mode is calculated that likelihood is calculated as well. So we might as well just return the saved neg_log_likelihood_
			}
			else {//not CalcModePostRandEff_already_done
				if (InitializeModeCovMat) {
					//We reset the initial modes to 0. This is done to avoid that different calls to EvalLaplaceApproxNegLogLikelihood lead to (very small) differences.
					for (const auto& cluster_i : unique_clusters_) {
						likelihood_[cluster_i]->InitializeModeAvec();
					}
					const vec_t cov_pars_vec = Eigen::Map<const vec_t>(cov_pars, num_cov_par_);
					SetCovParsComps(cov_pars_vec);
					if (redetermine_neighbors_vecchia) {
						if (ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(true)) {
							RedetermineNearestNeighborsVecchiaInducingPointsFITC(true);//called if gp_approx_ == "vecchia" or  gp_approx_ == "full_scale_vecchia" and neighbors are selected based on correlations and not distances or gp_approx_ == "fitc" with ard kernel
						}
					}
					CalcCovFactor(true, 1.);
				}//end InitializeModeCovMat
				negll = -CalcModePostRandEffCalcMLL(fixed_effects, true);
			}//end not CalcModePostRandEff_already_done
		}//end EvalLaplaceApproxNegLogLikelihood

		/*!
		* \brief Calculate the value of the (approximate) negative log-marginal likelihood
		* \param y_data Response variable data
		* \param cov_pars Values for covariance parameters of RE components
		* \param[out] negll Approximate negative marginal log-likelihood
		* \param fixed_effects Fixed effects component of location parameter
		* \param InitializeModeCovMat If true, posterior mode is initialized to 0 and the covariance matrix is calculated. Otherwise, existing values are used
		* \param CalcModePostRandEff_already_done If true, it is assumed that the posterior mode of the random effects has already been calculated
		*/
		void EvalNegLogLikelihood(const double* y_data,
			const double* cov_pars,
			double& negll,
			const double* fixed_effects,
			bool InitializeModeCovMat,
			bool CalcModePostRandEff_already_done) {
			if (gauss_likelihood_) {
				EvalNegLogLikelihoodGauss(y_data, cov_pars, fixed_effects, negll,
					false, false, false, true);
			}
			else {
				EvalLaplaceApproxNegLogLikelihood(y_data, cov_pars, negll, fixed_effects,
					InitializeModeCovMat, CalcModePostRandEff_already_done, true);
			}			
		}//end EvalNegLogLikelihood

		/*!
		* \brief Print out current parameters when trace / logging is activated for convergence monitoring
		* \param cov_pars_in Covariance parameters on transformed scale
		* \param beta Regression coefficients on transformed scale
		* \param aux_pars Additional parameters for the likelihood
		* \param print_cov_aux_pars If true, cov_pars and aux_pars are printed
		*/
		void PrintTraceParameters(const vec_t& cov_pars_in,
			const vec_t& beta,
			const double* aux_pars,
			bool print_cov_aux_pars) {
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			vec_t cov_pars_orig, beta_orig;
			if (Log::GetLevelRE() == LogLevelRE::Debug) { // do transformation only if log level Debug is active
				if (print_cov_aux_pars) {
					TransformBackCovPars(cov_pars, cov_pars_orig);
					for (int i = 0; i < (int)cov_pars.size(); ++i) {
						Log::REDebug("cov_pars[%d]: %g", i, cov_pars_orig[i]);
					}
				}
				if (has_covariates_) {
					if (scale_covariates_) {
						TransformBackCoef(beta, beta_orig);
					}
					else {
						beta_orig = beta;
					}
					for (int i = 0; i < std::min((int)beta.size(), NUM_COEF_PRINT_TRACE_); ++i) {
						Log::REDebug("beta[%d]: %g", i, beta_orig[i]);
					}
					if (has_covariates_ && beta.size() > NUM_COEF_PRINT_TRACE_) {
						Log::REDebug("Note: only the first %d linear regression coefficients are shown ", NUM_COEF_PRINT_TRACE_);
					}
				}
				if (estimate_aux_pars_ && print_cov_aux_pars) {
					SetAuxPars(aux_pars);//hack to avoid that wrong parameters are displayed for likelihoods when some parameters are not estimated (e.g., the 'df' parameter for a 't' likelihood)
					const double* aux_pars_print = GetAuxPars();
					vec_t aux_pars_print_orig(NumAuxPars());
					BackTransformAuxPars(aux_pars_print, aux_pars_print_orig.data());
					for (int i = 0; i < NumAuxPars(); ++i) {
						Log::REDebug("%s: %g", likelihood_[unique_clusters_[0]]->GetNameAuxPars(i), aux_pars_print_orig[i]);
					}
				}
			}
		}//end PrintTraceParameters

		/*!
		* \brief Calculate gradient wrt fixed effects F (for GPBoost algorithm) and write on input (for Gaussian data, the gradient is Psi^-1*y (=y_aux))
		* \param[out] y Input response data and output gradient written on it.
		*		For the GPBoost algorithm for Gaussian data, the input is F - y where F is the fitted value of the ensemble at the training data and y the response data.
		*		For the GPBoost algorithm for non-Gaussian data, this input is ignored as the response data has been set before.
		*		The gradient (Psi^-1*y for Gaussian data) is then written on it as output. y needs to be of length num_data_
		* \param fixed_effects Fixed effects component F of location parameter (only used for non-Gaussian data). For Gaussian data, this is ignored (and can be set to nullptr)
		* \param calc_cov_factor If true, the covariance matrix is factorized, otherwise the existing factorization is used
		* \param cov_pars_in Covariance parameters
		*/
		void CalcGradientF(double* y,
			const double* fixed_effects,
			bool calc_cov_factor,
			const vec_t& cov_pars_in) {
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			//1. Factorize covariance matrix
			if (calc_cov_factor) {
				SetCovParsComps(cov_pars);
				CalcCovFactor(true, 1.);
				if (!gauss_likelihood_) {//not gauss_likelihood_
					CalcModePostRandEffCalcMLL(fixed_effects, true);
				}//end gauss_likelihood_
			}//end calc_cov_factor
			//2. Calculate gradient
			if (gauss_likelihood_) {//Gaussian data
				SetY(y);
				CalcYAux(cov_pars[0], false);
				GetYAux(y);
			}
			else {//not gauss_likelihood_
				CalcGradFLaplace(y, fixed_effects);
			}
		}// end CalcGradientF

		/*!
		* \brief Set the data used for making predictions (useful if the same data is used repeatedly, e.g., in validation of GPBoost)
		* \param num_data_pred Number of data points for which predictions are made
		* \param cluster_ids_data_pred IDs / labels indicating independent realizations of Gaussian processes (same values = same process realization) for which predictions are to be made
		* \param re_group_data_pred Labels of group levels for the grouped random effects in column-major format (i.e. first the levels for the first effect, then for the second, etc.). Every group label needs to end with the null character '\0'
		* \param re_group_rand_coef_data_pred Covariate data for grouped random coefficients
		* \param gp_coords_data_pred Coordinates (features) for Gaussian process
		* \param gp_rand_coef_data_pred Covariate data for Gaussian process random coefficients
		* \param covariate_data_pred Covariate data (=independent variables, features) for prediction
		* \param vecchia_pred_type Type of Vecchia approximation for making predictions. "order_obs_first_cond_obs_only" = observed data is ordered first and neighbors are only observed points, "order_obs_first_cond_all" = observed data is ordered first and neighbors are selected among all points (observed + predicted), "order_pred_first" = predicted data is ordered first for making predictions, "latent_order_obs_first_cond_obs_only"  = Vecchia approximation for the latent process and observed data is ordered first and neighbors are only observed points, "latent_order_obs_first_cond_all"  = Vecchia approximation for the latent process and observed data is ordered first and neighbors are selected among all points
		* \param num_neighbors_pred The number of neighbors used in the Vecchia approximation for making predictions (-1 means that the value already set at initialization is used)
		* \param cg_delta_conv_pred Tolerance level for L2 norm of residuals for checking convergence in conjugate gradient algorithm when being used for prediction
		* \param nsim_var_pred Number of samples when simulation is used for calculating predictive variances
		* \param rank_pred_approx_matrix_lanczos Rank of the matrix for approximating predictive covariances obtained using the Lanczos algorithm
		*/
		void SetPredictionData(int num_data_pred,
			const data_size_t* cluster_ids_data_pred,
			const char* re_group_data_pred,
			const double* re_group_rand_coef_data_pred,
			double* gp_coords_data_pred,
			const double* gp_rand_coef_data_pred,
			const double* covariate_data_pred,
			const char* vecchia_pred_type,
			int num_neighbors_pred,
			double cg_delta_conv_pred,
			int nsim_var_pred,
			int rank_pred_approx_matrix_lanczos) {
			if (!(gp_coords_data_pred == nullptr && re_group_data_pred == nullptr && re_group_rand_coef_data_pred == nullptr
				&& cluster_ids_data_pred == nullptr && gp_rand_coef_data_pred == nullptr && covariate_data_pred == nullptr)) {
				CHECK(num_data_pred > 0);
				num_data_pred_ = num_data_pred;
			}
			if (cluster_ids_data_pred != nullptr) {
				cluster_ids_data_pred_ = std::vector<data_size_t>(cluster_ids_data_pred, cluster_ids_data_pred + num_data_pred);
			}
			if (re_group_data_pred != nullptr) {
				//For grouped random effecst: create matrix 're_group_levels_pred' (vector of vectors, dimension: num_group_variables_ x num_data_) with strings of group levels from characters in 'const char* re_group_data_pred'
				re_group_levels_pred_ = std::vector<std::vector<re_group_t>>(num_group_variables_, std::vector<re_group_t>(num_data_pred));
				ConvertCharToStringGroupLevels(num_data_pred, num_group_variables_, re_group_data_pred, re_group_levels_pred_);
			}
			if (re_group_rand_coef_data_pred != nullptr) {
				re_group_rand_coef_data_pred_ = std::vector<double>(re_group_rand_coef_data_pred, re_group_rand_coef_data_pred + num_data_pred * num_re_group_rand_coef_);
			}
			if (gp_coords_data_pred != nullptr) {
				gp_coords_data_pred_ = std::vector<double>(gp_coords_data_pred, gp_coords_data_pred + num_data_pred * dim_gp_coords_);
			}
			if (gp_rand_coef_data_pred != nullptr) {
				gp_rand_coef_data_pred_ = std::vector<double>(gp_rand_coef_data_pred, gp_rand_coef_data_pred + num_data_pred * num_gp_rand_coef_);
			}
			if (covariate_data_pred != nullptr) {
				covariate_data_pred_ = std::vector<double>(covariate_data_pred, covariate_data_pred + num_data_pred * num_covariates_);
			}
			if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
				if (vecchia_pred_type != nullptr) {
					SetVecchiaPredType(vecchia_pred_type);
				}
				if (num_neighbors_pred > 0) {
					num_neighbors_pred_ = num_neighbors_pred;
				}
			}
			if (nsim_var_pred > 0) {
				nsim_var_pred_ = nsim_var_pred;
				nsim_var_pred_has_been_set_ = true;
			}
			if (matrix_inversion_method_ == "iterative") {
				if (cg_delta_conv_pred > 0) {
					cg_delta_conv_pred_ = cg_delta_conv_pred;
				}
				if (rank_pred_approx_matrix_lanczos > 0) {
					rank_pred_approx_matrix_lanczos_ = rank_pred_approx_matrix_lanczos;
				}
				SetPropertiesLikelihood();
			}
		}//end SetPredictionData

		/*!
		* \brief Make predictions: calculate conditional mean and variances or covariance matrix
		*		 Note: You should pre-allocate memory for out_predict
		*			   Its length is equal to num_data_pred if only the conditional mean is predicted (predict_cov_mat==false && predict_var==false)
		*			   or num_data_pred * (1 + num_data_pred) if the predictive covariance matrix is also calculated (predict_cov_mat==true)
		*			   or num_data_pred * 2 if predictive variances are also calculated (predict_var==true)
		* \param cov_pars_pred Covariance parameters of components
		* \param y_obs Response variable for observed data
		* \param num_data_pred Number of data points for which predictions are made
		* \param[out] out_predict Predictive/conditional mean at prediciton points followed by the predictive covariance matrix in column-major format (if predict_cov_mat==true) or the predictive variances (if predict_var==true)
		* \param calc_cov_factor If true, the covariance matrix of the observed data is factorized otherwise a previously done factorization is used (default=true)
		* \param predict_cov_mat If true, the predictive/conditional covariance matrix is calculated (default=false) (predict_var and predict_cov_mat cannot be both true)
		* \param predict_var If true, the predictive/conditional variances are calculated (default=false) (predict_var and predict_cov_mat cannot be both true)
		* \param predict_response If true, the response variable (label) is predicted, otherwise the latent random effects
		* \param sample_posterior If true, posterior samples are generated
		* \param sample_prior If true, prior samples are generated
		* \param num_post_samples Number of posterior samples
		* \param num_prior_samples Number of prior samples
		* \param covariate_data_pred Covariate data (=independent variables, features) for prediction
		* \param coef_pred Coefficients for linear covariates
		* \param cluster_ids_data_pred IDs / labels indicating independent realizations of Gaussian processes (same values = same process realization) for which predictions are to be made
		* \param re_group_data_pred Labels of group levels for the grouped random effects in column-major format (i.e. first the levels for the first effect, then for the second, etc.). Every group label needs to end with the null character '\0'
		* \param re_group_rand_coef_data_pred Covariate data for grouped random coefficients
		* \param gp_coords_data_pred Coordinates (features) for Gaussian process
		* \param gp_rand_coef_data_pred Covariate data for Gaussian process random coefficients
		* \param use_saved_data If true, saved data is used and some arguments are ignored
		* \param fixed_effects Fixed effects component of location parameter for observed data (only used for non-Gaussian likelihoods)
		* \param fixed_effects_pred Fixed effects component of location parameter for predicted data (only used for non-Gaussian likelihoods)
		*/
		void Predict(const double* cov_pars_pred,
			const double* y_obs,
			data_size_t num_data_pred,
			double* out_predict,
			bool calc_cov_factor,
			bool predict_cov_mat,
			bool predict_var,
			bool predict_response,
			bool sample_posterior,
			bool sample_prior,
			int num_post_samples,
			int num_prior_samples,
			const double* covariate_data_pred,
			const double* coef_pred,
			const data_size_t* cluster_ids_data_pred,
			const char* re_group_data_pred,
			const double* re_group_rand_coef_data_pred,
			double* gp_coords_data_pred,
			const double* gp_rand_coef_data_pred,
			bool use_saved_data,
			const double* fixed_effects,
			const double* fixed_effects_pred) {
			//First check whether previously set data should be used and load it if required
			std::vector<std::vector<re_group_t>> re_group_levels_pred, re_group_levels_pred_orig;//Matrix with group levels for the grouped random effects (re_group_levels_pred[j] contains the levels for RE number j)
			// Note: re_group_levels_pred_orig is only used for the case (only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_)
			//			since then re_group_levels_pred is over-written for every cluster and the original data thus needs to be saved
			if (use_saved_data) {
				if (num_data_pred > 0) {
					CHECK(num_data_pred == num_data_pred_);
				}
				else {
					num_data_pred = num_data_pred_;
				}
				re_group_levels_pred = re_group_levels_pred_;
				if (cluster_ids_data_pred_.empty()) {
					cluster_ids_data_pred = nullptr;
				}
				else {
					cluster_ids_data_pred = cluster_ids_data_pred_.data();
				}
				if (re_group_rand_coef_data_pred_.empty()) {
					re_group_rand_coef_data_pred = nullptr;
				}
				else {
					re_group_rand_coef_data_pred = re_group_rand_coef_data_pred_.data();
				}
				if (gp_coords_data_pred_.empty()) {
					gp_coords_data_pred = nullptr;
				}
				else {
					gp_coords_data_pred = gp_coords_data_pred_.data();
				}
				if (gp_rand_coef_data_pred_.empty()) {
					gp_rand_coef_data_pred = nullptr;
				}
				else {
					gp_rand_coef_data_pred = gp_rand_coef_data_pred_.data();
				}
				if (covariate_data_pred_.empty()) {
					covariate_data_pred = nullptr;
				}
				else {
					covariate_data_pred = covariate_data_pred_.data();
				}
			}// end use_saved_data
			else {
				if (re_group_data_pred != nullptr) {
					//For grouped random effects: create matrix 're_group_levels_pred' (vector of vectors, dimension: num_group_variables_ x num_data_) with strings of group levels from characters in 'const char* re_group_data_pred'
					re_group_levels_pred = std::vector<std::vector<re_group_t>>(num_group_variables_, std::vector<re_group_t>(num_data_pred));
					ConvertCharToStringGroupLevels(num_data_pred, num_group_variables_, re_group_data_pred, re_group_levels_pred);
				}
			}
			if (only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
				re_group_levels_pred_orig = re_group_levels_pred;
			}
			//Some checks including whether required data is present
			if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
				CHECK(num_neighbors_pred_ > 0);
				CHECK(cg_delta_conv_pred_ > 0.);
				CHECK(nsim_var_pred_ > 0);
				CHECK(rank_pred_approx_matrix_lanczos_ > 0);
			}
			if ((int)re_group_levels_pred.size() == 0 && num_group_variables_ > 0) {
				Log::REFatal("Missing grouping data ('group_data_pred') for grouped random effects foDEr making predictions");
			}
			if (re_group_rand_coef_data_pred == nullptr && num_re_group_rand_coef_ > 0) {
				Log::REFatal("Missing covariate data ('re_group_rand_coef_data_pred') for random coefficients "
					"for grouped random effects for making predictions");
			}
			if (gp_coords_data_pred == nullptr && num_gp_ > 0) {
				Log::REFatal("Missing coordinates data ('gp_coords_pred') for Gaussian process for making predictions");
			}
			if (gp_rand_coef_data_pred == nullptr && num_gp_rand_coef_ > 0) {
				Log::REFatal("Missing covariate data ('gp_rand_coef_data_pred') for random coefficients for Gaussian process for making predictions");
			}
			if (cluster_ids_data_pred == nullptr && num_clusters_ > 1) {
				Log::REFatal("Missing cluster_id data ('cluster_ids_pred') for making predictions");
			}
			CHECK(num_data_pred > 0);
			if (!gauss_likelihood_ && predict_response && predict_cov_mat) {
				Log::REFatal("Calculation of the predictive covariance matrix is not supported "
					"when predicting the response variable (label) for non-Gaussian likelihoods");
			}
			if (predict_cov_mat && predict_var) {
				Log::REFatal("Calculation of both the predictive covariance matrix and variances is not supported. "
					"Choose one of these option (predict_cov_mat or predict_var)");
			}
			CHECK(cov_pars_pred != nullptr);
			if (has_covariates_) {
				if (covariate_data_pred == nullptr) {
					Log::REFatal("Covariate data 'X_pred' not provided ");
				}
				CHECK(coef_pred != nullptr);
			}
			else {
				if (covariate_data_pred != nullptr) {
					Log::REFatal("Covariate data 'X_pred' provided but model has no linear regresion covariates ");
				}
			}
			if (sample_posterior && sample_prior) {
				Log::REFatal("Cannot sample from prior and posterior simultaneously ");
			}
			vec_t y_zero_dummy;
			if (sample_prior) {
				if (predict_cov_mat || predict_var) {
					Log::REFatal("Cannot calculate predictive (co-)variances if sampling from prior ");
				}
				y_zero_dummy = vec_t::Zero(num_data_);
				y_obs = y_zero_dummy.data();
			}
			if (y_obs == nullptr) {
				if (!y_has_been_set_ && !(sample_prior && !sample_posterior)) {
					Log::REFatal("Response variable data is not provided and has not been set before");
				}
			}
			else {
				// Check response variable data
				if (LightGBM::Common::HasNAOrInf(y_obs, num_data_)) {
					Log::REFatal("NaN or Inf in response variable / label");
				}
			}
			if (num_data_pred > 10000 && predict_cov_mat) {
				double num_mem_d = ((double)num_data_pred) * ((double)num_data_pred);
				int mem_size = (int)(num_mem_d * 8. / 1000000.);
				Log::REWarning("The covariance matrix can be very large for large sample sizes which might lead to memory limitations. "
					"In your case (n = %d), the covariance needs at least approximately %d mb of memory. ", num_data_pred, mem_size);
			}
			// Initialize linear predictor related terms and covariance parameters
			vec_t coef, mu;//mu = linear regression predictor
			if (has_covariates_) {//calculate linear regression term
				coef = Eigen::Map<const vec_t>(coef_pred, num_covariates_ * num_sets_re_);
				den_mat_t X_pred = Eigen::Map<const den_mat_t>(covariate_data_pred, num_data_pred, num_covariates_);
				mu = vec_t(num_sets_re_ * num_data_pred);
				for (int igp = 0; igp < num_sets_re_; ++igp) {
					mu.segment(num_data_pred * igp, num_data_pred) = X_pred * (coef.segment(num_covariates_ * igp, num_covariates_));
				}
			}
			vec_t cov_pars = Eigen::Map<const vec_t>(cov_pars_pred, num_cov_par_);
			//Set up cluster IDs
			std::map<data_size_t, int> num_data_per_cluster_pred;
			std::map<data_size_t, std::vector<int>> data_indices_per_cluster_pred;
			std::vector<data_size_t> unique_clusters_pred;
			data_size_t num_clusters_pred;
			SetUpClusterIds(num_data_pred, cluster_ids_data_pred, num_data_per_cluster_pred,
				data_indices_per_cluster_pred, unique_clusters_pred, num_clusters_pred);
			//Check whether predictions are made for existing clusters or if only for new independet clusters predictions are made
			bool pred_for_observed_data = false;
			for (const auto& cluster_i : unique_clusters_pred) {
				if (std::find(unique_clusters_.begin(), unique_clusters_.end(), cluster_i) != unique_clusters_.end()) {
					pred_for_observed_data = true;
					break;
				}
			}
			//Factorize covariance matrix and calculate Psi^{-1}y_obs or calculate Laplace approximation (if required)
			if (pred_for_observed_data) {
				const double* fixed_effects_ptr = nullptr;
				if (fixed_effects != nullptr) {
					fixed_effects_ptr = fixed_effects;
				}
				else if (has_fixed_effects_) {
					fixed_effects_ptr = fixed_effects_.data();
				}
				SetYCalcCovCalcYAuxForPred(cov_pars, coef, y_obs, calc_cov_factor, fixed_effects_ptr, false, sample_prior);
			}
			bool predict_var_or_response = predict_var || (!gauss_likelihood_ && predict_response && likelihood_[unique_clusters_[0]]->NeedPredLatentVarForResponseMean()); //variance needs to be available for response prediction for most non-Gaussian likelihoods
			// Loop over different clusters to calculate predictions
			for (const auto& cluster_i : unique_clusters_pred) {

				//Case 1: no data observed for this Gaussian process with ID 'cluster_i'
				if (std::find(unique_clusters_.begin(), unique_clusters_.end(), cluster_i) == unique_clusters_.end()) {
					if (num_sets_re_ > 1) {
						Log::REFatal("Predict function is not implemented for new clusters (in 'cluster_ids_pred') for heterodescadistic models ");
					}
					if (grouped_RE_and_vecchia_GP_) {
						Log::REFatal("Predict function is not implemented for new clusters (in 'cluster_ids_pred') for grouped random effects and a Vecchia-approximated GP ");
					}
					if (sample_posterior) {
						Log::REFatal("Posterior sampling is not implemented for new clusters (in 'cluster_ids_pred') ");
					}
					T_mat psi;
					std::vector<std::shared_ptr<RECompBase<T_mat>>> re_comps_cluster_i;
					int num_REs_pred = num_data_per_cluster_pred[cluster_i];
					if (gp_approx_ == "vecchia" && gauss_likelihood_ && predict_var && num_REs_pred > 10000) {
						Log::REWarning("Calculation of (only) predictive variances is currently not optimized for the Vecchia approximation, "
							"and this might takes a lot of time and/or memory.");
					}
					//Calculate covariance matrix if needed
					if (predict_cov_mat || predict_var || predict_response) {
						if (gp_approx_ == "vecchia") {
							//TODO: move this code out into another function for better readability
							std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp_clus0 = re_comps_vecchia_[unique_clusters_[0]][0][0];
							// Initialize RE components
							std::vector<std::vector<int>> nearest_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<den_mat_t> dist_obs_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<den_mat_t> dist_between_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<Triplet_t> entries_init_B_cluster_i;
							std::vector<std::vector<den_mat_t>> z_outer_z_obs_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_vecchia_cluster_i;
							bool only_one_GP_calculations_on_RE_scale_dummy = false;
							CreateREComponentsVecchia(num_data_pred, dim_gp_coords_, data_indices_per_cluster_pred, cluster_i,
								num_data_per_cluster_pred, gp_coords_data_pred,
								gp_rand_coef_data_pred, re_comps_vecchia_cluster_i,
								nearest_neighbors_cluster_i, dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i,
								entries_init_B_cluster_i, z_outer_z_obs_neighbors_cluster_i, only_one_GP_calculations_on_RE_scale_dummy, has_duplicates_coords_,
								"none", num_neighbors_pred_, vecchia_neighbor_selection_, false, rng_, num_gp_rand_coef_, num_gp_total_, num_comps_total_, gauss_likelihood_,
								re_comp_gp_clus0->CovFunctionName(), re_comp_gp_clus0->CovFunctionShape(), re_comp_gp_clus0->CovFunctionTaperRange(), re_comp_gp_clus0->CovFunctionTaperShape(),
								gp_approx_ == "tapering", save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, nearest_neighbors_determined_, GPU_use_, grouped_RE_and_vecchia_GP_);//TODO: maybe also use ordering for making predictions? (need to check that there are not errors)
							for (int j = 0; j < num_comps_total_; ++j) {
								const vec_t pars = cov_pars.segment(ind_par_[j], ind_par_[j + 1] - ind_par_[j]);
								re_comps_vecchia_cluster_i[j]->SetCovPars(pars);
							}
							if (re_comp_gp_clus0->RedetermineVecchiaNeighborsInTransformedSpace() || vecchia_neighbor_selection_ == "correlation") {//determine nearest neighbors when using correlation-based approach
								UpdateNearestNeighbors(re_comps_vecchia_cluster_i, nearest_neighbors_cluster_i,
									entries_init_B_cluster_i, num_neighbors_, vecchia_neighbor_selection_, rng_,
									has_duplicates_coords_, false, gauss_likelihood_, gp_approx_, chol_ip_cross_cov_[cluster_i][0],
									dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i, save_distances_isotropic_cov_fct_Vecchia_, GPU_use_);
								nearest_neighbors_determined_ = true;
							}
							// Calculate a Cholesky factor
							sp_mat_t B_cluster_i;
							sp_mat_t D_inv_cluster_i;
							std::vector<sp_mat_t> B_grad_cluster_i, D_grad_cluster_i;//not used, but needs to be passed to function
							CalcCovFactorGradientVecchia(num_data_per_cluster_pred[cluster_i], true, false, re_comps_vecchia_cluster_i,
								re_comps_cross_cov_[cluster_i][0], re_comps_ip_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0],
								nearest_neighbors_cluster_i, dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i,
								entries_init_B_cluster_i, z_outer_z_obs_neighbors_cluster_i,
								B_cluster_i, D_inv_cluster_i, B_grad_cluster_i, D_grad_cluster_i, sigma_ip_inv_cross_cov_T_[cluster_i][0],
								sigma_ip_grad_sigma_ip_inv_cross_cov_T_[cluster_i][0],
								true, 1., false, num_gp_total_, gauss_likelihood_, save_distances_isotropic_cov_fct_Vecchia_, gp_approx_,
								nullptr, estimate_cov_par_index_, nearest_neighbors_determined_, false);
							//Calculate Psi
							sp_mat_t D_sqrt(num_data_per_cluster_pred[cluster_i], num_data_per_cluster_pred[cluster_i]);
							D_sqrt.setIdentity();
							D_sqrt.diagonal().array() = D_inv_cluster_i.diagonal().array().pow(-0.5);
							sp_mat_t B_inv_D_sqrt;
							TriangularSolve<sp_mat_t, sp_mat_t, sp_mat_t>(B_cluster_i, D_sqrt, B_inv_D_sqrt, false);
							psi = B_inv_D_sqrt * B_inv_D_sqrt.transpose();
						}//end gp_approx_ == "vecchia"
						else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
							std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp_clus0 = re_comps_ip_[unique_clusters_[0]][0][0];
							psi = T_mat(num_REs_pred, num_REs_pred);
							if (gauss_likelihood_ && predict_response) {
								psi.setIdentity();//nugget effect
							}
							else {
								psi.setZero();
							}
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_ip_cluster_i;
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cross_cov_cluster_i;
							std::vector<std::shared_ptr<RECompGP<T_mat>>> re_comps_resid_cluster_i;
							CreateREComponentsFITC_FSA(num_data_pred, data_indices_per_cluster_pred, cluster_i, gp_coords_data_pred,
								re_comp_gp_clus0->CovFunctionName(), re_comp_gp_clus0->CovFunctionShape(), re_comp_gp_clus0->CovFunctionTaperRange(), re_comp_gp_clus0->CovFunctionTaperShape(),
								re_comps_ip_cluster_i, re_comps_cross_cov_cluster_i, re_comps_resid_cluster_i, true);
							for (int j = 0; j < num_comps_total_; ++j) {
								const vec_t pars = cov_pars.segment(ind_par_[j], ind_par_[j + 1] - ind_par_[j]);
								re_comps_ip_cluster_i[j]->SetCovPars(pars);
								re_comps_cross_cov_cluster_i[j]->SetCovPars(pars);
								re_comps_ip_cluster_i[j]->CalcSigma();
								re_comps_cross_cov_cluster_i[j]->CalcSigma();
								den_mat_t sigma_ip_stable = *(re_comps_ip_cluster_i[j]->GetZSigmaZt());
								sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
								chol_den_mat_t chol_fact_sigma_ip;
								chol_fact_sigma_ip.compute(sigma_ip_stable);
								den_mat_t cross_cov = *(re_comps_cross_cov_cluster_i[j]->GetZSigmaZt());
								den_mat_t sigma_interim = cross_cov * chol_fact_sigma_ip.solve(cross_cov.transpose());
								ConvertTo_T_mat_FromDense<T_mat>(sigma_interim, psi);
								//psi = cross_cov * chol_fact_sigma_ip.solve(cross_cov.transpose());
								if (gp_approx_ == "full_scale_tapering") {
									re_comps_resid_cluster_i[j]->SetCovPars(pars);
									re_comps_resid_cluster_i[j]->CalcSigma();
									re_comps_resid_cluster_i[j]->SubtractMatFromSigmaForResidInFullScale(psi);// Subtract predictive process covariance
									re_comps_resid_cluster_i[j]->ApplyTaper();
									psi += *(re_comps_resid_cluster_i[j]->GetZSigmaZt());
								}
								else {
									vec_t FITC_Diag = vec_t::Zero(cross_cov.rows());
									if (re_comps_cross_cov_[cluster_i][0][j]->VarianceOnDiagonal()) {
										FITC_Diag.array() += sigma_ip_stable.coeffRef(0, 0);
									}
									else {
#pragma omp parallel for schedule(static)
										for (int ii = 0; ii < re_comps_cross_cov_cluster_i[j]->GetNumUniqueREs(); ++ii) {
											FITC_Diag[ii] += re_comps_cross_cov_cluster_i[j]->GetZSigmaZtij(ii, ii);//uses coords of all data points (not IPs) to calculate Sigma[ii,ii]
										}
									}
									FITC_Diag -= psi.diagonal();
									psi += FITC_Diag.asDiagonal();
								}
							}
						}//end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
						else if (gp_approx_ == "full_scale_vecchia") {
							std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp_clus0 = re_comps_ip_[unique_clusters_[0]][0][0];
							psi = T_mat(num_REs_pred, num_REs_pred);
							if (gauss_likelihood_ && predict_response) {
								psi.setIdentity();//nugget effect
							}
							else {
								psi.setZero();
							}
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_ip_cluster_i;
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cross_cov_cluster_i;
							std::vector<std::shared_ptr<RECompGP<T_mat>>> re_comps_resid_cluster_i;
							if (vecchia_ordering_ == "random" || vecchia_ordering_ == "time_random_space") {
								std::shuffle(data_indices_per_cluster_pred[cluster_i].begin(), data_indices_per_cluster_pred[cluster_i].end(), rng_);
							}
							CreateREComponentsFITC_FSA(num_data_pred, data_indices_per_cluster_pred, cluster_i, gp_coords_data_pred,
								re_comp_gp_clus0->CovFunctionName(), re_comp_gp_clus0->CovFunctionShape(), re_comp_gp_clus0->CovFunctionTaperRange(), re_comp_gp_clus0->CovFunctionTaperShape(),
								re_comps_ip_cluster_i, re_comps_cross_cov_cluster_i, re_comps_resid_cluster_i, true);
							for (int j = 0; j < num_comps_total_; ++j) {
								const vec_t pars = cov_pars.segment(ind_par_[j], ind_par_[j + 1] - ind_par_[j]);
								re_comps_ip_cluster_i[j]->SetCovPars(pars);
								re_comps_cross_cov_cluster_i[j]->SetCovPars(pars);
								re_comps_ip_cluster_i[j]->CalcSigma();
								re_comps_cross_cov_cluster_i[j]->CalcSigma();
								den_mat_t sigma_ip_stable = *(re_comps_ip_cluster_i[j]->GetZSigmaZt());
								sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
								chol_den_mat_t chol_fact_sigma_ip;
								chol_fact_sigma_ip.compute(sigma_ip_stable);
								den_mat_t cross_cov = *(re_comps_cross_cov_cluster_i[j]->GetZSigmaZt());
								den_mat_t sigma_interim = cross_cov * chol_fact_sigma_ip.solve(cross_cov.transpose());
								ConvertTo_T_mat_FromDense<T_mat>(sigma_interim, psi); // for all T_mat? see ConvertTo_T_mat_FromDense from Pascal
								//psi = cross_cov * chol_fact_sigma_ip.solve(cross_cov.transpose());
							}
							re_comp_gp_clus0 = re_comps_vecchia_[unique_clusters_[0]][0][0];
							// Initialize RE components
							std::vector<std::vector<int>> nearest_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<den_mat_t> dist_obs_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<den_mat_t> dist_between_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<Triplet_t> entries_init_B_cluster_i;
							std::vector<std::vector<den_mat_t>> z_outer_z_obs_neighbors_cluster_i(num_data_per_cluster_pred[cluster_i]);
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_vecchia_cluster_i;
							bool only_one_GP_calculations_on_RE_scale_dummy = false;
							CreateREComponentsVecchia(num_data_pred, dim_gp_coords_, data_indices_per_cluster_pred, cluster_i,
								num_data_per_cluster_pred, gp_coords_data_pred,
								gp_rand_coef_data_pred, re_comps_vecchia_cluster_i,
								nearest_neighbors_cluster_i, dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i,
								entries_init_B_cluster_i, z_outer_z_obs_neighbors_cluster_i, only_one_GP_calculations_on_RE_scale_dummy, has_duplicates_coords_,
								"none", num_neighbors_pred_, vecchia_neighbor_selection_, false, rng_, num_gp_rand_coef_, num_gp_total_, num_comps_total_, gauss_likelihood_,
								re_comp_gp_clus0->CovFunctionName(), re_comp_gp_clus0->CovFunctionShape(), re_comp_gp_clus0->CovFunctionTaperRange(), re_comp_gp_clus0->CovFunctionTaperShape(),
								gp_approx_ == "tapering", save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, nearest_neighbors_determined_, GPU_use_, grouped_RE_and_vecchia_GP_);//TODO: maybe also use ordering for making predictions? (need to check that there are not errors)
							for (int j = 0; j < num_comps_total_; ++j) {
								const vec_t pars = cov_pars.segment(ind_par_[j], ind_par_[j + 1] - ind_par_[j]);
								re_comps_vecchia_cluster_i[j]->SetCovPars(pars);
							}
							if (re_comp_gp_clus0->RedetermineVecchiaNeighborsInTransformedSpace() || vecchia_neighbor_selection_ == "residual_correlation") {//determine nearest neighbors when using correlation-based approach
								UpdateNearestNeighbors(re_comps_vecchia_cluster_i, nearest_neighbors_cluster_i,
									entries_init_B_cluster_i, num_neighbors_, vecchia_neighbor_selection_, rng_,
									has_duplicates_coords_, false, gauss_likelihood_, gp_approx_, chol_ip_cross_cov_[cluster_i][0],
									dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i, save_distances_isotropic_cov_fct_Vecchia_, GPU_use_);
								nearest_neighbors_determined_ = true;
							}
							// Calculate a Cholesky factor
							sp_mat_t B_cluster_i;
							sp_mat_t D_inv_cluster_i;
							std::vector<sp_mat_t> B_grad_cluster_i, D_grad_cluster_i;//not used, but needs to be passed to function
							CalcCovFactorGradientVecchia(num_data_per_cluster_pred[cluster_i], true, false, re_comps_vecchia_cluster_i,
								re_comps_cross_cov_[cluster_i][0], re_comps_ip_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0],
								nearest_neighbors_cluster_i, dist_obs_neighbors_cluster_i, dist_between_neighbors_cluster_i,
								entries_init_B_cluster_i, z_outer_z_obs_neighbors_cluster_i,
								B_cluster_i, D_inv_cluster_i, B_grad_cluster_i, D_grad_cluster_i, sigma_ip_inv_cross_cov_T_[cluster_i][0],
								sigma_ip_grad_sigma_ip_inv_cross_cov_T_[cluster_i][0],
								true, 1., false, num_gp_total_, gauss_likelihood_, save_distances_isotropic_cov_fct_Vecchia_, gp_approx_,
								nullptr, estimate_cov_par_index_, nearest_neighbors_determined_, false);
							//Calculate Psi
							sp_mat_t D_sqrt(num_data_per_cluster_pred[cluster_i], num_data_per_cluster_pred[cluster_i]);
							D_sqrt.setIdentity();
							D_sqrt.diagonal().array() = D_inv_cluster_i.diagonal().array().pow(-0.5);
							sp_mat_t B_inv_D_sqrt;
							TriangularSolve<sp_mat_t, sp_mat_t, sp_mat_t>(B_cluster_i, D_sqrt, B_inv_D_sqrt, false);
							sp_mat_t B_inv_D_B_inv = B_inv_D_sqrt * B_inv_D_sqrt.transpose();
							den_mat_t sigma_interim = (den_mat_t)B_inv_D_B_inv;
							T_mat psi_interim;
							ConvertTo_T_mat_FromDense<T_mat>(sigma_interim, psi_interim);
							psi += psi_interim;
						}
						else if (gp_approx_ == "none") {
							string_t cov_fct = "";
							double cov_fct_shape = 0., cov_fct_taper_range = 0., cov_fct_taper_shape = 0.;
							if (num_gp_ > 0) {
								std::shared_ptr<RECompGP<T_mat>> re_comp_gp_clus0 = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[unique_clusters_[0]][0][ind_intercept_gp_]);
								cov_fct = re_comp_gp_clus0->CovFunctionName();
								cov_fct_shape = re_comp_gp_clus0->CovFunctionShape();
								cov_fct_taper_range = re_comp_gp_clus0->CovFunctionTaperRange();
								cov_fct_taper_shape = re_comp_gp_clus0->CovFunctionTaperShape();
							}
							CreateREComponents(num_data_pred, data_indices_per_cluster_pred, cluster_i,
								re_group_levels_pred, num_data_per_cluster_pred, re_group_rand_coef_data_pred,
								gp_coords_data_pred, gp_rand_coef_data_pred, true,
								cov_fct, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape,
								re_comps_cluster_i);
							if (only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_) {
								num_REs_pred = re_comps_cluster_i[0]->GetNumUniqueREs();
							}
							else {
								num_REs_pred = num_data_per_cluster_pred[cluster_i];
							}
							psi = T_mat(num_REs_pred, num_REs_pred);
							if (gauss_likelihood_ && predict_response) {
								psi.setIdentity();//nugget effect
							}
							else {
								psi.setZero();
							}
							for (int j = 0; j < num_comps_total_; ++j) {
								const vec_t pars = cov_pars.segment(ind_par_[j], ind_par_[j + 1] - ind_par_[j]);
								re_comps_cluster_i[j]->SetCovPars(pars);
								re_comps_cluster_i[j]->CalcSigma();
								psi += (*(re_comps_cluster_i[j]->GetZSigmaZt().get()));
							}

						}//end gp_approx_ == "none"
						if (gauss_likelihood_) {
							psi *= cov_pars[0];//back-transform
						}
					}//end calculation of covariance matrix
					// Add external fixed_effects
					vec_t mean_pred_id = vec_t::Zero(num_data_per_cluster_pred[cluster_i]);
					if (fixed_effects_pred != nullptr) {//add externaly provided fixed effects
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
							mean_pred_id[i] += fixed_effects_pred[data_indices_per_cluster_pred[cluster_i][i]];
						}
					}
					// Add linear regression predictor
					if (has_covariates_) {
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
							mean_pred_id[i] += mu[data_indices_per_cluster_pred[cluster_i][i]];
						}
					}
					vec_t var_pred_id;
					if (predict_var_or_response) {
						var_pred_id = psi.diagonal();
					}
					// Map from predictions from random effects scale b to "data scale" Zb
					if (only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_) {
						if (predict_var_or_response) {
							vec_t var_pred_id_on_RE_scale = var_pred_id;
							var_pred_id = vec_t(num_data_per_cluster_pred[cluster_i]);
#pragma omp parallel for schedule(static)
							for (data_size_t i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								var_pred_id[i] = var_pred_id_on_RE_scale[(re_comps_cluster_i[0]->random_effects_indices_of_data_)[i]];
							}
						}
						if (predict_cov_mat) {
							T_mat cov_mat_pred_id_on_RE_scale = psi;
							sp_mat_t Zpred(num_data_per_cluster_pred[cluster_i], num_REs_pred);
							std::vector<Triplet_t> triplets(num_data_per_cluster_pred[cluster_i]);
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								triplets[i] = Triplet_t(i, (re_comps_cluster_i[0]->random_effects_indices_of_data_)[i], 1.);
							}
							Zpred.setFromTriplets(triplets.begin(), triplets.end());
							psi = Zpred * cov_mat_pred_id_on_RE_scale * Zpred.transpose();
						}
					}//end only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_
					// Transform to response scale for non-Gaussian likelihoods if needed
					if (!gauss_likelihood_ && predict_response) {
						vec_t dummy, dummy2;
						likelihood_[unique_clusters_[0]]->PredictResponse(mean_pred_id, var_pred_id, dummy, dummy2,  predict_var);
					}
					// Write on output
#pragma omp parallel for schedule(static)
					for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
						out_predict[data_indices_per_cluster_pred[cluster_i][i]] = mean_pred_id[i];
					}
					// Write covariance / variance on output
					if (!predict_response || gauss_likelihood_) {//this is not done if predict_response==true for non-Gaussian likelihoods 
						if (predict_cov_mat) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {//column index
								for (int j = 0; j < num_data_per_cluster_pred[cluster_i]; ++j) {//row index
									out_predict[data_indices_per_cluster_pred[cluster_i][i] * num_data_pred + data_indices_per_cluster_pred[cluster_i][j] + num_data_pred] = psi.coeff(j, i);
								}
							}
						}//end predict_cov_mat
						if (predict_var) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_pred[cluster_i][i] + num_data_pred] = var_pred_id[i];
							}
						}//end predict_var
					}//end !predict_response || gauss_likelihood_
					else { // predict_response && !gauss_likelihood_
						if (predict_var) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_pred[cluster_i][i] + num_data_pred] = var_pred_id[i];
							}
						}//end predict_var
					}//end write covariance / variance on output

				}//end cluster_i with no observed data
				else {
					//Case 2: there exists observed data for this cluster_i
					den_mat_t gp_coords_mat_pred;
					std::vector<data_size_t> random_effects_indices_of_data_pred;
					int num_REs_pred = num_data_per_cluster_pred[cluster_i];
					if (num_gp_ > 0) {
						std::vector<double> gp_coords_pred;
						for (int j = 0; j < dim_gp_coords_; ++j) {
							for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
								gp_coords_pred.push_back(gp_coords_data_pred[j * num_data_pred + id]);
							}
						}
						gp_coords_mat_pred = Eigen::Map<den_mat_t>(gp_coords_pred.data(), num_data_per_cluster_pred[cluster_i], dim_gp_coords_);
					}
					if (only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
						// Determine unique group levels per cluster and create map which maps every data point per cluster to a group level
						random_effects_indices_of_data_pred = std::vector<data_size_t>(num_data_per_cluster_pred[cluster_i]);
						std::vector<re_group_t> re_group_levels_pred_unique;
						std::map<re_group_t, int> map_group_label_index_pred;
						int num_group_pred = 0;
						int ii = 0;
						for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
							if (map_group_label_index_pred.find(re_group_levels_pred_orig[0][id]) == map_group_label_index_pred.end()) {
								map_group_label_index_pred.insert({ re_group_levels_pred_orig[0][id], num_group_pred });
								re_group_levels_pred_unique.push_back(re_group_levels_pred_orig[0][id]);
								random_effects_indices_of_data_pred[ii] = num_group_pred;
								num_group_pred += 1;
							}
							else {
								random_effects_indices_of_data_pred[ii] = map_group_label_index_pred[re_group_levels_pred_orig[0][id]];
							}
							ii += 1;
						}
						re_group_levels_pred[0] = re_group_levels_pred_unique;
						num_REs_pred = (int)re_group_levels_pred[0].size();
					}//end only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_
					else if (only_one_GP_calculations_on_RE_scale_) {
						random_effects_indices_of_data_pred = std::vector<data_size_t>(num_data_per_cluster_pred[cluster_i]);
						std::vector<int> uniques;//unique points
						std::vector<int> unique_idx;//used for constructing incidence matrix Z_ if there are duplicates
						DetermineUniqueDuplicateCoordsFast(gp_coords_mat_pred, num_data_per_cluster_pred[cluster_i], uniques, unique_idx);
#pragma omp for schedule(static)
						for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
							random_effects_indices_of_data_pred[i] = unique_idx[i];
						}
						den_mat_t gp_coords_mat_pred_unique = gp_coords_mat_pred(uniques, Eigen::all);
						gp_coords_mat_pred = gp_coords_mat_pred_unique;
						num_REs_pred = (int)gp_coords_mat_pred.rows();
					}//end only_one_GP_calculations_on_RE_scale_
					// Initialize predictive mean and covariance
					std::map<int, vec_t> mean_pred_id;//mean_pred_id[1] = predictive mean for variance parameter in heteroscedastic models
					for (int igp = 0; igp < num_sets_re_; ++igp) {
						if (only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_ ||
							only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
							mean_pred_id[igp] = vec_t(num_REs_pred);
						}
						else {
							mean_pred_id[igp] = vec_t(num_data_per_cluster_pred[cluster_i]);
						}
					}
					T_mat cov_mat_pred_id;
					if (num_sets_re_ > 1) {
						if (predict_cov_mat) {
							Log::REFatal("Predictive covariance matrices are not supported for num_sets_re_ > 1");
						}
						if (sample_posterior) {
							Log::REFatal("Posterior sampling is not supported for num_sets_re_ > 1");
						}
						if (sample_prior) {
							Log::REFatal("Prior sampling is not supported for num_sets_re_ > 1");
						}
					}
					std::map<int, vec_t> var_pred_id;//var_pred_id[1] = predictive variance for variance parameter in heteroscedastic models
					den_mat_t post_samples_id;// sample from posterior without the mean, the mean is added at then end
					den_mat_t prior_samples_id;// sample from prior
					std::map<int, sp_mat_t> Bpo, Bp; // used only if gp_approx_ == "vecchia" && !gauss_likelihood_
					std::map<int, vec_t> Dp;
					int num_gp_pred = predict_var_or_response ? num_sets_re_ : 1;
					// Calculate predictions
					if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
						if (gp_approx_ == "vecchia" && sample_prior) {
							B_rm_[cluster_i][0] = sp_mat_rm_t(B_[cluster_i][0]);
							D_inv_rm_[cluster_i][0] = sp_mat_rm_t(D_inv_[cluster_i][0]);
						}
						std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp = re_comps_vecchia_[cluster_i][0][0];
						den_mat_t cov_mat_pred_vecchia_id;
						if (gauss_likelihood_) {
							int num_data_tot = num_data_per_cluster_[cluster_i] + num_data_per_cluster_pred[cluster_i];
							double num_mem_d = ((double)num_neighbors_pred_) * ((double)num_neighbors_pred_) * (double)(num_data_tot)+(double)(num_neighbors_pred_) * (double)(num_data_tot);
							int mem_size = (int)(num_mem_d * 8. / 1000000.);
							if (mem_size > 4000) {
								Log::REDebug("The current implementation of the Vecchia approximation needs a lot of memory if the number of neighbors is large. "
									"In your case (nb. of neighbors = %d, nb. of observations = %d, nb. of predictions = %d), "
									"this needs at least approximately %d mb of memory.",
									num_neighbors_pred_, num_data_per_cluster_[cluster_i], num_data_per_cluster_pred[cluster_i], mem_size);
							}
							den_mat_t gp_coords_mat_ip, cross_cov_pred_ip;
							if (vecchia_pred_type_ == "order_obs_first_cond_obs_only") {
								if (gp_approx_ == "full_scale_vecchia") {
									std::shared_ptr<RECompGP<den_mat_t>> re_comp_cross_cov_cluster_i_pred_ip = std::dynamic_pointer_cast<RECompGP<den_mat_t>>(re_comps_cross_cov_[cluster_i][0][0]);
									gp_coords_mat_ip = re_comp_cross_cov_cluster_i_pred_ip->coords_ind_point_;
								}
								CalcPredVecchiaObservedFirstOrder(true, cluster_i, num_data_pred,
									re_comps_cross_cov_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_woodbury_[cluster_i], cross_cov_pred_ip,
									B_rm_[cluster_i][0], D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0], data_indices_per_cluster_pred,
									re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, gp_coords_mat_ip, num_neighbors_pred_, vecchia_neighbor_selection_,
									re_comps_vecchia_[cluster_i][0], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], gauss_likelihood_, rng_,
									predict_cov_mat, predict_var, sample_posterior, sample_prior, num_post_samples, num_prior_samples, post_samples_id, prior_samples_id, seed_rng_, cg_generator_counter_,
									mean_pred_id[0], cov_mat_pred_vecchia_id, var_pred_id[0], Bpo[0], Bp[0], Dp[0], save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, GPU_use_);
							}
							else if (vecchia_pred_type_ == "order_obs_first_cond_all") {
								if (gp_approx_ == "full_scale_vecchia") {
									std::shared_ptr<RECompGP<den_mat_t>> re_comp_cross_cov_cluster_i_pred_ip = std::dynamic_pointer_cast<RECompGP<den_mat_t>>(re_comps_cross_cov_[cluster_i][0][0]);
									gp_coords_mat_ip = re_comp_cross_cov_cluster_i_pred_ip->coords_ind_point_;
								}
								CalcPredVecchiaObservedFirstOrder(false, cluster_i, num_data_pred,
									re_comps_cross_cov_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_woodbury_[cluster_i], cross_cov_pred_ip,
									B_rm_[cluster_i][0], D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0], data_indices_per_cluster_pred,
									re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, gp_coords_mat_ip, num_neighbors_pred_, vecchia_neighbor_selection_,
									re_comps_vecchia_[cluster_i][0], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], gauss_likelihood_, rng_,
									predict_cov_mat, predict_var, sample_posterior, sample_prior, num_post_samples, num_prior_samples, post_samples_id, prior_samples_id, seed_rng_, cg_generator_counter_,
									mean_pred_id[0], cov_mat_pred_vecchia_id, var_pred_id[0], Bpo[0], Bp[0], Dp[0], save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, GPU_use_);
							}
							else if (vecchia_pred_type_ == "order_pred_first") {
								if (gp_approx_ == "full_scale_vecchia") {
									Log::REFatal("The full-scale Vecchia approximation is currently not implemented when prediction locations appear first in the ordering. Please use vecchia_pred_type = order_obs_first_cond_all or vecchia_pred_type = order_obs_first_cond_obs_only");
								}
								if (sample_posterior) {
									Log::REFatal("Posterior sampling is not implemented when prediction locations appear first in the ordering.");
								}
								if (sample_prior) {
									Log::REFatal("Prior sampling is not implemented when prediction locations appear first in the ordering.");
								}
								CalcPredVecchiaPredictedFirstOrder(cluster_i, num_data_pred, data_indices_per_cluster_pred,
									re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, num_neighbors_pred_, vecchia_neighbor_selection_,
									re_comps_vecchia_[cluster_i][0], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], rng_,
									predict_cov_mat, predict_var, mean_pred_id[0], cov_mat_pred_vecchia_id, var_pred_id[0], save_distances_isotropic_cov_fct_Vecchia_, GPU_use_);
							}
							else if (vecchia_pred_type_ == "latent_order_obs_first_cond_obs_only") {
								if (num_gp_rand_coef_ > 0) {
									Log::REFatal("The Vecchia approximation for latent process(es) is currently not implemented when having random coefficients");
								}
								if (gp_approx_ == "full_scale_vecchia") {
									Log::REFatal("The full-scale Vecchia approximation for latent process(es) is currently not implemented");
								}
								if (sample_posterior) {
									Log::REFatal("Posterior sampling is not implemented for latent process(es).");
								}
								if (sample_prior) {
									Log::REFatal("Prior sampling is not implemented for latent process(es).");
								}
								CalcPredVecchiaLatentObservedFirstOrder(true,
									re_comp_gp->coords_, gp_coords_mat_pred, num_neighbors_pred_, vecchia_neighbor_selection_,
									re_comps_vecchia_[cluster_i][0], y_[cluster_i], rng_,
									predict_cov_mat, predict_var, predict_response, mean_pred_id[0], cov_mat_pred_vecchia_id, var_pred_id[0], save_distances_isotropic_cov_fct_Vecchia_, GPU_use_);
								// Note: we use the function 'CalcPredVecchiaLatentObservedFirstOrder' instead of the function 'CalcPredVecchiaObservedFirstOrder' since 
								//	the current implementation cannot handle duplicate values in gp_coords (coordinates / input features) for Vecchia approximations
								//	for latent processes (as matrices that need to be inverted will be singular due to the duplicate values).
								//	The function 'CalcPredVecchiaLatentObservedFirstOrder' avoids this singularity problem by using incidence matrices Z and 
								//	and applying a Vecchia approximation to the GP of all unique gp_coords
							}
							else if (vecchia_pred_type_ == "latent_order_obs_first_cond_all") {
								if (num_gp_rand_coef_ > 0) {
									Log::REFatal("The Vecchia approximation for latent process(es) is currently not implemented when having random coefficients");
								}
								if (gp_approx_ == "full_scale_vecchia") {
									Log::REFatal("The full-scale Vecchia approximation for latent process(es) is currently not implemented");
								}
								if (sample_posterior) {
									Log::REFatal("Posterior sampling is not implemented for latent process(es).");
								}
								if (sample_prior) {
									Log::REFatal("Prior sampling is not implemented for latent process(es).");
								}
								CalcPredVecchiaLatentObservedFirstOrder(false,
									re_comp_gp->coords_, gp_coords_mat_pred, num_neighbors_pred_, vecchia_neighbor_selection_,
									re_comps_vecchia_[cluster_i][0], y_[cluster_i], rng_,
									predict_cov_mat, predict_var, predict_response, mean_pred_id[0], cov_mat_pred_vecchia_id, var_pred_id[0], save_distances_isotropic_cov_fct_Vecchia_, GPU_use_);
							}
							else {
								Log::REFatal("Prediction type '%s' is not supported for the Veccia approximation.", vecchia_pred_type_.c_str());
							}
							if (predict_var || predict_cov_mat) {
								// subtract nugget variance in case latent process is predicted
								if (!predict_response && (vecchia_pred_type_ == "order_obs_first_cond_obs_only" ||
									vecchia_pred_type_ == "order_obs_first_cond_all" ||
									vecchia_pred_type_ == "order_pred_first")) {
									if (predict_cov_mat) {
#pragma omp parallel for schedule(static)
										for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
											cov_mat_pred_vecchia_id.coeffRef(i, i) -= 1.;
										}
									}
									if (predict_var) {
#pragma omp parallel for schedule(static)
										for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
											var_pred_id[0][i] -= 1.;
										}
									}
								}
							}
						}//end gauss_likelihood_
						else {//not gauss_likelihood_
							const double* fixed_effects_cluster_i_ptr = nullptr;
							// Note that fixed_effects_cluster_i_ptr is not used since calc_mode == false
							// The mode has been calculated already before in the Predict() function above
							// mean_pred_id, var_pred_id, and cov_mat_pred_id are not calculated in 'CalcPredVecchiaObservedFirstOrder', only Bpo, Bp, and Dp for non-Gaussian likelihoods
							den_mat_t gp_coords_mat_ip, cross_cov_pred_ip;
							for (int igp = 0; igp < num_gp_pred; ++igp) {
								if (gp_approx_ == "full_scale_vecchia") {
									std::shared_ptr<RECompGP<den_mat_t>> re_comp_cross_cov_cluster_i_pred_ip = std::dynamic_pointer_cast<RECompGP<den_mat_t>>(re_comps_cross_cov_[cluster_i][0][0]);
									gp_coords_mat_ip = re_comp_cross_cov_cluster_i_pred_ip->coords_ind_point_;
									if (vecchia_pred_type_ == "latent_order_obs_first_cond_obs_only") {
										CalcPredVecchiaObservedFirstOrder(true, cluster_i, num_data_pred,
											re_comps_cross_cov_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0],
											chol_fact_sigma_woodbury_[cluster_i], cross_cov_pred_ip, B_rm_[cluster_i][0], D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0],
											data_indices_per_cluster_pred, re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, gp_coords_mat_ip, num_neighbors_pred_, vecchia_neighbor_selection_,
											re_comps_vecchia_[cluster_i][igp], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], gauss_likelihood_, rng_,
											false, false, sample_posterior, sample_prior, num_post_samples, num_prior_samples, post_samples_id, prior_samples_id, seed_rng_, cg_generator_counter_,
											mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp], Bpo[igp], Bp[igp], Dp[igp], save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, GPU_use_);
										likelihood_[cluster_i]->PredictLaplaceApproxFSVA(y_[cluster_i].data(), y_int_[cluster_i].data(), fixed_effects_cluster_i_ptr,
											B_[cluster_i][0], D_inv_[cluster_i][0], Bpo[igp], Bp[igp], Dp[igp], re_comps_ip_[cluster_i][0][0]->GetZSigmaZt(), re_comps_ip_preconditioner_[cluster_i][0],
											re_comps_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0],
											sigma_woodbury_[cluster_i], chol_fact_sigma_woodbury_[cluster_i], chol_ip_cross_cov_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0],
											re_comps_cross_cov_[cluster_i][0], cross_cov_pred_ip, B_T_D_inv_B_cross_cov_[cluster_i][0], D_inv_B_cross_cov_[cluster_i][0], 
											sample_posterior, num_post_samples, post_samples_id, mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp],
											predict_cov_mat, predict_var_or_response, false, true, GPU_use_);
									}
									else if (vecchia_pred_type_ == "latent_order_obs_first_cond_all") {
										CalcPredVecchiaObservedFirstOrder(false, cluster_i, num_data_pred,
											re_comps_cross_cov_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_woodbury_[cluster_i], cross_cov_pred_ip,
											B_rm_[cluster_i][0], D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0],
											data_indices_per_cluster_pred, re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, gp_coords_mat_ip, num_neighbors_pred_, vecchia_neighbor_selection_,
											re_comps_vecchia_[cluster_i][igp], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], gauss_likelihood_, rng_,
											false, false, sample_posterior, sample_prior, num_post_samples, num_prior_samples, post_samples_id, prior_samples_id, seed_rng_, cg_generator_counter_,
											mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp], Bpo[igp], Bp[igp], Dp[igp], save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, GPU_use_);
										likelihood_[cluster_i]->PredictLaplaceApproxFSVA(y_[cluster_i].data(), y_int_[cluster_i].data(), fixed_effects_cluster_i_ptr,
											B_[cluster_i][0], D_inv_[cluster_i][0], Bpo[igp], Bp[igp], Dp[igp], re_comps_ip_[cluster_i][0][0]->GetZSigmaZt(), re_comps_ip_preconditioner_[cluster_i][0],
											re_comps_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0],
											sigma_woodbury_[cluster_i], chol_fact_sigma_woodbury_[cluster_i], chol_ip_cross_cov_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0],
											re_comps_cross_cov_[cluster_i][0], cross_cov_pred_ip, B_T_D_inv_B_cross_cov_[cluster_i][0], D_inv_B_cross_cov_[cluster_i][0], 
											sample_posterior, num_post_samples, post_samples_id, mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp],
											predict_cov_mat, predict_var_or_response, false, false, GPU_use_);
									}
									else {
										Log::REFatal("Prediction type '%s' is not supported for the Veccia approximation.", vecchia_pred_type_.c_str());
									}
								}
								else if (gp_approx_ == "vecchia") {
									if (vecchia_pred_type_ == "latent_order_obs_first_cond_obs_only") {
										CalcPredVecchiaObservedFirstOrder(true, cluster_i, num_data_pred,
											re_comps_cross_cov_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_woodbury_[cluster_i], cross_cov_pred_ip,
											B_rm_[cluster_i][0], D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0],
											data_indices_per_cluster_pred, re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, gp_coords_mat_ip, num_neighbors_pred_, vecchia_neighbor_selection_,
											re_comps_vecchia_[cluster_i][igp], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], gauss_likelihood_, rng_,
											false, false, sample_posterior, sample_prior, num_post_samples, num_prior_samples, post_samples_id, prior_samples_id, seed_rng_, cg_generator_counter_,
											mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp], Bpo[igp], Bp[igp], Dp[igp], save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, GPU_use_);
									}
									else if (vecchia_pred_type_ == "latent_order_obs_first_cond_all") {
										CalcPredVecchiaObservedFirstOrder(false, cluster_i, num_data_pred,
											re_comps_cross_cov_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_woodbury_[cluster_i], cross_cov_pred_ip,
											B_rm_[cluster_i][0], D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0],
											data_indices_per_cluster_pred, re_comp_gp->coords_, gp_coords_mat_pred, gp_rand_coef_data_pred, gp_coords_mat_ip, num_neighbors_pred_, vecchia_neighbor_selection_,
											re_comps_vecchia_[cluster_i][igp], num_gp_rand_coef_, num_gp_total_, y_[cluster_i], gauss_likelihood_, rng_,
											false, false, sample_posterior, sample_prior, num_post_samples, num_prior_samples, post_samples_id, prior_samples_id, seed_rng_, cg_generator_counter_,
											mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp], Bpo[igp], Bp[igp], Dp[igp], save_distances_isotropic_cov_fct_Vecchia_, gp_approx_, GPU_use_);
									}
									else {
										Log::REFatal("Prediction type '%s' is not supported for the Veccia approximation ", vecchia_pred_type_.c_str());
									}
									if (!grouped_RE_and_vecchia_GP_) {
										bool CondObsOnly = vecchia_pred_type_ == "latent_order_obs_first_cond_obs_only";
										likelihood_[cluster_i]->PredictLaplaceApproxVecchia(y_[cluster_i].data(), y_int_[cluster_i].data(), fixed_effects_cluster_i_ptr,
											B_[cluster_i], D_inv_[cluster_i], Bpo[igp], Bp[igp], Dp[igp],
											sample_posterior, num_post_samples, post_samples_id, 
											mean_pred_id[igp], cov_mat_pred_vecchia_id, var_pred_id[igp],
											predict_cov_mat, predict_var_or_response, false, CondObsOnly, re_comps_ip_preconditioner_[cluster_i][0],
											re_comps_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0],
											igp, cluster_i, this);
									}
								}//end gp_approx_ == "vecchia"
							}// end loop over num_gp_pred
						}//end not gauss_likelihood_
						if (predict_cov_mat) {
							ConvertTo_T_mat_FromDense(cov_mat_pred_vecchia_id, cov_mat_pred_id);
						}
					}//end (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia")
					else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
						if (sample_posterior) {
							Log::REFatal("Posterior sampling is not implemented for gp_approx = '%s' ", gp_approx_.c_str());
						}
						if (sample_prior) {
							Log::REFatal("Prior sampling is not implemented for gp_approx = '%s' ", gp_approx_.c_str());
						}
						CalcPredFITC_FSA(cluster_i, gp_coords_mat_pred, predict_cov_mat,
							predict_var_or_response, predict_response, mean_pred_id[0], cov_mat_pred_id, var_pred_id[0], nsim_var_pred_, cg_delta_conv_pred_);
					}
					if (gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" &&
						gp_approx_ != "full_scale_vecchia" && gp_approx_ != "vecchia" || grouped_RE_and_vecchia_GP_) {
						if (grouped_RE_and_vecchia_GP_) {
							CHECK(num_gp_pred == 1);
							CHECK(gp_approx_ == "vecchia");
						}
						bool VecchiaCondObsOnly = vecchia_pred_type_ == "latent_order_obs_first_cond_obs_only";
						CalcPred(cluster_i, num_data_pred, num_data_per_cluster_pred, data_indices_per_cluster_pred,
							re_group_levels_pred, re_group_rand_coef_data_pred, gp_coords_mat_pred, gp_rand_coef_data_pred,
							predict_cov_mat, predict_var_or_response, predict_response, sample_posterior, sample_prior, num_post_samples, num_prior_samples,
							mean_pred_id[0], cov_mat_pred_id, var_pred_id[0], post_samples_id, prior_samples_id, Bpo[0], Bp[0], Dp[0], VecchiaCondObsOnly);
					}
					for (int igp = 0; igp < num_gp_pred; ++igp) {
						//map from predictions from random effects scale b to "data scale" Zb
						if (only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_ ||
							only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
							vec_t mean_pred_id_on_RE_scale = mean_pred_id[igp];
							mean_pred_id[igp] = vec_t(num_data_per_cluster_pred[cluster_i]);
#pragma omp parallel for schedule(static)
							for (data_size_t i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								mean_pred_id[igp][i] = mean_pred_id_on_RE_scale[random_effects_indices_of_data_pred[i]];
							}
							if (predict_var_or_response) {
								vec_t var_pred_id_on_RE_scale = var_pred_id[igp];
								var_pred_id[igp] = vec_t(num_data_per_cluster_pred[cluster_i]);
#pragma omp parallel for schedule(static)
								for (data_size_t i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
									var_pred_id[igp][i] = var_pred_id_on_RE_scale[random_effects_indices_of_data_pred[i]];
								}
							}
							if (sample_posterior) {
								den_mat_t post_samples_id_on_RE_scale;
								post_samples_id_on_RE_scale.swap(post_samples_id);
								post_samples_id.resize(num_data_per_cluster_pred[cluster_i], num_post_samples);
#pragma omp parallel for schedule(static)
								for (int is = 0; is < num_post_samples; ++is) {
									for (data_size_t i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
										post_samples_id(i, is) = post_samples_id_on_RE_scale(random_effects_indices_of_data_pred[i], is);
									}
								}
							}
							if (sample_prior) {
								den_mat_t prior_samples_id_on_RE_scale;
								prior_samples_id_on_RE_scale.swap(prior_samples_id);
								prior_samples_id.resize(num_data_per_cluster_[cluster_i], num_prior_samples);
								if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
#pragma omp parallel for schedule(static)
										for (int is = 0; is < num_prior_samples; ++is) {
											for (data_size_t i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
												prior_samples_id(i, is) = prior_samples_id_on_RE_scale((re_comps_vecchia_[cluster_i][0][0]->random_effects_indices_of_data_)[i], is);
											}
										}
								}
								else {
#pragma omp parallel for schedule(static)
									for (int is = 0; is < num_prior_samples; ++is) {
										for (data_size_t i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
											prior_samples_id(i, is) = prior_samples_id_on_RE_scale((re_comps_[cluster_i][0][0]->random_effects_indices_of_data_)[i], is);
										}
									}
								}
							}
							if (predict_cov_mat) {
								T_mat cov_mat_pred_id_on_RE_scale = cov_mat_pred_id;
								sp_mat_t Zpred(num_data_per_cluster_pred[cluster_i], num_REs_pred);
								std::vector<Triplet_t> triplets(num_data_per_cluster_pred[cluster_i]);
#pragma omp parallel for schedule(static)
								for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
									triplets[i] = Triplet_t(i, random_effects_indices_of_data_pred[i], 1.);
								}
								Zpred.setFromTriplets(triplets.begin(), triplets.end());
								cov_mat_pred_id = Zpred * cov_mat_pred_id_on_RE_scale * Zpred.transpose();
							}
						}//end only_one_GP_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_
						// Add externaly provided fixed effects
						if (fixed_effects_pred != nullptr) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								mean_pred_id[igp][i] += fixed_effects_pred[data_indices_per_cluster_pred[cluster_i][i] + num_data_pred * igp];
							}
						}
						// Add linear regression predictor
						if (has_covariates_) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								mean_pred_id[igp][i] += mu[data_indices_per_cluster_pred[cluster_i][i] + num_data_pred * igp];
							}
						}
					}//end loop over num_gp_pred
					if (gauss_likelihood_) {
						if (predict_var) {
							var_pred_id[0] *= cov_pars[0];
						}
						if (predict_cov_mat) {
							cov_mat_pred_id *= cov_pars[0];
						}
						if (sample_posterior) {
							post_samples_id *= std::sqrt(cov_pars[0]);
						}
						if (sample_prior) {
							prior_samples_id *= std::sqrt(cov_pars[0]);
						}
					}
					if (sample_posterior) {
						const vec_t& mu_loc = mean_pred_id.at(0);
						post_samples_id.colwise() += mu_loc;
					}
					if (!gauss_likelihood_ && predict_response) {
						if (sample_posterior) {
							Log::REFatal("Posterior sampling for the option 'predict_response' is currently not yet implemented for this likelihood, set predict_response = false");
						}
						if (sample_prior) {
							Log::REFatal("Prior sampling for the option 'predict_response' is currently not yet implemented for this likelihood, set predict_response = false");
						}
						likelihood_[unique_clusters_[0]]->PredictResponse(mean_pred_id[0], var_pred_id[0], mean_pred_id[1], var_pred_id[1], predict_var);
					}

					// Write on output
					int idx_start_post_sample = 0;
					if (!sample_prior) {
						idx_start_post_sample += num_data_pred;
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
							out_predict[data_indices_per_cluster_pred[cluster_i][i]] = mean_pred_id[0][i];
						}
					}
					// Write covariance / variance on output
					if (predict_cov_mat) {
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {//column index
							for (int j = 0; j < num_data_per_cluster_pred[cluster_i]; ++j) {//row index
								out_predict[data_indices_per_cluster_pred[cluster_i][i] * num_data_pred + data_indices_per_cluster_pred[cluster_i][j] + num_data_pred] = cov_mat_pred_id.coeff(j, i);
							}
						}
						idx_start_post_sample += num_data_pred * num_data_pred;
					}//end predict_cov_mat
					if (predict_var) {
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
							out_predict[data_indices_per_cluster_pred[cluster_i][i] + num_data_pred] = var_pred_id[0][i];
						}
						idx_start_post_sample += num_data_pred;
					}//end predict_var
					if (sample_posterior) {
						CHECK(post_samples_id.cols() == num_post_samples);
						CHECK(post_samples_id.rows() == num_data_per_cluster_pred[cluster_i]);
#pragma omp parallel for schedule(static)
						for (int is = 0; is < num_post_samples; ++is) {
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_pred[cluster_i][i] + num_data_pred * is + idx_start_post_sample] = post_samples_id(i, is);
							}
						}
						idx_start_post_sample += num_data_pred * num_post_samples;
					}
					if (sample_prior) {
						CHECK(prior_samples_id.cols() == num_prior_samples);
#pragma omp parallel for schedule(static)
						for (int is = 0; is < num_prior_samples; ++is) {
							for (int i = 0; i < (int)prior_samples_id.rows(); ++i) {
								out_predict[data_indices_per_cluster_[cluster_i][i] + prior_samples_id.rows() * is + idx_start_post_sample] = prior_samples_id(i, is);
							}
						}
					}
				}//end cluster_i with data
			}//end loop over cluster
			//Set cross-covariances between different independent clusters to 0
			if (predict_cov_mat && unique_clusters_pred.size() > 1 && (!predict_response || gauss_likelihood_)) {
				for (const auto& cluster_i : unique_clusters_pred) {
					for (const auto& cluster_j : unique_clusters_pred) {
						if (cluster_i != cluster_j) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_data_per_cluster_pred[cluster_i]; ++i) {//column index
								for (int j = 0; j < num_data_per_cluster_pred[cluster_j]; ++j) {//row index
									out_predict[data_indices_per_cluster_pred[cluster_i][i] * num_data_pred + data_indices_per_cluster_pred[cluster_j][j] + num_data_pred] = 0.;
								}
							}
						}
					}
				}
			}
		}//end Predict

		/*!
		* \brief Predict ("estimate") training data random effects
		* \param cov_pars_pred Covariance parameters of components
		* \param coef_pred Coefficients for linear covariates
		* \param y_obs Response variable for observed data
		* \param[out] out_predict Predicted training data random effects and variances if calc_var
		* \param calc_cov_factor If true, the covariance matrix of the observed data is factorized otherwise a previously done factorization is used
		* \param fixed_effects Fixed effects component of location parameter for observed data (only used for non-Gaussian likelihoods)
		* \param calc_var If true, variances are also calculated
		*/
		void PredictTrainingDataRandomEffects(const double* cov_pars_pred,
			const double* coef_pred,
			const double* y_obs,
			double* out_predict,
			bool calc_cov_factor,
			const double* fixed_effects,
			bool calc_var) {
			if (linear_kernel_use_woodbury_identity_) {
				Log::REFatal("PredictTrainingDataRandomEffects() is not implemented for the option 'linear_kernel_use_woodbury_identity_' ");
			}
			if (grouped_RE_and_vecchia_GP_ && calc_var) {
				Log::REFatal("PredictTrainingDataRandomEffects() is not implemented for variance when having grouped random effects and a Vecchia approximation ");
			}
			//Some checks
			CHECK(cov_pars_pred != nullptr);
			if (has_covariates_) {
				CHECK(coef_pred != nullptr);
			}
			if (y_obs == nullptr) {
				if (!y_has_been_set_) {
					Log::REFatal("Response variable data is not provided and has not been set before");
				}
			}
			vec_t cov_pars = Eigen::Map<const vec_t>(cov_pars_pred, num_cov_par_);
			vec_t coef;
			if (has_covariates_) {
				coef = Eigen::Map<const vec_t>(coef_pred, num_covariates_);
			}
			if (gauss_likelihood_ && (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia")) {
				calc_cov_factor = true;//recalculate Vecchia approximation since it might have been done (saved in B_[0]) with a different nugget effect if calc_std_dev == true in CalcStdDevCovPar
			}
			const double* fixed_effects_ptr = nullptr;
			if (fixed_effects != nullptr) {
				fixed_effects_ptr = fixed_effects;
			}
			else if (has_fixed_effects_) {
				fixed_effects_ptr = fixed_effects_.data();
			}
			SetYCalcCovCalcYAuxForPred(cov_pars, coef, y_obs, calc_cov_factor, fixed_effects_ptr, true, false);
			// Loop over different clusters to calculate predictions
			for (const auto& cluster_i : unique_clusters_) {
				if (gauss_likelihood_) {
					if (gp_approx_ == "vecchia") {
						if (num_comps_total_ > 1) {
							Log::REFatal("PredictTrainingDataRandomEffects() is not implemented for the Vecchia approximation "
								"when having multiple GPs / random coefficient GPs ");
						}
#pragma omp parallel for schedule(static)// Write mean on output
						for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
							out_predict[data_indices_per_cluster_[cluster_i][i]] = y_[cluster_i][i] - y_aux_[cluster_i][i];
						}
						if (calc_var) {
							sp_mat_t B_inv(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
							sp_mat_t M_aux = B_[cluster_i][0].cwiseProduct(D_inv_[cluster_i][0] * B_[cluster_i][0]);
#pragma omp parallel for schedule(static)// Write on output
							for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * num_comps_total_] = cov_pars[0] * (1. - M_aux.col(i).sum());
							}
						}
					}
					else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						Log::REFatal("PredictTrainingDataRandomEffects() is currently not implemented for the '%s' approximation. Call the predict() function instead ", gp_approx_.c_str());
					}
					else {
						int cn = 0;//component number counter
						const vec_t* y_aux = &(y_aux_[cluster_i]);
						vec_t mean_pred_id, var_pred_id;
						if (calc_var) {
							var_pred_id = vec_t(num_data_per_cluster_[cluster_i]);
						}
						//Grouped random effects
						for (int j = 0; j < num_re_group_total_; ++j) {
							double sigma = re_comps_[cluster_i][0][cn]->cov_pars_[0];
							if (use_woodbury_identity_ && num_re_group_total_ == 1) {
								if (re_comps_[cluster_i][0][cn]->IsRandCoef()) {
									Log::REFatal("PredictTrainingDataRandomEffects() is not implemented when having only one grouped random coefficient effect ");
								}
								mean_pred_id = vec_t(num_data_per_cluster_[cluster_i]);
								int num_re = re_comps_[cluster_i][0][cn]->GetNumUniqueREs();
								vec_t ZtYAux(num_re);
								CalcZtVGivenIndices(num_data_per_cluster_[cluster_i], num_re,
									re_comps_[cluster_i][0][cn]->random_effects_indices_of_data_.data(), (*y_aux).data(), ZtYAux.data(), true);
#pragma omp parallel for schedule(static)
								for (data_size_t i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									mean_pred_id[i] = sigma * ZtYAux[(re_comps_[cluster_i][0][0]->random_effects_indices_of_data_)[i]];
								}
								if (calc_var) {
									vec_t M_aux = (ZtZ_[cluster_i].diagonal().array() / sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array()).matrix();
									vec_t M_aux2 = (ZtZ_[cluster_i].diagonal().array() - M_aux.array().square()).matrix();
									vec_t M_aux3 = (sigma - (sigma * sigma * M_aux2.array())).matrix();
									vec_t M_aux4 = M_aux3.array().sqrt().matrix();
									sp_mat_t M_aux5 = M_aux4.asDiagonal() * Zt_[cluster_i];
#pragma omp parallel for schedule(static)
									for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
										var_pred_id[i] = cov_pars[0] * M_aux5.col(i).squaredNorm();
									}
								}
							}//end use_woodbury_identity_ && num_re_group_total_ == 1
							else {
								sp_mat_t* Z_j = re_comps_[cluster_i][0][cn]->GetZ();
								sp_mat_t Z_base_j;
								if (re_comps_[cluster_i][0][cn]->IsRandCoef()) {
									Z_base_j = *Z_j;
#pragma omp parallel for schedule(static)
									for (int k = 0; k < Z_base_j.outerSize(); ++k) {
										for (sp_mat_t::InnerIterator it(Z_base_j, k); it; ++it) {
											it.valueRef() = 1.;
										}
									}
									mean_pred_id = sigma * Z_base_j * ((*Z_j).transpose() * (*y_aux));
								}
								else {
									mean_pred_id = sigma * (*Z_j) * ((*Z_j).transpose() * (*y_aux));
								}
								if (calc_var) {
									sp_mat_t ZjtZj = (*Z_j).transpose() * (*Z_j);
									if (use_woodbury_identity_) {
										if (matrix_inversion_method_ == "iterative") {
											Log::REFatal("PredictTrainingDataRandomEffects() is currently not implemented for matrix_inversion_method_ == '%s' and likelihood == 'Gaussian'. Call the predict() function instead.",
												matrix_inversion_method_.c_str());
										} //end iterative
										else { //start Cholesky
											T_mat M_aux;
											if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
												TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), P_ZtZj_[cluster_i][cn], M_aux, false);
											}
											else {
												TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), ZtZj_[cluster_i][cn], M_aux, false);
											}
											T_mat M_aux3 = sigma * sigma * (M_aux.transpose() * M_aux - ZjtZj);
											M_aux3.diagonal().array() += sigma;
											if (re_comps_[cluster_i][0][cn]->IsRandCoef()) {
												Z_j = &Z_base_j;
											}
											T_mat M_aux4 = (*Z_j) * M_aux3;
#pragma omp parallel for schedule(static)
											for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
												var_pred_id[i] = cov_pars[0] * (*Z_j).row(i).cwiseProduct(M_aux4.row(i)).sum();
											}
										} //end Cholesky
									}//end use_woodbury_identity_
									else {//!use_woodbury_identity_
										T_mat M_aux;
										TriangularSolveGivenCholesky<T_chol, T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i], *Z_j, M_aux, false);
										T_mat M_aux2 = (*Z_j) * M_aux.transpose();
										if (re_comps_[cluster_i][0][cn]->IsRandCoef()) {
											Z_j = &Z_base_j;
										}
#pragma omp parallel for schedule(static)
										for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
											var_pred_id[i] = cov_pars[0] * (sigma - sigma * sigma * M_aux2.row(i).squaredNorm());
										}
									}//end !use_woodbury_identity_
								}//end calc_var
							}//end !(use_woodbury_identity_ && num_re_group_total_ == 1)
#pragma omp parallel for schedule(static)// Write on output
							for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn] = mean_pred_id[i];
							}
							if (calc_var) {
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn + num_data_ * num_comps_total_] = var_pred_id[i];
								}
							}
							cn += 1;
						}//end gouped random effects
						//Gaussian process
						if (num_gp_ > 0) {
							std::shared_ptr<RECompGP<T_mat>> re_comp_base = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][cn]);
							sp_mat_t* Z_j = nullptr, * Z_base_j = nullptr;
							for (int j = 0; j < num_gp_total_; ++j) {
								double sigma = re_comps_[cluster_i][0][cn]->cov_pars_[0];
								std::shared_ptr<RECompGP<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][cn]);
								if (re_comp->IsRandCoef() || re_comp_base->HasZ()) {
									Z_j = re_comp->GetZ();
									if (re_comp_base->HasZ()) {
										Z_base_j = re_comp_base->GetZ();
									}
								}
								if (re_comp_base->HasZ()) {
									mean_pred_id = (*Z_base_j) * ((re_comp->sigma_) * ((*Z_j).transpose() * (*y_aux)));
								}
								else if (re_comp->IsRandCoef()) {
									mean_pred_id = (re_comp->sigma_) * ((*Z_j).transpose() * (*y_aux));
								}
								else {
									mean_pred_id = (re_comp->sigma_) * (*y_aux);
								}
								if (calc_var) {
									T_mat M_aux, M_aux2, M_aux3;
									if (re_comp_base->HasZ()) {
										M_aux = (*Z_j) * (re_comp->sigma_) * (*Z_base_j).transpose();
									}
									else if (re_comp->IsRandCoef()) {
										M_aux = (*Z_j) * (re_comp->sigma_);
									}
									else {
										M_aux = re_comp->sigma_;
									}
									TriangularSolveGivenCholesky<T_chol, T_mat, T_mat, T_mat>(chol_facts_[cluster_i], M_aux, M_aux2, false);
#pragma omp parallel for schedule(static)
									for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
										var_pred_id[i] = cov_pars[0] * (sigma - M_aux2.col(i).squaredNorm());
									}
								}
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn] = mean_pred_id[i];
								}
								if (calc_var) {
#pragma omp parallel for schedule(static)// Write on output
									for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
										out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn + num_data_ * num_comps_total_] = var_pred_id[i];
									}
								}
								cn += 1;
							}//end loop over GP components
						}// end Gaussian process
					}// end not gp_approx_ == "vecchia"
				}//end gauss_likelihood_
				else {//not gauss_likelihood_
					const vec_t* mode = likelihood_[cluster_i]->GetMode();
					if (gp_approx_ == "vecchia") {
						vec_t var_pred_id;
						if (calc_var) {
							likelihood_[cluster_i]->CalcVarLaplaceApproxVecchia(var_pred_id, re_comps_cross_cov_preconditioner_[cluster_i][0]);
						}
						for (int igp = 0; igp < num_sets_re_; ++igp) {
							const int offset = num_data_ * igp + num_re_group_total_ * num_data_ * num_sets_re_;
							const int num_re_comp = grouped_RE_and_vecchia_GP_ ? cum_num_rand_eff_[cluster_i][num_re_group_total_] : 0;
							const int offset_mode = (likelihood_[cluster_i]->GetDimModePerSetsRE()) * igp + num_re_comp;
							if (only_one_GP_calculations_on_RE_scale_) {//there are duplicates
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + offset] = (*mode)[(re_comps_vecchia_[cluster_i][0][0]->random_effects_indices_of_data_)[i] + offset_mode];
								}
							}
							else {//no duplicates
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + offset] = (*mode)[i + offset_mode];
								}
							}
							if (calc_var) {
								int offset_var_out = num_data_ * num_sets_re_ + num_data_ * igp;
								if (only_one_GP_calculations_on_RE_scale_) {//there are duplicates
#pragma omp parallel for schedule(static)// Write on output
									for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
										out_predict[data_indices_per_cluster_[cluster_i][i] + offset_var_out] = var_pred_id[(re_comps_vecchia_[cluster_i][0][0]->random_effects_indices_of_data_)[i] + offset_mode];
									}
								}
								else {//no duplicates
#pragma omp parallel for schedule(static)// Write on output
									for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
										out_predict[data_indices_per_cluster_[cluster_i][i] + offset_var_out] = var_pred_id[i + offset_mode];
									}
								}
							}//end calc_var
						}// end loop over num_sets_re_
					}//end gp_approx_ == "vecchia"
					else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						Log::REFatal("PredictTrainingDataRandomEffects() is currently not implemented for the '%s' approximation. Call the predict() function instead ", gp_approx_.c_str());
					}
					if ((use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) || grouped_RE_and_vecchia_GP_) {//note: this must be an 'if' and not an 'else if' since it should be run if gp_approx_ == "vecchia" and grouped_RE_and_vecchia_GP_
						CHECK(gp_approx_ == "none" || (gp_approx_ == "vecchia" && grouped_RE_and_vecchia_GP_));
						CHECK(num_gp_total_ == 0 || grouped_RE_and_vecchia_GP_);
						CHECK(num_re_group_total_ > 0);
						vec_t var_pred_all;
						for (int cn = 0; cn < num_re_group_total_; ++cn) {
							vec_t mean_pred_id;
							sp_mat_t* Z_j = re_comps_[cluster_i][0][cn]->GetZ();
							sp_mat_t Z_base_j;
							if (re_comps_[cluster_i][0][cn]->IsRandCoef()) {
								Z_base_j = *Z_j;
#pragma omp parallel for schedule(static)
								for (int k = 0; k < Z_base_j.outerSize(); ++k) {
									for (sp_mat_t::InnerIterator it(Z_base_j, k); it; ++it) {
										it.valueRef() = 1.;
									}
								}
								Z_j = &Z_base_j;
							}
							int num_re_comp = cum_num_rand_eff_[cluster_i][cn + 1] - cum_num_rand_eff_[cluster_i][cn];
							int start_re_comp = cum_num_rand_eff_[cluster_i][cn];
							mean_pred_id = (*Z_j) * (*mode).segment(start_re_comp, num_re_comp);
#pragma omp parallel for schedule(static)// Write on output
							for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn] = mean_pred_id[i];
							}
							if (calc_var) {
								if (cn == 0) {
									likelihood_[cluster_i]->CalcVarLaplaceApproxGroupedRE(var_pred_all);
								}
								vec_t var_pred_id = (*Z_j) * var_pred_all.segment(start_re_comp, num_re_comp);
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn + num_data_ * num_comps_total_] = var_pred_id[i];
								}
							}
						}
					}//end use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_
					else if (gp_approx_ != "vecchia" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" && gp_approx_ != "full_scale_vecchia") {
						if (only_one_grouped_RE_calculations_on_RE_scale_ || only_one_GP_calculations_on_RE_scale_) {
#pragma omp parallel for schedule(static)// Write on output
							for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
								out_predict[data_indices_per_cluster_[cluster_i][i]] = (*mode)[(re_comps_[cluster_i][0][0]->random_effects_indices_of_data_)[i]];
							}
							if (calc_var) {
								vec_t var_pred_id;
								if (only_one_GP_calculations_on_RE_scale_) {
									likelihood_[cluster_i]->CalcVarLaplaceApproxOnlyOneGPCalculationsOnREScale(ZSigmaZt_[cluster_i], var_pred_id);
								}
								else if (only_one_grouped_RE_calculations_on_RE_scale_) {
									likelihood_[cluster_i]->CalcVarLaplaceApproxOnlyOneGroupedRECalculationsOnREScale(var_pred_id);
								}
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_] = var_pred_id[(re_comps_[cluster_i][0][0]->random_effects_indices_of_data_)[i]];
								}
							}
						}//end only_one_grouped_RE_calculations_on_RE_scale_ || only_one_GP_calculations_on_RE_scale_
						else {//at least one GP and additional components
							CHECK(num_gp_total_ > 0);
							CHECK(!grouped_RE_and_vecchia_GP_ && !use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_)
								//Note: use the "general" prediction formula since the mode is calculated on the aggregate scale and not for every component separaretly (mode_.size() == num_data and not num_data * num_comps_total_)
								if (calc_var) {
									Log::REFatal("PredictTrainingDataRandomEffects(): calculating of variances is not implemented when having at least one GP and additional random effects ");
								}
							const vec_t* first_deriv = likelihood_[cluster_i]->GetFirstDerivLL();
							int cn = 0;//component number counter
							vec_t mean_pred_id;
							for (int j = 0; j < num_re_group_total_; ++j) {
								double sigma = re_comps_[cluster_i][0][cn]->cov_pars_[0];
								if (re_comps_[cluster_i][0][cn]->IsRandCoef()) {
									sp_mat_t* Z_j = re_comps_[cluster_i][0][cn]->GetZ();
									sp_mat_t Z_base_j = *Z_j;
#pragma omp parallel for schedule(static)
									for (int k = 0; k < Z_base_j.outerSize(); ++k) {
										for (sp_mat_t::InnerIterator it(Z_base_j, k); it; ++it) {
											it.valueRef() = 1.;
										}
									}
									mean_pred_id = sigma * Z_base_j * (*Z_j).transpose() * (*first_deriv);
								}
								else {
									sp_mat_t* Z_j = re_comps_[cluster_i][0][cn]->GetZ();
									mean_pred_id = sigma * (*Z_j) * (*Z_j).transpose() * (*first_deriv);
								}
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn] = mean_pred_id[i];
								}
								cn += 1;
							}//end loop over grouped RE
							//GPs 
							std::shared_ptr<RECompGP<T_mat>> re_comp_base = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][cn]);
							sp_mat_t* Z_j = nullptr, * Z_base_j = nullptr;
							for (int j = 0; j < num_gp_total_; ++j) {
								std::shared_ptr<RECompGP<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][cn]);
								if (re_comp->IsRandCoef() || re_comp_base->HasZ()) {
									Z_j = re_comp->GetZ();
									if (re_comp_base->HasZ()) {
										Z_base_j = re_comp_base->GetZ();
									}
								}
								if (re_comp_base->HasZ()) {
									mean_pred_id = (*Z_base_j) * (re_comp->sigma_) * (*Z_j).transpose() * (*first_deriv);
								}
								else if (re_comp->IsRandCoef()) {
									mean_pred_id = (re_comp->sigma_) * (*Z_j).transpose() * (*first_deriv);
								}
								else {
									mean_pred_id = (re_comp_base->sigma_) * (*first_deriv);
								}
#pragma omp parallel for schedule(static)// Write on output
								for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
									out_predict[data_indices_per_cluster_[cluster_i][i] + num_data_ * cn] = mean_pred_id[i];
								}
								cn += 1;
							}//end loop over GPs
						}//end at least one GP and additional components
					}//end gp_approx_ != "vecchia" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" && gp_approx_ != "full_scale_vecchia"
				}//end not gauss_likelihood_
			}//end loop over cluster
		}//end PredictTrainingDataRandomEffects

		/*!
		* \brief Find "reasonable" default values for the initial values of the covariance parameters (on transformed scale)
		*		 Note: You should pre-allocate memory for optim_cov_pars (length = number of covariance parameters)
		* \param y_data Response variable data
		* \param fixed_effects Additional fixed effects that are added to the linear predictor (= offset)
		* \param[out] init_cov_pars Initial values for covariance parameters of RE components
		*/
		void FindInitCovPar(const double* y_data,
			const double* fixed_effects,
			double* init_cov_pars) {
			const double* y_data_ptr;
			data_size_t num_data;
			if (y_data == nullptr) {
				y_data_ptr = y_[unique_clusters_[0]].data();
				num_data = (data_size_t)y_[unique_clusters_[0]].size();
			}
			else {
				y_data_ptr = y_data;
				num_data = num_data_;
			}
			double mean = 0;
			double var = 0;
			int ind_par;
			double init_marg_var = 1.;
			if (gauss_likelihood_ || likelihood_[unique_clusters_[0]]->GetLikelihood() == "gaussian" ||
				likelihood_[unique_clusters_[0]]->GetLikelihood() == "gaussian_heteroscedastic") {
				CHECK(num_data > 0);
				//determine initial value for nugget effect
				if (fixed_effects == nullptr) {
#pragma omp parallel for schedule(static) reduction(+:mean)
					for (int i = 0; i < num_data; ++i) {
						mean += y_data_ptr[i];
					}
				}
				else {
#pragma omp parallel for schedule(static) reduction(+:mean)
					for (int i = 0; i < num_data; ++i) {
						mean += y_data_ptr[i] - fixed_effects[i];
					}
				}
				mean /= num_data;
				if (fixed_effects == nullptr) {
#pragma omp parallel for schedule(static) reduction(+:var)
					for (int i = 0; i < num_data; ++i) {
						var += (y_data_ptr[i] - mean) * (y_data_ptr[i] - mean);
					}
				}
				else {
#pragma omp parallel for schedule(static) reduction(+:var)
					for (int i = 0; i < num_data; ++i) {
						var += (y_data_ptr[i] - fixed_effects[i] - mean) * (y_data_ptr[i] - fixed_effects[i] - mean);
					}
				}
				var /= (num_data - 1);
			}
			if (gauss_likelihood_){
				init_cov_pars[0] = var / 2;
				ind_par = 1;
			}//end Gaussian data
			else {//non-Gaussian likelihoods
				ind_par = 0;
				if (likelihood_[unique_clusters_[0]]->GetLikelihood() == "gaussian" ||
					likelihood_[unique_clusters_[0]]->GetLikelihood() == "gaussian_heteroscedastic") {
					init_marg_var = var / 2;
				}
				else if (optimizer_cov_pars_ == "nelder_mead") {
					init_marg_var = 0.1;
				}
				//TODO: find better initial values depending on the likelihood (e.g., poisson, gamma, etc.)
			}
			init_marg_var /= num_comps_total_;
			if ((gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") && !grouped_RE_and_vecchia_GP_) {//Neither distances nor coordinates are saved for random coefficient GPs in the Vecchia approximation -> cannot find initial parameters -> just copy the ones from the intercept GP
				// find initial values for intercept process
				int num_par_j = ind_par_[1] - ind_par_[0];
				vec_t pars = vec_t(num_par_j);
				re_comps_vecchia_[unique_clusters_[0]][0][0]->FindInitCovPar(rng_, pars, init_marg_var);
				for (int jj = 0; jj < num_par_j; ++jj) {
					init_cov_pars[ind_par] = pars[jj];
					ind_par++;
				}
				//set the same values to random coefficient processes
				for (int j = 1; j < num_gp_total_; ++j) {
					num_par_j = ind_par_[j + 1] - ind_par_[j];
					for (int jj = 0; jj < num_par_j; ++jj) {
						init_cov_pars[ind_par] = pars[jj];
						ind_par++;
					}
				}
			}
			else {// not gp_approx_ == "vecchia" 
				for (int j = 0; j < num_comps_total_; ++j) {
					int num_par_j = ind_par_[j + 1] - ind_par_[j];
					vec_t pars = vec_t(num_par_j);
					if (grouped_RE_and_vecchia_GP_ && j == (num_comps_total_ - 1)) {
						re_comps_vecchia_[unique_clusters_[0]][0][0]->FindInitCovPar(rng_, pars, init_marg_var);
					}
					else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
						re_comps_ip_[unique_clusters_[0]][0][j]->FindInitCovPar(rng_, pars, init_marg_var);
					}
					else {
						re_comps_[unique_clusters_[0]][0][j]->FindInitCovPar(rng_, pars, init_marg_var);
					}
					for (int jj = 0; jj < num_par_j; ++jj) {
						init_cov_pars[ind_par] = pars[jj];
						ind_par++;
					}
				}
			}
			if (num_sets_re_ > 1) {// set initial values for other parameters that are modeled using REs / GPs (use the same values)
				CHECK(num_sets_re_ == 2); // check whether this makes sense if other models with num_sets_re_> 1 are implemented in the future
				for (int igp = 1; igp < num_sets_re_; ++igp) {
					ind_par = 0;
					for (int j = 0; j < num_comps_total_; ++j) {
						int num_par_j = ind_par_[j + 1] - ind_par_[j];
						for (int jj = 0; jj < num_par_j; ++jj) {
							if (jj == 0) {
								init_cov_pars[num_cov_par_per_set_re_ * igp + ind_par] = std::log((1. + std::sqrt(1. + 2. * init_cov_pars[ind_par])) / 2.) / 2.;// a mean-zero log-normal variale has variance (e^sigma2 - 1) * e^sigma2. We solve this for (e^sigma2 - 1) * e^sigma2 = marg_var / 2 and divide by 2
							}
							else {
								init_cov_pars[num_cov_par_per_set_re_ * igp + ind_par] = init_cov_pars[ind_par];
							}
							ind_par++;
						}
					}
				}
			}// end num_sets_re_ > 1
		}//end FindInitCovPar

		int num_cov_par() {
			return(num_cov_par_);
		}

		/*!
		* \brief Calculate the leaf values when performing a Newton update step after the tree structure has been found in tree-boosting
		*    Note: only used in GPBoost for combined Gaussian process tree-boosting (this is called from 'objective_function_->NewtonUpdateLeafValues'). It is assumed that 'CalcYAux' has been called before (from 'objective_function_->GetGradients').
		* \param data_leaf_index Leaf index for every data point (array of size num_data)
		* \param num_leaves Number of leaves
		* \param[out] leaf_values Leaf values when performing a Newton update step (array of size num_leaves)
		* \param marg_variance The marginal variance. Default = 1. Can be used to multiply values by it since Newton updates do not depend on it but 'CalcYAux' might have been called using marg_variance!=1.
		*/
		void NewtonUpdateLeafValues(const int* data_leaf_index,
			const int num_leaves,
			double* leaf_values,
			double marg_variance) {
			if (!gauss_likelihood_) {
				Log::REFatal("Newton updates for leaf values is only supported for Gaussian data");
			}
			CHECK(y_aux_has_been_calculated_);//y_aux_ has already been calculated when calculating the gradient for finding the tree structure from 'GetGradients' in 'regression_objetive.hpp'
			den_mat_t HTPsiInvH(num_leaves, num_leaves);
			vec_t HTYAux(num_leaves);
			HTPsiInvH.setZero();
			HTYAux.setZero();
			for (const auto& cluster_i : unique_clusters_) {
				//Entries for matrix H_cluster_i = incidence matrix H that relates tree leaves to observations for cluster_i
				std::vector<Triplet_t> entries_H_cluster_i(num_data_per_cluster_[cluster_i]);
#pragma omp parallel for schedule(static)
				for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
					entries_H_cluster_i[i] = Triplet_t(i, data_leaf_index[data_indices_per_cluster_[cluster_i][i]], 1.);
				}
				den_mat_t HTPsiInvH_cluster_i;
				if (gp_approx_ == "vecchia") {
					sp_mat_t H_cluster_i(num_data_per_cluster_[cluster_i], num_leaves);//row major format is needed for Vecchia approx.
					H_cluster_i.setFromTriplets(entries_H_cluster_i.begin(), entries_H_cluster_i.end());
					HTYAux -= H_cluster_i.transpose() * y_aux_[cluster_i];//minus sign since y_aux_ has been calculated on the gradient = F-y (and not y-F)
					sp_mat_t BH = B_[cluster_i][0] * H_cluster_i;
					HTPsiInvH_cluster_i = den_mat_t(BH.transpose() * D_inv_[cluster_i][0] * BH);
				}
				else {
					sp_mat_t H_cluster_i(num_data_per_cluster_[cluster_i], num_leaves);
					H_cluster_i.setFromTriplets(entries_H_cluster_i.begin(), entries_H_cluster_i.end());
					HTYAux -= H_cluster_i.transpose() * y_aux_[cluster_i];//minus sign since y_aux_ has been calculated on the gradient = F-y (and not y-F)
					if (use_woodbury_identity_) {
						T_mat MInvSqrtZtH;
						if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
							sp_mat_t ZtH_cluster_i = Zt_[cluster_i] * H_cluster_i;
							MInvSqrtZtH = sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().inverse().matrix().asDiagonal() * ZtH_cluster_i;
						}
						else {
							if (matrix_inversion_method_ == "iterative") {
								Log::REFatal("Newton update step for the tree leaves after the gradient step is currently not implemented for matrix_inversion_method_ == '%s'.",
									matrix_inversion_method_.c_str());
							} //end iterative
							else { //start Cholesky
								sp_mat_t ZtH_cluster_i;
								if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
									ZtH_cluster_i = P_Zt_[cluster_i] * H_cluster_i;
								}
								else {
									ZtH_cluster_i = Zt_[cluster_i] * H_cluster_i;
								}
								TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), ZtH_cluster_i, MInvSqrtZtH, false);
							}
						}
						HTPsiInvH_cluster_i = H_cluster_i.transpose() * H_cluster_i - MInvSqrtZtH.transpose() * MInvSqrtZtH;
					}
					else {
						T_mat PsiInvSqrtH;
						TriangularSolveGivenCholesky<T_chol, T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i], H_cluster_i, PsiInvSqrtH, false);
						HTPsiInvH_cluster_i = PsiInvSqrtH.transpose() * PsiInvSqrtH;
					}
				}
				HTPsiInvH += HTPsiInvH_cluster_i;
			}
			HTYAux *= marg_variance;
			vec_t new_leaf_values = HTPsiInvH.llt().solve(HTYAux);
			for (int i = 0; i < num_leaves; ++i) {
				leaf_values[i] = new_leaf_values[i];
			}
		}//end NewtonUpdateLeafValues

		/*!
		* \brief Apply a momentum step
		* \param it Iteration number
		* \param pars Parameters
		* \param pars_lag1 Parameters from last iteration
		* \param[out] pars_acc Accelerated parameters
		* \param nesterov_acc_rate Nesterov acceleration speed
		* \param nesterov_schedule_version Which version of Nesterov schedule should be used. Default = 0
		* \param exclude_first_log_scale If true, no momentum is applied to the first value and the momentum step is done on the log-scale for the other values. Default = true
		* \param momentum_offset Number of iterations for which no mometum is applied in the beginning
		* \param log_scale If true, the momentum step is done on the log-scale
		*/
		static void ApplyMomentumStep(int it,
			vec_t& pars,
			vec_t& pars_lag1,
			vec_t& pars_acc,
			double nesterov_acc_rate,
			int nesterov_schedule_version,
			bool exclude_first_log_scale,
			int momentum_offset,
			bool log_scale) {
			double mu = NesterovSchedule(it, nesterov_schedule_version, nesterov_acc_rate, momentum_offset);
			int num_par = (int)pars.size();
			if (exclude_first_log_scale) {
				pars_acc[0] = pars[0];
				pars_acc.segment(1, num_par - 1) = ((mu + 1.) * (pars.segment(1, num_par - 1).array().log()) - mu * (pars_lag1.segment(1, num_par - 1).array().log())).exp().matrix();//Momentum is added on the log scale
			}
			else {
				if (log_scale) {
					pars_acc = ((mu + 1.) * (pars.array().log()) - mu * (pars_lag1.array().log())).exp().matrix();
				}
				else {
					pars_acc = (mu + 1) * pars - mu * pars_lag1;
				}
			}
		}// end ApplyMomentumStep

		/*!
		* \brief Indicates whether inducing points or/and correlation-based nearest neighbors for Vecchia approximation should be updated
		* \param force_redermination If true, inducing points/neighbors are redetermined if applicaple irrespective of num_iter_
		* \return redetermine_nn True, if inducing points/nearest neighbors have been redetermined
		*/
		bool ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(bool force_redermination) {
			redetermine_vecchia_neighbors_ = false;
			redetermine_inducing_points_ = false;
			bool num_iter_redetermine = (((num_iter_ + 1) & num_iter_) == 0) || num_iter_ == 0 || force_redermination;//(num_iter_ + 1) is power of 2 or 0 
			// Note that convergence of internal optimizers is not checked in iterations with redetermine_nn if convergence_criterion_ == "relative_change_in_log_likelihood"
			if (gp_approx_ == "vecchia") {
				std::shared_ptr<RECompGP<den_mat_t>> re_comp = re_comps_vecchia_[unique_clusters_[0]][0][0];
				if ((re_comp->RedetermineVecchiaNeighborsInTransformedSpace() || vecchia_neighbor_selection_ == "correlation") && num_iter_redetermine) {
					redetermine_vecchia_neighbors_ = true;
				}
			}
			else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
				std::shared_ptr<RECompGP<den_mat_t>> re_comp = re_comps_cross_cov_[unique_clusters_[0]][0][0];
				if (re_comp->UseScaledCoordinates() && num_iter_redetermine) {//note: it is usually better to always redetermine inducing points in the transformed space due to numerical instabilities
					redetermine_inducing_points_ = true;
				}
			}
			else if (gp_approx_ == "full_scale_vecchia") {
				std::shared_ptr<RECompGP<den_mat_t>> re_comp = re_comps_vecchia_[unique_clusters_[0]][0][0];
				if (re_comp->UseScaledCoordinates() && num_iter_redetermine) {
					redetermine_inducing_points_ = true;
					}
				if ((re_comp->RedetermineVecchiaNeighborsInTransformedSpace() || vecchia_neighbor_selection_ == "residual_correlation") && num_iter_redetermine) {
					redetermine_vecchia_neighbors_ = true;
				}
			}
			return(redetermine_inducing_points_ || redetermine_vecchia_neighbors_);
		}//end ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC

		/*!
		* \brief Redetermine inducing points or/and correlation-based nearest neighbors for Vecchia approximation
		* \param force_redermination If true, inducing points/neighbors are redetermined if applicaple irrespective of num_iter_
		*/
		void RedetermineNearestNeighborsVecchiaInducingPointsFITC(bool force_redermination) {
			CHECK(ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(force_redermination));
			// redetermine inducing points
			if (redetermine_inducing_points_) {				
				if (gp_approx_ == "full_scale_vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
					int num_ind_points = num_ind_points_;
					for (const auto& cluster_i : unique_clusters_) {
						std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_ip_cluster_i;
						std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cross_cov_cluster_i;
						std::shared_ptr<RECompGP<den_mat_t>> re_comp = std::dynamic_pointer_cast<RECompGP<den_mat_t>>(re_comps_cross_cov_[cluster_i][0][0]);
						if (re_comp->UseScaledCoordinates()) {
							// Scale Coordinates
							den_mat_t coords_scaled;
							vec_t pars = re_comp->CovPars();
							den_mat_t coords_all = re_comp->GetCoords();
							re_comp->ScaleCoordinates(pars, coords_all, coords_scaled);
							// Determine inducing points on unique locataions
							den_mat_t gp_coords_all_unique;
							std::vector<int> uniques;//unique points
							std::vector<int> unique_idx;//not used
							DetermineUniqueDuplicateCoordsFast(coords_scaled, num_data_per_cluster_[cluster_i], uniques, unique_idx);
							if ((data_size_t)uniques.size() == num_data_per_cluster_[cluster_i]) {//no multiple observations at the same locations -> no incidence matrix needed
								gp_coords_all_unique = coords_scaled;
							}
							else {
								gp_coords_all_unique = coords_scaled(uniques, Eigen::all);
								if ((int)gp_coords_all_unique.rows() < num_ind_points) {
									Log::REFatal("Cannot have more inducing points than unique coordinates for '%s' approximation ", gp_approx_.c_str());
								}
							}
							std::vector<int> indices;
							den_mat_t gp_coords_ip_mat;
							if (ind_points_selection_ == "cover_tree") {
								Log::REDebug("Starting cover tree algorithm for determining inducing points ");
								CoverTree(gp_coords_all_unique, cover_tree_radius_, rng_, gp_coords_ip_mat);
								Log::REDebug("Inducing points have been determined ");
								num_ind_points = (int)gp_coords_ip_mat.rows();
							}
							else if (ind_points_selection_ == "random") {
								if (gp_approx_ == "full_scale_vecchia" && !gauss_likelihood_) {
									Log::REFatal("Method '%s' is not supported for finding inducing points in the full-scale-vecchia approximation for non-Gaussian data", ind_points_selection_.c_str());//due to numerical instabilities for non-Gaussian likelihoods
								}
								SampleIntNoReplaceSort((int)gp_coords_all_unique.rows(), num_ind_points, rng_, indices);
								gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_unique.cols());
								for (int j = 0; j < num_ind_points; ++j) {
									gp_coords_ip_mat.row(j) = gp_coords_all_unique.row(indices[j]);
								}
							}
							else if (ind_points_selection_ == "kmeans++") {
								// Initialize
								int max_it_kmeans = 1000;
								den_mat_t gp_coords_ip_mat_scaled;
								// Start with inducing points from last redetermination
								re_comp->ScaleCoordinates(pars, gp_coords_ip_mat_, gp_coords_ip_mat_scaled);
								CHECK((int)gp_coords_ip_mat_scaled.rows() == num_ind_points);
								CHECK(gp_coords_ip_mat_scaled.cols() == gp_coords_all_unique.cols());
								kmeans_plusplus(gp_coords_all_unique, num_ind_points, rng_, gp_coords_ip_mat_scaled, max_it_kmeans, true);
								gp_coords_ip_mat = gp_coords_ip_mat_scaled;
							}
							else if (ind_points_selection_ == "space_time_kmeans++") {
								gp_coords_ip_mat = gp_coords_ip_mat_;
							}
							else {
								Log::REFatal("Method '%s' is not supported for redetrmine inducing points. Use '%s' when using an ARD covariance function ",
									ind_points_selection_.c_str(), "kmeans++");
							}
							den_mat_t coords_ip_rescaled;
							vec_t pars_inv = pars.cwiseInverse();
							re_comp->ScaleCoordinates(pars_inv, gp_coords_ip_mat, coords_ip_rescaled);
							gp_coords_ip_mat_.resize(coords_ip_rescaled.rows(), coords_ip_rescaled.cols());
							gp_coords_ip_mat_ = coords_ip_rescaled;
							gp_coords_all_unique.resize(0, 0);
							std::shared_ptr<RECompGP<den_mat_t>> gp_ip(new RECompGP<den_mat_t>(
								coords_ip_rescaled, re_comp->CovFunctionName(), re_comp->CovFunctionShape(), re_comp->CovFunctionTaperRange(), re_comp->CovFunctionTaperShape(), false, false, true, false, false, true));
							if (gp_ip->HasDuplicatedCoords()) {
								Log::REFatal("Duplicates found in inducing points / low-dimensional knots ");
							}
							re_comps_ip_cluster_i.push_back(gp_ip);
							if (!(gp_approx_ == "full_scale_vecchia")) {
								only_one_GP_calculations_on_RE_scale_ = false;
								has_duplicates_coords_ = only_one_GP_calculations_on_RE_scale_;
							}
							re_comps_cross_cov_cluster_i.push_back(std::shared_ptr<RECompGP<den_mat_t>>(new RECompGP<den_mat_t>(
								coords_all, coords_ip_rescaled, re_comp->CovFunctionName(), re_comp->CovFunctionShape(), re_comp->CovFunctionTaperRange(), re_comp->CovFunctionTaperShape(), false, false, only_one_GP_calculations_on_RE_scale_)));
							re_comps_ip_[cluster_i][0] = re_comps_ip_cluster_i;
							re_comps_cross_cov_[cluster_i][0] = re_comps_cross_cov_cluster_i;
							re_comps_ip_cluster_i[0]->SetCovPars(pars);
							re_comps_cross_cov_cluster_i[0]->SetCovPars(pars);
							re_comps_ip_cluster_i[0]->CalcSigma();
							re_comps_cross_cov_cluster_i[0]->CalcSigma();
							den_mat_t sigma_ip_stable = *(re_comps_ip_cluster_i[0]->GetZSigmaZt());
							sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
							chol_fact_sigma_ip_[cluster_i][0].compute(sigma_ip_stable);
							//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_[cluster_i][0],
							//	(*(re_comps_cross_cov_cluster_i[0]->GetZSigmaZt())).transpose(), chol_ip_cross_cov_[cluster_i][0], false);
							GPBoost::solve_lower_triangular(chol_fact_sigma_ip_[cluster_i][0],
								(*(re_comps_cross_cov_cluster_i[0]->GetZSigmaZt())).transpose(), chol_ip_cross_cov_[cluster_i][0], GPU_use_);
							if (gp_approx_ == "full_scale_vecchia") {
								if (fitc_piv_chol_preconditioner_rank_ == num_ind_points_) {
									fitc_piv_chol_preconditioner_rank_ = num_ind_points;
								}
							}
							num_ind_points_ = num_ind_points;
							if (num_ll_evaluations_ > 0) {
								Log::REDebug("Inducing points redetermined after iteration number %d ", num_iter_ + 1);
							}
						}//end re_comp->UseScaledCoordinates()
					}//end loop cluster_i
				}//end if (gp_approx_ == "full_scale_vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering")
				if (cg_preconditioner_type_ == "fitc" && matrix_inversion_method_ == "iterative") {
					// redetermine inducing points for FITC preconditioner
					if (gp_approx_ == "fitc" || ((gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") && (gauss_likelihood_ && !vecchia_latent_approx_gaussian_))) {
						Log::REFatal("'iterative' methods are not implemented for gp_approx = '%s' and the chosen likelihood ", gp_approx_.c_str());
					}
					int num_ind_points = fitc_piv_chol_preconditioner_rank_;
					if (gp_approx_ == "full_scale_tapering" || (fitc_piv_chol_preconditioner_rank_ == num_ind_points_ && gp_approx_ != "vecchia") || ind_points_selection_ == "space_time_kmeans++") {
						for (const auto& cluster_i : unique_clusters_) {
							re_comps_ip_preconditioner_[cluster_i][0] = re_comps_ip_[cluster_i][0];
							re_comps_cross_cov_preconditioner_[cluster_i][0] = re_comps_cross_cov_[cluster_i][0];
							chol_fact_sigma_ip_preconditioner_[cluster_i][0] = chol_fact_sigma_ip_[cluster_i][0];
							chol_ip_cross_cov_preconditioner_[cluster_i] = chol_ip_cross_cov_[cluster_i];
						}
						fitc_piv_chol_preconditioner_rank_ = num_ind_points_;
					}
					else {
						for (const auto& cluster_i : unique_clusters_) {
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_ip_cluster_i;
							std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cross_cov_cluster_i;
							std::shared_ptr<RECompGP<den_mat_t>> re_comp = re_comps_cross_cov_[cluster_i][0][0];
							// Scale Coordinates
							den_mat_t coords_scaled;
							den_mat_t coords_all = re_comp->GetCoords();
							vec_t pars = re_comp->CovPars();
							if (re_comp->UseScaledCoordinates()) {
								re_comp->ScaleCoordinates(pars, coords_all, coords_scaled);
							}
							else {
								coords_scaled = coords_all;
							}
							// Determine inducing points on unique locataions
							den_mat_t gp_coords_all_unique;
							std::vector<int> uniques;//unique points
							std::vector<int> unique_idx;//not used
							DetermineUniqueDuplicateCoordsFast(coords_scaled, num_data_per_cluster_[cluster_i], uniques, unique_idx);
							if ((data_size_t)uniques.size() == num_data_per_cluster_[cluster_i]) {//no multiple observations at the same locations -> no incidence matrix needed
								gp_coords_all_unique = coords_scaled;
							}
							else {
								gp_coords_all_unique = coords_scaled(uniques, Eigen::all);
								if ((int)gp_coords_all_unique.rows() < num_ind_points) {
									Log::REFatal("Cannot have more inducing points than unique coordinates for '%s' approximation ", gp_approx_.c_str());
								}
							}
							std::vector<int> indices;
							den_mat_t gp_coords_ip_mat;
							if (ind_points_selection_ == "cover_tree") {
								Log::REDebug("Starting cover tree algorithm for determining inducing points ");
								CoverTree(gp_coords_all_unique, cover_tree_radius_, rng_, gp_coords_ip_mat);
								Log::REDebug("Inducing points have been determined ");
								num_ind_points = (int)gp_coords_ip_mat.rows();
							}
							else if (ind_points_selection_ == "random") {
								SampleIntNoReplaceSort((int)gp_coords_all_unique.rows(), num_ind_points, rng_, indices);
								gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_unique.cols());
								for (int j = 0; j < num_ind_points; ++j) {
									gp_coords_ip_mat.row(j) = gp_coords_all_unique.row(indices[j]);
								}
							}
							else if (ind_points_selection_ == "kmeans++") {
								// Initialize
								int max_it_kmeans = 1000;
								den_mat_t gp_coords_ip_mat_scaled;
								// Start with inducing points from last redetermination
								if (ind_points_determined_for_preconditioner_) {
									if (re_comp->UseScaledCoordinates()) {
										re_comp->ScaleCoordinates(pars, gp_coords_ip_mat_preconditioner_, gp_coords_ip_mat_scaled);
									}
									else {
										gp_coords_ip_mat_scaled = gp_coords_ip_mat_preconditioner_;
									}
									CHECK((int)gp_coords_ip_mat_scaled.rows() == num_ind_points);
									CHECK(gp_coords_ip_mat_scaled.cols() == gp_coords_all_unique.cols());
									kmeans_plusplus(gp_coords_all_unique, num_ind_points, rng_, gp_coords_ip_mat_scaled, max_it_kmeans, true);
									gp_coords_ip_mat = gp_coords_ip_mat_scaled;
								}
								else {
									gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_unique.cols());
									Log::REDebug("Starting kmeans++ algorithm for determining inducing points ");
									kmeans_plusplus(gp_coords_all_unique, num_ind_points, rng_, gp_coords_ip_mat, max_it_kmeans, false);
									Log::REDebug("Inducing points have been determined ");
									ind_points_determined_for_preconditioner_ = true;
								}
							}
							else {
								Log::REFatal("Method '%s' is not supported for redetrmine inducing points. Use '%s' when using an ARD covariance function ",
									ind_points_selection_.c_str(), "kmeans++");
							}
							den_mat_t coords_ip_rescaled;
							// Start with inducing points from last redetermination
							if (re_comp->UseScaledCoordinates()) {
								vec_t pars_inv = pars.cwiseInverse();
								re_comp->ScaleCoordinates(pars_inv, gp_coords_ip_mat, coords_ip_rescaled);
							}
							else {
								coords_ip_rescaled = gp_coords_ip_mat;
							}
							gp_coords_ip_mat_preconditioner_.resize(coords_ip_rescaled.rows(), coords_ip_rescaled.cols());
							gp_coords_ip_mat_preconditioner_ = coords_ip_rescaled;
							gp_coords_all_unique.resize(0, 0);
							std::shared_ptr<RECompGP<den_mat_t>> gp_ip(new RECompGP<den_mat_t>(
								coords_ip_rescaled, re_comp->CovFunctionName(), re_comp->CovFunctionShape(), re_comp->CovFunctionTaperRange(), re_comp->CovFunctionTaperShape(), false, false, true, false, false, true));
							if (gp_ip->HasDuplicatedCoords()) {
								Log::REFatal("Duplicates found in inducing points / low-dimensional knots ");
							}
							re_comps_ip_cluster_i.push_back(gp_ip);
							if (!(gp_approx_ == "full_scale_vecchia")) {
								only_one_GP_calculations_on_RE_scale_ = false;
								has_duplicates_coords_ = only_one_GP_calculations_on_RE_scale_;
							}
							re_comps_cross_cov_cluster_i.push_back(std::shared_ptr<RECompGP<den_mat_t>>(new RECompGP<den_mat_t>(
								coords_all, coords_ip_rescaled, re_comp->CovFunctionName(), re_comp->CovFunctionShape(), re_comp->CovFunctionTaperRange(), re_comp->CovFunctionTaperShape(), false, false, only_one_GP_calculations_on_RE_scale_)));
							re_comps_ip_preconditioner_[cluster_i][0] = re_comps_ip_cluster_i;
							re_comps_cross_cov_preconditioner_[cluster_i][0] = re_comps_cross_cov_cluster_i;
							re_comps_ip_cluster_i[0]->SetCovPars(pars);
							re_comps_cross_cov_cluster_i[0]->SetCovPars(pars);
							re_comps_ip_cluster_i[0]->CalcSigma();
							re_comps_cross_cov_cluster_i[0]->CalcSigma();
							den_mat_t sigma_ip_stable = *(re_comps_ip_cluster_i[0]->GetZSigmaZt());
							sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
							chol_fact_sigma_ip_preconditioner_[cluster_i][0].compute(sigma_ip_stable);
							//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_preconditioner_[cluster_i][0],
							//	(*(re_comps_cross_cov_cluster_i[0]->GetZSigmaZt())).transpose(), chol_ip_cross_cov_preconditioner_[cluster_i][0], false);
							GPBoost::solve_lower_triangular(chol_fact_sigma_ip_preconditioner_[cluster_i][0],
								(*(re_comps_cross_cov_cluster_i[0]->GetZSigmaZt())).transpose(), chol_ip_cross_cov_preconditioner_[cluster_i][0], GPU_use_);
						}
						fitc_piv_chol_preconditioner_rank_ = num_ind_points;
					}
					if (num_ll_evaluations_ > 0) {
						Log::REDebug("Inducing points for preconditioner redetermined after iteration number %d ", num_iter_ + 1);
					}
				}//end if (cg_preconditioner_type_ == "fitc" && matrix_inversion_method_ == "iterative")
			}//end redetermine inducing points
			if (redetermine_vecchia_neighbors_) { // redetermine Vecchia neighbors
				if (chol_ip_cross_cov_.empty() && gp_approx_ == "full_scale_vecchia" && vecchia_neighbor_selection_ == "residual_correlation") {
					CalcSigmaComps();
				}
				if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
					for (const auto& cluster_i : unique_clusters_) {
						for (int igp = 0; igp < num_sets_re_; ++igp) {
							// redetermine nearest neighbors for models for which neighbors are selected based on correlations / scaled distances
							UpdateNearestNeighbors(re_comps_vecchia_[cluster_i][igp], nearest_neighbors_[cluster_i][igp],
								entries_init_B_[cluster_i][igp], num_neighbors_, vecchia_neighbor_selection_, rng_,
								has_duplicates_coords_, true, gauss_likelihood_, gp_approx_, chol_ip_cross_cov_[cluster_i][0],
								dist_obs_neighbors_[cluster_i][0], dist_between_neighbors_[cluster_i][0], save_distances_isotropic_cov_fct_Vecchia_, GPU_use_);
							nearest_neighbors_determined_ = true;
							if (!gauss_likelihood_) {
								likelihood_[cluster_i]->SetCholFactPatternAnalyzedFalse();
							}
						}
					}
					if (num_ll_evaluations_ > 0) {
						Log::REDebug("Nearest neighbors redetermined after iteration number %d ", num_iter_ + 1);
					}
				}
			}//end redetermine_vecchia_neighbors_
			redetermine_vecchia_neighbors_ = false;
			redetermine_inducing_points_ = false;
		}//end RedetermineNearestNeighborsVecchiaInducingPointsFITC

		/*!
		* \brief Get the maximal step length along a direction such that relative changes of covariance and auxiliary parameters are not larger than 'MAX_GRADIENT_UPDATE_LOG_SCALE_'
		* \param neg_step_dir Negative step direction for making updates. E.g., neg_step_dir = grad for gradient descent and neg_step_dir = FI^-1 * grad for Fisher scoring (="natural" gradient)
		*/
		double MaximalLearningRateCovAuxPars(const vec_t& neg_step_dir) const {
			double max_abs_neg_step_dir = 0.;
			for (int ip = 0; ip < (int)neg_step_dir.size(); ++ip) {
				if (std::abs(neg_step_dir[ip]) > max_abs_neg_step_dir) {
					max_abs_neg_step_dir = std::abs(neg_step_dir[ip]);
				}
			}
			return(MAX_GRADIENT_UPDATE_LOG_SCALE_ / max_abs_neg_step_dir);
		}//end MaximalLearningRateCovAuxPars

		/*!
		* \brief Get the maximal step length along a direction such that the change in linear regression coefficients is not overly large (monitor relative change in linear predictor)
		* \param beta Current / lag1 value of beta
		* \param neg_step_dir Negative step direction for making updates
		*/
		double MaximalLearningRateCoef(const vec_t& beta,
			const vec_t& neg_step_dir) const {
			vec_t lp_change = vec_t(num_data_ * num_sets_re_);
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				lp_change.segment(igp * num_data_, num_data_) = X_ * (neg_step_dir.segment(num_covariates_ * igp, num_covariates_));
			}
			vec_t lp_lag1 = vec_t(num_data_ * num_sets_re_);
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				lp_lag1.segment(igp * num_data_, num_data_) = X_ * (beta.segment(num_covariates_ * igp, num_covariates_));
			}
			double mean_lp_change = 0., mean_lp_lag1 = 0., var_lp_change = 0, cov_lp_lag1_lp_change = 0;
#pragma omp parallel for schedule(static) reduction(+:mean_lp_change, mean_lp_lag1, var_lp_change, cov_lp_lag1_lp_change)
			for (data_size_t i = 0; i < num_data_ * num_sets_re_; ++i) {
				mean_lp_change += lp_change[i];
				mean_lp_lag1 += lp_lag1[i];
				var_lp_change += lp_change[i] * lp_change[i];
				cov_lp_lag1_lp_change += lp_change[i] * lp_lag1[i];
			}
			mean_lp_change /= num_data_;
			mean_lp_lag1 /= num_data_;
			var_lp_change /= num_data_;
			cov_lp_lag1_lp_change /= num_data_;
			var_lp_change -= mean_lp_change * mean_lp_change;
			cov_lp_lag1_lp_change -= mean_lp_change * mean_lp_lag1;
			double max_lr_mu = C_mu_ * C_MAX_CHANGE_COEF_ / std::abs(mean_lp_change);
			double max_lr_var = (std::abs(cov_lp_lag1_lp_change) +
				std::sqrt(cov_lp_lag1_lp_change * cov_lp_lag1_lp_change + 4 * var_lp_change * C_sigma2_ * C_MAX_CHANGE_COEF_)) /
				2 / var_lp_change;
			return(std::min({ max_lr_mu , max_lr_var }));
		}//end MaximalLearningRateCoef

		/*!
		* \brief Calculate a Vecchia approximation for an observable variable where the error variance is given by 'add_diagonal'
		* \param cluster_i Cluster index
		* \param[out] B_vecchia Matrix A = I - B (= Cholesky factor of inverse covariance) for Vecchia approximation
		* \param[out] D_inv_vecchia Diagonal matrices D^-1 for Vecchia approximation
		* \param add_diagonal Vector of (additional) observation specific nugget / error variance added to the diagonal
		*/
		void CalcVecchiaApproxLatentAddDiagonal(data_size_t cluster_i,
			sp_mat_t& B_vecchia,
			sp_mat_t& D_inv_vecchia,
			const double* add_diagonal) {
			if (num_sets_re_ > 1) {
				Log::REFatal("CalcVecchiaApproxLatentAddDiagonal: not implemented if num_sets_re_ > 1");
			}
			CHECK(!gauss_likelihood_);
			int igp = 0;
			data_size_t num_re_cluster_i = re_comps_vecchia_[cluster_i][igp][0]->GetNumUniqueREs();
			std::vector<sp_mat_t> B_grad_cluster_i, D_grad_cluster_i;//not used, but needs to be passed to function
			CalcCovFactorGradientVecchia(num_re_cluster_i, true, false, re_comps_vecchia_[cluster_i][igp],
				re_comps_cross_cov_[cluster_i][0], re_comps_ip_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0], nearest_neighbors_[cluster_i][igp],
				dist_obs_neighbors_[cluster_i][igp], dist_between_neighbors_[cluster_i][igp],
				entries_init_B_[cluster_i][igp], z_outer_z_obs_neighbors_[cluster_i][igp],
				B_vecchia, D_inv_vecchia, B_grad_cluster_i, D_grad_cluster_i,
				sigma_ip_inv_cross_cov_T_[cluster_i][0], sigma_ip_grad_sigma_ip_inv_cross_cov_T_[cluster_i][0], true, 1.,
				true, num_gp_total_, gauss_likelihood_, save_distances_isotropic_cov_fct_Vecchia_, gp_approx_,
				add_diagonal, estimate_cov_par_index_, nearest_neighbors_determined_, false);
		}//end CalcVecchiaApproxLatentAddDiagonal


	private:

		// RESPONSE DATA
		/*! \brief Number of data points */
		data_size_t num_data_;
		/*! \brief If true, the response variables have a Gaussian likelihood, otherwise not */
		bool gauss_likelihood_ = true;
		/*! \brief Likelihood objects */
		std::map<data_size_t, std::unique_ptr<Likelihood<T_mat, T_chol>>> likelihood_;
		/*! \brief Additional shape parameter for likelihood (e.g., degrees of freedom for t-distribution) */
		double likelihood_additional_param_;
		/*! \brief Used for checking convergence in mode finding algorithm for non-Gaussian likelihoods (terminate if relative change in Laplace approx. is below this value) */
		double delta_conv_mode_finding_ = 1e-8;
		/*! \brief Value of negative log-likelihood or approximate marginal negative log-likelihood for non-Gaussian likelihoods */
		double neg_log_likelihood_;
		/*! \brief Value of negative log-likelihood or approximate marginal negative log-likelihood for non-Gaussian likelihoods of previous iteration in optimization used for convergence checking */
		double neg_log_likelihood_lag1_;
		/*! \brief Value of negative log-likelihood or approximate marginal negative log-likelihood for non-Gaussian likelihoods after linear regression coefficients are update (this equals neg_log_likelihood_lag1_ if there are no regression coefficients). This is used for step-size checking for the covariance parameters */
		double neg_log_likelihood_after_lin_coef_update_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: data y */
		std::map<data_size_t, vec_t> y_;
		/*! \brief Copy of response data (used only for Gaussian data and if there are also linear covariates since then y_ is modified during the optimization algorithm and this contains the original data) */
		vec_t y_vec_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: data y of integer type (used only for non-Gaussian likelihood) */
		std::map<data_size_t, vec_int_t> y_int_;
		// Note: the response variable data is saved in y_ / y_int_ (depending on the likelihood type) for Gaussian data with no covariates and for all non-Gaussian likelihoods.
		//			For Gaussian data with covariates, the response variables is saved in y_vec_ and y_ is replaced by y - X * beta during the optimization
		/*! \brief Key: labels of independent realizations of REs/GPs, value: Psi^-1*y_ (used for various computations) */
		std::map<data_size_t, vec_t> y_aux_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: Psi^-1*y_ (from last iteration) */
		std::map<data_size_t, vec_t> last_y_aux_;
		/*! \brief Psi^-1*X_ (from last iteration) */
		std::map<data_size_t, den_mat_t> last_psi_inv_X_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z) (used for various computations when use_woodbury_identity_==true) */
		std::map<data_size_t, vec_t> y_tilde_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: Z * L ^ -T * L ^ -1 * Z ^ T * y, L = chol(Sigma^-1 + Z^T * Z) (used for various computations when use_woodbury_identity_==true) */
		std::map<data_size_t, vec_t> y_tilde2_;
		/*! \brief Indicates whether y_aux_ has been calculated */
		bool y_aux_has_been_calculated_ = false;
		/*! \brief If true, the response variable data has been set (otherwise y_ is empty) */
		bool y_has_been_set_ = false;
		/*! \brief True if there are weights */
		bool has_weights_ = false;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: weights */
		std::map<data_size_t, vec_t> weights_;
		/*! \brief A learning rate for the likelihood for generalized Bayesian inference (only non-Gaussian likelihoods) */
		double likelihood_learning_rate_ = 1.;

		// GROUPED RANDOM EFFECTS
		/*! \brief Number of grouped (intercept) random effects components */
		data_size_t num_re_group_ = 0;
		/*! \brief Number of categorical grouping variables that define grouped random effects (can be different from num_re_group_ in case intercept effects are dropped when having random coefficients)  */
		data_size_t num_group_variables_ = 0;
		/*! \brief Number of grouped random coefficients components */
		data_size_t num_re_group_rand_coef_ = 0;
		/*! \brief Indices that relate every random coefficients to a "base" intercept grouped random effect component. Counting starts at 1 (and ends at the number of base intercept random effects). Length of vector = num_re_group_rand_coef_. */
		std::vector<int> ind_effect_group_rand_coef_;
		/*! \brief Indicates whether intercept random effects are dropped (only for random coefficients). If drop_intercept_group_rand_effect_[k] is true, the intercept random effect number k is dropped. Only random effects with random slopes can be dropped. Length of vector = num_re_group_. */
		std::vector<bool> drop_intercept_group_rand_effect_;
		/*! \brief Total number of grouped random effects components (random intercepts plus random coefficients (slopes)) */
		data_size_t num_re_group_total_ = 0;

		// GAUSSIAN PROCESS
		/*! \brief 1 if there is a Gaussian process 0 otherwise */
		data_size_t num_gp_ = 0;
		/*! \brief Number of random coefficient GPs */
		data_size_t num_gp_rand_coef_ = 0;
		/*! \brief Total number of GPs (random intercepts plus random coefficients) */
		data_size_t num_gp_total_ = 0;
		/*! \brief Index in the vector of random effect components (in the values of 're_comps_') of the intercept GP associated with the random coefficient GPs */
		int ind_intercept_gp_;
		/*! \brief Dimension of the coordinates (=number of features) for Gaussian process */
		int dim_gp_coords_ = 2;//required to save since it is needed in the Predict() function when predictions are made for new independent realizations of GPs
		/*! \brief If true, there are duplicates in coords among the neighbors (currently only for non-Gaussian likelihoods for some GP approximations) */
		bool has_duplicates_coords_ = false;
		/*! \brief Type of GP-approximation for handling large data */
		string_t gp_approx_ = "none";
		/*! \brief List of supported optimizers for covariance parameters */
		const std::set<string_t> SUPPORTED_GP_APPROX_{ "none", "vecchia", "tapering", "fitc", "full_scale_tapering", "full_scale_vecchia" };
		/*! \brief True if 'gp_approx_' has been changed internally from 'fitc' to 'full_scale_tapering' for handling duplicate coordinates */
		bool gp_approx_changed_to_FSA_from_FITC_ = false;
		/*! \brief How to calculate predictive variances and covariances for "full_scale_tapering" when using the cholesky decomposition */
		string_t calc_pred_cov_var_FSA_cholesky_ = "stochastic_stable";//"exact" (direct calculation), "exact_stable" (using a numerically stable version, but potentially large memory and time footpringt), "stochastic_stable" (using a numerically stable version and simulations to reduce memory and time footpringt)
		/*! \brief If true, the Vecchia approximation is done for the latent process for Gaussian likelihoods */
		bool vecchia_latent_approx_gaussian_ = false;

		// RANDOM EFFECT / GP COMPONENTS
		/*! \brief Outer key: independent realizations of REs/GPs over "clusters", inner key: set index of REs / GPs  for multiple parameters (e.g. for heteroscedastic GP), values: vectors with individual RE/GP components */
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompBase<T_mat>>>>> re_comps_;
		/*! \brief Indices of parameters of RE components in global parameter vector cov_pars. ind_par_[i] and ind_par_[i+1]-1 are the indices of the first and last parameter of component number i (counting starts at i = 0) */
		std::vector<data_size_t> ind_par_;
		/*! \brief Number of covariance parameters */
		data_size_t num_cov_par_;
		/*! \brief Total number of random effect components (grouped REs plus other GPs) */
		data_size_t num_comps_total_ = 0;
		/*! \brief Number of sets of random effects / GPs for different parameters with REs / GPs. This is larger than 1, e.g., heteroscedastic models */
		int num_sets_re_ = 1;
		/*! \brief Number of covariance parameters per sets of random effects / GPs. This is larger than 1, e.g., heteroscedastic models */
		int num_cov_par_per_set_re_;
		/*! \brief Number of fixed effects sets (>1 e.g. for heteroscedastic models) */
		int num_sets_fixed_effects_ = 1;

		// SPECIAL CASES OF RE MODELS FOR FASTER CALCULATIONS
		/*! \brief If true, the Woodbury, Sherman and Morrison matrix inversion formula is used for calculating the inverse of the covariance matrix (only used if there are only grouped REs and no Gaussian processes) */
		bool use_woodbury_identity_ = false;
		/*! \brief True if there is only one grouped random effect component, and (all) calculations are done on the b-scale instead of the Zb-scale (this flag is only used for non-Gaussian likelihoods) */
		bool only_one_grouped_RE_calculations_on_RE_scale_ = false;
		/*! \brief True if there is only one grouped random effect component for Gaussian data, can calculations for predictions (only) are done on the b-scale instead of the Zb-scale (this flag is only used for Gaussian likelihoods) */
		bool only_one_grouped_RE_calculations_on_RE_scale_for_prediction_ = false;
		/*! \brief True if there is only one GP random effect component, and calculations are done on the b-scale instead of the Zb-scale (only for non-Gaussian likelihoods) */
		bool only_one_GP_calculations_on_RE_scale_ = false;
		/*! \brief True if there is only one GP random effect component, and calculations are done on the b-scale instead of the Zb-scale (only for non-Gaussian likelihoods) */
		bool Vecchia_calculations_on_RE_scale_ = false;
		/*! \brief If true, the Woodbury formula is used for calculating the inverse of the covariance matrix (only if cov_function = "linear" and there is only one GP) */
		bool linear_kernel_use_woodbury_identity_ = false;
		/*! \brief If true, this is an iid model without a random effects / GP component */
		bool iid_model_ = false;
		/*! \brief If true, grouped REs are used together with a Vecchia-approximated GP */
		bool grouped_RE_and_vecchia_GP_ = false;

		// COVARIANCE MATRIX AND CHOLESKY FACTORS OF IT
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Cholesky decomposition of covariance matrices */
		std::map<data_size_t, T_chol> chol_facts_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: Square root of diagonal of matrix Sigma^-1 + Zt * Z  (used only if there is only one grouped random effect and ZtZ is diagonal) */
		std::map<data_size_t, vec_t> sqrt_diag_SigmaI_plus_ZtZ_;
		/*! \brief Indicates whether the covariance matrix has been factorized or not */
		bool covariance_matrix_has_been_factorized_ = false;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Idendity matrices used for calculation of inverse covariance matrix */
		std::map<data_size_t, T_mat> Id_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Permuted idendity matrices used for calculation of inverse covariance matrix when Cholesky factors have a permutation matrix */
		std::map<data_size_t, T_mat> P_Id_;
		/*! \brief Indicates whether a symbolic decomposition for calculating the Cholesky factor of the covariance matrix has been done or not (only for sparse matrices) */
		bool chol_fact_pattern_analyzed_ = false;
		/*! \brief Collects inverse covariance matrices Psi^{-1} (usually not saved, but used e.g. in Fisher scoring without the Vecchia approximation) */
		std::map<data_size_t, T_mat> psi_inv_;
		/*! \brief Inverse covariance matrices Sigma^-1 of random effects. This is only used if use_woodbury_identity_==true (if there are only grouped REs) */
		std::map<data_size_t, sp_mat_t> SigmaI_;
		/*! \brief Pointer to covariance matrix of the random effects (sum of all components). This is only used for non-Gaussian likelihoods and if use_woodbury_identity_==false. In the Gaussian case this needs not be saved */
		std::map<data_size_t, std::shared_ptr<T_mat>> ZSigmaZt_;

		// COVARIATE DATA FOR LINEAR REGRESSION TERM
		/*! \brief If true, the model linearly incluses covariates */
		bool has_covariates_ = false;
		/*! \brief Number of covariates */
		int num_covariates_;
		/*! \brief Covariate data */
		den_mat_t X_;
		/*! \brief Number of coefficients that are printed out when trace / logging is activated */
		const int NUM_COEF_PRINT_TRACE_ = 5;
		/*! \brief True if X_ contains an intercept column */
		bool has_intercept_ = false;
		/*! \brief Index of the intercept column in X_ */
		int intercept_col_;
		/*! \brief If true, X_ and the linear regression covariates are scaled */
		bool scale_covariates_;
		/*! \brief Location transformation if covariate data is scaled */
		vec_t loc_transf_;
		/*! \brief Scale transformation if covariate data is scaled */
		vec_t scale_transf_;
		/*! \brief Linear regression coefficients */
		vec_t beta_;
		/*! \brief Linear regression coefficients of previous iteration (used only by external optimizers) */
		vec_t beta_lag1_;
		/*! \brief If true, there are additional external offsets */
		bool has_fixed_effects_ = false;
		/*! \brief Fixed effects / offset */
		vec_t fixed_effects_;

		/*! \brief Variance of idiosyncratic error term (nugget effect) */
		double sigma2_ = 1.;//initialize with 1. to avoid valgrind false positives in EvalLLforLBFGSpp() in optim_utils.h
		/*! \brief Previous value of variance of idiosyncratic error term (used only by external optimizers) */
		double sigma2_lag1_ = 1.;
		/*! \brief Quadratic form y^T Psi^-1 y (saved for avoiding double computations when profiling out sigma2 for Gaussian data) */
		double yTPsiInvy_;
		/*! \brief Determinant of Psi (to avoid double computations) */
		double log_det_Psi_;
		// True if parameter optimization in 'OptimLinRegrCoefCovPar' is running 
		bool optimization_running_currently_ = false;
		// Auxiliary variable to save covariance parameters that were set for the first time. This is use to avoid the fact that the marginal variance changes when not estimating it due to the reparametrization with the nugget effect for gaussian likelihoods
		vec_t cov_pars_set_first_time_;

		// OPTIMIZER PROPERTIES
		/*! \brief Optimizer for covariance parameters. Internal default values are set in 'InitializeOptimSettings' */
		string_t optimizer_cov_pars_;
		/*! \brief true if 'optimizer_coef_' has been set */
		bool optimizer_cov_pars_has_been_set_ = false;
		/*! \brief List of supported optimizers for covariance parameters */
		const std::set<string_t> SUPPORTED_OPTIM_COV_PAR_{ "gradient_descent", "fisher_scoring", "newton", "nelder_mead",
			"adam", "lbfgs", "lbfgs_not_profile_out_nugget", "lbfgs_linesearch_nocedal_wright" }; // "adam" is only experimental and not fully supported, "bfgs_optim_lib" is no longer supported (12.11.2024)
		/*! \brief Convergence criterion for terminating the 'OptimLinRegrCoefCovPar' optimization algorithm */
		string_t convergence_criterion_ = "relative_change_in_log_likelihood";
		/*! \brief List of supported convergence criteria used for terminating the optimization algorithm */
		const std::set<string_t> SUPPORTED_CONV_CRIT_{ "relative_change_in_parameters", "relative_change_in_log_likelihood" };
		/*! \brief Maximal number of iterations for covariance parameter and linear regression parameter estimation */
		int max_iter_ = 1000;
		/*!
		\brief Convergence tolerance for covariance and linear regression coefficient estimation.
			The algorithm stops if the relative change in either the (approximate) log-likelihood or the parameters is below this value.
			For "bfgs_optim_lib", the L2 norm of the gradient is used instead of the relative change in the log-likelihood.
			If delta_rel_conv_init_ < 0, internal default values are set in 'InitializeOptimSettings'
		*/
		double delta_rel_conv_;
		/*! \brief Initial convergence tolerance (to remember as default values for delta_rel_conv_ are different for 'nelder_mead' vs. other optimizers and the optimization might get restarted) */
		double delta_rel_conv_init_ = -1;
		/*! \brief Learning rate for covariance parameters. If lr_cov_init_ < 0, internal default values are set in 'InitializeOptimSettings' */
		double lr_cov_;
		/*! \brief Initial learning rate for covariance parameters (to remember as lr_cov_ can be decreased) */
		double lr_cov_init_ = -1;
		/*! \brief True if 'lr_cov_' and other learning rates have been initialized, i.e., if 'InitializeOptimSettings' has been called */
		bool lr_have_been_initialized_ = false;
		/*! \brief Learning rate for covariance parameters after first iteration (to remember as lr_cov_ can be decreased) */
		double lr_cov_after_first_iteration_;
		/*! \brief Learning rate for covariance parameters after first optimization iteration in the first boosting iteration (only for the GPBoost algorithm) */
		double lr_cov_after_first_optim_boosting_iteration_;
		/*! \brief Learning rate for auxiliary parameters for non-Gaussian likelihoods (e.g., shape of a gamma likelihood) */
		double lr_aux_pars_;
		/*! \brief Initial learning rate for auxiliary parameters for non-Gaussian likelihoods (e.g., shape of a gamma likelihood) */
		double lr_aux_pars_init_ = 0.1;
		/*! \brief Learning rate for auxiliary parameters after first iteration (to remember as lr_cov_ can be decreased) */
		double lr_aux_pars_after_first_iteration_ = 0.1;
		/*! \brief Learning rate for auxiliary parameters after first optimization iteration in the first boosting iteration (only for the GPBoost algorithm) */
		double lr_aux_pars_after_first_optim_boosting_iteration_ = 0.1;
		/*! \brief Indicates whether Nesterov acceleration is used in the gradient descent for finding the covariance parameters (only used for "gradient_descent") */
		bool use_nesterov_acc_ = true;
		/*! \brief Acceleration rate for covariance parameters for Nesterov acceleration (only relevant if use_nesterov_acc and nesterov_schedule_version == 0) */
		double acc_rate_cov_ = 0.5;
		/*! \brief Number of iterations for which no mometum is applied in the beginning (only relevant if use_nesterov_acc) */
		int momentum_offset_ = 2;
		/*! \brief Select Nesterov acceleration schedule 0 or 1 */
		int nesterov_schedule_version_ = 0;
		/*! \brief Optimizer for linear regression coefficients (The default = "wls" is changed to "gradient_descent" for non-Gaussian likelihoods upon initialization). See the constructor REModelTemplate() for the default values which depend on whether the likelihood is Gaussian or not */
		string_t optimizer_coef_;
		/*! \brief List of supported optimizers for regression coefficients for Gaussian likelihoods */
		const std::set<string_t> SUPPORTED_OPTIM_COEF_GAUSS_{ "gradient_descent", "wls", "nelder_mead", "bfgs_optim_lib", "adam", "lbfgs", "lbfgs_not_profile_out_nugget", "lbfgs_linesearch_nocedal_wright" };
		/*! \brief List of supported optimizers for regression coefficients for non-Gaussian likelihoods */
		const std::set<string_t> SUPPORTED_OPTIM_COEF_NONGAUSS_{ "gradient_descent", "nelder_mead", "bfgs_optim_lib", "adam", "lbfgs", "lbfgs_linesearch_nocedal_wright" };
		/*! \brief Learning rate for fixed-effect linear coefficients */
		double lr_coef_;
		/*! \brief Initial learning rate for fixed-effect linear coefficients (to remember as lr_coef_ can be decreased) */
		double lr_coef_init_ = 0.1;
		/*! \brief Learning rate for fixed-effect linear coefficients after first iteration (to remember as lr_coef_ can be decreased) */
		double lr_coef_after_first_iteration_ = 0.1;
		/*! \brief Learning rate for linear regression coefficients (actually the learning rate interpreted as a coefficient) after first optimization iteration in the first boosting iteration (only for the GPBoost algorithm) */
		double lr_coef_after_first_optim_boosting_iteration_ = 0.1;
		/*! \brief Acceleration rate for coefficients for Nesterov acceleration (only relevant if use_nesterov_acc and nesterov_schedule_version == 0) */
		double acc_rate_coef_ = 0.5;
		/*! \brief Maximal number of steps for which learning rate shrinkage is done for gradient-based optimization of covariance parameters and regression coefficients */
		int max_number_lr_shrinkage_steps_ = 30;
		/*! \brief Maximal number of steps for which learning rate shrinkage is done for gradient-based optimization of covariance parameters and regression coefficients */
		const int MAX_NUMBER_LR_SHRINKAGE_STEPS_DEFAULT_ = 30;
		/*! \brief Learning rate shrinkage factor for gradient-based optimization of covariance parameters and regression coefficients */
		double LR_SHRINKAGE_FACTOR_ = 0.5;
		/*! \brief Threshold value for a learning rate below which a learning rate might be increased again (only in case there are also regression coefficients and for gradient descent optimization of covariance parameters and regression coefficients) */
		double lr_is_small_threshold_cov_ = 1e-6;
		/*! \brief Threshold value for a learning rate below which a learning rate might be increased again (only in case there are also regression coefficients and for gradient descent optimization of covariance parameters and regression coefficients) */
		double lr_is_small_threshold_coef_ = 1e-6;
		/*! \brief Threshold value for a learning rate below which a learning rate might be increased again (only in case there are also regression coefficients and for gradient descent optimization of covariance parameters and regression coefficients) */
		double lr_is_small_threshold_aux_ = 1e-6;
		/*! \brief Threshold value for relative change in parameters below which a learning rate might be increased again (only in case there are also regression coefficients and for gradient descent optimization of covariance parameters and regression coefficients) */
		double LR_IS_SMALL_REL_CHANGE_IN_PARS_THRESHOLD_ = 1e-4;
		/*! \brief Threshold value for relative change in other parameters above which a learning rate is again set to its initial value (only in case there are also regression coefficients and for gradient descent optimization of covariance parameters and regression coefficients) */
		double MIN_REL_CHANGE_IN_OTHER_PARS_FOR_RESETTING_LR_ = 1e-2;
		/*! \brief true if 'optimizer_coef_' has been set */
		bool coef_optimizer_has_been_set_ = false;
		/*! \brief List of optimizers which use upstream extern optimizer libraries */
		const std::set<string_t> OPTIM_EXTERNAL_{ "nelder_mead", "bfgs_optim_lib", "adam", "lbfgs", "lbfgs_linesearch_nocedal_wright" };
		/*! \brief List of xternal optimizers which support the "wls" option for optimizer_coef_ */
		const std::set<string_t> OPTIM_EXTERNAL_SUPPORT_WLS_{ "lbfgs", "lbfgs_linesearch_nocedal_wright" };
		/*! \brief If true, any additional parameters for non-Gaussian likelihoods are also estimated (e.g., shape parameter of gamma likelihood) */
		bool estimate_aux_pars_ = false;
		/*! \brief True if the function 'SetOptimConfig' has been called */
		bool set_optim_config_has_been_called_ = false;
		/*! \brief If true, the covariance parameters or linear coefficients were updated for the first time with gradient descent*/
		bool first_update_ = false;
		/*! \brief Number of likelihood evaluations during optimization */
		int num_ll_evaluations_ = 0;
		/*! \brief Number of iterations during optimization */
		int num_iter_ = 0;
		/*! \brief True, if 'OptimLinRegrCoefCovPar' has been called */
		bool model_has_been_estimated_ = false;
		/*! \brief Maximal relative change for covariance parameters in one iteration */
		int MAX_REL_CHANGE_GRADIENT_UPDATE_ = 100; // allow maximally a change by a factor of 'MAX_REL_CHANGE_GRADIENT_UPDATE_' in one iteration
		/*! \brief Maximal value of gradient updates on log-scale for covariance parameters */
		double MAX_GRADIENT_UPDATE_LOG_SCALE_ = std::log((double)MAX_REL_CHANGE_GRADIENT_UPDATE_);
		/*! \brief Maximal relative change for for auxiliary parameters in one iteration */
		int MAX_REL_CHANGE_GRADIENT_UPDATE_AUX_PARS_ = 100;
		/*! \brief Maximal value of gradient updates on log-scale for auxiliary parameters */
		double MAX_GRADIENT_UPDATE_LOG_SCALE_AUX_PARS_ = std::log((double)MAX_REL_CHANGE_GRADIENT_UPDATE_AUX_PARS_);
		/*! \brief Constant C_mu used for checking whether step sizes for linear regression coefficients are clearly too large */
		double C_mu_;
		/*! \brief Constant C_sigma2_ used for checking whether step sizes for linear regression coefficients are clearly too large */
		double C_sigma2_;
		/*! \brief Constant used for checking whether step sizes for linear regression coefficients are clearly too large */
		double C_MAX_CHANGE_COEF_ = 10.;
		/*! \brief True if covariance parameters and potential auxiliary parameters have been etimated before in a previous boosting iteration (applies only to the GPBoost algorithm) */
		bool cov_pars_have_been_estimated_once_ = false;
		/*! \brief True if covariance parameters and potential auxiliary parameters have been etimated in the most recent call (applies only to the GPBoost algorithm) */
		bool cov_pars_have_been_estimated_during_last_call_ = false;
		/*! \brief True if regression coefficients (actually the learning rate interpreted as a coefficient) have been etimated before in a previous boosting iteration (applies only to the GPBoost algorithm) */
		bool coef_have_been_estimated_once_ = false;
		/*! \brief True if 'lr_cov_' and 'lr_aux_pars_ have been doubled in the first optimization iteration (num_iter_ == 0) (applies only to the GPBoost algorithm) */
		bool learning_rates_have_been_doubled_in_first_iteration_ = false;
		/*! \brief True if 'lr_coef_' hase been doubled in the first optimization iteration (num_iter_ == 0) (applies only to the GPBoost algorithm) */
		bool learning_rate_coef_have_been_doubled_in_first_iteration_ = false;
		/*! \brief If true, Armijo's condition is used to check whether there is sufficient decrease in the negative log-likelighood (otherwise it is only checked for a decrease) */
		bool armijo_condition_ = true;
		/*! \brief Constant c for Armijo's condition. Needs to be in (0,1) */
		double c_armijo_ = 1e-4;
		/*! \brief Constant c for Armijo's condition for the Nesterov momentum part. Needs to be in (0,1) */
		double c_armijo_mom_ = 1e-4;
		/*! \brief Constant c for Armijo's condition. Needs to be in (0,1) */
		const double C_ARMIJO_DEFAULT_ = 1e-4;
		/*! \brief Constant c for Armijo's condition for the Nesterov momentum part. Needs to be in (0,1) */
		const double C_ARMIJO_MOM_DEFAULT_ = 1e-4;
		/*! \brief Directional derivative wrt covariance parameters for Armijo / Wolfe condition */
		double dir_deriv_armijo_cov_pars_;
		/*! \brief Directional derivativet wrt auxiliary coefficients for Armijo / Wolfe condition */
		double dir_deriv_armijo_aux_pars_;
		/*! \brief Directional derivative linear regression coefficients for Armijo / Wolfe condition */
		double dir_deriv_armijo_coef_;
		/*! \brief Momentum term directional derivative wrt covariance parameters for Armijo / Wolfe condition */
		double mom_dir_deriv_armijo_cov_pars_;
		/*! \brief Momentum term directional derivativet wrt auxiliary coefficients for Armijo / Wolfe condition */
		double mom_dir_deriv_armijo_aux_pars_;
		/*! \brief Momentum term directional derivative linear regression coefficients for Armijo / Wolfe condition */
		double mom_dir_deriv_armijo_coef_;
		/*! \brief If true, the initial learning rates in every iteration are set such that there is a constant first order change */
		bool learning_rate_constant_first_order_change_ = false;
		/*! \brief If true, the learning rates are reset to initial values in every iteration (only for gradient_descent) */
		bool reset_learning_rate_every_iteration_ = false;
		/*! \brief If true, the learning rates can be increased again in latter iterations after they have been decreased (only for gradient_descent) */
		bool increase_learning_rate_again_ = false;
		/*! \brief If true, the learning rates have been descreased (only for gradient_descent) */
		bool learning_rate_decreased_first_time_ = false;
		/*! \brief If true, the learning rates have been increased after they have been descreased (only for gradient_descent) */
		bool learning_rate_increased_after_descrease_ = false;
		/*! \brief If true, the learning rates have been descreased again after they have been increased (only for gradient_descent) */
		bool learning_rate_decreased_after_increase_ = false;
		/*! \brief Threshold value, for relative change in the log-likelihood, below which learning rates are increased again for gradient descent */
		double INCREASE_LR_CHANGE_LL_THRESHOLD_ = 1e-3;
		/*! \brief If true, the nugget effect is profiled out for Gaussian likelihoods (=use closed-form expression for error / nugget variance) */
		bool profile_out_error_variance_ = false;
		/*! \brief Approximation to the Hessian matrix for LBFGS saved here for reuse */
		LBFGSpp::BFGSMat<double> m_bfgs_;
		// Indicates which parameters are estimated (>0) and which not (<= 0)
		std::vector<int> estimate_cov_par_index_;
		// True if estimate_cov_par_index_ has been set in 'SetOptimConfig()'
		bool estimate_cov_par_index_has_been_set_ = false;
		// Number of corrections to approximate the inverse Hessian matrix for the lbfgs optimizer
		int m_lbfgs_ = 6;

		// MATRIX INVERSION PROPERTIES
		/*! \brief Matrix inversion method */
		string_t matrix_inversion_method_ = "";
		string_t matrix_inversion_method_user_provided_ = "";
		/*! \brief Supported matrix inversion methods */
		const std::set<string_t> SUPPORTED_MATRIX_INVERSION_METHODS_{ "cholesky", "iterative" };
		/*! \brief Maximal number of iterations for conjugate gradient algorithm */
		int cg_max_num_it_ = 1000;
		/*! \brief Maximal number of iterations for conjugate gradient algorithm when being run as Lanczos algorithm for tridiagonalization */
		int cg_max_num_it_tridiag_ = 1000;
		/*! \brief Tolerance level for L2 norm of residuals for checking convergence in conjugate gradient algorithm when being used for parameter estimation */
		double cg_delta_conv_ = 1e-2;
		/*! \brief Tolerance level for L2 norm of residuals for checking convergence in conjugate gradient algorithm when being used for prediction */
		double cg_delta_conv_pred_ = 1e-3;
		/*! \brief Threshold to avoid numerical instability in the CG: If the L1-norm of the rhs is below the defined threshold the CG is not executed and a vector of 0's is returned */
		const double THRESHOLD_ZERO_RHS_CG_ = 1.0e-100;
		/*! \brief Number of samples when simulation is used for calculating predictive variances */
		int nsim_var_pred_ = 1000;
		/*! \brief Number of random vectors (e.g., Rademacher) for stochastic approximation of the trace of a matrix */
		int num_rand_vec_trace_ = 50;
		/*! \brief If true, random vectors (e.g., Rademacher) for stochastic approximation of the trace of a matrix are sampled only once at the beginning of Newton's method for finding the mode in the Laplace approximation and are then reused in later trace approximations, otherwise they are sampled every time a trace is calculated */
		bool reuse_rand_vec_trace_ = true;
		/*! \brief Seed number to generate random vectors (e.g., Rademacher) */
		int seed_rand_vec_trace_ = 1;
		/*! If the seed of the random number generator cg_generator_ is set, cg_generator_seeded_ is set to true*/
		bool cg_generator_seeded_ = false;
		/*! \brief General RNG seet */
		int seed_rng_ = 1;
		/*! Indicates if we observe a NAN or Inf value in a conjugate gradient iteration */
		bool NaN_found = false;
		/*! Random number generator used in iterative methods */
		RNG_t cg_generator_;
		/*! See counter for parallel RNG */
		uint64_t cg_generator_counter_ = 0;
		/*! Matrix of random Rademacher vectors (u_1,...,u_t), where u_i is of dimension n & Cov(u_i) = I*/
		std::map<data_size_t, den_mat_t> rand_vec_probe_;
		/*! Matrix of random Rademacher vectors (u_1,...,u_t), where u_i is of dimension n & Cov(u_i) = I or Cov(u_i) = P*/
		std::map<data_size_t, den_mat_t> rand_vec_probe_P_;
		/*! Matrix of random Rademacher vectors (u_1,...,u_t), where u_i is of dimension m & Cov(u_i) = I*/
		std::map<data_size_t, den_mat_t> rand_vec_probe_low_rank_;
		/*! Matrix of random Rademacher vectors for Fisher information (u_1,...,u_t), where u_i is of dimension n & Cov(u_i) = I*/
		std::map<data_size_t, den_mat_t> rand_vec_fisher_info_;
		/*! If reuse_rand_vec_trace_ is true and random probe vectors have been generated for the first time, then saved_rand_vec_ is set to true  */
		std::map<data_size_t, bool> saved_rand_vec_;
		/*! If reuse_rand_vec_trace_ is true and random probe vectors have been generated for the first time, then saved_rand_vec_ is set to true  */
		std::map<data_size_t, bool> saved_rand_vec_fisher_info_;
		/*! CG solution for Rademacher vectors:	Sigma^-1 (u_1,...,u_t) */
		std::map<data_size_t, den_mat_t> solution_for_trace_;
		/*! \brief Type of preconditioner used for conjugate gradient algorithms */
		string_t cg_preconditioner_type_;
		/*! \brief List of supported preconditioners for conjugate gradient algorithms for Gaussian likelihoods and gp_approx = "full_scale_tapering" */
		const std::set<string_t> SUPPORTED_PRECONDITIONERS_GAUSS_FSA_{  "fitc", "none"};
		/*! \brief List of supported preconditioners for conjugate gradient algorithms for non-Gaussian likelihoods and gp_approx = "vecchia" */
		const std::set<string_t> SUPPORTED_PRECONDITIONERS_NONGAUSS_VECCHIA_{ "vadu", "pivoted_cholesky", "fitc", "incomplete_cholesky", "vecchia_response"};
		/*! \brief List of supported preconditioners for conjugate gradient algorithms for grouped random effects */
		const std::set<string_t> SUPPORTED_PRECONDITIONERS_GROUPED_RE_{ "ssor", "incomplete_cholesky", "diagonal", "none"};
		/*! \brief List of supported preconditioners for conjugate gradient algorithms for non-Gaussian likelihoods and gp_approx = "full_scale_vecchia" */
		const std::set<string_t> SUPPORTED_PRECONDITIONERS_NONGAUSS_VIF_{ "fitc", "vifdu", "none" };
		/*! \brief List of supported preconditioners for conjugate gradient algorithms for grouped random effects */
		const std::set<string_t> SUPPORTED_PRECONDITIONERS_VECCHIA_GROUPED_RE_{ "ssor", "incomplete_cholesky", "diagonal", "none" };
		/*! \brief true if 'cg_preconditioner_type_' has been set */
		bool cg_preconditioner_type_has_been_set_ = false;
		/*! \brief true if 'fitc_piv_chol_preconditioner_rank_' has been set */
		bool fitc_piv_chol_preconditioner_rank_has_been_set_ = false;
		/*! \brief true if 'nsim_var_pred_' has been set */
		bool nsim_var_pred_has_been_set_ = false;
		/*! \brief Rank of the FITC and pivoted Cholesky decomposition preconditioners for iterative methods for Vecchia and VIF approximations */
		int fitc_piv_chol_preconditioner_rank_ = 200;
		int default_fitc_preconditioner_rank_ = 200;
		int default_piv_chol_preconditioner_rank_ = 50;
		/*! \brief Rank of the matrix for approximating predictive covariances obtained using the Lanczos algorithm */
		int rank_pred_approx_matrix_lanczos_ = 1000;
		/*! \brief Number of CG steps when the CG method was last run */
		int num_cg_steps_last_ = 0;
		/*! \brief Number of CG steps when the CG method was last run for SLQ */
		int num_cg_steps_tridiag_last_ = 0;

		// WOODBURY IDENTITY FOR GROUPED RANDOM EFFECTS ONLY
		/*! \brief Collects matrices Z^T (only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects, otherwise these matrices are saved only in the indepedent RE components) */
		std::map<data_size_t, sp_mat_t> Zt_;
		/*! \brief Collects matrices Z^TZ (only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects, otherwise these matrices are saved only in the indepedent RE components) */
		std::map<data_size_t, sp_mat_t> ZtZ_;
		/*! \brief Collects vectors Z^Ty (only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects) */
		std::map<data_size_t, vec_t> Zty_;
		/*! \brief Cumulative number of random effects for components (usually not saved, only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects, otherwise these matrices are saved only in the indepedent RE components) */
		std::map<data_size_t, std::vector<data_size_t>> cum_num_rand_eff_;//The random effects of component j start at cum_num_rand_eff_[0][j]+1 and end at cum_num_rand_eff_[0][j+1]
		/*! \brief Sum of squared entries of Z_j for every random effect component (usually not saved, only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects && gauss_likelihood_) */
		std::map<data_size_t, std::vector<double>> Zj_square_sum_;
		/*! \brief Collects matrices Z^T * Z_j for every random effect component (usually not saved, only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects) */
		std::map<data_size_t, std::vector<sp_mat_t>> ZtZj_;
		/*! \brief Collects matrices L^-1 * Z^T * Z_j for every random effect component (usually not saved, only saved when use_woodbury_identity_=true i.e. when there are only grouped random effects and when Fisher scoring is done) */
		std::map<data_size_t, std::vector<T_mat>> LInvZtZj_;
		/*! \brief Permuted matrices Zt_ when Cholesky factors have a permutation matrix */
		std::map<data_size_t, sp_mat_t> P_Zt_;
		/*! \brief Permuted matrices ZtZj_ when Cholesky factors have a permutation matrix */
		std::map<data_size_t, std::vector<sp_mat_t>> P_ZtZj_;
		/*! \brief Collects matrices Z_j^T * Z_k (usually not saved, only if use_woodbury_identity_ and Fisher scoring is used) */
		std::map<data_size_t, std::vector<T_mat>> Zjt_Zk_;
		/*! \brief Indicates whether Zjt_Zk_ has been saved */
		bool Zjt_Zk_saved_ = false;
		/*! \brief Collects matrices (Z_j^T * Z_k).squaredNorm() (usually not saved, only if use_woodbury_identity_ and Fisher scoring is used) */
		std::map<data_size_t, std::vector<double>> Zjt_Zk_squaredNorm_;

		// ITERATIVE METHODS for GROUPED RANDOM EFFECTS
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: Matrix Sigma^-1 + Z^T Z */
		std::map<data_size_t, sp_mat_rm_t> SigmaI_plus_ZtZ_rm_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: (Sigma^-1 + Z^T * Z)^-1 * Z^T * y_ (from last iteration) */
		std::map<data_size_t, vec_t> last_MInvZty_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: (Sigma^-1 + Z^T * Z)^-1 * Z^T * X (from last iteration) */
		std::map<data_size_t, den_mat_t> last_MInvZtX_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For SSOR preconditioner - lower.triangular(Sigma^-1 + Z^T Z) times diag(Sigma^-1 + Z^T Z)^(-0.5)*/
		std::map<data_size_t, sp_mat_rm_t> P_SSOR_L_D_sqrt_inv_rm_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For SSOR preconditioner - diag(Sigma^-1 + Z^T Z)^(-1)*/
		std::map<data_size_t, vec_t> P_SSOR_D_inv_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For SSOR preconditioner for K=2 - diag(Sigma^-1 + Z^T Z)^(-1)[1:n_1]*/
		std::map<data_size_t, vec_t> P_SSOR_D1_inv_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For SSOR preconditioner for K=2 - diag(Sigma^-1 + Z^T Z)^(-1)[(n_1+1):n_2]*/
		std::map<data_size_t, vec_t> P_SSOR_D2_inv_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For SSOR preconditioner for K=2 - (Sigma^-1 + Z^T Z)[(n_1+1):n_2,1:n_1]*/
		std::map<data_size_t, sp_mat_rm_t> P_SSOR_B_rm_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For ZIC preconditioner - sparse cholesky factor L of matrix L L^T = (Sigma^-1 + Z^T Z)*/
		std::map<data_size_t, sp_mat_rm_t> L_SigmaI_plus_ZtZ_rm_;
		/*! \brief  Key: labels of independent realizations of REs/GPs, values: For diagonal preconditioner - diag(Sigma^-1 + Z^T Z)^(-1)*/
		std::map<data_size_t, vec_t> SigmaI_plus_ZtZ_inv_diag_;
		/*! \brief Key: labels of independent realizations of REs/GPs, value: P^(-1) z_i, where z_i ~ N(0, P), for later reuse in the calculation of the Fisher Information (only saved when Fisher scoring is done)*/
		std::map<data_size_t, den_mat_t> PI_RV_;

		// VECCHIA APPROXIMATION for GP
		/*! \brief If true, a memory optimized version of the Vecchia approximation is used (at the expense of being slightly slower). THiS IS CURRENTLY NOT IMPLEMENTED */
		bool vecchia_approx_optim_memory = false;
		/*! \brief The number of neighbors used in the Vecchia approximation */
		int num_neighbors_;
		/*! \brief Ordering used in the Vecchia approximation. "none" = no ordering, "random" = random ordering */
		string_t vecchia_ordering_ = "random";
		/*! \brief List of supported options for orderings of the Vecchia approximation */
		const std::set<string_t> SUPPORTED_VECCHIA_ORDERING_{ "none", "random", "time", "time_random_space" };
		/*! \brief The way how neighbors are selected */
		string_t vecchia_neighbor_selection_ = "nearest";//"nearest", "correlation", "residual_correlation"
		/*! \brief The number of neighbors used in the Vecchia approximation for making predictions */
		int num_neighbors_pred_;
		/*!
		* \brief Ordering used in the Vecchia approximation for making predictions
		* "order_obs_first_cond_obs_only" = observed data is ordered first and neighbors are only observed points & Vecchia approximation is done for observable process (only for Gaussian likelihoods)
		* "order_obs_first_cond_all" = observed data is ordered first and neighbors are selected among all points (observed + predicted) & Vecchia approximation is done for observable process (only for Gaussian likelihoods)
		* "order_pred_first" = predicted data is ordered first for making prediction & Vecchia approximation is done for observable process (only for Gaussian likelihoods)
		* "latent_order_obs_first_cond_obs_only" = observed data is ordered first and neighbors are only observed points & Vecchia approximation is done for latent process
		* "latent_order_obs_first_cond_all" = observed data is ordered first and neighbors are selected among all points (observed + predicted) & Vecchia approximation is done for latent process
		* See the constructor REModelTemplate() for the default values which depend on whether the likelihood is Gaussian or not
		*/
		string_t vecchia_pred_type_; //This is saved and not simply set in the prediction function since it needs to be used repeatedly in the GPBoost algorithm when making predictions in "regression_metric.hpp" and the way predictions are done for the Vecchia approximation should be decoupled from the boosting algorithm
		/*! \brief List of supported options for prediction with a Vecchia approximation for Gaussian likelihoods */
		const std::set<string_t> SUPPORTED_VECCHIA_PRED_TYPES_GAUSS_{ "order_obs_first_cond_obs_only",
			"order_obs_first_cond_all", "order_pred_first",
			"latent_order_obs_first_cond_obs_only", "latent_order_obs_first_cond_all" };
		/*! \brief List of supported options for prediction with a Vecchia approximation for non-Gaussian likelihoods */
		const std::set<string_t> SUPPORTED_VECCHIA_PRED_TYPES_NONGAUSS_{ "latent_order_obs_first_cond_obs_only",
			"latent_order_obs_first_cond_all", "order_obs_first_cond_obs_only", "order_obs_first_cond_all" };
		/*! \brief Collects indices of nearest neighbors (used for Vecchia approximation) */
		std::map<int, std::map<int, std::vector<std::vector<int>>>> nearest_neighbors_;
		// True, if nearest neighbors have been determined
		bool nearest_neighbors_determined_ = false;
		/*! \brief Distances between locations and their nearest neighbors (this is used only if the Vecchia approximation is used, otherwise the distances are saved directly in the base GP component) */
		std::map<int, std::map<int, std::vector<den_mat_t>>> dist_obs_neighbors_;
		/*! \brief Distances between nearest neighbors for all locations (this is used only if the Vecchia approximation is used, otherwise the distances are saved directly in the base GP component) */
		std::map<int, std::map<int, std::vector<den_mat_t>>> dist_between_neighbors_;//Note: this contains duplicate information (i.e. distances might be saved reduntly several times). But there is a trade-off between storage and computational speed. I currently don't see a way for saving unique distances without copying them when using them.
		/*! \brief Outer product of covariate vector at observations and neighbors with itself. First index = cluster, second index = data point i, third index = GP number j (this is used only if the Vecchia approximation is used, this is handled saved directly in the GP component using Z_) */
		std::map<int, std::map<int, std::vector<std::vector<den_mat_t>>>> z_outer_z_obs_neighbors_;
		/*! \brief Collects matrices B = I - A (=Cholesky factor of inverse covariance) for Vecchia approximation */
		std::map<int, std::map<int, sp_mat_t>> B_;
		/*! \brief Collects diagonal matrices D^-1 for Vecchia approximation */
		std::map<int, std::map<int, sp_mat_t>> D_inv_;
		/*! \brief Collects derivatives of matrices B ( = derivative of matrix -A) for Vecchia approximation */
		std::map<int, std::map<int, std::vector<sp_mat_t>>> B_grad_;
		/*! \brief Collects derivatives of matrices D for Vecchia approximation */
		std::map<int, std::map<int, std::vector<sp_mat_t>>> D_grad_;
		/*! \brief Triplets for initializing the matrices B */
		std::map<int, std::map<int, std::vector<Triplet_t>>> entries_init_B_;
		/*! \brief If true, the function 'SetVecchiaPredType' has been called and vecchia_pred_type_ has been set */
		bool vecchia_pred_type_has_been_set_ = false;
		/*! \brief If true, a stochastic trace approximation is used to calculate the Fisher information for a Vecchia approximation for Gaussian likelihoods */
		bool use_stochastic_trace_for_Fisher_information_Vecchia_ = true;
		/*! \brief If true, distances among points and neighbors are saved for Vecchia approximations for isotropic covariance functions (currently disables to save memory) */
		bool save_distances_isotropic_cov_fct_Vecchia_ = false;
		/*! \brief Outer key: independent realizations of REs/GPs over "clusters", inner key: set index of REs / GPs  for multiple parameters (e.g. for heteroscedastic GP), values: vectors with Vecchia GP components */
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompGP<den_mat_t>>>>> re_comps_vecchia_;
		/*! \brief Indicates whether the matrices A and D_inv for the Vecchia approximation have last been calculated on the transformed scale or not */
		bool cov_factor_vecchia_calculated_on_transf_scale_;
		/*! \brief Row-major matrix of the Veccia-matrix B*/
		std::map<int, std::map<int, sp_mat_rm_t>> B_rm_;
		/*! \brief Row-major matrix of the Veccia-matrix D_inv*/
		std::map<int, std::map<int, sp_mat_rm_t>> D_inv_rm_;
		/*! \brief Row-major matrix of B^T D^(-1)*/
		std::map<int, std::map<int, sp_mat_rm_t>> B_t_D_inv_rm_;
		/*! \brief Matrix of D^(-1) B Sigma_nm*/
		std::map<int, std::map<int, den_mat_t>> D_inv_B_cross_cov_;
		/*! \brief Matrix of B Sigma_nm*/
		std::map<int, std::map<int, den_mat_t>> B_cross_cov_;
		/*! \brief Matrix of B^T D^(-1) B Sigma_nm*/
		std::map<int, std::map<int, den_mat_t>> B_T_D_inv_B_cross_cov_;
		/*! \brief Matrix of Sigma_m^(-1) Sigma_mn*/
		std::map<int, std::map<int, den_mat_t>> sigma_ip_inv_cross_cov_T_;
		/*! \brief Matrix of grad(Sigma_m) Sigma_m^(-1) Sigma_mn*/
		std::map<int, std::map<int, std::vector<den_mat_t>>> sigma_ip_grad_sigma_ip_inv_cross_cov_T_;
		/*! \brief If true, correlation-based nearest neighbors for Vecchia approximation are updated */
		bool redetermine_vecchia_neighbors_ = false;
		/*! \brief If true, inducing points are updated */
		bool redetermine_inducing_points_ = false;


		// PREDICTIVE PROCESS AND FULL SCALE APPROXIMATION FOR GP
		/*! \brief Method for choosing inducing points */
		string_t ind_points_selection_ = "kmeans++";
		/*! \brief List of supported inducing point methods*/
		const std::set<string_t> SUPPORTED_METHOD_INDUCING_POINTS_{ "random", "kmeans++", "cover_tree", "space_time_kmeans++" };
		/*! \brief Number of inducing points */
		int num_ind_points_;
		/*! \brief Coordinates of inducing points. Used for redetermine inducing points for kmeans++ algo*/
		den_mat_t gp_coords_ip_mat_;
		/*! \brief Coordinates of inducing points of preconditioner. Used for redetermine inducing points for kmeans++ algo*/
		den_mat_t gp_coords_ip_mat_preconditioner_;
		/*! \brief Radius (= "spatial resolution") for the cover tree algorithm */
		double cover_tree_radius_;
		/*! \brief Outer key: independent realizations of REs/GPs over "clusters", inner key: set index of REs / GPs  for multiple parameters (e.g. for heteroscedastic GP), values: vectors with inducing points GP components */
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompGP<den_mat_t>>>>> re_comps_ip_;
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompGP<den_mat_t>>>>> re_comps_ip_preconditioner_;
		/*! \brief Outer key: independent realizations of REs/GPs over "clusters", inner key: set index of REs / GPs  for multiple parameters (e.g. for heteroscedastic GP), values: vectors with cross-covariance GP components */
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompGP<den_mat_t>>>>> re_comps_cross_cov_;
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompGP<den_mat_t>>>>> re_comps_cross_cov_preconditioner_;
		/*! \brief Outer key: independent realizations of REs/GPs over "clusters", inner key: set index of REs / GPs  for multiple parameters (e.g. for heteroscedastic GP), values: vectors with residual GP components with sparse (tapered) covariances */
		std::map<int, std::map<int, std::vector<std::shared_ptr<RECompGP<T_mat>>>>> re_comps_resid_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Cholesky decompositions of inducing points matrix sigma_ip */
		std::map<int, std::map<int, chol_den_mat_t>> chol_fact_sigma_ip_;
		std::map<int, std::map<int, chol_den_mat_t>> chol_fact_sigma_ip_preconditioner_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Inverse of Cholesky factor of inducing points matrix sigma_ip times cross-covariance */
		std::map<int, std::map<int, den_mat_t>> chol_ip_cross_cov_;
		std::map<int, std::map<int, den_mat_t>> chol_ip_cross_cov_preconditioner_;
		bool ind_points_determined_for_preconditioner_ = false;
		std::map<data_size_t, den_mat_t> sigma_inv_sigma_grad_rand_vec_;
		std::map<data_size_t, den_mat_t> sigma_grad_sigma_inv_rand_vec_;

		/*! \brief Key: labels of independent realizations of REs/GPs, values: diagonal of fully independent training conditional for predictive process */
		std::map<data_size_t, vec_t> fitc_resid_diag_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Cholesky decompositions of residual covariance matrix */
		std::map<data_size_t, T_chol> chol_fact_resid_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Matrix sigma_ip + cross_cov^T * sigma_resid^-1 * cross_cov used in Woodbury identity */
		std::map<data_size_t, den_mat_t> sigma_woodbury_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Cholesky decompositions of matrix sigma_ip + cross_cov^T * sigma_resid^-1 * cross_cov used in Woodbury identity */
		std::map<data_size_t, chol_den_mat_t> chol_fact_sigma_woodbury_;
		///*! \brief Key: labels of independent realizations of REs/GPs, values: Cholesky decompositions of matrix I + sigma_ip^(-1/2) * cross_cov^T * sigma_resid^-1 * cross_cov * sigma_ip^(-T/2) used in stable version for determinant in Woodbury identity */
		//std::map<data_size_t, chol_den_mat_t> chol_fact_sigma_woodbury_stable_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Diagonal of residual covariance matrix (Preconditioner) */
		std::map<data_size_t, vec_t> diagonal_approx_preconditioner_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Inverse of diagonal of residual covariance matrix (Preconditioner) */
		std::map<data_size_t, vec_t> diagonal_approx_inv_preconditioner_;
		/*! \brief Key: labels of independent realizations of REs/GPs, values: Cholesky decompositions of matrix sigma_ip + cross_cov^T * D^-1 * cross_cov used in Woodbury identity where D is given by the Preconditioner */
		std::map<data_size_t, chol_den_mat_t> chol_fact_woodbury_preconditioner_;

		// CLUSTERs of INDEPENDENT REALIZATIONS
		/*! \brief Keys: Labels of independent realizations of REs/GPs, values: vectors with indices for data points */
		std::map<data_size_t, std::vector<int>> data_indices_per_cluster_;
		/*! \brief Keys: Labels of independent realizations of REs/GPs, values: number of data points per independent realization */
		std::map<data_size_t, int> num_data_per_cluster_;
		/*! \brief Number of independent realizations of the REs/GPs */
		data_size_t num_clusters_;
		/*! \brief Unique labels of independent realizations */
		std::vector<data_size_t> unique_clusters_;

		// PREDICTION
		/*! \brief Cluster IDs for prediction */
		std::vector<data_size_t> cluster_ids_data_pred_;
		/*! \brief Levels of grouped RE for prediction */
		std::vector<std::vector<re_group_t>> re_group_levels_pred_;
		/*! \brief Covariate data for grouped random RE for prediction */
		std::vector<double> re_group_rand_coef_data_pred_;
		/*! \brief Coordinates for GP for prediction */
		std::vector<double> gp_coords_data_pred_;
		/*! \brief Covariate data for random GP for prediction */
		std::vector<double> gp_rand_coef_data_pred_;
		/*! \brief Covariate data for linear regression term */
		std::vector<double> covariate_data_pred_;
		/*! \brief Number of prediction points */
		data_size_t num_data_pred_;

		/*! Random number generator */
		RNG_t rng_;

		/*! Use GPU */
		bool GPU_use_ = false;

		/*! \brief Nesterov schedule */
		static double NesterovSchedule(int iter,
			int momentum_schedule_version,
			double nesterov_acc_rate,
			int momentum_offset) {
			if (iter < momentum_offset) {
				return(0.);
			}
			else {
				if (momentum_schedule_version == 0) {
					return(nesterov_acc_rate);
				}
				else if (momentum_schedule_version == 1) {
					return(1. - (3. / (6. + iter)));
				}
				else {
					Log::REFatal("NesterovSchedule: version = %d is not supported ", momentum_schedule_version);
				}
			}
			return(0.);
		}//end NesterovSchedule

		/*! \brief mutex for threading safe call */
		std::mutex mutex_;

		/*! \brief Constructs identity matrices if sparse matrices are used (used for calculating inverse covariance matrix) */
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<sp_mat_t, T_aux>::value || std::is_same<sp_mat_rm_t, T_aux>::value>::type* = nullptr >
		void ConstructI(data_size_t cluster_i) {
			int dim_I = use_woodbury_identity_ ? cum_num_rand_eff_[cluster_i][num_comps_total_] : num_data_per_cluster_[cluster_i];
			T_mat I(dim_I, dim_I);//identity matrix for calculating precision matrix
			I.setIdentity();
			I.makeCompressed();
			Id_.insert({ cluster_i, I });
		}
		/*! \brief Constructs identity matrices if dense matrices are used (used for calculating inverse covariance matrix) */
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<den_mat_t, T_aux>::value>::type* = nullptr >
		void ConstructI(data_size_t cluster_i) {
			int dim_I = use_woodbury_identity_ ? cum_num_rand_eff_[cluster_i][num_comps_total_] : num_data_per_cluster_[cluster_i];
			den_mat_t I(dim_I, dim_I);//identity matrix for calculating precision matrix
			I.setIdentity();
			Id_.insert({ cluster_i, I });
		}

		/*!
		* \brief Set response variable data y_ (and calculate Z^T * y if  use_woodbury_identity_ == true)
		* \param y_data Response variable data
		*/
		void SetY(const double* y_data) {
			if (gauss_likelihood_) {
				if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {
					y_[unique_clusters_[0]] = Eigen::Map<const vec_t>(y_data, num_data_);
				}
				else {
					for (const auto& cluster_i : unique_clusters_) {
						y_[cluster_i] = vec_t(num_data_per_cluster_[cluster_i]);
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							y_[cluster_i][j] = y_data[data_indices_per_cluster_[cluster_i][j]];
						}
					}
				}
				if (use_woodbury_identity_) {
					CalcZtY();
				}
			}//end gauss_likelihood_
			else {//not gauss_likelihood_
				(*likelihood_[unique_clusters_[0]]).template CheckY<double>(y_data, num_data_);
				if (likelihood_[unique_clusters_[0]]->label_type() == "int") {
					for (const auto& cluster_i : unique_clusters_) {
						y_int_[cluster_i] = vec_int_t(num_data_per_cluster_[cluster_i]);
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							y_int_[cluster_i][j] = static_cast<int>(y_data[data_indices_per_cluster_[cluster_i][j]]);
						}
					}
				}
				else if (likelihood_[unique_clusters_[0]]->label_type() == "double") {
					for (const auto& cluster_i : unique_clusters_) {
						y_[cluster_i] = vec_t(num_data_per_cluster_[cluster_i]);
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							y_[cluster_i][j] = y_data[data_indices_per_cluster_[cluster_i][j]];
						}
					}
				}
			}//end not gauss_likelihood_
			y_has_been_set_ = true;
		}

		/*!
		* \brief Set response variable data y_ if data is of type float (used for GPBoost algorithm since labels are float)
		* \param y_data Response variable data
		*/
		void SetY(const float* y_data) {
			if (gauss_likelihood_) {
				Log::REFatal("SetY is not implemented for Gaussian data and lables of type float (since it is not needed)");
			}//end gauss_likelihood_
			else {//not gauss_likelihood_
				(*likelihood_[unique_clusters_[0]]).template CheckY<float>(y_data, num_data_);
				if (likelihood_[unique_clusters_[0]]->label_type() == "int") {
					for (const auto& cluster_i : unique_clusters_) {
						y_int_[cluster_i] = vec_int_t(num_data_per_cluster_[cluster_i]);
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							y_int_[cluster_i][j] = static_cast<int>(y_data[data_indices_per_cluster_[cluster_i][j]]);
						}
					}
				}
				else if (likelihood_[unique_clusters_[0]]->label_type() == "double") {
					for (const auto& cluster_i : unique_clusters_) {
						y_[cluster_i] = vec_t(num_data_per_cluster_[cluster_i]);
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							y_[cluster_i][j] = static_cast<double>(y_data[data_indices_per_cluster_[cluster_i][j]]);
						}
					}
				}
			}
			y_has_been_set_ = true;
		}//end SetY

		/*!
		* \brief Return (last used) response variable data
		* \param[out] y Response variable data (memory needs to be preallocated)
		*/
		void GetY(double* y) {
			if (!y_has_been_set_) {
				Log::REFatal("Respone variable data has not been set");
			}
			if (gauss_likelihood_) {
#pragma omp parallel for schedule(static)
				for (data_size_t i = 0; i < num_data_; ++i) {
					y[i] = y_vec_[i];
				}
			}
			else if (likelihood_[unique_clusters_[0]]->label_type() == "double") {
				for (const auto& cluster_i : unique_clusters_) {
#pragma omp parallel for schedule(static)
					for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
						y[data_indices_per_cluster_[cluster_i][i]] = y_[cluster_i][i];
					}
				}
			}
			else if (likelihood_[unique_clusters_[0]]->label_type() == "int") {
				for (const auto& cluster_i : unique_clusters_) {
#pragma omp parallel for schedule(static)
					for (int i = 0; i < num_data_per_cluster_[cluster_i]; ++i) {
						y[data_indices_per_cluster_[cluster_i][i]] = y_int_[cluster_i][i];
					}
				}
			}
		}//end GetY

		/*!
		* \brief Return covariate data
		* \param[out] covariate_data covariate data
		*/
		void GetCovariateData(double* covariate_data) {
			if (!has_covariates_) {
				Log::REFatal("Model does not have covariates for a linear predictor");
			}
#pragma omp parallel for schedule(static)
			for (int i = 0; i < num_data_ * num_covariates_; ++i) {
				covariate_data[i] = X_.data()[i];
			}
		}

		/*!
		* \brief Return offset data
		* \param[out] fixed_effects offset data
		*/
		void GetOffsetData(double* fixed_effects) {
			if (!has_fixed_effects_) {
				Log::REFatal("Model does not have an offset term ");
			}
#pragma omp parallel for schedule(static)
			for (int i = 0; i < num_data_ * num_sets_re_; ++i) {
				fixed_effects[i] = fixed_effects_[i];
			}
		}

		/*!
		* \brief Set offset data
		* \param fixed_effects offset data
		*/
		void SetOffsetData(const double* fixed_effects) {//note 'offset' is otherwise called 'fixed_effects' in this file for historical reasons ...
			has_fixed_effects_ = true;
			fixed_effects_ = Eigen::Map<const vec_t>(fixed_effects, num_data_ * num_sets_re_);
		}

		/*!
		* \brief Calculate Z^T*y (use only when use_woodbury_identity_ == true)
		*/
		void CalcZtY() {
			for (const auto& cluster_i : unique_clusters_) {
				Zty_[cluster_i] = Zt_[cluster_i] * y_[cluster_i];
			}
		}

		/*!
		* \brief Get y_aux = Psi^-1*y
		* \param[out] y_aux Psi^-1*y (=y_aux_). Array needs to be pre-allocated of length num_data_
		*/
		void GetYAux(double* y_aux) {
			CHECK(y_aux_has_been_calculated_);
			if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {
#pragma omp parallel for schedule(static)
				for (int j = 0; j < num_data_; ++j) {
					y_aux[j] = y_aux_[unique_clusters_[0]][j];
				}
			}
			else {
				for (const auto& cluster_i : unique_clusters_) {
#pragma omp parallel for schedule(static)
					for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
						y_aux[data_indices_per_cluster_[cluster_i][j]] = y_aux_[cluster_i][j];
					}
				}
			}
		}

		/*!
		* \brief Get y_aux = Psi^-1*y
		* \param[out] y_aux Psi^-1*y (=y_aux_). This vector needs to be pre-allocated of length num_data_
		*/
		void GetYAux(vec_t& y_aux) {
			CHECK(y_aux_has_been_calculated_);
			if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {
				y_aux = y_aux_[unique_clusters_[0]];
			}
			else {
				for (const auto& cluster_i : unique_clusters_) {
					y_aux(data_indices_per_cluster_[cluster_i]) = y_aux_[cluster_i];
				}
			}
		}

		/*!
		* \brief Calculate Cholesky decomposition
		* \param psi Covariance matrix for which the Cholesky decomposition is calculated
		* \param cluster_i Cluster index for which the Cholesky factor is calculated
		*/
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<sp_mat_t, T_aux>::value || std::is_same<sp_mat_rm_t, T_aux>::value>::type* = nullptr >
		void CalcChol(const T_mat& psi, data_size_t cluster_i) {
			if (!chol_fact_pattern_analyzed_) {
				chol_facts_[cluster_i].analyzePattern(psi);
				if (cluster_i == unique_clusters_.back()) {
					chol_fact_pattern_analyzed_ = true;
				}
				if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
					P_Id_[cluster_i] = chol_facts_[cluster_i].permutationP() * Id_[cluster_i];
					P_Id_[cluster_i].makeCompressed();
					if (use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) {
						P_Zt_[cluster_i] = chol_facts_[cluster_i].permutationP() * Zt_[cluster_i];
						std::vector<sp_mat_t> P_ZtZj_cluster_i(num_comps_total_);
						for (int j = 0; j < num_comps_total_; ++j) {
							P_ZtZj_cluster_i[j] = chol_facts_[cluster_i].permutationP() * ZtZj_[cluster_i][j];
						}
						P_ZtZj_[cluster_i] = P_ZtZj_cluster_i;
					}
				}
			}
			chol_facts_[cluster_i].factorize(psi);
		}
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<den_mat_t, T_aux>::value>::type* = nullptr >
		void CalcChol(const den_mat_t& psi, data_size_t cluster_i) {
			chol_facts_[cluster_i].compute(psi);
		}

		/*!
		* \brief Calculate Cholesky decomposition of residual process in full scale approximation
		* \param psi Covariance matrix for which the Cholesky decomposition is calculated
		* \param cluster_i Cluster index for which the Cholesky factor is calculated
		*/
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<sp_mat_t, T_aux>::value || std::is_same<sp_mat_rm_t, T_aux>::value>::type* = nullptr >
		void CalcCholFSAResid(const T_mat& psi, data_size_t cluster_i) {
			if (!chol_fact_pattern_analyzed_) {
				chol_fact_resid_[cluster_i].analyzePattern(psi);
				if (cluster_i == unique_clusters_.back()) {
					chol_fact_pattern_analyzed_ = true;
				}
			}
			chol_fact_resid_[cluster_i].factorize(psi);
		}
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<den_mat_t, T_aux>::value>::type* = nullptr >
		void CalcCholFSAResid(const den_mat_t& psi, data_size_t cluster_i) {
			chol_fact_resid_[cluster_i].compute(psi);
		}

		/*!
		* \brief Caclulate Psi^(-1) if sparse matrices are used
		* \param psi_inv[out] Inverse covariance matrix
		* \param cluster_i Cluster index for which Psi^(-1) is calculated
		* \param only_at_non_zeroes_of_psi If true, psi_inv is calculated only at non-zero entries of psi, e.g., since it is used for calculating gradients afterwards
		*/
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<sp_mat_t, T_aux>::value || std::is_same<sp_mat_rm_t, T_aux>::value>::type* = nullptr >
		void CalcPsiInv(T_mat& psi_inv, data_size_t cluster_i, bool only_at_non_zeros_of_psi) {
			if (gp_approx_ == "vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
				Log::REFatal("'CalcPsiInv': this is not implemented for the approximation '%s' ", gp_approx_.c_str());
			}
			if (use_woodbury_identity_) {
				sp_mat_t MInvSqrtZt;
				if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
					MInvSqrtZt = sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().inverse().matrix().asDiagonal() * Zt_[cluster_i];
				}
				else {
					T_mat L_inv;
					if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
						TriangularSolve<T_mat, T_mat, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), P_Id_[cluster_i], L_inv, false);
					}
					else {
						TriangularSolve<T_mat, T_mat, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), Id_[cluster_i], L_inv, false);
					}
					MInvSqrtZt = L_inv * Zt_[cluster_i];
				}
				if (only_at_non_zeros_of_psi) {
					CalcZSigmaZt(psi_inv, cluster_i);//find out sparsity pattern where psi_inv is needed for gradient
					CalcLtLGivenSparsityPattern<T_mat>(MInvSqrtZt, psi_inv, true);
					psi_inv *= -1.;
				}
				else {
					psi_inv = -MInvSqrtZt.transpose() * MInvSqrtZt;//this is slow since n can be large (O(n^2*m)) (but its usually not run, only when calculating the Fisher information)
				}
				psi_inv.diagonal().array() += 1.0;
			}
			else {
				T_mat L_inv;
				if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
					TriangularSolve<T_mat, T_mat, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), P_Id_[cluster_i], L_inv, false);
				}
				else {
					TriangularSolve<T_mat, T_mat, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), Id_[cluster_i], L_inv, false);
				}
				if (only_at_non_zeros_of_psi) {
					//find out sparsity pattern where psi_inv is needed for gradient
					if (num_re_group_total_ == 0) {
						std::shared_ptr<T_mat> psi = re_comps_[cluster_i][0][0]->GetZSigmaZt();
						psi_inv = *psi;
					}
					else {
						CalcZSigmaZt(psi_inv, cluster_i);
					}
					CalcLtLGivenSparsityPattern<T_mat>(L_inv, psi_inv, true);
				}
				else {
					psi_inv = L_inv.transpose() * L_inv;//Note: this is the computational bottleneck for large data when psi=ZSigmaZt and its Cholesky factor is sparse (but its usually not run, only when calculating the Fisher information)
				}
			}
		}// end CalcPsiInv for sparse matrices
		template <class T_aux = T_mat, typename std::enable_if <std::is_same<den_mat_t, T_aux>::value>::type* = nullptr >
		void CalcPsiInv(den_mat_t& psi_inv, data_size_t cluster_i, bool) {
			if (gp_approx_ == "vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
				Log::REFatal("'CalcPsiInv': this is not implemented for the approximation '%s' ", gp_approx_.c_str());
			}
			if (use_woodbury_identity_) {//typically currently not called as use_woodbury_identity_ is only true for grouped REs only i.e. sparse matrices
				den_mat_t MInvSqrtZt;
				if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
					MInvSqrtZt = sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().inverse().matrix().asDiagonal() * Zt_[cluster_i];
				}
				else {
					TriangularSolve<den_mat_t, sp_mat_t, den_mat_t>(chol_facts_[cluster_i].CholFactMatrix(), Zt_[cluster_i], MInvSqrtZt, false);
				}
				psi_inv = -MInvSqrtZt.transpose() * MInvSqrtZt;
				psi_inv.diagonal().array() += 1.0;
			}
			else {
				den_mat_t L_inv;
				TriangularSolve<den_mat_t, den_mat_t, den_mat_t>(chol_facts_[cluster_i].CholFactMatrix(), Id_[cluster_i], L_inv, false);
				psi_inv = L_inv.transpose() * L_inv;
			}
		}// end CalcPsiInv for dense matrices

		/*!
		* \brief Caclulate X^TPsi^(-1)X
		* \param X Covariate data matrix X
		* \param[out] XT_psi_inv_X X^TPsi^(-1)X
		*/
		void CalcXTPsiInvX(const den_mat_t& X, den_mat_t& XT_psi_inv_X) {
			CHECK(gauss_likelihood_);
			if (num_clusters_ == 1 && (gp_approx_ != "vecchia" || vecchia_ordering_ == "none") && gp_approx_ != "full_scale_tapering" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_vecchia") {//only one cluster / idependent GP realization
				if (gp_approx_ == "vecchia") {
					den_mat_t BX = B_[unique_clusters_[0]][0] * X;
					XT_psi_inv_X = BX.transpose() * D_inv_[unique_clusters_[0]][0] * BX;
				}
				else {
					if (use_woodbury_identity_) {
						den_mat_t ZtX = Zt_[unique_clusters_[0]] * X;
						if (matrix_inversion_method_ == "cholesky") {
							den_mat_t MInvSqrtZtX;
							if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
								MInvSqrtZtX = sqrt_diag_SigmaI_plus_ZtZ_[unique_clusters_[0]].array().inverse().matrix().asDiagonal() * ZtX;
							}
							else {
								TriangularSolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_facts_[unique_clusters_[0]], ZtX, MInvSqrtZtX, false);
							}
							XT_psi_inv_X = X.transpose() * X - MInvSqrtZtX.transpose() * MInvSqrtZtX;
						}//end cholesky
						else if (matrix_inversion_method_ == "iterative") {
							den_mat_t MInvZtX;
							//Use last solution as initial guess
							if (num_iter_ > 0 && optimizer_coef_ == "wls") {
								MInvZtX = last_MInvZtX_[unique_clusters_[0]];
							}
							else {
								MInvZtX.resize(cum_num_rand_eff_[unique_clusters_[0]][num_comps_total_], X.cols());
							}
							//Reduce max. number of iterations for the CG in first update
							int cg_max_num_it = cg_max_num_it_;
							if (first_update_) {
								cg_max_num_it = (int)round(cg_max_num_it_ / 3);
							}
							CGRandomEffectsMat(SigmaI_plus_ZtZ_rm_[unique_clusters_[0]], ZtX, MInvZtX, NaN_found,
								cum_num_rand_eff_[unique_clusters_[0]][num_comps_total_], (int)X.cols(),
								cg_max_num_it, cg_delta_conv_, cg_preconditioner_type_,
								L_SigmaI_plus_ZtZ_rm_[unique_clusters_[0]], P_SSOR_L_D_sqrt_inv_rm_[unique_clusters_[0]]);
							last_MInvZtX_[unique_clusters_[0]] = MInvZtX;
							if (NaN_found) {
								Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
							}
							XT_psi_inv_X = X.transpose() * X - ZtX.transpose() * MInvZtX;
						}//end iterative
						else {
							Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
						}
					}
					else {
						den_mat_t MInvSqrtX;
						TriangularSolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_facts_[unique_clusters_[0]], X, MInvSqrtX, false);
						XT_psi_inv_X = MInvSqrtX.transpose() * MInvSqrtX;
					}
				}
			}//end only one cluster / idependent GP realization
			else {//more than one cluster and order of samples matters
				XT_psi_inv_X = den_mat_t(X.cols(), X.cols());
				XT_psi_inv_X.setZero();
				den_mat_t BX, psi_inv_X;
				for (const auto& cluster_i : unique_clusters_) {
					den_mat_t X_cluster_i = X(data_indices_per_cluster_[cluster_i], Eigen::all);
					if (gp_approx_ == "vecchia") {
						BX = B_[cluster_i][0] * X_cluster_i;
						XT_psi_inv_X += BX.transpose() * D_inv_[cluster_i][0] * BX;
					}
					else if (gp_approx_ == "full_scale_tapering" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia") {
						const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr();
						if (matrix_inversion_method_ == "cholesky") {
							if (gp_approx_ == "fitc") {
								den_mat_t cross_covT_X = (*cross_cov).transpose() * (fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * X_cluster_i);
								den_mat_t sigma_woodbury_I_cross_covT_X = chol_fact_sigma_woodbury_[cluster_i].solve(cross_covT_X);
								cross_covT_X.resize(0, 0);
								den_mat_t cross_cov_sigma_woodbury_I_cross_covT_X = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * ((*cross_cov) * sigma_woodbury_I_cross_covT_X);
								sigma_woodbury_I_cross_covT_X.resize(0, 0);
								psi_inv_X = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * X_cluster_i - cross_cov_sigma_woodbury_I_cross_covT_X;
							}
							else {
								den_mat_t sigma_resid_I_X(num_data_per_cluster_[cluster_i], X_cluster_i.cols());
								if (gp_approx_ == "full_scale_tapering") {
									sigma_resid_I_X = chol_fact_resid_[cluster_i].solve(X_cluster_i);
								}
								else {
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < X_cluster_i.cols(); ++i) {
										sigma_resid_I_X.col(i) = B_t_D_inv_rm_[cluster_i][0] * (B_rm_[cluster_i][0] * X_cluster_i.col(i));
									}
								}
								den_mat_t cross_covT_sigma_resid_I_X = (*cross_cov).transpose() * sigma_resid_I_X;
								den_mat_t sigma_woodbury_I_cross_covT_sigma_resid_I_X = chol_fact_sigma_woodbury_[cluster_i].solve(cross_covT_sigma_resid_I_X);
								cross_covT_sigma_resid_I_X.resize(0, 0);
								den_mat_t cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X = (*cross_cov) * sigma_woodbury_I_cross_covT_sigma_resid_I_X;
								sigma_woodbury_I_cross_covT_sigma_resid_I_X.resize(0, 0);
								den_mat_t sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X(num_data_per_cluster_[cluster_i], X_cluster_i.cols());
								if (gp_approx_ == "full_scale_tapering") {
									sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X = chol_fact_resid_[cluster_i].solve(cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X);
								}
								else {
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < X_cluster_i.cols(); ++i) {
										sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X.col(i) = B_t_D_inv_rm_[cluster_i][0] * (B_rm_[cluster_i][0] * cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X.col(i));
									}
								}
								psi_inv_X = sigma_resid_I_X - sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_X;
							}
						}
						else {
							//Use last solution as initial guess
							if (num_iter_ > 0 && optimizer_coef_ == "wls") {
								psi_inv_X = last_psi_inv_X_[cluster_i];
							}
							else {
								psi_inv_X.resize(num_data_per_cluster_[cluster_i], X_cluster_i.cols());
								psi_inv_X.setZero();
							}
							//Reduce max. number of iterations for the CG in first update
							int cg_max_num_it = cg_max_num_it_;
							if (first_update_) {
								cg_max_num_it = (int)round(cg_max_num_it_ / 3);
							}
							std::shared_ptr<T_mat> sigma_resid = re_comps_resid_[cluster_i][0][0]->GetZSigmaZt();
							if (cg_preconditioner_type_ == "fitc") {
								const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
								CGFSA_MULTI_RHS<T_mat>(*sigma_resid, (*cross_cov_preconditioner), chol_ip_cross_cov_[cluster_i][0], X_cluster_i, psi_inv_X,
									NaN_found, num_data_per_cluster_[cluster_i], (int)X_cluster_i.cols(), cg_max_num_it, cg_delta_conv_,
									cg_preconditioner_type_, chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
							}
							else {
								CGFSA_MULTI_RHS<T_mat>(*sigma_resid, (*cross_cov), chol_ip_cross_cov_[cluster_i][0], X_cluster_i, psi_inv_X,
									NaN_found, num_data_per_cluster_[cluster_i], (int)X_cluster_i.cols(), cg_max_num_it, cg_delta_conv_,
									cg_preconditioner_type_, chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
							}
							last_psi_inv_X_[cluster_i] = psi_inv_X;
							if (NaN_found) {
								Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
							}
						}
						XT_psi_inv_X += X_cluster_i.transpose() * psi_inv_X;
					}//end gp_approx_ == "full_scale_tapering" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia"
					else {//not gp_approx_ == "vecchia" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia"
						if (use_woodbury_identity_) {
							den_mat_t ZtX = Zt_[cluster_i] * X_cluster_i;
							if (matrix_inversion_method_ == "cholesky") {
								den_mat_t MInvSqrtZtX;
								if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
									MInvSqrtZtX = sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().inverse().matrix().asDiagonal() * ZtX;
								}
								else {
									TriangularSolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_facts_[cluster_i], ZtX, MInvSqrtZtX, false);
								}
								XT_psi_inv_X += (X_cluster_i).transpose() * X_cluster_i - MInvSqrtZtX.transpose() * MInvSqrtZtX;
							}//end cholesky
							else if (matrix_inversion_method_ == "iterative") {
								den_mat_t MInvZtX;
								//Use last solution as initial guess
								if (num_iter_ > 0 && optimizer_coef_ == "wls") {
									MInvZtX = last_MInvZtX_[cluster_i];
								}
								else {
									MInvZtX.resize(cum_num_rand_eff_[cluster_i][num_comps_total_], X_cluster_i.cols());
								}
								//Reduce max. number of iterations for the CG in first update
								int cg_max_num_it = cg_max_num_it_;
								if (first_update_) {
									cg_max_num_it = (int)round(cg_max_num_it_ / 3);
								}
								CGRandomEffectsMat(SigmaI_plus_ZtZ_rm_[cluster_i], ZtX, MInvZtX, NaN_found,
									cum_num_rand_eff_[cluster_i][num_comps_total_], (int)X_cluster_i.cols(),
									cg_max_num_it, cg_delta_conv_, cg_preconditioner_type_,
									L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i]);
								last_MInvZtX_[cluster_i] = MInvZtX;
								if (NaN_found) {
									Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
								}
								XT_psi_inv_X = X.transpose() * X - ZtX.transpose() * MInvZtX;
							}//end iterative
							else {
								Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
							}
						}
						else {
							den_mat_t MInvSqrtX;
							TriangularSolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_facts_[cluster_i], X_cluster_i, MInvSqrtX, false);
							XT_psi_inv_X += MInvSqrtX.transpose() * MInvSqrtX;
						}
					}
				}//end not gp_approx_ == "vecchia" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_vecchia"
			}//end more than one cluster
		}//end CalcXTPsiInvX

		/*!
		* \brief Initialize data structures for handling independent realizations of the Gaussian processes
		* \param num_data Number of data points
		* \param cluster_ids_data IDs / labels indicating independent realizations of Gaussian processes (same values = same process realization)
		* \param[out] num_data_per_cluster Keys: labels of independent clusters, values: number of data points per independent realization
		* \param[out] data_indices_per_cluster Keys: labels of independent clusters, values: vectors with indices for data points that belong to the every cluster
		* \param[out] unique_clusters Unique labels of independent realizations
		* \param[out] num_clusters Number of independent clusters
		*/
		void SetUpClusterIds(data_size_t num_data,
			const data_size_t* cluster_ids_data,
			std::map<data_size_t, int>& num_data_per_cluster,
			std::map<data_size_t, std::vector<int>>& data_indices_per_cluster,
			std::vector<data_size_t>& unique_clusters,
			data_size_t& num_clusters) {
			if (cluster_ids_data != nullptr) {
				for (int i = 0; i < num_data; ++i) {
					if (num_data_per_cluster.find(cluster_ids_data[i]) == num_data_per_cluster.end()) {//first occurrence of cluster_ids_data[i]
						unique_clusters.push_back(cluster_ids_data[i]);
						num_data_per_cluster.insert({ cluster_ids_data[i], 1 });
						std::vector<int> id;
						id.push_back(i);
						data_indices_per_cluster.insert({ cluster_ids_data[i], id });
					}
					else {
						num_data_per_cluster[cluster_ids_data[i]] += 1;
						data_indices_per_cluster[cluster_ids_data[i]].push_back(i);
					}
				}
				num_clusters = (data_size_t)unique_clusters.size();
			}
			else {
				unique_clusters.push_back(0);
				num_data_per_cluster.insert({ 0, num_data });
				num_clusters = 1;
				std::vector<int> gp_id_vec(num_data);
				for (int i = 0; i < num_data; ++i) {
					gp_id_vec[i] = i;
				}
				data_indices_per_cluster.insert({ 0, gp_id_vec });
			}
		}//end SetUpClusterIds

		/*!
		* \brief Convert characters in 'const char* re_group_data' to matrix (num_re_group x num_data) with strings of group labels
		* \param num_data Number of data points
		* \param num_re_group Number of grouped random effects
		* \param re_group_data Labels of group levels for the grouped random effects in column-major format (i.e. first the levels for the first effect, then for the second, etc.). Every group label needs to end with the null character '\0'
		* \param[out] Matrix of dimension num_re_group x num_data with strings of group labels for levels of grouped random effects
		*/
		void ConvertCharToStringGroupLevels(data_size_t num_data,
			data_size_t num_re_group,
			const char* re_group_data,
			std::vector<std::vector<re_group_t>>& re_group_levels) {
			int char_start = 0;
			for (int ire = 0; ire < num_re_group; ++ire) {
				for (int id = 0; id < num_data; ++id) {
					int number_chars = 0;
					while (re_group_data[char_start + number_chars] != '\0') {
						number_chars++;
					}
					re_group_levels[ire][id] = std::string(re_group_data + char_start);
					char_start += number_chars + 1;
				}
			}
		}

		/*!
		* \brief Initialize likelihoods
		* \param likelihood Likelihood name
		*/
		void InitializeLikelihoods(const string_t& likelihood) {
			string_t likelihood_parse = likelihood;
			if (vecchia_latent_approx_gaussian_ && likelihood == "gaussian") {
				likelihood_parse = "gaussian_use_likelihoods";
			}
			for (const auto& cluster_i : unique_clusters_) {
				if ((gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") && !grouped_RE_and_vecchia_GP_) {
					likelihood_[cluster_i] = std::unique_ptr<Likelihood<T_mat, T_chol>>(new Likelihood<T_mat, T_chol>(likelihood_parse,
						/* num_data = */ num_data_per_cluster_[cluster_i],
						/* num_re = */ re_comps_vecchia_[cluster_i][0][0]->GetNumUniqueREs(),
						/* has_SigmaI_mode = */ false,
						/* use_random_effects_indices_of_data = */ only_one_GP_calculations_on_RE_scale_,
						re_comps_vecchia_[cluster_i][0][0]->random_effects_indices_of_data_.data(),
						nullptr,
						likelihood_additional_param_,
						has_weights_, weights_[cluster_i].data(), likelihood_learning_rate_, false));
				}
				else if (gp_approx_ == "fitc") {
					likelihood_[cluster_i] = std::unique_ptr<Likelihood<T_mat, T_chol>>(new Likelihood<T_mat, T_chol>(likelihood_parse,
						/* num_data = */ num_data_per_cluster_[cluster_i],
						/* num_re = */ re_comps_cross_cov_[cluster_i][0][0]->GetNumUniqueREs(),
						/* has_SigmaI_mode = */ true,
						/* use_random_effects_indices_of_data = */ only_one_GP_calculations_on_RE_scale_,
						re_comps_cross_cov_[cluster_i][0][0]->random_effects_indices_of_data_.data(),
						nullptr,
						likelihood_additional_param_,
						has_weights_, weights_[cluster_i].data(), likelihood_learning_rate_, false));
				}
				else if ((use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) || grouped_RE_and_vecchia_GP_) {
					likelihood_[cluster_i] = std::unique_ptr<Likelihood<T_mat, T_chol>>(new Likelihood<T_mat, T_chol>(likelihood_parse,
						/* num_data = */ num_data_per_cluster_[cluster_i],
						/* num_re = */ cum_num_rand_eff_[cluster_i][num_comps_total_],
						/* has_SigmaI_mode = */ false,
						/* use_random_effects_indices_of_data = */ false,
						nullptr,
						&(Zt_[cluster_i]),
						likelihood_additional_param_,
						has_weights_, weights_[cluster_i].data(), likelihood_learning_rate_, false));
				}
				else if (only_one_grouped_RE_calculations_on_RE_scale_) {
					likelihood_[cluster_i] = std::unique_ptr<Likelihood<T_mat, T_chol>>(new Likelihood<T_mat, T_chol>(likelihood_parse,
						/* num_data = */ num_data_per_cluster_[cluster_i],
						/* num_re = */ re_comps_[cluster_i][0][0]->GetNumUniqueREs(),
						/* has_SigmaI_mode = */ false,
						/* use_random_effects_indices_of_data = */ true,
						re_comps_[cluster_i][0][0]->random_effects_indices_of_data_.data(),
						nullptr,
						likelihood_additional_param_,
						has_weights_, weights_[cluster_i].data(), likelihood_learning_rate_, true));
				}
				else if (only_one_GP_calculations_on_RE_scale_ && gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") {
					likelihood_[cluster_i] = std::unique_ptr<Likelihood<T_mat, T_chol>>(new Likelihood<T_mat, T_chol>(likelihood_parse,
						/* num_data = */ num_data_per_cluster_[cluster_i],
						/* num_re = */ re_comps_[cluster_i][0][0]->GetNumUniqueREs(),
						/* has_SigmaI_mode = */ true,
						/* use_random_effects_indices_of_data = */ true,
						re_comps_[cluster_i][0][0]->random_effects_indices_of_data_.data(),
						nullptr,
						likelihood_additional_param_,
						has_weights_, weights_[cluster_i].data(), likelihood_learning_rate_, false));
				}
				else {//!only_one_GP_calculations_on_RE_scale_ && gp_approx_ == "none"
					likelihood_[cluster_i] = std::unique_ptr<Likelihood<T_mat, T_chol>>(new Likelihood<T_mat, T_chol>(likelihood_parse,
						/* num_data = */ num_data_per_cluster_[cluster_i],
						/* num_re = */ num_data_per_cluster_[cluster_i],
						/* has_SigmaI_mode = */ true,
						/* use_random_effects_indices_of_data = */ false,
						nullptr,
						nullptr,
						likelihood_additional_param_,
						has_weights_, weights_[cluster_i].data(), likelihood_learning_rate_, false));
				}
				if (!gauss_likelihood_) {
					likelihood_[cluster_i]->InitializeModeAvec();
				}
			}
			num_sets_re_ = likelihood_[unique_clusters_[0]]->GetNumSetsRE();
			num_sets_fixed_effects_ = num_sets_re_;
			if (num_sets_re_ > 1) {
				if (!(gp_approx_ == "vecchia" && num_gp_ == 1 && num_comps_total_ == 1)) {
					Log::REFatal("likelihood = '%s' is currently only supported for GPs with a 'vecchia' approximation ", (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
				}
				CHECK(num_sets_re_ == 2);
				for (const auto& cluster_i : unique_clusters_) {
					if (gp_approx_ == "vecchia") {
						nearest_neighbors_[cluster_i][1] = nearest_neighbors_[cluster_i][0];
						dist_obs_neighbors_[cluster_i][1] = dist_obs_neighbors_[cluster_i][0];
						dist_between_neighbors_[cluster_i][1] = dist_between_neighbors_[cluster_i][0];
						entries_init_B_[cluster_i][1] = entries_init_B_[cluster_i][0];
						z_outer_z_obs_neighbors_[cluster_i][1] = z_outer_z_obs_neighbors_[cluster_i][0];
						for (const auto& ptr : re_comps_vecchia_[cluster_i][0]) {
							re_comps_vecchia_[cluster_i][1].push_back(std::make_shared<RECompGP<den_mat_t>>(*ptr));
						}
					}//end gp_approx_ == "vecchia"
					else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						for (const auto& ptr : re_comps_ip_[cluster_i][0]) {
							re_comps_ip_[cluster_i][1].push_back(std::make_shared<RECompGP<den_mat_t>>(*ptr));
						}
						for (const auto& ptr : re_comps_cross_cov_[cluster_i][0]) {
							re_comps_cross_cov_[cluster_i][1].push_back(std::make_shared<RECompGP<den_mat_t>>(*ptr));
						}
						for (const auto& ptr : re_comps_resid_[cluster_i][0]) {
							re_comps_resid_[cluster_i][1].push_back(std::make_shared<RECompGP<T_mat>>(*ptr));
						}
						if (gp_approx_ == "full_scale_vecchia") {
							nearest_neighbors_[cluster_i][1] = nearest_neighbors_[cluster_i][0];
							dist_obs_neighbors_[cluster_i][1] = dist_obs_neighbors_[cluster_i][0];
							dist_between_neighbors_[cluster_i][1] = dist_between_neighbors_[cluster_i][0];
							entries_init_B_[cluster_i][1] = entries_init_B_[cluster_i][0];
							z_outer_z_obs_neighbors_[cluster_i][1] = z_outer_z_obs_neighbors_[cluster_i][0];
							for (const auto& ptr : re_comps_vecchia_[cluster_i][0]) {
								re_comps_vecchia_[cluster_i][1].push_back(std::make_shared<RECompGP<den_mat_t>>(*ptr));
							}
						}
					}
					else {
						for (const auto& ptr : re_comps_[cluster_i][0]) {
							re_comps_[cluster_i][1].push_back(ptr->clone());
						}
					}
				}
			}
		}//end InitializeLikelihoods

		/*!
		* \brief Function that determines
		*		(i) the indices (in ind_par_) of the covariance parameters of every random effect component in the vector of all covariance parameter
		*		(ii) the total number of covariance parameters
		*/
		void DetermineCovarianceParameterIndicesNumCovPars() {
			// Determine ind_par_ and num_cov_par_
			ind_par_ = std::vector<data_size_t>();
			//First re_comp has either index 0 or 1 (the latter if there is an nugget effect for Gaussian data)
			if (gauss_likelihood_) {
				num_cov_par_ = 1;
				ind_par_.push_back(1);
			}
			else {
				num_cov_par_ = 0;
				ind_par_.push_back(0);
			}
			//Add indices of parameters of individual components in joint parameter vector
			if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
				for (int j = 0; j < (int)re_comps_ip_[unique_clusters_[0]][0].size(); ++j) {
					ind_par_.push_back(ind_par_.back() + re_comps_ip_[unique_clusters_[0]][0][j]->NumCovPar());//end points of parameter indices of components
					num_cov_par_ += re_comps_ip_[unique_clusters_[0]][0][j]->NumCovPar();
				}
			}
			else if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
				for (int j = 0; j < (int)re_comps_vecchia_[unique_clusters_[0]][0].size(); ++j) {
					ind_par_.push_back(ind_par_.back() + re_comps_vecchia_[unique_clusters_[0]][0][j]->NumCovPar());//end points of parameter indices of components
					num_cov_par_ += re_comps_vecchia_[unique_clusters_[0]][0][j]->NumCovPar();
				}
			}
			else {
				for (int j = 0; j < (int)re_comps_[unique_clusters_[0]][0].size(); ++j) {
					ind_par_.push_back(ind_par_.back() + re_comps_[unique_clusters_[0]][0][j]->NumCovPar());//end points of parameter indices of components
					num_cov_par_ += re_comps_[unique_clusters_[0]][0][j]->NumCovPar();
				}
				if (grouped_RE_and_vecchia_GP_) {
					ind_par_.push_back(ind_par_.back() + re_comps_vecchia_[unique_clusters_[0]][0][0]->NumCovPar());//end points of parameter indices of components
					num_cov_par_ += re_comps_vecchia_[unique_clusters_[0]][0][0]->NumCovPar();
				}
			}
			num_cov_par_per_set_re_ = num_cov_par_;
			if (num_sets_re_ > 1) {
				num_cov_par_ *= num_sets_re_;
			}
			CHECK(ind_par_.size() == num_comps_total_ + 1);
		}//end DetermineCovarianceParameterIndicesNumCovPars

		/*!
		* \brief Function that determines whether to use special options for estimation and prediction for certain special cases of random effects models
		*/
		void DetermineSpecialCasesModelsEstimationPrediction(std::string& cov_fct) {
			chol_fact_pattern_analyzed_ = false;
			// Decide whether to use the Woodbury identity (i.e. do matrix inversion on the b scale and not the Zb scale) for grouped random effects models only
			if (num_re_group_total_ > 0 && num_gp_total_ == 0) {
				use_woodbury_identity_ = true;//Faster to use Woodbury identity since the dimension of the random effects is typically much smaller than the number of data points
				//Note: the use of the Woodburry identity is currently only implemented for grouped random effects (which is also the only use of it). 
				//		If this should be applied to GPs in the future, adaptions need to be made e.g. in the calculations of the gradient (see y_tilde2_)
			}
			else {
				use_woodbury_identity_ = false;
			}
			// Define options for faster calculations for special cases of RE models (these options depend on the type of likelihood)
			only_one_GP_calculations_on_RE_scale_ = (num_gp_total_ == 1 && num_comps_total_ == 1 && !gauss_likelihood_ && gp_approx_ == "none") || Vecchia_calculations_on_RE_scale_;//If there is only one GP, we do calculations on the b-scale instead of Zb-scale (only for non-Gaussian likelihoods)
			only_one_grouped_RE_calculations_on_RE_scale_ = num_re_group_total_ == 1 && num_comps_total_ == 1 && !gauss_likelihood_;//If there is only one grouped RE, we do (all) calculations on the b-scale instead of the Zb-scale (this flag is only used for non-Gaussian likelihoods)
			only_one_grouped_RE_calculations_on_RE_scale_for_prediction_ = num_re_group_total_ == 1 && num_comps_total_ == 1 && gauss_likelihood_;//If there is only one grouped RE, we do calculations for prediction on the b-scale instead of the Zb-scale (this flag is only used for Gaussian likelihoods)
			if (num_gp_total_ == 1 && num_comps_total_ == 1) {
				if (cov_fct == "linear" && gp_approx_ == "none") {
					use_woodbury_identity_ = true;
					linear_kernel_use_woodbury_identity_ = true;
					only_one_GP_calculations_on_RE_scale_ = false;
				}
				else if (cov_fct == "linear_no_woodbury") {
					cov_fct = "linear";
				}
			}
			if (num_re_group_total_ > 0 && num_gp_total_ > 0 && gp_approx_ == "vecchia") {
				grouped_RE_and_vecchia_GP_ = true;
				gauss_likelihood_ = false;
			}
		}//end DetermineSpecialCasesModelsEstimationPrediction

		void SetDefaultMatrixInversionMethod() {
			if (matrix_inversion_method_user_provided_ == "default") {
				if (UseIterativeByDefault()) {
					CHECK(CanUseIterative());
					matrix_inversion_method_ = "iterative";
				}
				else {
					matrix_inversion_method_ = "cholesky";
				}
			}
			else {
				matrix_inversion_method_ = matrix_inversion_method_user_provided_;
			}
		}//end SetDefaultMatrixInversionMethod

		/*!
		* \brief Function that set default values for several parameters if they were not initialized
		*/
		void InitializeDefaultSettings() {
			if (!vecchia_pred_type_has_been_set_) {
				if (gauss_likelihood_) {
					vecchia_pred_type_ = "order_obs_first_cond_obs_only";
				}
				else {
					vecchia_pred_type_ = "latent_order_obs_first_cond_obs_only";
				}
			}
			if (!set_optim_config_has_been_called_ && NumAuxPars() > 0) {
				if (!gauss_likelihood_) {
					estimate_aux_pars_ = true;;
				}
				else {
					estimate_aux_pars_ = false;
				}
			}
			if (!cg_preconditioner_type_has_been_set_) {
				if (use_woodbury_identity_ && ((num_re_group_total_ > 1) || linear_kernel_use_woodbury_identity_) || grouped_RE_and_vecchia_GP_) {
					cg_preconditioner_type_ = "ssor";
				}
				else if (gauss_likelihood_ && gp_approx_ == "full_scale_tapering") {
					cg_preconditioner_type_ = "fitc";
				}
				else if (!gauss_likelihood_ && gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
					cg_preconditioner_type_ = "vadu";
				}
				else if (!gauss_likelihood_ && gp_approx_ == "full_scale_vecchia") {
					cg_preconditioner_type_ = "fitc";
				}
				CheckPreconditionerType();
			}
			if (!fitc_piv_chol_preconditioner_rank_has_been_set_) {
				if (cg_preconditioner_type_ == "fitc") {
					fitc_piv_chol_preconditioner_rank_ = default_fitc_preconditioner_rank_;
				} else if (cg_preconditioner_type_ == "pivoted_cholesky") {
					fitc_piv_chol_preconditioner_rank_ = default_piv_chol_preconditioner_rank_;
				}
			}
			if (!nsim_var_pred_has_been_set_) {
				if (use_woodbury_identity_ && ((num_re_group_total_ > 1) || linear_kernel_use_woodbury_identity_) || grouped_RE_and_vecchia_GP_) {
					nsim_var_pred_ = 500;
				}
				else if (gauss_likelihood_ && gp_approx_ == "full_scale_tapering") {
					nsim_var_pred_ = 1000;
				}
				else if (!gauss_likelihood_ && gp_approx_ == "vecchia") {
					nsim_var_pred_ = 1000;
				}
				else if (!gauss_likelihood_ && gp_approx_ == "full_scale_vecchia") {
					nsim_var_pred_ = 100;
				}
			}
			if (!estimate_cov_par_index_has_been_set_) {
				estimate_cov_par_index_ = std::vector<int>(num_cov_par_, 1);
			}
		}//end InitializeDefaultSettings

		/*!
		* \brief Initialize matrices Z_ and related quantities used, e.g. when use_woodbury_identity_==true
		*/
		void InitializeMatricesForUseWoodburyIdentity() {
			CHECK(use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_ || grouped_RE_and_vecchia_GP_);
			Zt_ = std::map<data_size_t, sp_mat_t>();
			cum_num_rand_eff_ = std::map<data_size_t, std::vector<data_size_t>>();
			if (gauss_likelihood_) {
				Zj_square_sum_ = std::map<data_size_t, std::vector<double>>();
				ZtZ_ = std::map<data_size_t, sp_mat_t>();
				ZtZj_ = std::map<data_size_t, std::vector<sp_mat_t>>();
			}
			for (const auto& cluster_i : unique_clusters_) {
				std::vector<data_size_t> cum_num_rand_eff_cluster_i(num_comps_total_ + 1);
				cum_num_rand_eff_cluster_i[0] = 0;
				if (linear_kernel_use_woodbury_identity_) {
					std::shared_ptr<RECompGP<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][ind_intercept_gp_]);
					den_mat_t coords = re_comp->GetCoords();
					Zt_.insert({ cluster_i, coords.transpose().sparseView()});
					cum_num_rand_eff_cluster_i[1] = re_comp->GetDimCoords();
					cum_num_rand_eff_.insert({ cluster_i, cum_num_rand_eff_cluster_i });
					if (gauss_likelihood_) {
						ZtZ_.insert({ cluster_i, (coords.transpose() * coords).sparseView() });
						std::vector<double> Zj_square_sum_cluster_i = { coords.squaredNorm() };
						Zj_square_sum_.insert({ cluster_i, Zj_square_sum_cluster_i });
						std::vector<sp_mat_t> ZtZj_cluster_i = { ZtZ_[cluster_i] };
						ZtZj_.insert({ cluster_i, ZtZj_cluster_i });
					}
				}//end linear_kernel_use_woodbury_identity_
				else {// only grouped REs
					//Determine number of rows and non-zero entries of Z
					int non_zeros = 0;
					int ncols = 0;
					for (int j = 0; j < num_re_group_total_; ++j) {
						sp_mat_t* Z_j = re_comps_[cluster_i][0][j]->GetZ();
						ncols += (int)Z_j->cols();
						non_zeros += (int)Z_j->nonZeros();
						cum_num_rand_eff_cluster_i[j + 1] = ncols;
					}
					if (grouped_RE_and_vecchia_GP_) {
						CHECK(num_sets_re_ == 1);
						ncols += re_comps_vecchia_[cluster_i][0][0]->GetNumUniqueREs();
						non_zeros += re_comps_vecchia_[cluster_i][0][0]->NumData();
						cum_num_rand_eff_cluster_i[num_re_group_total_ + 1] = ncols;
					}
					cum_num_rand_eff_.insert({ cluster_i, cum_num_rand_eff_cluster_i });
					//Create matrix Z and calculate sum(Z_j^2) = trace(Z_j^T * Z_j)
					std::vector<Triplet_t> triplets;
					triplets.reserve(non_zeros);
					std::vector<double> Zj_square_sum_cluster_i(num_re_group_total_);
					int ncol_prev = 0;
					for (int j = 0; j < num_re_group_total_; ++j) {
						sp_mat_t* Z_j = re_comps_[cluster_i][0][j]->GetZ();
						for (int k = 0; k < Z_j->outerSize(); ++k) {
							for (sp_mat_t::InnerIterator it(*Z_j, k); it; ++it) {
								triplets.emplace_back(it.row(), ncol_prev + it.col(), it.value());
							}
						}
						ncol_prev += (int)Z_j->cols();
						if (gauss_likelihood_) Zj_square_sum_cluster_i[j] = Z_j->squaredNorm();
					}
					if (grouped_RE_and_vecchia_GP_) {
						CHECK(num_sets_re_ == 1);
						CHECK((re_comps_vecchia_[cluster_i][0][0]->random_effects_indices_of_data_).size() == re_comps_vecchia_[cluster_i][0][0]->NumData());
						for (int i = 0; i < re_comps_vecchia_[cluster_i][0][0]->NumData(); ++i) {
							triplets.emplace_back(i, ncol_prev + (re_comps_vecchia_[cluster_i][0][0]->random_effects_indices_of_data_)[i], 1.);
						}						
						ncol_prev += re_comps_vecchia_[cluster_i][0][0]->GetNumUniqueREs();
					}
					sp_mat_t Z_cluster_i(num_data_per_cluster_[cluster_i], ncols);
					Z_cluster_i.setFromTriplets(triplets.begin(), triplets.end());
					sp_mat_t Zt_cluster_i = Z_cluster_i.transpose();
					Zt_.insert({ cluster_i, Zt_cluster_i });
					if (gauss_likelihood_) {
						Zj_square_sum_.insert({ cluster_i, Zj_square_sum_cluster_i });
						sp_mat_t ZtZ_cluster_i = Zt_cluster_i * Z_cluster_i;
						ZtZ_.insert({ cluster_i, ZtZ_cluster_i });
						//Calculate Z^T * Z_j
						std::vector<sp_mat_t> ZtZj_cluster_i(num_comps_total_);
						for (int j = 0; j < num_re_group_total_; ++j) {
							sp_mat_t* Z_j = re_comps_[cluster_i][0][j]->GetZ();
							ZtZj_cluster_i[j] = Zt_cluster_i * (*Z_j);
						}
						ZtZj_.insert({ cluster_i, ZtZj_cluster_i });						
					}//end gauss_likelihood_
				}//end !linear_kernel_use_woodbury_identity_
			}
		}//end InitializeMatricesForUseWoodburyIdentity

		/*!
		* \brief Initialize identity matrices required for Gaussian data
		*/
		void InitializeIdentityMatricesForGaussianData() {
			if (gauss_likelihood_ && gp_approx_ != "vecchia" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" && gp_approx_ != "full_scale_vecchia") {
				for (const auto& cluster_i : unique_clusters_) {
					ConstructI(cluster_i);//Idendity matrices needed for computing inverses of covariance matrices used in gradient descent for Gaussian data
				}
			}
		}

		/*!
		* \brief Function that checks the compatibility of the chosen special options for estimation and prediction for certain special cases of random effects models
		*/
		void CheckCompatibilitySpecialOptions() {
			if (SUPPORTED_MATRIX_INVERSION_METHODS_.find(matrix_inversion_method_) == SUPPORTED_MATRIX_INVERSION_METHODS_.end()) {
				Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
			}
			if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
				CHECK(num_ind_points_ > 0);
			}
			if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
				CHECK(num_neighbors_ > 0);
			}
			//Some checks
			if (only_one_GP_calculations_on_RE_scale_ && use_woodbury_identity_) {
				Log::REFatal("Cannot set both 'only_one_GP_calculations_on_RE_scale_' and 'use_woodbury_identity_' to 'true'");
			}
			if (only_one_GP_calculations_on_RE_scale_ && only_one_grouped_RE_calculations_on_RE_scale_) {
				Log::REFatal("Cannot set both 'only_one_GP_calculations_on_RE_scale_' and 'only_one_grouped_RE_calculations_on_RE_scale_' to 'true'");
			}
			if (gp_approx_ != "none" && gp_approx_ != "vecchia" && num_re_group_total_ > 0) {
				Log::REFatal("The GP approximation '%s' can currently not be used when there are grouped random effects ", gp_approx_.c_str());
			}
			if (only_one_GP_calculations_on_RE_scale_) {//only_one_GP_calculations_on_RE_scale_
				if (gauss_likelihood_) {
					Log::REFatal("Option 'only_one_GP_calculations_on_RE_scale_' is currently not implemented for Gaussian data");
				}
				if (gp_approx_ != "vecchia" && gp_approx_ != "fitc" && gp_approx_ != "none") {
					Log::REFatal("Option 'only_one_GP_calculations_on_RE_scale_' is currently not implemented for the approximation '%s' ", gp_approx_.c_str());
				}
				CHECK(num_gp_total_ == 1);
				CHECK(num_comps_total_ == 1);
				CHECK(num_re_group_total_ == 0);
			}
			if (only_one_grouped_RE_calculations_on_RE_scale_) {//only_one_grouped_RE_calculations_on_RE_scale_
				if (gauss_likelihood_) {
					Log::REFatal("Option 'only_one_grouped_RE_calculations_on_RE_scale_' is currently not implemented for Gaussian data");
				}
				CHECK(gp_approx_ == "none");
				CHECK(num_gp_total_ == 0);
				CHECK(num_comps_total_ == 1);
				CHECK(num_re_group_total_ == 1);
			}
			if (only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {//only_one_grouped_RE_calculations_on_RE_scale_for_prediction_
				CHECK(gp_approx_ == "none");
				CHECK(num_gp_total_ == 0);
				CHECK(num_comps_total_ == 1);
				CHECK(num_re_group_total_ == 1);
				if (!gauss_likelihood_) {
					Log::REFatal("Option 'only_one_grouped_RE_calculations_on_RE_scale_for_prediction_' is currently only effective for Gaussian data");
				}
			}
			if (use_woodbury_identity_) {//use_woodbury_identity_
				if (gauss_likelihood_ && only_one_grouped_RE_calculations_on_RE_scale_) {
					Log::REFatal("Cannot enable 'only_one_grouped_RE_calculations_on_RE_scale_' if 'use_woodbury_identity_' is enabled for Gaussian data");
				}
				if (linear_kernel_use_woodbury_identity_) {
					CHECK(num_gp_total_ == 1);
					CHECK(num_comps_total_ == num_gp_total_);
					CHECK(num_re_group_total_ == 0);
				}
				else {
					CHECK(num_gp_total_ == 0);
					CHECK(num_comps_total_ == num_re_group_total_);
				}
			}
			if (linear_kernel_use_woodbury_identity_) {
				CHECK(num_gp_total_ == 1);
			}
			if (gp_approx_ == "full_scale_tapering" && !gauss_likelihood_) {
				Log::REFatal("Approximation '%s' is currently not supported for non-Gaussian likelihoods ", gp_approx_.c_str());
			}
			if (matrix_inversion_method_ == "iterative") {
				if (!CanUseIterative()) {
					if (use_woodbury_identity_ && num_re_group_total_ == 1) {
						Log::REFatal("Cannot use matrix_inversion_method = 'iterative' if there is only a single-level grouped random effects. "
							"Use matrix_inversion_method = 'cholesky' instead (this is very fast). Iterative methods are for multiple grouped random effects ");
					}
					else {
						Log::REFatal("Cannot use matrix_inversion_method = 'iterative' if gp_approx = '%s' and likelihood = '%s'. Use matrix_inversion_method = 'cholesky' instead ",
							gp_approx_.c_str(), (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
					}
				}
			}
			CHECK((int)estimate_cov_par_index_.size() == num_cov_par_);
			if (grouped_RE_and_vecchia_GP_) {
				CHECK(num_gp_total_ == 1);//currently only for one GP implemented
				CHECK(num_re_group_total_ > 0);
				CHECK(!use_woodbury_identity_);
				CHECK(gp_approx_ == "vecchia");
				CHECK(!only_one_GP_calculations_on_RE_scale_);
				CHECK(!only_one_grouped_RE_calculations_on_RE_scale_);
				CHECK(!gauss_likelihood_);
			}
		}//end CheckCompatibilitySpecialOptions

		bool CanUseIterative() const {
			bool can_use_iter = ((gp_approx_ == "full_scale_vecchia" || gp_approx_ == "vecchia") && !gauss_likelihood_) ||
				(gp_approx_ == "full_scale_tapering" && gauss_likelihood_) ||
				(use_woodbury_identity_ && (num_re_group_total_ > 1 || linear_kernel_use_woodbury_identity_));
			return can_use_iter;
		}

		bool UseIterativeByDefault() {
			bool use_iter = ((gp_approx_ == "full_scale_vecchia" || gp_approx_ == "vecchia") && !gauss_likelihood_) ||
				(gp_approx_ == "full_scale_tapering" && gauss_likelihood_ && !gp_approx_changed_to_FSA_from_FITC_) ||
				(use_woodbury_identity_ && num_re_group_total_ > 1);// note: do not use iterative method for a linear kernel (linear_kernel_use_woodbury_identity_) by default as is can be slower for low-dimensional problems
			if (likelihood_.size() > 0) {
				if (likelihood_[unique_clusters_[0]]->GetLikelihood() == "asymmetric_laplace" && likelihood_[unique_clusters_[0]]->UseFisherForModeFinding()) {
					use_iter = false;// note: iterative methods are slower than Cholesky since mode finding often requires many iterations and the Fisher information is constant and the Cholesky decompostion has to be calculated only once
				}
			}
			if (use_iter) {
				CHECK(CanUseIterative());
			}
			return use_iter;
		}

		/*! \brief Check whether preconditioner is supported */
		void CheckPreconditionerType() {
			if (matrix_inversion_method_ == "iterative") {
				if (use_woodbury_identity_ && (num_re_group_total_ > 1 || linear_kernel_use_woodbury_identity_)) {
					if (SUPPORTED_PRECONDITIONERS_GROUPED_RE_.find(cg_preconditioner_type_) == SUPPORTED_PRECONDITIONERS_GROUPED_RE_.end()) {
						Log::REFatal("Preconditioner type '%s' is not supported for grouped random effects ",
							cg_preconditioner_type_.c_str(), gp_approx_.c_str(), (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
					}
				}
				else if (gauss_likelihood_ && gp_approx_ == "full_scale_tapering") {
					if (SUPPORTED_PRECONDITIONERS_GAUSS_FSA_.find(cg_preconditioner_type_) == SUPPORTED_PRECONDITIONERS_GAUSS_FSA_.end()) {
						Log::REFatal("Preconditioner type '%s' is not supported for gp_approx = '%s' and likelihood = '%s' ",
							cg_preconditioner_type_.c_str(), gp_approx_.c_str(), (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
					}
				}
				else if (!gauss_likelihood_ && gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
					if (SUPPORTED_PRECONDITIONERS_NONGAUSS_VECCHIA_.find(cg_preconditioner_type_) == SUPPORTED_PRECONDITIONERS_NONGAUSS_VECCHIA_.end()) {
						Log::REFatal("Preconditioner type '%s' is not supported for gp_approx = '%s' and likelihood = '%s' ",
							cg_preconditioner_type_.c_str(), gp_approx_.c_str(), (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
					}
				}
				else if (!gauss_likelihood_ && gp_approx_ == "full_scale_vecchia") {
					if (SUPPORTED_PRECONDITIONERS_NONGAUSS_VIF_.find(cg_preconditioner_type_) == SUPPORTED_PRECONDITIONERS_NONGAUSS_VIF_.end()) {
						Log::REFatal("Preconditioner type '%s' is not supported for gp_approx = '%s' (VIF approximation) and likelihood = '%s' ",
							cg_preconditioner_type_.c_str(), gp_approx_.c_str(), (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
					}
				}
				else if (grouped_RE_and_vecchia_GP_) {
					if (SUPPORTED_PRECONDITIONERS_VECCHIA_GROUPED_RE_.find(cg_preconditioner_type_) == SUPPORTED_PRECONDITIONERS_VECCHIA_GROUPED_RE_.end()) {
						Log::REFatal("Preconditioner type '%s' is not supported for gp_approx = '%s' in combination with grouped random effects ",
							cg_preconditioner_type_.c_str(), gp_approx_.c_str());
					}
				}
			}
		}//end CheckPreconditionerType

		static string_t ParsePreconditionerAlias(const string_t& type) {
			if (type == "VADU" || type == "vecchia_approximation_with_diagonal_update" || type == "Sigma_inv_plus_BtWB") {
				return "vadu";
			}
			else if (type == "VIFDU" || type == "Bt_Sigma_inv_plus_W_B") {
				return "vifdu";
			}
			else if (type == "piv_chol" || type == "piv_chol_on_Sigma") {
				return "pivoted_cholesky";
			}
			else if (type == "ZIRC" || type == "zirc" || type == "ZIC" || type == "zic" ||
				type == "zero_infill_incomplete_cholesky" ||
				type == "zero_fillin_incomplete_cholesky" || type == "zero_fill_in_incomplete_cholesky" ||
				type == "zero_fill-in_incomplete_cholesky" || type == "zero_infill_incomplete_cholesky" ||
				type == "zero_fillin_incomplete_reverse_cholesky" || type == "zero_fill_in_incomplete_reverse_cholesky" ||
				type == "zero_fill-in_incomplete_reverse_cholesky" || type == "zero_infill_incomplete_reverse_cholesky") {
				return "incomplete_cholesky";
			}
			else if (type == "SSOR"|| type == "symmetric_successive_over_relaxation") {
				return "ssor";
			}
			else if (type == "FITC" || type == "predictive_process_plus_diagonal") {
				return "fitc";
			}
			else if (type == "diagonal" || type == "diag" || type == "Diagonal" || type == "Diag") {
				return "diagonal";
			}
			else if (type == "vecchia_observable" || type == "vecchia") {
				return "vecchia_response";
			}
			return type;
		}

		/*! \brief Set properties for likelihoods.h (matrix inversion properties, choices for iterative methods, etc.) */
		void SetPropertiesLikelihood() {
			if (!gauss_likelihood_) {
				for (const auto& cluster_i : unique_clusters_) {
					likelihood_[cluster_i]->SetPropertiesLikelihood(matrix_inversion_method_,
						cg_max_num_it_, cg_max_num_it_tridiag_, cg_delta_conv_, cg_delta_conv_pred_,
						num_rand_vec_trace_, reuse_rand_vec_trace_, seed_rand_vec_trace_,
						cg_preconditioner_type_, fitc_piv_chol_preconditioner_rank_, rank_pred_approx_matrix_lanczos_, nsim_var_pred_,
						delta_conv_mode_finding_);
				}
			}
		}//end SetPropertiesLikelihood

		/*!
		* \brief Initialize individual component models and collect them in a containter
		* \param num_data Number of data points
		* \param data_indices_per_cluster Keys: Labels of independent realizations of REs/GPs, values: vectors with indices for data points
		* \param cluster_i Index / label of the realization of the Gaussian process for which the components should be constructed
		* \param Group levels for every grouped random effect
		* \param num_data_per_cluster Keys: Labels of independent realizations of REs/GPs, values: number of data points per independent realization
		* \param re_group_rand_coef_data Covariate data for grouped random coefficients
		* \param gp_coords_data Coordinates (features) for Gaussian process
		* \param gp_rand_coef_data Covariate data for Gaussian process random coefficients
		* \param calculateZZt If true, the matrix Z*Z^T is calculated for grouped random effects and saved (usually not needed if Woodbury identity is used)
		* \param cov_fct Type of covariance function
		* \param cov_fct_shape Shape parameter of covariance function (=smoothness parameter for Matern and Wendland covariance. This parameter is irrelevant for some covariance functions such as the exponential or Gaussian
		* \param cov_fct_taper_range Range parameter of the Wendland covariance function and Wendland correlation taper function. We follow the notation of Bevilacqua et al. (2019, AOS)
		* \param cov_fct_taper_shape Shape parameter of the Wendland covariance function and Wendland correlation taper function. We follow the notation of Bevilacqua et al. (2019, AOS)
		* \param[out] re_comps_cluster_i Container that collects the individual component models
		*/
		void CreateREComponents(data_size_t num_data,
			std::map<data_size_t, std::vector<int>>& data_indices_per_cluster,
			data_size_t cluster_i,
			std::vector<std::vector<re_group_t>>& re_group_levels,
			std::map<data_size_t, int>& num_data_per_cluster,
			const double* re_group_rand_coef_data,
			const double* gp_coords_data,
			const double* gp_rand_coef_data,
			bool calculateZZt,
			string_t cov_fct,
			double cov_fct_shape,
			double cov_fct_taper_range,
			double cov_fct_taper_shape,
			std::vector<std::shared_ptr<RECompBase<T_mat>>>& re_comps_cluster_i) {
			//Grouped random effects
			if (num_re_group_ > 0) {
				for (int j = 0; j < num_re_group_; ++j) {
					std::vector<re_group_t> group_data;
					for (const auto& id : data_indices_per_cluster[cluster_i]) {
						group_data.push_back(re_group_levels[j][id]);
					}
					re_comps_cluster_i.push_back(std::shared_ptr<RECompGroup<T_mat>>(new RECompGroup<T_mat>(group_data, calculateZZt,
						!only_one_grouped_RE_calculations_on_RE_scale_)));
				}
				//Random slope grouped random effects
				if (num_re_group_rand_coef_ > 0) {
					for (int j = 0; j < num_re_group_rand_coef_; ++j) {
						std::vector<double> rand_coef_data;
						for (const auto& id : data_indices_per_cluster[cluster_i]) {
							rand_coef_data.push_back(re_group_rand_coef_data[j * num_data + id]);
						}
						std::shared_ptr<RECompGroup<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGroup<T_mat>>(re_comps_cluster_i[ind_effect_group_rand_coef_[j] - 1]);//Subtract -1 since ind_effect_group_rand_coef[j] starts counting at 1 not 0
						re_comps_cluster_i.push_back(std::shared_ptr<RECompGroup<T_mat>>(new RECompGroup<T_mat>(
							re_comp->random_effects_indices_of_data_.data(), re_comp->num_data_, re_comp->map_group_label_index_,
							re_comp->num_group_, rand_coef_data, calculateZZt)));
					}
					// drop some intercept random effects (if specified)
					int num_droped = 0;
					for (int j = 0; j < num_re_group_; ++j) {
						if (drop_intercept_group_rand_effect_[j]) {
							re_comps_cluster_i.erase(re_comps_cluster_i.begin() + j);
							num_droped += 1;
						}
					}
					num_re_group_ -= num_droped;
					num_re_group_total_ -= num_droped;
					num_comps_total_ -= num_droped;
				}
			}//end num_re_group_ > 0
			//GPs
			if (num_gp_ > 0) {
				std::vector<double> gp_coords;
				for (int j = 0; j < dim_gp_coords_; ++j) {
					for (const auto& id : data_indices_per_cluster[cluster_i]) {
						gp_coords.push_back(gp_coords_data[j * num_data + id]);
					}
				}
				den_mat_t gp_coords_mat = Eigen::Map<den_mat_t>(gp_coords.data(), num_data_per_cluster[cluster_i], dim_gp_coords_);
				bool use_Z_for_duplicates = (gp_approx_ == "none");
				re_comps_cluster_i.push_back(std::shared_ptr<RECompGP<T_mat>>(new RECompGP<T_mat>(
					gp_coords_mat, cov_fct, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape,
					gp_approx_ == "tapering", false, true, use_Z_for_duplicates, only_one_GP_calculations_on_RE_scale_, true)));
				//Random slope GPs
				if (num_gp_rand_coef_ > 0) {
					for (int j = 0; j < num_gp_rand_coef_; ++j) {
						std::vector<double> rand_coef_data;
						for (const auto& id : data_indices_per_cluster[cluster_i]) {
							rand_coef_data.push_back(gp_rand_coef_data[j * num_data + id]);
						}
						std::shared_ptr<RECompGP<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_cluster_i[ind_intercept_gp_]);
						re_comps_cluster_i.push_back(std::shared_ptr<RECompGP<T_mat>>(new RECompGP<T_mat>(
							re_comp->dist_, re_comp->has_Z_, &re_comp->Z_, rand_coef_data,
							cov_fct, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape,
							re_comp->GetTaperMu(), gp_approx_ == "tapering", false, dim_gp_coords_)));
					}
				}
			}//end num_gp_ > 0)
		}//end CreateREComponents

		/*!
		* \brief Initialize individual component models and collect them in a containter
		* \param num_data Number of data points
		* \param data_indices_per_cluster Keys: Labels of independent realizations of REs/GPs, values: vectors with indices for data points
		* \param cluster_i Index / label of the realization of the Gaussian process for which the components should be constructed
		* \param gp_coords_data Coordinates (features) for Gaussian process
		* \param cov_fct Type of covariance function
		* \param cov_fct_shape Shape parameter of covariance function (=smoothness parameter for Matern and Wendland covariance. This parameter is irrelevant for some covariance functions such as the exponential or Gaussian
		* \param cov_fct_taper_range Range parameter of the Wendland covariance function and Wendland correlation taper function. We follow the notation of Bevilacqua et al. (2019, AOS)
		* \param cov_fct_taper_shape Shape parameter of the Wendland covariance function and Wendland correlation taper function. We follow the notation of Bevilacqua et al. (2019, AOS)
		* \param[out] re_comps_ip_cluster_i Inducing point GP for predictive process
		* \param[out] re_comps_cross_cov_cluster_i Cross-covariance GP for predictive process
		* \param[out] re_comps_resid_cluster_i Residual GP component for full scale approximation
		* \param for_prediction_new_cluster
		*/
		void CreateREComponentsFITC_FSA(data_size_t num_data,
			std::map<data_size_t, std::vector<int>>& data_indices_per_cluster,
			data_size_t cluster_i,
			const double* gp_coords_data,
			string_t cov_fct,
			double cov_fct_shape,
			double cov_fct_taper_range,
			double cov_fct_taper_shape,
			std::vector<std::shared_ptr<RECompGP<den_mat_t>>>& re_comps_ip_cluster_i,
			std::vector<std::shared_ptr<RECompGP<den_mat_t>>>& re_comps_cross_cov_cluster_i,
			std::vector<std::shared_ptr<RECompGP<T_mat>>>& re_comps_resid_cluster_i,
			bool for_prediction_new_cluster) {
			int num_ind_points = num_ind_points_;
			if (for_prediction_new_cluster) {
				num_ind_points = std::min(num_ind_points_, num_data_per_cluster_[cluster_i]);
			}
			if (gp_approx_ == "fitc") {
				if (num_data_per_cluster_[cluster_i] < num_ind_points) {
					Log::REFatal("Cannot have more inducing points than data points for '%s' approximation ", gp_approx_.c_str());
				}
			}
			else if (gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
				if (num_data_per_cluster_[cluster_i] <= num_ind_points) {
					Log::REFatal("Need to have less inducing points (currently num_ind_points = %d) than data points (%d) if gp_approx = '%s' ",
						num_ind_points, num_data_per_cluster_[cluster_i], gp_approx_.c_str());
				}
			}
			CHECK(num_gp_ > 0);
			std::vector<double> gp_coords_all;
			for (int j = 0; j < dim_gp_coords_; ++j) {
				for (const auto& id : data_indices_per_cluster[cluster_i]) {
					gp_coords_all.push_back(gp_coords_data[j * num_data + id]);
				}
			}
			den_mat_t gp_coords_all_mat = Eigen::Map<den_mat_t>(gp_coords_all.data(), num_data_per_cluster_[cluster_i], dim_gp_coords_);
			// Determine inducing points on unique locataions
			den_mat_t gp_coords_all_unique;
			std::vector<int> uniques;//unique points
			std::vector<int> unique_idx;//not used
			DetermineUniqueDuplicateCoordsFast(gp_coords_all_mat, num_data_per_cluster_[cluster_i], uniques, unique_idx);
			if ((data_size_t)uniques.size() == num_data_per_cluster_[cluster_i]) {//no multiple observations at the same locations -> no incidence matrix needed
				gp_coords_all_unique = gp_coords_all_mat;
			}
			else {//there are multiple observations at the same locations
				if (gp_approx_ == "fitc" && gauss_likelihood_) {
					Log::REWarning("There are duplicate coordinates. Currently, this is not well handled when 'gp_approx = fitc' and 'likelihood = gaussian'. "
						"For this reason, 'gp_approx' is internally changed to 'full_scale_tapering' with a very small taper range. "
						"Note that this is just a technical trick that results in an equivalent model and you don't need to do something ");
					gp_approx_changed_to_FSA_from_FITC_ = true;
					gp_approx_ = "full_scale_tapering";
					cov_fct_taper_range = 1e-8;
				}
				gp_coords_all_unique = gp_coords_all_mat(uniques, Eigen::all);
				if ((int)gp_coords_all_unique.rows() < num_ind_points) {
					Log::REFatal("Cannot have more inducing points than unique coordinates for '%s' approximation ", gp_approx_.c_str());
				}
			}
			std::vector<int> indices;
			den_mat_t gp_coords_ip_mat;
			if (ind_points_selection_ == "cover_tree") {
				Log::REDebug("Starting cover tree algorithm for determining inducing points ");
				CoverTree(gp_coords_all_unique, cover_tree_radius_, rng_, gp_coords_ip_mat);
				Log::REDebug("Inducing points have been determined ");
				num_ind_points = (int)gp_coords_ip_mat.rows();
			}
			else if (ind_points_selection_ == "random") {
				if (gp_approx_ == "full_scale_vecchia" && !gauss_likelihood_) {
					Log::REFatal("Method '%s' is not supported for finding inducing points in the full-scale-vecchia approximation for non-Gaussian data", ind_points_selection_.c_str());
				}
				SampleIntNoReplaceSort((int)gp_coords_all_unique.rows(), num_ind_points, rng_, indices);
				gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_mat.cols());
				for (int j = 0; j < num_ind_points; ++j) {
					gp_coords_ip_mat.row(j) = gp_coords_all_unique.row(indices[j]);
				}
			}
			else if (ind_points_selection_ == "kmeans++") {
				gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_mat.cols());
				int max_it_kmeans = 1000;
				Log::REDebug("Starting kmeans++ algorithm for determining inducing points ");
				kmeans_plusplus(gp_coords_all_unique, num_ind_points, rng_, gp_coords_ip_mat, max_it_kmeans, false);
				Log::REDebug("Inducing points have been determined ");
			}
			else if (ind_points_selection_ == "space_time_kmeans++") {
				Log::REDebug("Starting space-time separated kmeans++ algorithm for determining inducing points ");

				gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_mat.cols());

				vec_t col = gp_coords_all_unique.col(0);
				// Frequency count using unordered_map
				std::unordered_map<double, int> freq;
				for (int i = 0; i < col.size(); ++i) {
					freq[col[i]]++;
				}
				int num_unique_time_points = (int)freq.size();
				int average_number_of_location = ((int)col.size()) / num_unique_time_points;
				

				int num_ind_points_space = std::min(std::max((int)std::sqrt((double)(num_ind_points * average_number_of_location) / (double)num_unique_time_points), 1), num_ind_points);
				int num_ind_points_time = std::min(std::max((int)std::sqrt((double)(num_ind_points * num_unique_time_points) / (double)average_number_of_location), 1), num_ind_points);

				num_ind_points = num_ind_points_time * num_ind_points_space;
				Log::REInfo("Number of time ip %i number of space ip %i and total %i", num_ind_points_time, num_ind_points_space, num_ind_points);
				// kmeans++ for time
				int max_it_kmeans = 1000;
				den_mat_t gp_coords_ip_mat_time(num_ind_points_time, 1);
				kmeans_plusplus(gp_coords_all_unique.col(0), num_ind_points_time, rng_, gp_coords_ip_mat_time, max_it_kmeans, false);
				// kmeans++ for space
				den_mat_t gp_coords_ip_mat_space(num_ind_points_space, gp_coords_all_unique.cols() - 1);
				kmeans_plusplus(gp_coords_all_unique.block(0, 1, gp_coords_all_unique.rows(), gp_coords_all_unique.cols() - 1),
					num_ind_points_space, rng_, gp_coords_ip_mat_space, max_it_kmeans, false);
				// Construct irregular grid
				int c = (int)gp_coords_ip_mat_space.cols();
				// Output matrix: n * r rows, 1 + c columns
				gp_coords_ip_mat.resize(num_ind_points, c + 1);
				for (int i = 0; i < num_ind_points_time; ++i) {
					// Fill first column with v[i], repeated r times
					gp_coords_ip_mat.block(i * num_ind_points_space, 0, num_ind_points_space, 1).setConstant(gp_coords_ip_mat_time.coeffRef(i, 0));
					// Fill the remaining columns with B
					gp_coords_ip_mat.block(i * num_ind_points_space, 1, num_ind_points_space, c) = gp_coords_ip_mat_space;
				}
				gp_coords_ip_mat_.resize(num_ind_points, gp_coords_all_mat.cols());
				num_ind_points_ = num_ind_points;
				Log::REDebug("Inducing points have been determined ");

			}
			else {
				Log::REFatal("Method '%s' is not supported for finding inducing points ", ind_points_selection_.c_str());
			}
			if (gp_approx_ == "full_scale_vecchia" && !gauss_likelihood_) {
				int count = 0;
				den_mat_t gp_coords_ip_mat_interim;
				std::vector<int> ind_coin;
				std::vector<int> ind_all(gp_coords_ip_mat.rows());
				std::iota(std::begin(ind_all), std::end(ind_all), 0);
				for (int i = 0; i < gp_coords_ip_mat.rows(); i++) {
					for (int j = 0; j < gp_coords_all_mat.rows(); j++) {
						if ((gp_coords_ip_mat.row(i).array() == gp_coords_all_mat.row(j).array()).all()) {
							count += 1;
							ind_coin.push_back(i);
							break;
						}
					}
				}
				if (count > 0) {
					std::vector<int> diff;
					std::set_difference(ind_all.begin(), ind_all.end(), ind_coin.begin(), ind_coin.end(),
						std::inserter(diff, diff.begin()));
					Log::REWarning("%d inducing points are removed since they coincide with data points. If this is a problem, please use less inducing points or a different method ('ind_points_selection') for selecting the inducing points ", count);
					gp_coords_ip_mat_interim = gp_coords_ip_mat(diff, Eigen::all);
					gp_coords_ip_mat.resize(gp_coords_ip_mat_interim.rows(), gp_coords_ip_mat_interim.cols());
					gp_coords_ip_mat = gp_coords_ip_mat_interim;
					num_ind_points = (int)gp_coords_ip_mat.rows();
					num_ind_points_ = num_ind_points;
				}
			}
			gp_coords_ip_mat_ = gp_coords_ip_mat;
			gp_coords_all_unique.resize(0, 0);
			std::shared_ptr<RECompGP<den_mat_t>> gp_ip(new RECompGP<den_mat_t>(
				gp_coords_ip_mat, cov_fct, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape,
				false, false, true, false, false, true));
			if (gp_ip->HasDuplicatedCoords()) {
				Log::REFatal("Duplicates found in inducing points / low-dimensional knots ");
			}
			re_comps_ip_cluster_i.push_back(gp_ip);
			std::shared_ptr<RECompGP<den_mat_t>> re_comp_ip = std::dynamic_pointer_cast<RECompGP<den_mat_t>>(re_comps_ip_cluster_i[0]);
			if (!(gp_approx_ == "full_scale_vecchia")) {
				only_one_GP_calculations_on_RE_scale_ = num_gp_total_ == 1 && num_comps_total_ == 1 && !gauss_likelihood_ && re_comp_ip->HasIsotropicCovFct();
				has_duplicates_coords_ = only_one_GP_calculations_on_RE_scale_;
			}
			re_comps_cross_cov_cluster_i.push_back(std::shared_ptr<RECompGP<den_mat_t>>(new RECompGP<den_mat_t>(
				gp_coords_all_mat, gp_coords_ip_mat, cov_fct, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape, false, false, only_one_GP_calculations_on_RE_scale_)));
			if (gp_approx_ == "full_scale_tapering") {
				re_comps_resid_cluster_i.push_back(std::shared_ptr<RECompGP<T_mat>>(new RECompGP<T_mat>(
					gp_coords_all_mat, cov_fct, cov_fct_shape, cov_fct_taper_range, cov_fct_taper_shape,
					true, true, true, false, false, true)));
			}
			//Random slope GPs
			if (num_gp_rand_coef_ > 0) {
				Log::REFatal("Random coefficients are currently not supported for '%s' approximation ", ind_points_selection_.c_str());
			}
		}//end CreateREComponentsFITC_FSA

		/*!
		* \brief Function that makes sure that the marginal variance parameters are held fix when they are not estimated but the nugget effect changes during optimization for gaussian likelihoods
		* \param cov_pars Covariance parameters
		* \param[out] cov_pars_out Covariance parameters
		*/
		void MaybeKeepVarianceConstant(const vec_t& cov_pars,
			vec_t& cov_pars_out) {
			cov_pars_out = cov_pars;
			if (gauss_likelihood_ && optimization_running_currently_) {
				CHECK((int)cov_pars_set_first_time_.size() == num_cov_par_);
				for (int igp = 0; igp < num_sets_re_; ++igp) {
					for (int j = 0; j < num_comps_total_; ++j) {
						if ((estimate_cov_par_index_[0] > 0) && (estimate_cov_par_index_[ind_par_[j]] <= 0)) {
							// gaussian likelihood, the nugget effect is estimated, but this marginal variance parameter not
							// -> avoid that the marginal variance changes due to reparametrizing / factoring out the nugget effect variance
							cov_pars_out[ind_par_[j]] = cov_pars_set_first_time_[ind_par_[j]] * cov_pars_set_first_time_[0] / cov_pars[0];
						}
					}
				}
			}
		}//end MaybeKeepVarianceConstant

		/*!
		* \brief Set the covariance parameters of the components
		* \param cov_pars Covariance parameters
		*/
		void SetCovParsComps(const vec_t& cov_pars_in) {
			CHECK(cov_pars_in.size() == num_cov_par_);
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			if (gauss_likelihood_) {
				sigma2_ = cov_pars[0];
			}
			for (const auto& cluster_i : unique_clusters_) {
				for (int igp = 0; igp < num_sets_re_; ++igp) {
					for (int j = 0; j < num_comps_total_; ++j) {
						const vec_t pars = cov_pars.segment(ind_par_[j] + igp * num_cov_par_per_set_re_, ind_par_[j + 1] - ind_par_[j]);
						if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
							re_comps_ip_[cluster_i][igp][j]->SetCovPars(pars);
							re_comps_cross_cov_[cluster_i][igp][j]->SetCovPars(pars);
							if (gp_approx_ == "full_scale_tapering") {
								re_comps_resid_[cluster_i][igp][j]->SetCovPars(pars);
							}
							if (gp_approx_ == "full_scale_vecchia") {
								re_comps_vecchia_[cluster_i][igp][j]->SetCovPars(pars);
							}
						}
						else if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
							re_comps_vecchia_[cluster_i][igp][j]->SetCovPars(pars);
						}
						else if (grouped_RE_and_vecchia_GP_ && j == (num_comps_total_ - 1)) {
							re_comps_vecchia_[cluster_i][igp][0]->SetCovPars(pars);
						}
						else {
							re_comps_[cluster_i][igp][j]->SetCovPars(pars);
						}
					}
				}//end loop over num_sets_re_
			}//end loop over unique_clusters_
		}//end SetCovParsComps

		/*!
		* \brief Calculate the total variance of all random effects
		*		Note: for random coefficients processes, we ignore the covariates and simply use the marginal variance for simplicity (this function is used for calling 'FindInitialIntercept' for non-Gaussian likelihoods)
		* \param cov_pars_in Covariance parameters
		* \param ind_set_re Conuter for number of GPs / REs (e.g. for heteroscedastic GPs)
		*/
		double GetTotalVarComps(const vec_t& cov_pars_in,
			int ind_set_re) {
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			CHECK(cov_pars.size() == num_cov_par_);
			if (ind_set_re > 0) {
				CHECK(ind_set_re <= num_sets_re_);
			}
			vec_t cov_pars_orig;
			TransformBackCovPars(cov_pars, cov_pars_orig);
			double tot_var = 0.;
			for (int j = 0; j < num_comps_total_; ++j) {
				tot_var += cov_pars_orig[ind_par_[j] + ind_set_re * num_cov_par_per_set_re_];
			}
			if (gauss_likelihood_) {
				tot_var += cov_pars_orig[0];
			}
			return(tot_var);
		}

		/*!
		* \brief Transform the covariance parameters to the scale on which the optimization is done
		* \param cov_pars Covariance parameters on orginal scale
		* \param[out] cov_pars_trans Covariance parameters on transformed scale
		*/
		void TransformCovPars(const vec_t& cov_pars,
			vec_t& cov_pars_trans) {
			CHECK(cov_pars.size() == num_cov_par_);
			cov_pars_trans = vec_t(num_cov_par_);
			if (gauss_likelihood_) {
				cov_pars_trans[0] = cov_pars[0];
			}
			double nugget_var = gauss_likelihood_ ? cov_pars[0] : 1.;
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				for (int j = 0; j < num_comps_total_; ++j) {
					const vec_t pars = cov_pars.segment(ind_par_[j] + igp * num_cov_par_per_set_re_, ind_par_[j + 1] - ind_par_[j]);
					vec_t pars_trans = pars;
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						re_comps_ip_[unique_clusters_[0]][igp][j]->TransformCovPars(nugget_var, pars, pars_trans);
					}
					else if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
						re_comps_vecchia_[unique_clusters_[0]][igp][j]->TransformCovPars(nugget_var, pars, pars_trans);
					}
					else if (grouped_RE_and_vecchia_GP_ && j == (num_comps_total_ - 1)) {
						re_comps_vecchia_[unique_clusters_[0]][igp][0]->TransformCovPars(nugget_var, pars, pars_trans);
					}
					else {
						re_comps_[unique_clusters_[0]][igp][j]->TransformCovPars(nugget_var, pars, pars_trans);
					}
					cov_pars_trans.segment(ind_par_[j] + igp * num_cov_par_per_set_re_, ind_par_[j + 1] - ind_par_[j]) = pars_trans;
				}
			}
		}//end TransformCovPars

		/*!
		* \brief Back-transform the covariance parameters to the original scale
		* \param cov_pars Covariance parameters on transformed scale
		* \param[out] cov_pars_orig Covariance parameters on orginal scale
		*/
		void TransformBackCovPars(const vec_t& cov_pars,
			vec_t& cov_pars_orig) {
			CHECK(cov_pars.size() == num_cov_par_);
			cov_pars_orig = vec_t(num_cov_par_);
			if (gauss_likelihood_) {
				cov_pars_orig[0] = cov_pars[0];
			}
			double nugget_var = gauss_likelihood_ ? cov_pars[0] : 1.;
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				for (int j = 0; j < num_comps_total_; ++j) {
					const vec_t pars = cov_pars.segment(ind_par_[j] + igp * num_cov_par_per_set_re_, ind_par_[j + 1] - ind_par_[j]);
					vec_t pars_orig = pars;
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						re_comps_ip_[unique_clusters_[0]][igp][j]->TransformBackCovPars(nugget_var, pars, pars_orig);
					}
					else if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
						re_comps_vecchia_[unique_clusters_[0]][igp][j]->TransformBackCovPars(nugget_var, pars, pars_orig);
					}
					else if (grouped_RE_and_vecchia_GP_ && j == (num_comps_total_ - 1)) {
						re_comps_vecchia_[unique_clusters_[0]][igp][0]->TransformBackCovPars(nugget_var, pars, pars_orig);
					}
					else {
						re_comps_[unique_clusters_[0]][igp][j]->TransformBackCovPars(nugget_var, pars, pars_orig);
					}
					cov_pars_orig.segment(ind_par_[j] + igp * num_cov_par_per_set_re_, ind_par_[j + 1] - ind_par_[j]) = pars_orig;
				}
			}
		}//end TransformBackCovPars

		/*!
		* \brief Make a warning of some parameters are e.g. too large
		* \param cov_pars Covariance parameters (on transformed scale)
		*/
		void CovarianceParameterRangeWarning(const vec_t& cov_pars) {
			CHECK(cov_pars.size() == num_cov_par_);
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				for (int j = 0; j < num_comps_total_; ++j) {
					const vec_t pars = cov_pars.segment(ind_par_[j] + igp * num_cov_par_per_set_re_, ind_par_[j + 1] - ind_par_[j]);
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						re_comps_ip_[unique_clusters_[0]][igp][j]->CovarianceParameterRangeWarning(pars);
					}
					else if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
						re_comps_vecchia_[unique_clusters_[0]][igp][j]->CovarianceParameterRangeWarning(pars);
					}
					else if (grouped_RE_and_vecchia_GP_ && j == (num_comps_total_ - 1)) {
						re_comps_vecchia_[unique_clusters_[0]][igp][0]->CovarianceParameterRangeWarning(pars);
					}
					else {
						re_comps_[unique_clusters_[0]][igp][j]->CovarianceParameterRangeWarning(pars);
					}
				}
			}
		}//end CovarianceParameterRangeWarning

		/*!
		* \brief Transform the linear regression coefficients to the scale on which the optimization is done
		* \param beta Regression coefficients on orginal scale
		* \param[out] beta_trans Regression coefficients on transformed scale
		*/
		void TransformCoef(const vec_t& beta,
			vec_t& beta_trans) {
			CHECK(loc_transf_.size() == beta.size() / num_sets_re_);
			CHECK(scale_transf_.size() == beta.size() / num_sets_re_);
			beta_trans = beta;
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				for (int icol = 0; icol < num_covariates_; ++icol) {
					if (!has_intercept_ || icol != intercept_col_) {
						if (has_intercept_) {
							beta_trans[igp * num_covariates_ + intercept_col_] += beta_trans[igp * num_covariates_ + icol] * loc_transf_[icol];
						}
						beta_trans[igp * num_covariates_ + icol] *= scale_transf_[icol];
					}
				}
				if (has_intercept_) {
					beta_trans[igp * num_covariates_ + intercept_col_] *= scale_transf_[intercept_col_];
				}
			}
		}

		/*!
		* \brief Back-transform linear regression coefficients back to original scale
		* \param beta Regression coefficients on transformed scale
		* \param[out] beta_orig Regression coefficients on orginal scale
		*/
		void TransformBackCoef(const vec_t& beta,
			vec_t& beta_orig) {
			CHECK(loc_transf_.size() == beta.size() / num_sets_re_);
			CHECK(scale_transf_.size() == beta.size() / num_sets_re_);
			beta_orig = beta;
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				if (has_intercept_) {
					beta_orig[igp * num_covariates_ + intercept_col_] /= scale_transf_[intercept_col_];
				}
				for (int icol = 0; icol < num_covariates_; ++icol) {
					if (!has_intercept_ || icol != intercept_col_) {
						beta_orig[igp * num_covariates_ + icol] /= scale_transf_[icol];
						if (has_intercept_) {
							beta_orig[igp * num_covariates_ + intercept_col_] -= beta_orig[igp * num_covariates_ + icol] * loc_transf_[icol];
						}
					}
				}
			}
		}

		/*!
		* \brief Transform the auxiliary parameters to the scale on which the optimization is done (if there are any)
		* \param aux_pars_orig Auxiliary parameters on orginal scale
		* \param[out] aux_pars_trans Auxiliary parameters on transformed scale
		*/
		void TransformAuxPars(const double* aux_pars_orig,
			double* aux_pars_trans) {
			likelihood_[unique_clusters_[0]]->TransformAuxPars(aux_pars_orig, aux_pars_trans);
		}//end TransformAuxPars

		/*!
		* \brief Back-transform the auxiliary parameters to the scale on which the optimization is done (if there are any)
		* \param aux_pars_trans Auxiliary parameters on transformed scale
		* \param[out] aux_pars_orig Auxiliary parameters on orginal scale
		*/
		void BackTransformAuxPars(const double* aux_pars_trans,
			double* aux_pars_orig) {
			likelihood_[unique_clusters_[0]]->BackTransformAuxPars(aux_pars_trans, aux_pars_orig);
		}//end BackTransformAuxPars

		/*!
		* \brief Calculate covariance matrices of the components and some auxiliary quantities for some approximations
		*/
		void CalcSigmaComps() {
			CHECK(gp_approx_ != "vecchia");
			for (const auto& cluster_i : unique_clusters_) {
				for (int j = 0; j < num_comps_total_; ++j) {
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						re_comps_ip_[cluster_i][0][j]->CalcSigma();
						re_comps_cross_cov_[cluster_i][0][j]->CalcSigma();
						den_mat_t sigma_ip_stable = *(re_comps_ip_[cluster_i][0][j]->GetZSigmaZt());
						sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
						chol_fact_sigma_ip_[cluster_i][0].compute(sigma_ip_stable);
						const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][j]->GetSigmaPtr();
						if (gp_approx_ == "fitc") {
							den_mat_t sigma_ip_Ihalf_sigma_cross_covT = (*cross_cov).transpose();
							//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_[cluster_i][0],
							//	sigma_ip_Ihalf_sigma_cross_covT, sigma_ip_Ihalf_sigma_cross_covT, false);
							GPBoost::solve_lower_triangular(chol_fact_sigma_ip_[cluster_i][0],
								sigma_ip_Ihalf_sigma_cross_covT, sigma_ip_Ihalf_sigma_cross_covT, GPU_use_);
							if (gauss_likelihood_) {
								fitc_resid_diag_[cluster_i] = vec_t::Ones(re_comps_cross_cov_[cluster_i][0][0]->GetNumUniqueREs());//add nugget effect variance
							}
							else {
								fitc_resid_diag_[cluster_i] = vec_t::Zero(re_comps_cross_cov_[cluster_i][0][0]->GetNumUniqueREs());
							}
							//add marginal variance
							if (re_comps_cross_cov_[cluster_i][0][j]->VarianceOnDiagonal()) {
								fitc_resid_diag_[cluster_i].array() += sigma_ip_stable.coeffRef(0, 0);
							}
							else {
#pragma omp parallel for schedule(static)
								for (int ii = 0; ii < re_comps_cross_cov_[cluster_i][0][j]->GetNumUniqueREs(); ++ii) {
									fitc_resid_diag_[cluster_i][ii] += re_comps_cross_cov_[cluster_i][0][j]->GetZSigmaZtij(ii, ii);//uses coords of all data points (not IPs) to calculate Sigma[ii,ii]
								}
							}
#pragma omp parallel for schedule(static)
							for (int ii = 0; ii < re_comps_cross_cov_[cluster_i][0][0]->GetNumUniqueREs(); ++ii) {
								fitc_resid_diag_[cluster_i][ii] -= sigma_ip_Ihalf_sigma_cross_covT.col(ii).array().square().sum();
							}
						}
						else if (gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
							// Subtract predictive process covariance
							chol_ip_cross_cov_[cluster_i][0] = (*cross_cov).transpose();
							//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_[cluster_i][0],
							//	chol_ip_cross_cov_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0], false);
							GPBoost::solve_lower_triangular(chol_fact_sigma_ip_[cluster_i][0],
								chol_ip_cross_cov_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0], GPU_use_);
							if (gp_approx_ == "full_scale_tapering") {
								re_comps_resid_[cluster_i][0][j]->CalcSigma();
								re_comps_resid_[cluster_i][0][j]->SubtractPredProcFromSigmaForResidInFullScale(chol_ip_cross_cov_[cluster_i][0], true);
								re_comps_resid_[cluster_i][0][j]->ApplyTaper();
								if (gauss_likelihood_) {
									re_comps_resid_[cluster_i][0][j]->AddConstantToDiagonalSigma(1.);//add nugget effect variance
								}
							}
						}
					}//end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
					else if (!use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_ && !linear_kernel_use_woodbury_identity_) {
						re_comps_[cluster_i][0][j]->CalcSigma();
					}
				}
			}
			if (!gauss_likelihood_ && (gp_approx_ == "none" || gp_approx_ == "tapering")) {
				// Calculate the covariance matrix 'ZSigmaZt_' if !use_woodbury_identity_ or the inverse covariance matrix 'SigmaI_' if use_woodbury_identity_
				if (!only_one_grouped_RE_calculations_on_RE_scale_) {//Nothing to calculate if only_one_grouped_RE_calculations_on_RE_scale_
					if (use_woodbury_identity_) {
						for (const auto& cluster_i : unique_clusters_) {
							CalcSigmaOrInverseUseWoodbury(SigmaI_[cluster_i], cluster_i, true);
						}
					}
					else {
						for (const auto& cluster_i : unique_clusters_) {
							if (num_comps_total_ == 1) {//no need to sum up different components
								ZSigmaZt_[cluster_i] = re_comps_[cluster_i][0][0]->GetZSigmaZt();
							}
							else {
								T_mat ZSigmaZt;
								CalcZSigmaZt(ZSigmaZt, cluster_i);
								ZSigmaZt_[cluster_i] = std::make_shared<T_mat>(ZSigmaZt);
							}
						}
					}
				}
			}//end !gauss_likelihood_
		}//end CalcSigmaComps

		/*!
		* \brief Construct covariance matrix Sigma or inverse covariance matrix Sigma^-1 if use_woodbury_identity_ (e.g., if there are only grouped random effecs). This is a diagonal matrix
		* \param[out] SigmaI Covariance matrix or inverse covariance matrix of random effects (a diagonal matrix)
		* \param cluster_i Cluster index for which SigmaI is constructed
		* \param inverse If true, the inverse covariance matrix is calculated
		*/
		void CalcSigmaOrInverseUseWoodbury(sp_mat_t& SigmaI,
			data_size_t cluster_i,
			bool inverse) {
			CHECK(!only_one_grouped_RE_calculations_on_RE_scale_);
			const data_size_t num_re = linear_kernel_use_woodbury_identity_ ? num_comps_total_ : num_re_group_total_;
			if (grouped_RE_and_vecchia_GP_) {
				CHECK(num_re == num_re_group_total_);
			}
			std::vector<Triplet_t> triplets(cum_num_rand_eff_[cluster_i][num_re]);
			for (int j = 0; j < num_re; ++j) {
				double sigmaI = re_comps_[cluster_i][0][j]->cov_pars_[0];
				if (inverse) {
					sigmaI = 1.0 / sigmaI;
				}
#pragma omp parallel for schedule(static)
				for (int i = cum_num_rand_eff_[cluster_i][j]; i < cum_num_rand_eff_[cluster_i][j + 1]; ++i) {
					triplets[i] = Triplet_t(i, i, sigmaI);
				}
			}
			SigmaI = sp_mat_t(cum_num_rand_eff_[cluster_i][num_re], cum_num_rand_eff_[cluster_i][num_re]);
			SigmaI.setFromTriplets(triplets.begin(), triplets.end());
		}

		/*!
		* \brief Set initial values for some of the optimizer parameters
		* Internal default values are used if the corresponding parameters have not been set
		*/
		void OptimParamsSetInitialValues() {
			SetInitialValueLRCov();
			SetInitialValueDeltaRelConv();
		}//end OptimParamsSetInitialValues

		/*!
		* \brief Initialitze learning rates and convergence tolerance
		* \param reuse_learning_rates_from_previous_call If true, the learning rates for the covariance and potential auxiliary parameters are kept at the values from a previous call and not re-initialized (can only be set to true if called_in_GPBoost_algorithm is true)
		*/
		void InitializeOptimSettings(bool reuse_learning_rates_from_previous_call) {
			if (!optimizer_cov_pars_has_been_set_) {
				optimizer_cov_pars_ = "lbfgs";
			}
			if (!coef_optimizer_has_been_set_) {
				if (gauss_likelihood_) {
					optimizer_coef_ = "wls";
				}
				else {
					optimizer_coef_ = "lbfgs";
				}
			}
			if (reuse_learning_rates_from_previous_call &&
				((cov_pars_have_been_estimated_once_ && optimizer_cov_pars_ == "gradient_descent") ||
					(coef_have_been_estimated_once_ && optimizer_coef_ == "gradient_descent" && has_covariates_))) {//different initial learning rates and optimizer settings for GPBoost algorithm in later boosting iterations
				CHECK(lr_have_been_initialized_);
				if (cov_pars_have_been_estimated_once_ && optimizer_cov_pars_ == "gradient_descent") {
					lr_cov_ = lr_cov_after_first_optim_boosting_iteration_;
					if (estimate_aux_pars_) {
						lr_aux_pars_ = lr_aux_pars_after_first_optim_boosting_iteration_;
					}
				}
				if (coef_have_been_estimated_once_ && optimizer_coef_ == "gradient_descent" && has_covariates_) {
					lr_coef_ = lr_coef_after_first_optim_boosting_iteration_;
				}
				c_armijo_ = 0.;
				c_armijo_mom_ = 0.;
				max_number_lr_shrinkage_steps_ = (int) MAX_NUMBER_LR_SHRINKAGE_STEPS_DEFAULT_ / 2;
			}
			else {
				lr_coef_ = lr_coef_init_;
				lr_aux_pars_ = lr_aux_pars_init_;
				lr_cov_ = lr_cov_init_;
				delta_rel_conv_ = delta_rel_conv_init_;
				lr_have_been_initialized_ = true;
				c_armijo_ = C_ARMIJO_DEFAULT_;
				c_armijo_mom_ = C_ARMIJO_MOM_DEFAULT_;
				max_number_lr_shrinkage_steps_ = MAX_NUMBER_LR_SHRINKAGE_STEPS_DEFAULT_;
			}
		}//end InitializeOptimSettings

		/*! * \brief Set initial values 'lr_cov_init_' for lr_cov_ */
		void SetInitialValueLRCov() {
			if (lr_cov_init_ < 0.) {//A value below 0 indicates that default values should be used
				if (optimizer_cov_pars_ == "gradient_descent") {
					lr_cov_init_ = 0.1;
				}
				else {
					lr_cov_init_ = 1.;
				}
				lr_cov_after_first_iteration_ = lr_cov_init_;
				lr_cov_after_first_optim_boosting_iteration_ = lr_cov_init_;
				if (estimate_aux_pars_) {
					lr_aux_pars_init_ = lr_cov_init_;
					lr_aux_pars_after_first_iteration_ = lr_cov_init_;
					lr_aux_pars_after_first_optim_boosting_iteration_ = lr_cov_init_;
				}
			}
		}//end SetInitialValueLRCov

		/*! * \brief Set initial values 'delta_rel_conv_init_' for delta_rel_conv_ */
		void SetInitialValueDeltaRelConv() {
			if (delta_rel_conv_init_ < 0) {
				if (optimizer_cov_pars_ == "nelder_mead") {
					delta_rel_conv_init_ = 1e-8;
				}
				else {
					delta_rel_conv_init_ = 1e-6;
				}
			}
		}//end SetInitialValueDeltaRelConv

		/*!
		* \brief Avoid too large learning rates for covariance parameters and aux_pars
		* \param neg_step_dir Negative step direction for making updates. E.g., neg_step_dir = grad for gradient descent and neg_step_dir = FI^-1 * grad for Fisher scoring (="natural" gradient)
		*/
		void AvoidTooLargeLearningRatesCovAuxPars(const vec_t& neg_step_dir) {
			int num_grad_cov_par = (int)neg_step_dir.size();
			if (estimate_aux_pars_) {
				num_grad_cov_par -= NumAuxPars();
			}
			double max_lr_cov = MaximalLearningRateCovAuxPars(neg_step_dir.segment(0, num_grad_cov_par));
			if (lr_cov_ > max_lr_cov) {
				lr_cov_ = max_lr_cov;
				Log::REDebug("GPModel: The learning rate for the covariance parameters has been decreased in iteration number %d since "
					"the gradient update on the log-scale would have been too large (change by more than a factor %d). New learning rate = %g",
					num_iter_ + 1, MAX_REL_CHANGE_GRADIENT_UPDATE_, lr_cov_);
			}
			if (estimate_aux_pars_) {
				double max_lr_caux_pars = MaximalLearningRateCovAuxPars(neg_step_dir.segment(num_grad_cov_par, NumAuxPars()));
				if (lr_aux_pars_ > max_lr_caux_pars) {
					lr_aux_pars_ = max_lr_caux_pars;
					Log::REDebug("GPModel: The learning rate for the auxiliary parameters has been decreased in iteration number %d since "
						"the gradient update on the log-scale would have been too large (change by more than a factor %d). New learning rate = %g",
						num_iter_ + 1, MAX_REL_CHANGE_GRADIENT_UPDATE_AUX_PARS_, lr_aux_pars_);
				}
			}
		}//end AvoidTooLargeLearningRatesCovAuxPars

		/*!
		* \brief Avoid too large learning rates for linear regression coefficients
		* \param beta Current / lag1 value of beta
		* \param neg_step_dir Negative step direction for making updates. E.g., neg_step_dir = grad for gradient descent and neg_step_dir = FI^-1 * grad for Fisher scoring (="natural" gradient)
		*/
		void AvoidTooLargeLearningRateCoef(const vec_t& beta,
			const vec_t& neg_step_dir) {
			double max_lr_coef = MaximalLearningRateCoef(beta, neg_step_dir);
			if (lr_coef_ > max_lr_coef) {
				lr_coef_ = max_lr_coef;
				Log::REDebug("GPModel: The learning rate for the regression coefficients has been decreased in iteration number %d since "
					"the current one would have implied a too large change in the mean and variance of the linear predictor relative to the data. New learning rate = %g",
					num_iter_ + 1, lr_coef_);
			}
		}//end AvoidTooLargeLearningRateCoef

		/*!
		* \brief Calculate the directional derivative for the Armijo condition (if armijo_condition_) and
		*		update learning rate such that there is a constant first order change (if learning_rate_constant_first_order_change_)
		* \param grad Gradient of covariance and additional auxiliary likelihood parameters
		* \param neg_step_dir Negative step direction for making updates. E.g., neg_step_dir = grad for gradient descent and neg_step_dir = FI^-1 * grad for Fisher scoring (="natural" gradient)
		* \param cov_aux_pars Covariance and additional auxiliary likelihood parameters
		* \param cov_pars_after_grad_aux Covariance and auxiliary parameters after gradient step and before momentum step (of previous iteration)
		* \param use_nesterov_acc If true, Nesterov acceleration is used
		*/
		void CalcDirDerivArmijoAndLearningRateConstChangeCovAuxPars(const vec_t& grad,
			const vec_t& neg_step_dir,
			const vec_t& cov_aux_pars,
			const vec_t& cov_pars_after_grad_aux,
			bool use_nesterov_acc) {
			if ((learning_rate_constant_first_order_change_ && num_iter_ > 0) || armijo_condition_) {
				CHECK(grad.size() == neg_step_dir.size());
				int num_grad_cov_par = (int)neg_step_dir.size();
				if (estimate_aux_pars_) {
					num_grad_cov_par -= NumAuxPars();
				}
				if (learning_rate_constant_first_order_change_ && num_iter_ > 0) {
					double dir_deriv_armijo_cov_pars_new = -(grad.segment(0, num_grad_cov_par)).dot(neg_step_dir.segment(0, num_grad_cov_par));
					lr_cov_ *= dir_deriv_armijo_cov_pars_ / dir_deriv_armijo_cov_pars_new;
					dir_deriv_armijo_cov_pars_ = dir_deriv_armijo_cov_pars_new;
					if (estimate_aux_pars_) {
						double dir_deriv_armijo_aux_pars_new = -(grad.segment(num_grad_cov_par, NumAuxPars())).dot(neg_step_dir.segment(num_grad_cov_par, NumAuxPars()));
						lr_aux_pars_ *= dir_deriv_armijo_aux_pars_ / dir_deriv_armijo_aux_pars_new;
						dir_deriv_armijo_aux_pars_ = dir_deriv_armijo_aux_pars_new;
					}
				}
				else if (armijo_condition_) {
					dir_deriv_armijo_cov_pars_ = -(grad.segment(0, num_grad_cov_par)).dot(neg_step_dir.segment(0, num_grad_cov_par));
					if (estimate_aux_pars_) {
						dir_deriv_armijo_aux_pars_ = -(grad.segment(num_grad_cov_par, NumAuxPars())).dot(neg_step_dir.segment(num_grad_cov_par, NumAuxPars()));
					}
				}
				if (armijo_condition_ && use_nesterov_acc) {
					vec_t delta_pars = cov_aux_pars.array().log().matrix() - cov_pars_after_grad_aux.array().log().matrix();//gradient steps / update is done on log-scale
					vec_t delta_cov_pars;
					if (profile_out_error_variance_) {
						delta_cov_pars = delta_pars.segment(1, num_grad_cov_par);
					}
					else {
						delta_cov_pars = delta_pars.segment(0, num_grad_cov_par);
					}
					mom_dir_deriv_armijo_cov_pars_ = (grad.segment(0, num_grad_cov_par)).dot(delta_cov_pars);
					if (estimate_aux_pars_) {
						vec_t delta_aux_pars = delta_pars.segment(num_cov_par_, NumAuxPars());
						mom_dir_deriv_armijo_aux_pars_ = (grad.segment(num_grad_cov_par, NumAuxPars())).dot(delta_aux_pars);
					}
				}
				else {
					mom_dir_deriv_armijo_cov_pars_ = 0.;
					mom_dir_deriv_armijo_aux_pars_ = 0;
				}
			}
		}//CalcDirDerivArmijoAndLearningRateConstChangeCovAuxPars

		/*!
		* \brief Calculate the directional derivative for the Armijo condition (if armijo_condition_) and
		*			update the learning rate such that there is a constant first order change (if learning_rate_constant_first_order_change_) for regression coefficients
		* \param grad Gradient of linear regression coefficients
		* \param beta Linear regression coefficients
		* \param beta_after_grad_aux Linear regression coefficients after gradient step and before momentum step (of previous iteration)
		* \param use_nesterov_acc If true, Nesterov acceleration is used
		*/
		void CalcDirDerivArmijoAndLearningRateConstChangeCoef(const vec_t& grad,
			const vec_t& beta,
			const vec_t& beta_after_grad_aux,
			bool use_nesterov_acc) {
			if (learning_rate_constant_first_order_change_ && num_iter_ > 0) {
				double dir_deriv_armijo_coef_new = grad.squaredNorm();
				lr_coef_ *= dir_deriv_armijo_coef_ / dir_deriv_armijo_coef_new;
				dir_deriv_armijo_coef_ = dir_deriv_armijo_coef_new;
			}
			else if (armijo_condition_) {
				dir_deriv_armijo_coef_ = grad.squaredNorm();
			}
			if (armijo_condition_ && use_nesterov_acc) {
				mom_dir_deriv_armijo_coef_ = grad.dot(beta - beta_after_grad_aux);
			}
		}//end CalcDirDerivArmijoAndLearningRateConstChangeCoef

		/*!
		* \brief For the GPBoost algorithm, learning rates (lr_cov_ and lr_aux_pars_) are no reset to initial values in every boosting iteration
		*			but rather kept at their previous values. This can, however, sometimes imply that learning rates might become too small in later boosting iterations.
		* This function checks whether we should increase (double) the learing rates again and does the increase if necessary
		*/
		void PotentiallyIncreaseLearningRatesForGPBoostAlgorithm() {
			bool double_learning_rate = false;
			if (num_iter_ == 0) {
				if (!estimate_aux_pars_) {
					if ((-dir_deriv_armijo_cov_pars_ * lr_cov_) <= (delta_rel_conv_ * std::max(std::abs(neg_log_likelihood_lag1_), 1.))) {
						if ((-dir_deriv_armijo_cov_pars_ * lr_cov_init_) > std::max(std::abs(neg_log_likelihood_lag1_), 1.)) {
							// Increase the learning again if 
							//		(i) likely no change will be detected in the log-likelihood, i.e., convergence is achieved, (=first "if" condition) 
							//		(ii) but this is due a very small learning rate and for larger learning rates (i.e. lr_cov_init_) the log-likelihood would still change (=second "if" condition).
							//		In other words, if lr_cov_ is small but the directional derivative of the log-likelihood in the search direction (=first order change of log-likelihood) is large relative to the log-likelihood
							//		Note that (-dir_deriv_armijo_cov_pars_) is approximately equal to (neg_log_likelihood_lag1_ - neg_log_likelihood_) for small lr_cov_
							double_learning_rate = true;
						}
					}
				}
				else {//estimate_aux_pars_
					if ((-dir_deriv_armijo_cov_pars_ * lr_cov_ + -dir_deriv_armijo_aux_pars_ * lr_aux_pars_) <= (delta_rel_conv_ * std::max(std::abs(neg_log_likelihood_lag1_), 1.))) {
						if ((-dir_deriv_armijo_cov_pars_ * lr_cov_init_ + dir_deriv_armijo_aux_pars_ * lr_aux_pars_init_) >= std::max(std::abs(neg_log_likelihood_lag1_), 1.)) {
							double_learning_rate = true;
						}
					}
				}
			}//end num_iter_ == 0
			else if (num_iter_ == 1 && !learning_rates_have_been_doubled_in_first_iteration_) {//always increase the learning rate in the second iteration if more than one iteration is needed 
				double_learning_rate = true;
			}
			if (double_learning_rate) {
				if (2 * lr_cov_ <= lr_cov_init_) {
					lr_cov_ *= 2;
					if (num_iter_ == 0) {
						learning_rates_have_been_doubled_in_first_iteration_ = true;
					}
				}
				if (estimate_aux_pars_) {
					if (2 * lr_aux_pars_ <= lr_aux_pars_init_) {
						lr_aux_pars_ *= 2;
						if (num_iter_ == 0) {
							learning_rates_have_been_doubled_in_first_iteration_ = true;
						}
					}
				}
			}//end double_learning_rate
		}//end PotentiallyIncreaseLearningRatesForGPBoostAlgorithm

		/*!
		* \brief For the GPBoost algorithm, lr_coef_ is not reset to initial values in every boosting iteration (when finding an optimal learnin rate)
		*			but rather kept at their previous values. This can, however, sometimes imply that learning rates might become too small in later boosting iterations.
		* This function checks whether we should increase (double) the learing rate again and does the increase if necessary
		*/
		void PotentiallyIncreaseLearningRateCoefForGPBoostAlgorithm() {
			bool double_learning_rate = false;
			if (num_iter_ == 0) {
				if ((-dir_deriv_armijo_coef_ * lr_coef_) <= (delta_rel_conv_ * std::max(std::abs(neg_log_likelihood_lag1_), 1.))) {
					if ((-dir_deriv_armijo_coef_ * lr_coef_init_) > std::max(std::abs(neg_log_likelihood_lag1_), 1.)) {
						// Increase the learning again if 
						//		(i) likely no change will be detected in the log-likelihood, i.e., convergence is achieved, (=first "if" condition) 
						//		(ii) but this is due a very small learning rate and for larger learning rates (i.e. lr_coef_init_) the log-likelihood would still change (=second "if" condition).
						//		In other words, if lr_coef_ is small but the directional derivative of the log-likelihood in the search direction (=first order change of log-likelihood) is large relative to the log-likelihood
						//		Note that (-dir_deriv_armijo_coef_) is approximately equal to (neg_log_likelihood_lag1_ - neg_log_likelihood_) for small lr_coef_
						double_learning_rate = true;
					}
				}
			}//end num_iter_ == 0
			else if (num_iter_ == 1 && !learning_rate_coef_have_been_doubled_in_first_iteration_) {//always increase the learning rate in the second iteration if more than one iteration is needed 
				double_learning_rate = true;
			}
			if (double_learning_rate) {
				if (2 * lr_coef_ <= lr_coef_init_) {
					lr_coef_ *= 2;
					if (num_iter_ == 0) {
						learning_rate_coef_have_been_doubled_in_first_iteration_ = true;
					}
				}
			}//end double_learning_rate
		}//end PotentiallyIncreaseLearningRateCoefForGPBoostAlgorithm

		/*!
		* \brief Recaculate mode for Laplace approximation after reseting them to zero
		* \param fixed_effects Fixed effects component of location parameter
		*/
		void RecalculateModeLaplaceApprox(const double* fixed_effects) {
			if (!gauss_likelihood_) {
				//Reset the initial modes to 0. Otherwise, they can get stuck
				for (const auto& cluster_i : unique_clusters_) {
					likelihood_[cluster_i]->InitializeModeAvec();
				}
				CalcModePostRandEffCalcMLL(fixed_effects, false);
			}
		}//end RecalculateModeLaplaceApprox

		/*!
		* \brief Calculate the gradient of the Laplace-approximated negative log-likelihood with respect to the fixed effects F (only used for non-Gaussian likelihoods)
		* \param[out] grad_F Gradient of the Laplace-approximated negative log-likelihood with respect to the fixed effects F. This vector needs to be pre-allocated of length num_data_
		* \param fixed_effects Fixed effects component of location parameter
		*/
		void CalcGradFLaplace(double* grad_F,
			const double* fixed_effects) {
			const double* fixed_effects_cluster_i_ptr = nullptr;
			vec_t fixed_effects_cluster_i;
			for (const auto& cluster_i : unique_clusters_) {
				vec_t grad_F_cluster_i(num_data_per_cluster_[cluster_i] * num_sets_re_);
				//map fixed effects to clusters (if needed)
				if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {//only one cluster / independent realization and order of data does not matter
					fixed_effects_cluster_i_ptr = fixed_effects;
				}
				else if (fixed_effects != nullptr) {//more than one cluster and order of samples matters
					fixed_effects_cluster_i = vec_t(num_data_per_cluster_[cluster_i] * num_sets_re_);
					for (int igp = 0; igp < num_sets_re_; ++igp) {
#pragma omp parallel for schedule(static)
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							fixed_effects_cluster_i[j + num_data_per_cluster_[cluster_i] * igp] = fixed_effects[data_indices_per_cluster_[cluster_i][j] + num_data_ * igp];
						}
					}
					fixed_effects_cluster_i_ptr = fixed_effects_cluster_i.data();
				}
				if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
					likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxVecchia(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, B_[cluster_i], D_inv_[cluster_i], B_grad_[cluster_i], D_grad_[cluster_i],
						false, true, false, nullptr, grad_F_cluster_i, nullptr, false, num_comps_total_, false, re_comps_ip_preconditioner_[cluster_i][0],
						re_comps_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0],
						cluster_i, this, estimate_cov_par_index_, GPU_use_);
				}
				else if (gp_approx_ == "fitc") {
					likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxFITC(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, re_comps_ip_[cluster_i][0][0]->GetZSigmaZt(), chol_fact_sigma_ip_[cluster_i][0],
						re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr(), fitc_resid_diag_[cluster_i], re_comps_ip_[cluster_i][0], re_comps_cross_cov_[cluster_i][0],
						false, true, false, nullptr, grad_F_cluster_i, nullptr, false, false, estimate_cov_par_index_, GPU_use_);
				}
				else if (gp_approx_ == "full_scale_vecchia") {
					likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxFSVA(y_[cluster_i].data(),
						y_int_[cluster_i].data(), fixed_effects_cluster_i_ptr, chol_fact_sigma_ip_[cluster_i][0],
						chol_fact_sigma_woodbury_[cluster_i], chol_ip_cross_cov_[cluster_i][0], sigma_woodbury_[cluster_i],
						re_comps_ip_[cluster_i][0], re_comps_cross_cov_[cluster_i][0], B_[cluster_i][0], D_inv_[cluster_i][0],
						B_T_D_inv_B_cross_cov_[cluster_i][0], D_inv_B_cross_cov_[cluster_i][0], sigma_ip_inv_cross_cov_T_[cluster_i][0],
						B_grad_[cluster_i][0], D_grad_[cluster_i][0], false, true, false, nullptr, grad_F_cluster_i, nullptr,
						false, false, re_comps_ip_preconditioner_[cluster_i][0], re_comps_cross_cov_preconditioner_[cluster_i][0],
						chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0], estimate_cov_par_index_, GPU_use_);
				}
				else if ((use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) || grouped_RE_and_vecchia_GP_) {
					likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxGroupedRE(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, SigmaI_[cluster_i], grouped_RE_and_vecchia_GP_, B_[cluster_i][0], D_inv_[cluster_i][0], B_grad_[cluster_i][0], D_grad_[cluster_i][0], 
						cum_num_rand_eff_[cluster_i], false, true, false, nullptr, grad_F_cluster_i, nullptr, false, false, estimate_cov_par_index_);
				}
				else if (only_one_grouped_RE_calculations_on_RE_scale_) {
					likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxOnlyOneGroupedRECalculationsOnREScale(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, re_comps_[cluster_i][0][0]->cov_pars_[0],
						false, true, false, nullptr, grad_F_cluster_i, nullptr, false, false, estimate_cov_par_index_);
				}
				else {
					likelihood_[cluster_i]->CalcGradNegMargLikelihoodLaplaceApproxStable(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, ZSigmaZt_[cluster_i], re_comps_[cluster_i][0],
						false, true, false, nullptr, grad_F_cluster_i, nullptr, false, false, estimate_cov_par_index_);
				}
				//write on output
				if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {//only one cluster / independent realization and order of data does not matter
#pragma omp parallel for schedule(static)//write on output
					for (int j = 0; j < num_data_ * num_sets_re_; ++j) {
						grad_F[j] = grad_F_cluster_i[j];
					}
				}
				else {//more than one cluster and order of samples matters
					for (int igp = 0; igp < num_sets_re_; ++igp) {
#pragma omp parallel for schedule(static)
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							grad_F[data_indices_per_cluster_[cluster_i][j] + num_data_ * igp] = grad_F_cluster_i[j + num_data_per_cluster_[cluster_i] * igp];
						}
					}
				} // end more than one cluster
			}//end loop over cluster
		}//end CalcGradFLaplace

		/*!
		* \brief Update covariance and potential additional likelihood parameters, apply step size safeguard, factorize covariance matrix, and calculate new value of objective function
		* \param[out] cov_pars Covariance and additional auxiliary likelihood parameters
		* \param neg_step_dir Negative step direction for making updates. E.g., neg_step_dir = grad for gradient descent and neg_step_dir = FI^-1 * grad for Fisher scoring (="natural" gradient)
		* \param use_nesterov_acc If true, Nesterov acceleration is used
		* \param it Iteration number
		* \param[out] cov_pars_after_grad_aux Auxiliary variable used only if use_nesterov_acc == true (see the code below for a description)
		* \param[out] cov_pars_after_grad_aux_lag1 Auxiliary variable used only if use_nesterov_acc == true (see the code below for a description)
		* \param acc_rate_cov Nesterov acceleration speed
		* \param nesterov_schedule_version Which version of Nesterov schedule should be used. Default = 0
		* \param momentum_offset Number of iterations for which no mometum is applied in the beginning
		* \param fixed_effects Fixed effects component of location parameter
		*/
		void UpdateCovAuxPars(vec_t& cov_pars,
			const vec_t& neg_step_dir,
			bool use_nesterov_acc,
			int it,
			vec_t& cov_pars_after_grad_aux,
			vec_t& cov_pars_after_grad_aux_lag1,
			double acc_rate_cov,
			int nesterov_schedule_version,
			int momentum_offset,
			const double* fixed_effects) {
			if (use_nesterov_acc && nesterov_schedule_version == 1 && armijo_condition_) {
				Log::REFatal("Armijo condition backtracking is not implemented when nesterov_schedule_version = 1 ");
			}
			vec_t cov_pars_new(num_cov_par_ + NumAuxPars());
			if (profile_out_error_variance_) {
				cov_pars_new[0] = cov_pars[0];
			}
			double lr_cov = lr_cov_;
			double lr_aux_pars = lr_aux_pars_;
			bool decrease_found = false;
			bool halving_done = false;
			int num_grad_cov_par = (int)neg_step_dir.size();
			if (estimate_aux_pars_) {
				num_grad_cov_par -= NumAuxPars();
			}
			if (it == 0) {
				first_update_ = true;
			}
			else {
				first_update_ = false;
			}
			for (int ih = 0; ih < max_number_lr_shrinkage_steps_; ++ih) {
				vec_t update(neg_step_dir.size());
				update.segment(0, num_grad_cov_par) = lr_cov * neg_step_dir.segment(0, num_grad_cov_par);
				if (estimate_aux_pars_) {
					update.segment(num_grad_cov_par, NumAuxPars()) = lr_aux_pars * neg_step_dir.segment(num_grad_cov_par, NumAuxPars());
				}
				// Avoid to large steps on log-scale: updates on the log-scale in one Fisher scoring step are capped at a certain level
				// This is not done for gradient_descent since the learning rate is already adjusted accordingly in 'AvoidTooLargeLearningRatesCovAuxPars'
				if (optimizer_cov_pars_ == "fisher_scoring" || optimizer_cov_pars_ == "newton") {
					for (int ip = 0; ip < (int)update.size(); ++ip) {
						if (update[ip] > MAX_GRADIENT_UPDATE_LOG_SCALE_) {
							update[ip] = MAX_GRADIENT_UPDATE_LOG_SCALE_;
						}
						else if (update[ip] < -MAX_GRADIENT_UPDATE_LOG_SCALE_) {
							update[ip] = -MAX_GRADIENT_UPDATE_LOG_SCALE_;
						}
					}
				}
				if (profile_out_error_variance_) {
					cov_pars_new.segment(1, cov_pars.size() - 1) = (cov_pars.segment(1, cov_pars.size() - 1).array().log() - update.array()).exp().matrix();//make update on log-scale
				}
				else {
					cov_pars_new = (cov_pars.array().log() - update.array()).exp().matrix();//make update on log-scale
				}
				// Apply Nesterov acceleration
				if (use_nesterov_acc) {
					cov_pars_after_grad_aux = cov_pars_new;
					ApplyMomentumStep(it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1, cov_pars_new, acc_rate_cov,
						nesterov_schedule_version, profile_out_error_variance_, momentum_offset, true);
					// Note: (i) cov_pars_after_grad_aux and cov_pars_after_grad_aux_lag1 correspond to the parameters obtained after calculating the gradient before applying acceleration
					//		 (ii) cov_pars (below this) are the parameters obtained after applying acceleration (and cov_pars_lag1 is simply the value of the previous iteration)
					// We first apply a gradient step and then an acceleration step (and not the other way aroung) since this is computationally more efficient 
					//		(otherwise the covariance matrix needs to be factored twice: once for the gradient step (accelerated parameters) and once for calculating the
					//		 log-likelihood (non-accelerated parameters after gradient update) when checking for convergence at the end of an iteration. 
					//		However, performing the acceleration before or after the gradient update gives equivalent algorithms
				}
				if (estimate_aux_pars_) {
					SetAuxPars(cov_pars_new.data() + num_cov_par_);
				}
				CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
				// Safeguard against too large steps by halving the learning rate when the objective increases
				if (armijo_condition_) {
					double mu;
					if (use_nesterov_acc) {
						mu = NesterovSchedule(it, nesterov_schedule_version, acc_rate_cov, momentum_offset);
					}
					else {
						mu = 0.;
					}
					if (estimate_aux_pars_) {
						if ((neg_log_likelihood_ <= (neg_log_likelihood_after_lin_coef_update_ +
							c_armijo_ * lr_cov * dir_deriv_armijo_cov_pars_ + c_armijo_mom_ * mu * mom_dir_deriv_armijo_cov_pars_)) &&
							(neg_log_likelihood_ <= (neg_log_likelihood_after_lin_coef_update_ +
								c_armijo_ * lr_aux_pars * dir_deriv_armijo_aux_pars_ + c_armijo_mom_ * mu * mom_dir_deriv_armijo_aux_pars_))) {
							decrease_found = true;
						}
					}
					else {//!estimate_aux_pars_
						if (neg_log_likelihood_ <= (neg_log_likelihood_after_lin_coef_update_ +
							c_armijo_ * lr_cov * dir_deriv_armijo_cov_pars_ + c_armijo_mom_ * mu * mom_dir_deriv_armijo_cov_pars_)) {
							decrease_found = true;
						}
					}
				}
				else {
					if (neg_log_likelihood_ <= neg_log_likelihood_after_lin_coef_update_) {
						decrease_found = true;
					}
				}
				if (decrease_found) {
					break;
				}
				else {
					halving_done = true;
					learning_rate_decreased_first_time_ = true;
					if (learning_rate_increased_after_descrease_) {
						learning_rate_decreased_after_increase_ = true;
					}
					lr_cov *= LR_SHRINKAGE_FACTOR_;
					if (estimate_aux_pars_) {
						lr_aux_pars *= LR_SHRINKAGE_FACTOR_;
					}
					acc_rate_cov *= 0.5;
					if (!gauss_likelihood_) {
						// Reset mode to previous value since also parameters are discarded
						for (const auto& cluster_i : unique_clusters_) {
							likelihood_[cluster_i]->ResetModeToPreviousValue();
						}
					}
				}
			}//end loop over learnig rate halving procedure
			if (halving_done) {
				if ((optimizer_cov_pars_ == "fisher_scoring" || optimizer_cov_pars_ == "newton") &&
					!learning_rate_constant_first_order_change_) {
					Log::REDebug("GPModel covariance parameter estimation: No decrease in the objective function in iteration number %d. "
						"The learning rate has been decreased in this iteration ", it + 1);
				}
				else if (optimizer_cov_pars_ == "gradient_descent") {
					lr_cov_ = lr_cov; //permanently decrease learning rate (for Fisher scoring, this is not done. I.e., step halving is done newly in every iterarion of Fisher scoring)
					if (estimate_aux_pars_) {
						lr_aux_pars_ = lr_aux_pars;
						Log::REDebug("GPModel covariance and auxiliary parameter estimation: Learning rates have been decreased permanently in iteration number %d "
							"since with the previous learning rates, there was no decrease in the objective function. "
							"New learning rates: covariance parameters = %g, auxiliary parameters = %g ", it + 1, lr_cov_, lr_aux_pars_);
					}
					else {
						Log::REDebug("GPModel covariance parameter estimation: The learning rate has been decreased permanently in iteration number %d "
							"since with the previous learning rate, there was no decrease in the objective function. New learning rate = %g", it + 1, lr_cov_);
					}
				}
			}
			if (!decrease_found) {
				Log::REDebug("GPModel covariance parameter estimation: No decrease in the objective function in iteration number %d "
					"after the maximal number of halving steps (%d) ", it + 1, max_number_lr_shrinkage_steps_);
			}
			if (use_nesterov_acc) {
				cov_pars_after_grad_aux_lag1 = cov_pars_after_grad_aux;
			}
			cov_pars = cov_pars_new;
		}//end UpdateCovAuxPars

		////Alternative version with separate learning rate descreases for cov_pars and aux_pars
		////	-> not necessarily more efficient as more likelihood evaluations are needed
		//void UpdateCovAuxPars(vec_t& cov_pars,
		//	const vec_t& neg_step_dir,
		//	bool profile_out_marginal_variance,
		//	bool use_nesterov_acc,
		//	int it,
		//	vec_t& cov_pars_after_grad_aux,
		//	vec_t& cov_pars_after_grad_aux_lag1,
		//	double acc_rate_cov,
		//	int nesterov_schedule_version,
		//	int momentum_offset,
		//	const double* fixed_effects) {
		//	vec_t cov_pars_new(num_cov_par_);
		//	if (profile_out_marginal_variance) {
		//		cov_pars_new[0] = cov_pars[0];
		//	}
		//	double lr_cov = lr_cov_;
		//	double lr_aux_pars = lr_aux_pars_;
		//	bool decrease_found = false;
		//	bool halving_done = false;
		//	int num_grad_cov_par = (int)neg_step_dir.size();
		//	if (estimate_aux_pars_) {
		//		num_grad_cov_par -= NumAuxPars();
		//	}
		//	for (int ih = 0; ih < max_number_lr_shrinkage_steps_; ++ih) {
		//		if (ih > 0) {
		//			halving_done = true;
		//			//learning_rate_decreased_first_time_ = true;
		//			//if (learning_rate_increased_after_descrease_) {
		//			//	learning_rate_decreased_after_increase_ = true;
		//			//}
		//			lr_cov *= LR_SHRINKAGE_FACTOR_;
		//			acc_rate_cov *= 0.5;
		//			if (!gauss_likelihood_) {
		//				// Reset mode to previous value since also parameters are discarded
		//				for (const auto& cluster_i : unique_clusters_) {
		//					likelihood_[cluster_i]->ResetModeToPreviousValue();
		//				}
		//			}
		//		}// end ih > 0
		//		UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//			profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//			acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//		if (estimate_aux_pars_) {
		//			SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//		}
		//		CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
		//		if (estimate_aux_pars_ && ih > 0) {
		//			// Undo learning rate decrease for cov_pars and decrease learning rate for aux_pars and check whether this leads to a smaller log-likelihood
		//			lr_cov /= LR_SHRINKAGE_FACTOR_;
		//			lr_aux_pars *= LR_SHRINKAGE_FACTOR_;
		//			UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//				profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//				acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//			SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//			double neg_log_likelihood_cov_par_decrease = neg_log_likelihood_;
		//			if (!gauss_likelihood_) {
		//				for (const auto& cluster_i : unique_clusters_) {
		//					likelihood_[cluster_i]->ResetModeToPreviousValue();
		//				}
		//			}
		//			CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
		//			if (neg_log_likelihood_cov_par_decrease < neg_log_likelihood_) {
		//				lr_cov *= LR_SHRINKAGE_FACTOR_;
		//				lr_aux_pars /= LR_SHRINKAGE_FACTOR_;
		//				UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//					profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//					acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//				SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//				neg_log_likelihood_ = neg_log_likelihood_cov_par_decrease;
		//				if (neg_log_likelihood_ <= neg_log_likelihood_after_lin_coef_update_) {
		//					// the following needs only be done in case the for loop stops, otherwise it is done at the beginning of the next iteration
		//					if (!gauss_likelihood_) {
		//						for (const auto& cluster_i : unique_clusters_) {
		//							likelihood_[cluster_i]->ResetModeToPreviousValue();
		//						}
		//					}
		//					CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
		//				}
		//			}
		//		}// end estimate_aux_pars_ && ih > 0
		//		// Safeguard against too large steps by halving the learning rate when the objective increases
		//		if (neg_log_likelihood_ <= neg_log_likelihood_after_lin_coef_update_) {
		//			decrease_found = true;
		//			break;
		//		}
		//	}//end loop over learnig rate halving procedure
		//	if (halving_done) {
		//		if (optimizer_cov_pars_ == "fisher_scoring") {
		//			Log::REDebug("GPModel covariance parameter estimation: No decrease in the objective function in iteration number %d. "
		//				"The learning rate has been decreased in this iteration.", it + 1);
		//		}
		//		else if (optimizer_cov_pars_ == "gradient_descent") {
		//			lr_cov_ = lr_cov; //permanently decrease learning rate (for Fisher scoring, this is not done. I.e., step halving is done newly in every iterarion of Fisher scoring)
		//			if (estimate_aux_pars_) {
		//				lr_aux_pars_ = lr_aux_pars;
		//				Log::REDebug("GPModel covariance and auxiliary parameter estimation: Learning rates have been decreased permanently in iteration number %d "
		//					"since with the previous learning rates, there was no decrease in the objective function. "
		//					"New learning rates: covariance parameters = %g, auxiliary parameters = %g", it + 1, lr_cov_, lr_aux_pars_);
		//			}
		//			else {
		//				Log::REDebug("GPModel covariance parameter estimation: The learning rate has been decreased permanently in iteration number %d "
		//					"since with the previous learning rate, there was no decrease in the objective function. New learning rate = %g", it + 1, lr_cov_);
		//			}
		//		}
		//	}
		//	if (!decrease_found) {
		//		Log::REDebug("GPModel covariance parameter estimation: No decrease in the objective function in iteration number %d "
		//			"after the maximal number of halving steps (%d).", it + 1, max_number_lr_shrinkage_steps_);
		//	}
		//	if (use_nesterov_acc) {
		//		cov_pars_after_grad_aux_lag1 = cov_pars_after_grad_aux;
		//	}
		//	cov_pars = cov_pars_new;
		//}//end UpdateCovAuxPars

		//void UpdateCovAuxParsInternal(vec_t& cov_pars,
		//	vec_t& cov_pars_new,
		//	const vec_t& neg_step_dir,
		//	int num_grad_cov_par,
		//	double lr_cov,
		//	double lr_aux_pars,
		//	bool profile_out_marginal_variance,
		//	bool use_nesterov_acc,
		//	int it,
		//	vec_t& cov_pars_after_grad_aux,
		//	vec_t& cov_pars_after_grad_aux_lag1,
		//	double acc_rate_cov,
		//	int nesterov_schedule_version,
		//	int momentum_offset) {
		//	vec_t update(neg_step_dir.size());
		//	update.segment(0, num_grad_cov_par) = lr_cov * neg_step_dir.segment(0, num_grad_cov_par);
		//	if (estimate_aux_pars_) {
		//		update.segment(num_grad_cov_par, NumAuxPars()) = lr_aux_pars * neg_step_dir.segment(num_grad_cov_par, NumAuxPars());
		//	}
		//	// Avoid to large steps on log-scale: updates on the log-scale in one Fisher scoring step are capped at a certain level
		//	// This is not done for gradient_descent since the learning rate is already adjusted accordingly in 'AvoidTooLargeLearningRatesCovAuxPars'
		//	if (optimizer_cov_pars_ != "gradient_descent") {
		//		for (int ip = 0; ip < (int)update.size(); ++ip) {
		//			if (update[ip] > MAX_GRADIENT_UPDATE_LOG_SCALE_) {
		//				update[ip] = MAX_GRADIENT_UPDATE_LOG_SCALE_;
		//			}
		//			else if (update[ip] < -MAX_GRADIENT_UPDATE_LOG_SCALE_) {
		//				update[ip] = -MAX_GRADIENT_UPDATE_LOG_SCALE_;
		//			}
		//		}
		//	}
		//	if (profile_out_marginal_variance) {
		//		cov_pars_new.segment(1, cov_pars.size() - 1) = (cov_pars.segment(1, cov_pars.size() - 1).array().log() - update.array()).exp().matrix();//make update on log-scale
		//	}
		//	else {
		//		cov_pars_new = (cov_pars.array().log() - update.array()).exp().matrix();//make update on log-scale
		//	}
		//	// Apply Nesterov acceleration
		//	if (use_nesterov_acc) {
		//		cov_pars_after_grad_aux = cov_pars_new;
		//		ApplyMomentumStep(it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1, cov_pars_new, acc_rate_cov,
		//			nesterov_schedule_version, profile_out_marginal_variance, momentum_offset, true);
		//		// Note: (i) cov_pars_after_grad_aux and cov_pars_after_grad_aux_lag1 correspond to the parameters obtained after calculating the gradient before applying acceleration
		//		//		 (ii) cov_pars (below this) are the parameters obtained after applying acceleration (and cov_pars_lag1 is simply the value of the previous iteration)
		//		// We first apply a gradient step and then an acceleration step (and not the other way aroung) since this is computationally more efficient 
		//		//		(otherwise the covariance matrix needs to be factored twice: once for the gradient step (accelerated parameters) and once for calculating the
		//		//		 log-likelihood (non-accelerated parameters after gradient update) when checking for convergence at the end of an iteration. 
		//		//		However, performing the acceleration before or after the gradient update gives equivalent algorithms
		//	}
		//}//end UpdateCovAuxParsInternal

		////Alternative version where learning rates are decreased until 
		//// (i) a decrease in the log-likelihood compared to the previous one is found and 
		//// (ii) subsequent decreases are only small.
		//// Problem: this does not necessarily lead to less log-likelihood evaluations compared to the above version
		//void UpdateCovAuxPars(vec_t& cov_pars,
		//	const vec_t& neg_step_dir,
		//	bool profile_out_marginal_variance,
		//	bool use_nesterov_acc,
		//	int it,
		//	vec_t& cov_pars_after_grad_aux,
		//	vec_t& cov_pars_after_grad_aux_lag1,
		//	double acc_rate_cov,
		//	int nesterov_schedule_version,
		//	int momentum_offset,
		//	const double* fixed_effects) {
		//	vec_t cov_pars_new(num_cov_par_);
		//	if (profile_out_marginal_variance) {
		//		cov_pars_new[0] = cov_pars[0];
		//	}
		//	double lr_cov = lr_cov_;
		//	double lr_aux_pars = lr_aux_pars_;
		//	bool decrease_found = false;
		//	bool halving_done = false;
		//	double cur_lowest_neg_log_like = neg_log_likelihood_after_lin_coef_update_;// currently lowest negative log-likelihood
		//	bool lr_cov_last_decreased = true;// indicates whether lr_cov or lr_aux_pars has been last descreased
		//	int num_grad_cov_par = (int)neg_step_dir.size();
		//	if (estimate_aux_pars_) {
		//		num_grad_cov_par -= NumAuxPars();
		//	}
		//	for (int ih = 0; ih < max_number_lr_shrinkage_steps_; ++ih) {
		//		if (ih > 0) {
		//			learning_rate_decreased_first_time_ = true;
		//			if (learning_rate_increased_after_descrease_) {
		//				learning_rate_decreased_after_increase_ = true;
		//			}
		//			acc_rate_cov *= 0.5;
		//			if (ih == 1 || lr_cov_last_decreased || !decrease_found) {
		//				//lr_cov is decreased until there is a decrease found and subsequently only if if has been decreased last
		//				lr_cov *= LR_SHRINKAGE_FACTOR_;
		//				if (!gauss_likelihood_) {
		//					// Reset mode to previous value since also parameters are discarded
		//					for (const auto& cluster_i : unique_clusters_) {
		//						likelihood_[cluster_i]->ResetModeToPreviousValue();
		//					}
		//				}
		//				lr_cov_last_decreased = true;
		//			}
		//		}//end ih > 0
		//		UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//			profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//			acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//		if (estimate_aux_pars_) {
		//			SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//		}
		//		CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
		//		if (estimate_aux_pars_ && ih > 0 && (ih == 1 || !lr_cov_last_decreased || !decrease_found)) {
		//			//lr_aux_pars is decreased until there is a decrease found and subsequently only if if has been decreased last
		//			if (lr_cov_last_decreased) {
		//				// Undo decrease for 'lr_cov', decrease 'lr_aux_pars', and check whether this leads to a smaller negative log-likelihood
		//				lr_cov /= LR_SHRINKAGE_FACTOR_;
		//			}
		//			lr_aux_pars *= LR_SHRINKAGE_FACTOR_;
		//			UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//				profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//				acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//			SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//			double neg_log_likelihood_cov_par_decrease = neg_log_likelihood_;
		//			if (!gauss_likelihood_) {
		//				for (const auto& cluster_i : unique_clusters_) {
		//					likelihood_[cluster_i]->ResetModeToPreviousValue();
		//				}
		//			}
		//			CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
		//			if (lr_cov_last_decreased && neg_log_likelihood_cov_par_decrease < neg_log_likelihood_) {
		//				// Better to decrease 'lr_cov': undo decrease in 'lr_aux_pars'
		//				lr_cov *= LR_SHRINKAGE_FACTOR_;
		//				lr_aux_pars /= LR_SHRINKAGE_FACTOR_;
		//				UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//					profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//					acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//				SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//				neg_log_likelihood_ = neg_log_likelihood_cov_par_decrease;
		//			}
		//			else {
		//				lr_cov_last_decreased = false;
		//			}
		//		}// end estimate_aux_pars_ && ih > 0
		//		if (neg_log_likelihood_ <= neg_log_likelihood_after_lin_coef_update_) {
		//			decrease_found = true;
		//		}
		//		// Stop trying more decreases when decrease is only small
		//		if (((cur_lowest_neg_log_like - neg_log_likelihood_) <= 2.) && decrease_found) {
		//			if (ih > 1) {
		//				acc_rate_cov /= 0.5;
		//				if (lr_cov_last_decreased) {
		//					// Undo decrease for 'lr_cov'
		//					lr_cov /= LR_SHRINKAGE_FACTOR_;
		//				}
		//				else {
		//					lr_aux_pars /= LR_SHRINKAGE_FACTOR_;
		//					SetAuxPars(cov_pars_new.data() + num_cov_par_);
		//				}
		//				UpdateCovAuxParsInternal(cov_pars, cov_pars_new, neg_step_dir, num_grad_cov_par, lr_cov, lr_aux_pars,
		//					profile_out_marginal_variance, use_nesterov_acc, it, cov_pars_after_grad_aux, cov_pars_after_grad_aux_lag1,
		//					acc_rate_cov, nesterov_schedule_version, momentum_offset);
		//				if (!gauss_likelihood_) {
		//					for (const auto& cluster_i : unique_clusters_) {
		//						likelihood_[cluster_i]->ResetModeToPreviousValue();
		//					}
		//				}
		//				CalcCovFactorOrModeAndNegLL(cov_pars_new.segment(0, num_cov_par_), fixed_effects);
		//				halving_done = true;
		//			}
		//			break;// ll increases again after a decrease -> stop
		//		}
		//		if (neg_log_likelihood_ <= cur_lowest_neg_log_like) {
		//			cur_lowest_neg_log_like = neg_log_likelihood_;
		//		}
		//	}//end loop over learnig rate halving procedure
		//	if (halving_done) {
		//		if (optimizer_cov_pars_ == "fisher_scoring") {
		//			Log::REDebug("GPModel covariance parameter estimation: No decrease in the objective function in iteration number %d. "
		//				"The learning rate has been decreased in this iteration.", it + 1);
		//		}
		//		else if (optimizer_cov_pars_ == "gradient_descent") {
		//			lr_cov_ = lr_cov; //permanently decrease learning rate (for Fisher scoring, this is not done. I.e., step halving is done newly in every iterarion of Fisher scoring)
		//			if (estimate_aux_pars_) {
		//				lr_aux_pars_ = lr_aux_pars;
		//				Log::REDebug("GPModel covariance and auxiliary parameter estimation: Learning rates have been decreased permanently in iteration number %d "
		//					"since with the previous learning rates, there was no decrease in the objective function. "
		//					"New learning rates: covariance parameters = %g, auxiliary parameters = %g", it + 1, lr_cov_, lr_aux_pars_);
		//			}
		//			else {
		//				Log::REDebug("GPModel covariance parameter estimation: The learning rate has been decreased permanently in iteration number %d "
		//					"since with the previous learning rate, there was no decrease in the objective function. New learning rate = %g", it + 1, lr_cov_);
		//			}
		//		}
		//	}
		//	if (!decrease_found) {
		//		Log::REDebug("GPModel covariance parameter estimation: No decrease in the objective function in iteration number %d "
		//			"after the maximal number of halving steps (%d).", it + 1, max_number_lr_shrinkage_steps_);
		//	}
		//	if (use_nesterov_acc) {
		//		cov_pars_after_grad_aux_lag1 = cov_pars_after_grad_aux;
		//	}
		//	cov_pars = cov_pars_new;
		//}//end UpdateCovAuxPars

		/*!
		* \brief Update linear regression coefficients and apply step size safeguard
		* \param[out] beta Linear regression coefficients
		* \param grad Gradient
		* \param sigma2 Nugget / error term variance for Gaussian likelihoods
		* \param use_nesterov_acc If true, Nesterov acceleration is used
		* \param it Iteration number
		* \param[out] beta_after_grad_aux Auxiliary variable used only if use_nesterov_acc == true (see the code below for a description)
		* \param[out] beta_after_grad_aux_lag1 Auxiliary variable used only if use_nesterov_acc == true (see the code below for a description)
		* \param acc_rate_coef Nesterov acceleration speed
		* \param nesterov_schedule_version Which version of Nesterov schedule should be used. Default = 0
		* \param momentum_offset Number of iterations for which no mometum is applied in the beginning
		* \param fixed_effects External fixed effects
		* \param[out] fixed_effects_vec Fixed effects component of location parameter as sum of linear predictor and potentiall additional external fixed effects
		*/
		void UpdateLinCoef(vec_t& beta,
			const vec_t& grad,
			const double sigma2,
			bool use_nesterov_acc,
			int it,
			vec_t& beta_after_grad_aux,
			vec_t& beta_after_grad_aux_lag1,
			double acc_rate_coef,
			int nesterov_schedule_version,
			int momentum_offset,
			const double* fixed_effects,
			vec_t& fixed_effects_vec) {
			vec_t beta_new;
			double lr_coef = lr_coef_;
			bool decrease_found = false;
			bool halving_done = false;
			if (it == 0) {
				first_update_ = true;
			}
			else {
				first_update_ = false;
			}
			for (int ih = 0; ih < max_number_lr_shrinkage_steps_; ++ih) {
				beta_new = beta - lr_coef * grad;
				// Apply Nesterov acceleration
				if (use_nesterov_acc) {
					beta_after_grad_aux = beta_new;
					ApplyMomentumStep(it, beta_after_grad_aux, beta_after_grad_aux_lag1, beta_new, acc_rate_coef,
						nesterov_schedule_version, false, momentum_offset, false);
					//Note: use same version of Nesterov acceleration as for covariance parameters (see 'UpdateCovAuxPars')
				}
				UpdateFixedEffects(beta_new, fixed_effects, fixed_effects_vec);
				if (gauss_likelihood_) {
					EvalNegLogLikelihoodOnlyUpdateFixedEffects(sigma2, neg_log_likelihood_after_lin_coef_update_);
				}//end if gauss_likelihood_
				else {//non-Gaussian likelihoods
					neg_log_likelihood_after_lin_coef_update_ = -CalcModePostRandEffCalcMLL(fixed_effects_vec.data(), true);//calculate mode and approximate marginal likelihood
				}
				// Safeguard against too large steps by halving the learning rate when the objective increases
				if (armijo_condition_) {
					double mu;
					if (use_nesterov_acc) {
						mu = NesterovSchedule(it, nesterov_schedule_version, acc_rate_coef, momentum_offset);
					}
					else {
						mu = 0.;
					}
					if (neg_log_likelihood_after_lin_coef_update_ <= (neg_log_likelihood_lag1_ +
						c_armijo_ * lr_coef * dir_deriv_armijo_coef_ + c_armijo_mom_ * mu * mom_dir_deriv_armijo_coef_)) {
						decrease_found = true;
					}
				}
				else {
					if (neg_log_likelihood_after_lin_coef_update_ <= neg_log_likelihood_lag1_) {
						decrease_found = true;
					}
				}
				if (decrease_found) {
					break;
				}
				else {
					// Safeguard against too large steps by halving the learning rate
					halving_done = true;
					learning_rate_decreased_first_time_ = true;
					if (learning_rate_increased_after_descrease_) {
						learning_rate_decreased_after_increase_ = true;
					}
					lr_coef *= LR_SHRINKAGE_FACTOR_;
					acc_rate_coef *= 0.5;
					if (!gauss_likelihood_) {
						// Reset mode to previous value since also parameters are discarded
						for (const auto& cluster_i : unique_clusters_) {
							likelihood_[cluster_i]->ResetModeToPreviousValue();
						}
					}
				}
			}
			if (halving_done) {
				lr_coef_ = lr_coef; //permanently decrease learning rate
				Log::REDebug("GPModel: The learning rate for the regression coefficients has been decreased permanently since with the previous learning rate, "
					"there was no decrease in the objective function in iteration number %d. New learning rate = %g", it + 1, lr_coef_);
			}
			if (!decrease_found) {
				Log::REDebug("GPModel linear regression coefficient estimation: No decrease in the objective function "
					"in iteration number %d after the maximal number of halving steps (%d).", it + 1, max_number_lr_shrinkage_steps_);
			}
			if (use_nesterov_acc) {
				beta_after_grad_aux_lag1 = beta_after_grad_aux;
			}
			beta = beta_new;
		}//end UpdateLinCoef

		/*!
		* \brief Calculate the covariance matrix ZSigmaZt of the random effects (sum of all components)
		* \param[out] ZSigmaZt Covariance matrix ZSigmaZt
		* \param cluster_i Cluster index for which the covariance matrix is calculated
		*/
		void CalcZSigmaZt(T_mat& ZSigmaZt, data_size_t cluster_i) {
			ZSigmaZt = T_mat(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
			if (gauss_likelihood_) {
				ZSigmaZt.setIdentity();
			}
			else {
				ZSigmaZt.setZero();
			}
			for (int j = 0; j < num_comps_total_; ++j) {
				ZSigmaZt += (*(re_comps_[cluster_i][0][j]->GetZSigmaZt()));
			}
		}//end CalcZSigmaZt

		/*!
		* \brief Calculate the mode of the posterior of the latent random effects and the Laplace-approximated marginal log-likelihood. This function is only used for non-Gaussian likelihoods
		* \param fixed_effects Fixed effects component of location parameter
		* \param calc_mll If true the marginal log-likelihood is also calculated (only relevant for (gp_approx_ == "vecchia" || only grouped random effects) && matrix_inversion_method_ == "iterative")
		* \return Approximate marginal log-likelihood evaluated at the mode
		*/
		double CalcModePostRandEffCalcMLL(const double* fixed_effects,
			bool calc_mll) {
			double mll = 0.;
			double mll_cluster_i;
			const double* fixed_effects_cluster_i_ptr = nullptr;
			vec_t fixed_effects_cluster_i;
			for (const auto& cluster_i : unique_clusters_) {
				if (num_clusters_ == 1 && ((gp_approx_ != "vecchia" && gp_approx_ != "full_scale_vecchia") || vecchia_ordering_ == "none")) {//only one cluster / independent realization and order of data does not matter
					fixed_effects_cluster_i_ptr = fixed_effects;
				}
				else if (fixed_effects != nullptr) {//more than one cluster and order of samples matters
					fixed_effects_cluster_i = vec_t(num_data_per_cluster_[cluster_i] * num_sets_re_);
					//Note: this is quite inefficient as the mapping of the fixed_effects to the different clusters is done repeatedly for the same data. Could be saved if performance is an issue here
					for (int igp = 0; igp < num_sets_re_; ++igp) {
#pragma omp parallel for schedule(static)
						for (int j = 0; j < num_data_per_cluster_[cluster_i]; ++j) {
							fixed_effects_cluster_i[j + num_data_per_cluster_[cluster_i] * igp] = fixed_effects[data_indices_per_cluster_[cluster_i][j] + num_data_ * igp];
						}
					}
					fixed_effects_cluster_i_ptr = fixed_effects_cluster_i.data();
				}
				if (gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_) {
					den_mat_t Sigma_L_k;
					if (matrix_inversion_method_ == "iterative" && cg_preconditioner_type_ == "pivoted_cholesky") {
						//Do pivoted Cholesky decomposition for Sigma
						//TODO: only after cov-pars step, not after fixed-effect step
						PivotedCholsekyFactorizationSigma(re_comps_vecchia_[cluster_i][0][0].get(), Sigma_L_k, fitc_piv_chol_preconditioner_rank_, PIV_CHOL_STOP_TOL);
					}
					likelihood_[cluster_i]->FindModePostRandEffCalcMLLVecchia(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, B_[cluster_i], D_inv_[cluster_i],
						first_update_, Sigma_L_k, calc_mll, mll_cluster_i, re_comps_ip_preconditioner_[cluster_i][0],
						re_comps_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0],
						cluster_i, this);
				}
				else if (gp_approx_ == "full_scale_vecchia") {
					if (num_comps_total_ > 1) {
						Log::REFatal("'full_scale_vecchia' is currently not implemented when having more than one GP ");
					}
					likelihood_[cluster_i]->FindModePostRandEffCalcMLLFSVA(y_[cluster_i].data(), y_int_[cluster_i].data(), fixed_effects_cluster_i_ptr, *(re_comps_ip_[cluster_i][0][0]->GetZSigmaZt()),
						chol_fact_sigma_ip_[cluster_i][0], chol_fact_sigma_woodbury_[cluster_i], chol_ip_cross_cov_[cluster_i][0], re_comps_cross_cov_[cluster_i][0], sigma_woodbury_[cluster_i],
						B_[cluster_i][0], D_inv_[cluster_i][0], B_T_D_inv_B_cross_cov_[cluster_i][0], D_inv_B_cross_cov_[cluster_i][0], first_update_, calc_mll, mll_cluster_i,
						re_comps_ip_preconditioner_[cluster_i][0], re_comps_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_fact_sigma_ip_preconditioner_[cluster_i][0], GPU_use_);
				}
				else if (gp_approx_ == "fitc") {
					if (num_comps_total_ > 1) {
						Log::REFatal("'fitc' is currently not implemented when having more than one GP ");
					}
					likelihood_[cluster_i]->FindModePostRandEffCalcMLLFITC(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, re_comps_ip_[cluster_i][0][0]->GetZSigmaZt(),
						chol_fact_sigma_ip_[cluster_i][0], re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr(),
						fitc_resid_diag_[cluster_i], mll_cluster_i, GPU_use_);
				}
				else if ((use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) || grouped_RE_and_vecchia_GP_) {
					likelihood_[cluster_i]->FindModePostRandEffCalcMLLGroupedRE(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, SigmaI_[cluster_i], grouped_RE_and_vecchia_GP_, B_[cluster_i][0], D_inv_[cluster_i][0],
						first_update_, calc_mll, mll_cluster_i);
				}
				else if (only_one_grouped_RE_calculations_on_RE_scale_) {
					likelihood_[cluster_i]->FindModePostRandEffCalcMLLOnlyOneGroupedRECalculationsOnREScale(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, re_comps_[cluster_i][0][0]->cov_pars_[0], mll_cluster_i);
				}
				else {
					likelihood_[cluster_i]->FindModePostRandEffCalcMLLStable(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, ZSigmaZt_[cluster_i], mll_cluster_i);
					//Note: if(only_one_GP_calculations_on_RE_scale_), ZSigmaZt_[cluster_i] contains Sigma=Cov(b) and not Z*Sigma*Zt since has_Z_==false for this random effects component
				}
				mll += mll_cluster_i;
			}
			num_ll_evaluations_++;
			return(mll);
		}//CalcModePostRandEffCalcMLL

		/*!
		* \brief Calculate covariance matrices and, for "gaussian" likelihood, factorize them (either calculate a Cholesky factor or the inverse covariance matrix)
		* \param transf_scale If true, the derivatives are taken on the transformed scale otherwise on the original scale (only for Vecchia approximation)
		* \param nugget_var Nugget effect variance parameter sigma^2 (used only if gp_approx_ == "vecchia" and transf_scale == false to transform back, normally this is equal to one, since the variance paramter is modelled separately and factored out)
		*/
		void CalcCovFactor(bool transf_scale,
			double nugget_var) {
			if (gp_approx_ == "vecchia" || gp_approx_ == "full_scale_vecchia") {
				if (gp_approx_ == "full_scale_vecchia") {
					CalcSigmaComps();
				}
				CalcCovFactorVecchia(transf_scale, nugget_var);
				if (!gauss_likelihood_ && matrix_inversion_method_ == "iterative" && cg_preconditioner_type_ == "fitc") {
					Calc_FITC_Preconditioner_Vecchia();
				}
				if (gp_approx_ == "full_scale_vecchia" && !gauss_likelihood_) {
					CalcCovFactorFITC_FSA();
				}
				if (grouped_RE_and_vecchia_GP_) {
					CHECK(num_sets_re_ == 1);
					for (const auto& cluster_i : unique_clusters_) {
						CalcSigmaOrInverseUseWoodbury(SigmaI_[cluster_i], cluster_i, true);
					}
				}
			}
			if (gp_approx_ != "vecchia") {
				if (gp_approx_ != "full_scale_vecchia") {
					CalcSigmaComps();
				}
				if (gauss_likelihood_) {
					if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
						if (cg_preconditioner_type_ == "fitc" && matrix_inversion_method_ == "iterative") {
							if (gp_approx_ == "full_scale_tapering" || fitc_piv_chol_preconditioner_rank_ == num_ind_points_) {
								for (const auto& cluster_i : unique_clusters_) {
									re_comps_ip_preconditioner_[cluster_i][0] = re_comps_ip_[cluster_i][0];
									re_comps_cross_cov_preconditioner_[cluster_i][0] = re_comps_cross_cov_[cluster_i][0];
									chol_fact_sigma_ip_preconditioner_[cluster_i][0] = chol_fact_sigma_ip_[cluster_i][0];
									chol_ip_cross_cov_preconditioner_[cluster_i][0] = chol_ip_cross_cov_[cluster_i][0];
								}
							}
						}
						CalcCovFactorFITC_FSA();
					}//end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia"
					else {// gp_approx_ != "vecchia" && gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering"
						for (const auto& cluster_i : unique_clusters_) {
							if (use_woodbury_identity_) {//Use Woodburry matrix inversion formula: used only if there are only grouped REs
								if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
									CalcSigmaOrInverseUseWoodbury(SigmaI_[cluster_i], cluster_i, true);
									sqrt_diag_SigmaI_plus_ZtZ_[cluster_i] = (SigmaI_[cluster_i].diagonal().array() + ZtZ_[cluster_i].diagonal().array()).sqrt().matrix();
								}
								else {
									sp_mat_t SigmaI;
									CalcSigmaOrInverseUseWoodbury(SigmaI, cluster_i, true);
									if (matrix_inversion_method_ == "cholesky") {
										T_mat SigmaIplusZtZ = SigmaI + ZtZ_[cluster_i];
										CalcChol(SigmaIplusZtZ, cluster_i);
									}// end cholesky
									else if (matrix_inversion_method_ == "iterative") {// Contruct preconditioners and preprations
										SigmaI_plus_ZtZ_rm_[cluster_i] = sp_mat_rm_t(SigmaI + ZtZ_[cluster_i]);
										if (cg_preconditioner_type_ == "incomplete_cholesky") {
											ZeroFillInIncompleteCholeskyFactorization(SigmaI_plus_ZtZ_rm_[cluster_i], L_SigmaI_plus_ZtZ_rm_[cluster_i]);
										}
										else if (cg_preconditioner_type_ == "ssor") {
											P_SSOR_D_inv_[cluster_i] = SigmaI_plus_ZtZ_rm_[cluster_i].diagonal().cwiseInverse(); //store for gradient calculation
											vec_t P_SSOR_D_inv_sqrt = P_SSOR_D_inv_[cluster_i].cwiseSqrt(); //need to store this, otherwise slow
											sp_mat_rm_t P_SSOR_L_rm = SigmaI_plus_ZtZ_rm_[cluster_i].template triangularView<Eigen::Lower>();
											P_SSOR_L_D_sqrt_inv_rm_[cluster_i] = P_SSOR_L_rm * P_SSOR_D_inv_sqrt.asDiagonal();
										}
										else if (cg_preconditioner_type_ == "diagonal") {
											SigmaI_plus_ZtZ_inv_diag_[cluster_i] = SigmaI_plus_ZtZ_rm_[cluster_i].diagonal().cwiseInverse();
										}
										else if (cg_preconditioner_type_ != "none") {
											Log::REFatal("Preconditioner type '%s' is not supported ", cg_preconditioner_type_.c_str());
										}
									}//end iterative
									else {
										Log::REFatal("Matrix inversion method '%s' is not supported ", matrix_inversion_method_.c_str());
									}
								}
							}//end use_woodbury_identity_
							else {//not use_woodbury_identity_
								T_mat psi;
								CalcZSigmaZt(psi, cluster_i);
								CalcChol(psi, cluster_i);
							}//end not use_woodbury_identity_
						}//end loop cluster_i
					}
				}//end gauss_likelihood_
			}//end gp_approx_ != "vecchia"
			if (gauss_likelihood_) {
				covariance_matrix_has_been_factorized_ = true;
				num_ll_evaluations_++;//note: for non-Gaussian likelihoods, a call to 'CalcModePostRandEffCalcMLL' (=finding the mode for the Laplace approximation) is counted as a likelihood evaluation
			}
		}//end CalcCovFactor

		/*!
		* \brief Calculate matrices A and D_inv for Vecchia approximation
		* \param transf_scale If true, the derivatives are taken on the transformed scale otherwise on the original scale
		* \param nugget_var Nugget effect variance parameter sigma^2 (used only if transf_scale == false to transform back, normally this is equal to one, since the variance paramter is modelled separately and factored out)
		*/
		void CalcCovFactorVecchia(bool transf_scale,
			double nugget_var) {
			cov_factor_vecchia_calculated_on_transf_scale_ = transf_scale;
			for (int igp = 0; igp < num_sets_re_; ++igp) {
				for (const auto& cluster_i : unique_clusters_) {
					data_size_t num_re_cluster_i = re_comps_vecchia_[cluster_i][igp][0]->GetNumUniqueREs();
					CalcCovFactorGradientVecchia(num_re_cluster_i, true, false, re_comps_vecchia_[cluster_i][igp],
						re_comps_cross_cov_[cluster_i][0], re_comps_ip_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0], nearest_neighbors_[cluster_i][igp],
						dist_obs_neighbors_[cluster_i][igp], dist_between_neighbors_[cluster_i][igp],
						entries_init_B_[cluster_i][igp], z_outer_z_obs_neighbors_[cluster_i][igp],
						B_[cluster_i][igp], D_inv_[cluster_i][igp], B_grad_[cluster_i][igp], D_grad_[cluster_i][igp], sigma_ip_inv_cross_cov_T_[cluster_i][0],
						sigma_ip_grad_sigma_ip_inv_cross_cov_T_[cluster_i][0], transf_scale, nugget_var,
						gauss_likelihood_, num_gp_total_, gauss_likelihood_, save_distances_isotropic_cov_fct_Vecchia_, gp_approx_,
						nullptr, estimate_cov_par_index_, nearest_neighbors_determined_, false);
					if (gp_approx_ == "full_scale_vecchia") {
						//Convert to row-major for parallelization
						B_rm_[cluster_i][0] = sp_mat_rm_t(B_[cluster_i][0]);
						D_inv_rm_[cluster_i][0] = sp_mat_rm_t(D_inv_[cluster_i][0]);
						//B_t_D_inv_rm_[cluster_i][0] = B_rm_[cluster_i][0].transpose() * D_inv_rm_[cluster_i][0];
						GPBoost::spmatmul(sp_mat_rm_t(B_rm_[cluster_i][0].transpose()), D_inv_rm_[cluster_i][0], B_t_D_inv_rm_[cluster_i][0], GPU_use_);
					}
				}
			}
		}//end CalcCovFactorVecchia

		void Calc_FITC_Preconditioner_Vecchia() {
			CHECK(!gauss_likelihood_ && matrix_inversion_method_ == "iterative" && cg_preconditioner_type_ == "fitc");
			for (const auto& cluster_i : unique_clusters_) {
				std::shared_ptr<RECompGP<den_mat_t>> re_comp_gp_clus0 = re_comps_vecchia_[cluster_i][0][0];
				if (!ind_points_determined_for_preconditioner_) {
					std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_ip_cluster_i;
					std::vector<std::shared_ptr<RECompGP<den_mat_t>>> re_comps_cross_cov_cluster_i;
					int num_ind_points = fitc_piv_chol_preconditioner_rank_;
					den_mat_t gp_coords_all_mat = re_comp_gp_clus0->GetCoords();
					data_size_t num_data_vecchia = (data_size_t)gp_coords_all_mat.rows();
					if (num_data_per_cluster_[cluster_i] <= num_ind_points) {
						Log::REFatal("Need to have less inducing points (currently fitc_piv_chol_preconditioner_rank = %d) than data points (%d) for cg_preconditioner_type = '%s' ",
							num_ind_points, num_data_per_cluster_[cluster_i], cg_preconditioner_type_.c_str());
					}
					if (num_data_vecchia < num_data_per_cluster_[cluster_i]) {
						if ((int)num_data_vecchia < num_ind_points) {
							Log::REFatal("Cannot have more inducing points than unique coordinates for cg_preconditioner_type = '%s' ", cg_preconditioner_type_.c_str());
						}
					}
					// Determine inducing points on unique locataions
					den_mat_t gp_coords_all_unique;
					std::vector<int> uniques;//unique points
					std::vector<int> unique_idx;//not used
					DetermineUniqueDuplicateCoordsFast(gp_coords_all_mat, num_data_vecchia, uniques, unique_idx);
					if ((data_size_t)uniques.size() == num_data_vecchia) {//no multiple observations at the same locations -> no incidence matrix needed
						gp_coords_all_unique = gp_coords_all_mat;
					}
					else {//there are multiple observations at the same locations
						gp_coords_all_unique = gp_coords_all_mat(uniques, Eigen::all);
						if ((int)gp_coords_all_unique.rows() < num_ind_points) {
							Log::REFatal("Cannot have more inducing points than unique coordinates for cg_preconditioner_type = '%s' ", cg_preconditioner_type_.c_str());
						}
					}
					std::vector<int> indices;
					den_mat_t gp_coords_ip_mat;
					if (ind_points_selection_ == "cover_tree") {
						Log::REDebug("Starting cover tree algorithm for determining inducing points ");
						CoverTree(gp_coords_all_unique, cover_tree_radius_, rng_, gp_coords_ip_mat);
						Log::REDebug("Inducing points have been determined ");
						num_ind_points = (int)gp_coords_ip_mat.rows();
					}
					else if (ind_points_selection_ == "random") {
						SampleIntNoReplaceSort((int)gp_coords_all_unique.rows(), num_ind_points, rng_, indices);
						gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_mat.cols());
						for (int j = 0; j < num_ind_points; ++j) {
							gp_coords_ip_mat.row(j) = gp_coords_all_unique.row(indices[j]);
						}
					}
					else if (ind_points_selection_ == "kmeans++" || ind_points_selection_ == "space_time_kmeans++") {
						gp_coords_ip_mat.resize(num_ind_points, gp_coords_all_mat.cols());
						int max_it_kmeans = 1000;
						Log::REDebug("Starting kmeans++ algorithm for determining inducing points ");
						kmeans_plusplus(gp_coords_all_unique, num_ind_points, rng_, gp_coords_ip_mat, max_it_kmeans, false);
						Log::REDebug("Inducing points have been determined ");
					}
					else {
						Log::REFatal("Method '%s' is not supported for finding inducing points ", ind_points_selection_.c_str());
					}
					gp_coords_all_unique.resize(0, 0);
					std::shared_ptr<RECompGP<den_mat_t>> gp_ip(new RECompGP<den_mat_t>(
						gp_coords_ip_mat, re_comp_gp_clus0->CovFunctionName(), re_comp_gp_clus0->CovFunctionShape(), re_comp_gp_clus0->CovFunctionTaperRange(), re_comp_gp_clus0->CovFunctionTaperShape(),
						false, false, true, false, false, true));
					if (gp_ip->HasDuplicatedCoords()) {
						Log::REFatal("Duplicates found in inducing points / low-dimensional knots ");
					}
					re_comps_ip_cluster_i.push_back(gp_ip);
					bool only_one_GP_calculations_on_RE_scale_loc = num_gp_total_ == 1 && num_comps_total_ == 1 && !gauss_likelihood_;
					re_comps_cross_cov_cluster_i.push_back(std::shared_ptr<RECompGP<den_mat_t>>(new RECompGP<den_mat_t>(
						gp_coords_all_mat, gp_coords_ip_mat, re_comp_gp_clus0->CovFunctionName(), re_comp_gp_clus0->CovFunctionShape(), re_comp_gp_clus0->CovFunctionTaperRange(), re_comp_gp_clus0->CovFunctionTaperShape(),
						false, false, only_one_GP_calculations_on_RE_scale_loc)));
					re_comps_ip_preconditioner_[cluster_i][0] = re_comps_ip_cluster_i;
					re_comps_cross_cov_preconditioner_[cluster_i][0] = re_comps_cross_cov_cluster_i;
					ind_points_determined_for_preconditioner_ = true;
				}
				vec_t pars = re_comp_gp_clus0->CovPars();
				for (int j = 0; j < num_comps_total_; ++j) {
					re_comps_ip_preconditioner_[cluster_i][0][j]->SetCovPars(pars);
					re_comps_cross_cov_preconditioner_[cluster_i][0][j]->SetCovPars(pars);
					re_comps_ip_preconditioner_[cluster_i][0][j]->CalcSigma();
					re_comps_cross_cov_preconditioner_[cluster_i][0][j]->CalcSigma();
					den_mat_t sigma_ip_stable = *(re_comps_ip_preconditioner_[cluster_i][0][j]->GetZSigmaZt());
					sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
					chol_fact_sigma_ip_preconditioner_[cluster_i][0].compute(sigma_ip_stable);
					const den_mat_t* cross_cov_p = re_comps_cross_cov_preconditioner_[cluster_i][0][j]->GetSigmaPtr();
					chol_ip_cross_cov_preconditioner_[cluster_i][0] = (*cross_cov_p).transpose();
					//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_preconditioner_[cluster_i][0],
					//	chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], false);
					GPBoost::solve_lower_triangular(chol_fact_sigma_ip_preconditioner_[cluster_i][0],
						chol_ip_cross_cov_preconditioner_[cluster_i][0], chol_ip_cross_cov_preconditioner_[cluster_i][0], GPU_use_);
				}
			}//end loop over unique_clusters_
		}//end Calc_FITC_Preconditioner_Vecchia

		/*!
		* \brief Calculate gradients of matrices A and D_inv for Vecchia approximation
		* \param transf_scale If true, the derivatives are taken on the transformed scale otherwise on the original scale
		* \param nugget_var Nugget effect variance parameter sigma^2 (used only if transf_scale == false to transform back, normally this is equal to one, since the variance paramter is modelled separately and factored out)
		* \param calc_gradient_nugget If true, derivatives are also taken with respect to the nugget / noise variance
		*/
		void CalcGradientVecchia(bool transf_scale,
			double nugget_var,
			bool calc_gradient_nugget) {
			CHECK(cov_factor_vecchia_calculated_on_transf_scale_ == transf_scale);
			std::vector<int> estimate_cov_par_index_gp;
			if (grouped_RE_and_vecchia_GP_) {
				estimate_cov_par_index_gp = estimate_cov_par_index_;
				const int start = ind_par_[0];
				const int end_excl = ind_par_[num_re_group_total_];
				estimate_cov_par_index_gp.erase(estimate_cov_par_index_gp.begin() + start, estimate_cov_par_index_gp.begin() + end_excl);
			}
			else {
				estimate_cov_par_index_gp = estimate_cov_par_index_;
			}
			bool exclude_marg_var_grad = !gauss_likelihood_ && (re_comps_vecchia_[unique_clusters_[0]][0].size() == 1) &&
				!(gp_approx_ == "full_scale_vecchia");//gradient is not needed if there is only one GP for non-Gaussian likelihoods
			for (const auto& cluster_i : unique_clusters_) {
				for (int igp = 0; igp < num_sets_re_; ++igp) {
					data_size_t num_re_cluster_i = re_comps_vecchia_[cluster_i][igp][0]->GetNumUniqueREs();
					CalcCovFactorGradientVecchia(num_re_cluster_i, false, true, re_comps_vecchia_[cluster_i][igp],
						re_comps_cross_cov_[cluster_i][0], re_comps_ip_[cluster_i][0], chol_fact_sigma_ip_[cluster_i][0], chol_ip_cross_cov_[cluster_i][0], nearest_neighbors_[cluster_i][igp],
						dist_obs_neighbors_[cluster_i][igp], dist_between_neighbors_[cluster_i][igp],
						entries_init_B_[cluster_i][igp], z_outer_z_obs_neighbors_[cluster_i][igp],
						B_[cluster_i][igp], D_inv_[cluster_i][igp], B_grad_[cluster_i][igp], D_grad_[cluster_i][igp], sigma_ip_inv_cross_cov_T_[cluster_i][0],
						sigma_ip_grad_sigma_ip_inv_cross_cov_T_[cluster_i][0], transf_scale, nugget_var,
						calc_gradient_nugget, num_gp_total_, gauss_likelihood_, save_distances_isotropic_cov_fct_Vecchia_, gp_approx_,
						nullptr, estimate_cov_par_index_gp, nearest_neighbors_determined_, exclude_marg_var_grad);
				}
			}
		}//end CalcGradientVecchia

		/*!
		* \brief Calculate cholesky factor of Woodbury matrix for fitc and full scale approximations
		*/
		void CalcCovFactorFITC_FSA() {
			for (const auto& cluster_i : unique_clusters_) {
				// factorize matrix used in Woodbury identity
				if (matrix_inversion_method_ == "iterative") {
					if (gp_approx_ == "fitc") {
						Log::REFatal("'iterative' methods are not implemented for gp_approx = '%s'. Use 'cholesky' ", gp_approx_.c_str());
					}
					else if (gp_approx_ == "full_scale_vecchia") {
						const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr();
						den_mat_t sigma_ip_stable = *(re_comps_ip_[cluster_i][0][0]->GetZSigmaZt());
						sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
						if (gauss_likelihood_) {
							Log::REFatal("The iterative methods are not implemented for the Full-Scale-Vecchia approximation with Gaussian likelihood. Please use Cholesky.");
						}
						D_inv_B_cross_cov_[cluster_i][0].resize(num_data_per_cluster_[cluster_i], num_ind_points_);
						B_cross_cov_[cluster_i][0].resize(num_data_per_cluster_[cluster_i], num_ind_points_);
						B_T_D_inv_B_cross_cov_[cluster_i][0].resize(num_data_per_cluster_[cluster_i], num_ind_points_);
#pragma omp parallel for schedule(static)   
						for (int i = 0; i < num_ind_points_; ++i) {
							B_cross_cov_[cluster_i][0].col(i) = B_rm_[cluster_i][0] * (*cross_cov).col(i);
							D_inv_B_cross_cov_[cluster_i][0].col(i) = D_inv_rm_[cluster_i][0] * B_cross_cov_[cluster_i][0].col(i);
							B_T_D_inv_B_cross_cov_[cluster_i][0].col(i) = B_t_D_inv_rm_[cluster_i][0] * B_cross_cov_[cluster_i][0].col(i);
						}
						//den_mat_t sigma_woodbury = B_cross_cov_[cluster_i][0].transpose() * D_inv_B_cross_cov_[cluster_i][0];
						den_mat_t sigma_woodbury;
						GPBoost::matmul(B_cross_cov_[cluster_i][0].transpose(), D_inv_B_cross_cov_[cluster_i][0], sigma_woodbury, GPU_use_);
						sigma_woodbury += sigma_ip_stable;
						sigma_woodbury_[cluster_i] = sigma_woodbury;
						chol_fact_sigma_woodbury_[cluster_i].compute(sigma_woodbury);
					}
					else if (gp_approx_ == "full_scale_tapering") {
						if (cg_preconditioner_type_ == "fitc") {
							const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
							den_mat_t sigma_ip_stable_preconditioner = *(re_comps_ip_preconditioner_[cluster_i][0][0]->GetZSigmaZt());
							den_mat_t sigma_woodbury_preconditioner;// sigma_woodbury = sigma_ip + cross_cov^T * sigma_resid^-1 * cross_cov or for Preconditioner sigma_ip + cross_cov^T * D^-1 * cross_cov
							std::shared_ptr<T_mat> sigma_resid;
							sigma_resid = re_comps_resid_[cluster_i][0][0]->GetZSigmaZt();
							diagonal_approx_preconditioner_[cluster_i] = (*sigma_resid).diagonal();
							sigma_ip_stable_preconditioner.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
							diagonal_approx_inv_preconditioner_[cluster_i] = diagonal_approx_preconditioner_[cluster_i].cwiseInverse();
							//sigma_woodbury_preconditioner = (*cross_cov_preconditioner).transpose() * (diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * (*cross_cov_preconditioner));
							den_mat_t D_cross_cov = diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * (*cross_cov_preconditioner);
							GPBoost::matmul((*cross_cov_preconditioner).transpose(), D_cross_cov, sigma_woodbury_preconditioner, GPU_use_);
							sigma_woodbury_preconditioner += sigma_ip_stable_preconditioner;
							chol_fact_woodbury_preconditioner_[cluster_i].compute(sigma_woodbury_preconditioner);
						}
						else if (cg_preconditioner_type_ != "none") {
							Log::REFatal("Preconditioner type '%s' is not supported for gp_approx = '%s' and likelihood = '%s'",
								cg_preconditioner_type_.c_str(), gp_approx_.c_str(), (likelihood_[unique_clusters_[0]]->GetLikelihood()).c_str());
						}

					}
				}
				else if (matrix_inversion_method_ == "cholesky") {
					const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr();
					den_mat_t sigma_ip_stable = *(re_comps_ip_[cluster_i][0][0]->GetZSigmaZt());
					sigma_ip_stable.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;
					den_mat_t sigma_woodbury;// sigma_woodbury = sigma_ip + cross_cov^T * sigma_resid^-1 * cross_cov or for Preconditioner sigma_ip + cross_cov^T * D^-1 * cross_cov
					if (gp_approx_ == "fitc") {
						GPBoost::matmul((*cross_cov).transpose(), fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * (*cross_cov), sigma_woodbury, GPU_use_);
						//sigma_woodbury = ((*cross_cov).transpose() * fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal()) * (*cross_cov);
					}
					else if (gp_approx_ == "full_scale_tapering") {
						// factorize residual covariance matrix
						std::shared_ptr<T_mat> sigma_resid = re_comps_resid_[cluster_i][0][0]->GetZSigmaZt();
						CalcCholFSAResid(*sigma_resid, cluster_i);
						den_mat_t sigma_resid_Ihalf_cross_cov;
						//ApplyPermutationCholeskyFactor<den_mat_t, T_chol>(chol_fact_resid_[cluster_i], *cross_cov, sigma_resid_Ihalf_cross_cov, false);//DELETE_SOLVEINPLACE
						//chol_fact_resid_[cluster_i].matrixL().solveInPlace(sigma_resid_Ihalf_cross_cov);
						TriangularSolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_fact_resid_[cluster_i], *cross_cov, sigma_resid_Ihalf_cross_cov, false);
						sigma_woodbury = sigma_resid_Ihalf_cross_cov.transpose() * sigma_resid_Ihalf_cross_cov;
					}
					else if (gp_approx_ == "full_scale_vecchia") {
						D_inv_B_cross_cov_[cluster_i][0].resize(num_data_per_cluster_[cluster_i], num_ind_points_);
						B_cross_cov_[cluster_i][0].resize(num_data_per_cluster_[cluster_i], num_ind_points_);
						B_T_D_inv_B_cross_cov_[cluster_i][0].resize(num_data_per_cluster_[cluster_i], num_ind_points_);
#pragma omp parallel for schedule(static)   
						for (int i = 0; i < num_ind_points_; ++i) {
							B_cross_cov_[cluster_i][0].col(i) = B_rm_[cluster_i][0] * (*cross_cov).col(i);
							D_inv_B_cross_cov_[cluster_i][0].col(i) = D_inv_rm_[cluster_i][0] * B_cross_cov_[cluster_i][0].col(i);
							B_T_D_inv_B_cross_cov_[cluster_i][0].col(i) = B_t_D_inv_rm_[cluster_i][0] * B_cross_cov_[cluster_i][0].col(i);
						}
						GPBoost::matmul(B_cross_cov_[cluster_i][0].transpose(), D_inv_B_cross_cov_[cluster_i][0], sigma_woodbury, GPU_use_);
						//sigma_woodbury = B_cross_cov_[cluster_i][0].transpose() * D_inv_B_cross_cov_[cluster_i][0];
					}
					sigma_woodbury += sigma_ip_stable;

					if (gp_approx_ == "full_scale_vecchia") {
						sigma_woodbury_[cluster_i] = sigma_woodbury;
					}
					//// adding jitter to this Woodbury matrix changes the results too much without helping really (06.11.2024)
					//sigma_woodbury.diagonal().array() *= JITTER_MULT_IP_FITC_FSA;

					chol_fact_sigma_woodbury_[cluster_i].compute(sigma_woodbury);

					////alternative way for calculating determinants with Woodbury (does not solve numerical stability issue, 05.06.2024)
					//den_mat_t sigma_woodbury_stable = sigma_woodbury;
					//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_[cluster_i][0], sigma_woodbury_stable, sigma_woodbury_stable, false);
					//den_mat_t sigma_woodbury_stable_aux = sigma_woodbury_stable.transpose();
					//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_[cluster_i][0], sigma_woodbury_stable_aux, sigma_woodbury_stable_aux, false);
					//sigma_woodbury_stable = sigma_woodbury_stable_aux.transpose();
					//sigma_woodbury_stable.diagonal().array() += 1.;
					//chol_fact_sigma_woodbury_stable_[cluster_i].compute(sigma_woodbury_stable);
				}
				else {
					Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
				}
			}
		}//end CalcCovFactorFITC_FSA

		/*!
		* \brief Calculate Psi^-1*y (and save in y_aux_)
		* \param marg_variance The marginal variance. Default = 1.
		* \param store_ytilde2 If true && use_woodbury_identity_: y_tilde2_ = Z * (Sigma^(-1)+ Z^T Z)^(-1)  Z^T * y is stored for later covariance gradient calculations
		*/
		void CalcYAux(double marg_variance,
			bool store_ytilde2) {
			CHECK(gauss_likelihood_);
			for (const auto& cluster_i : unique_clusters_) {
				if (y_.find(cluster_i) == y_.end()) {
					Log::REFatal("Response variable data (y_) for random effects model has not been set. Call 'SetY' first ");
				}
				if (!covariance_matrix_has_been_factorized_) {
					Log::REFatal("Factorisation of covariance matrix has not been done. Call 'CalcCovFactor' first ");
				}
				if (gp_approx_ == "vecchia") {
					y_aux_[cluster_i] = B_[cluster_i][0].transpose() * D_inv_[cluster_i][0] * B_[cluster_i][0] * y_[cluster_i];
				}
				else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
					const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr();
					if (matrix_inversion_method_ == "cholesky") {
						if (gp_approx_ == "fitc") {
							vec_t cross_covT_y = (*cross_cov).transpose() * (fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * y_[cluster_i]);
							vec_t sigma_woodbury_I_cross_covT_y = chol_fact_sigma_woodbury_[cluster_i].solve(cross_covT_y);
							cross_covT_y.resize(0);
							vec_t cross_cov_sigma_woodbury_I_cross_covT_y = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * ((*cross_cov) * sigma_woodbury_I_cross_covT_y);
							sigma_woodbury_I_cross_covT_y.resize(0);
							y_aux_[cluster_i] = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * y_[cluster_i] - cross_cov_sigma_woodbury_I_cross_covT_y;
						}
						else if (gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
							vec_t sigma_resid_I_y;
							if (gp_approx_ == "full_scale_tapering") {
								sigma_resid_I_y = chol_fact_resid_[cluster_i].solve(y_[cluster_i]);
							}
							else {
								sigma_resid_I_y = B_t_D_inv_rm_[cluster_i][0] * (B_rm_[cluster_i][0] * y_[cluster_i]);
							}
							vec_t cross_covT_sigma_resid_I_y = (*cross_cov).transpose() * sigma_resid_I_y;
							vec_t sigma_woodbury_I_cross_covT_sigma_resid_I_y = chol_fact_sigma_woodbury_[cluster_i].solve(cross_covT_sigma_resid_I_y);
							cross_covT_sigma_resid_I_y.resize(0);
							vec_t cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y = (*cross_cov) * sigma_woodbury_I_cross_covT_sigma_resid_I_y;
							sigma_woodbury_I_cross_covT_sigma_resid_I_y.resize(0);
							vec_t sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y;
							if (gp_approx_ == "full_scale_tapering") {
								sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y = chol_fact_resid_[cluster_i].solve(cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y);
							}
							else {
								sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y = B_t_D_inv_rm_[cluster_i][0] * (B_rm_[cluster_i][0] * cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y);
							}
							cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y.resize(0);
							y_aux_[cluster_i] = sigma_resid_I_y - sigma_resid_I_cross_cov_sigma_woodbury_I_cross_covT_sigma_resid_I_y;
						}
					}
					else {
						//Use last solution as initial guess
						if (num_iter_ > 0 && last_y_aux_[cluster_i].size() > 0) {
							y_aux_[cluster_i] = last_y_aux_[cluster_i];
						}
						else {
							y_aux_[cluster_i] = vec_t::Zero(num_data_per_cluster_[cluster_i]);
						}
						//Reduce max. number of iterations for the CG in first update
						int cg_max_num_it = cg_max_num_it_;
						if (first_update_) {
							cg_max_num_it = (int)round(cg_max_num_it_ / 3);
						}
						std::shared_ptr<T_mat> sigma_resid = re_comps_resid_[cluster_i][0][0]->GetZSigmaZt();
						if (cg_preconditioner_type_ == "fitc") {
							const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
							CGFSA<T_mat>(*sigma_resid, *cross_cov_preconditioner, chol_ip_cross_cov_[cluster_i][0], y_[cluster_i], y_aux_[cluster_i],
								NaN_found, cg_max_num_it, cg_delta_conv_, THRESHOLD_ZERO_RHS_CG_, cg_preconditioner_type_,
								chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
						}
						else {
							CGFSA<T_mat>(*sigma_resid, *cross_cov, chol_ip_cross_cov_[cluster_i][0], y_[cluster_i], y_aux_[cluster_i],
								NaN_found, cg_max_num_it, cg_delta_conv_, THRESHOLD_ZERO_RHS_CG_, cg_preconditioner_type_,
								chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
						}
						last_y_aux_[cluster_i] = y_aux_[cluster_i];
						if (NaN_found) {
							Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
						}

					}
				}//end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
				else if (use_woodbury_identity_) {
					vec_t MInvZty;
					if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
						MInvZty = (Zty_[cluster_i].array() / sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().square()).matrix();
					}
					else {
						if (matrix_inversion_method_ == "cholesky") {
							MInvZty = chol_facts_[cluster_i].solve(Zty_[cluster_i]);
						}//end cholesky
						else if (matrix_inversion_method_ == "iterative") {
							//Use last solution as initial guess
							if (num_iter_ > 0 && last_MInvZty_[cluster_i].size() > 0) {
								MInvZty = last_MInvZty_[cluster_i];
							}
							else {
								MInvZty = vec_t::Zero(cum_num_rand_eff_[cluster_i][num_comps_total_]);
							}
							//Reduce max. number of iterations for the CG in first update
							int cg_max_num_it = cg_max_num_it_;
							if (first_update_) {
								cg_max_num_it = (int)round(cg_max_num_it_ / 3);
							}
							CGRandomEffectsVec(SigmaI_plus_ZtZ_rm_[cluster_i], Zty_[cluster_i], MInvZty, NaN_found, cg_max_num_it, cg_delta_conv_, false, THRESHOLD_ZERO_RHS_CG_, false, cg_preconditioner_type_,
								L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i], SigmaI_plus_ZtZ_inv_diag_[cluster_i], num_cg_steps_last_);
							last_MInvZty_[cluster_i] = MInvZty;
							if (NaN_found) {
								Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
							}
						}//end iterative
						else {
							Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
						}
					}
					if (store_ytilde2) {
						y_tilde2_[cluster_i] = Zt_[cluster_i].transpose() * MInvZty;
						y_aux_[cluster_i] = y_[cluster_i] - y_tilde2_[cluster_i];
					}
					else {
						y_aux_[cluster_i] = y_[cluster_i] - Zt_[cluster_i].transpose() * MInvZty;
					}
				}
				else {//not use_woodbury_identity_ || gp_approx_ == "vecchia" || gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
					y_aux_[cluster_i] = chol_facts_[cluster_i].solve(y_[cluster_i]);
				}
				if (!TwoNumbersAreEqual<double>(marg_variance, 1.)) {
					y_aux_[cluster_i] /= marg_variance;
				}
			}
			y_aux_has_been_calculated_ = true;
		}//end CalcYAux

		/*!
		* \brief Calculate y_tilde = L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z) (and save in y_tilde_)
		* \param also_calculate_ytilde2 If true, y_tilde2 = Z * L^-T * L^-1 * Z^T * y is also calculated
		*/
		void CalcYtilde(bool also_calculate_ytilde2) {
			CHECK(gauss_likelihood_);
			for (const auto& cluster_i : unique_clusters_) {
				if (y_.find(cluster_i) == y_.end()) {
					Log::REFatal("Response variable data (y_) for random effects model has not been set. Call 'SetY' first ");
				}
				if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
					y_tilde_[cluster_i] = (Zty_[cluster_i].array() / sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array()).matrix();
					if (also_calculate_ytilde2) {
						y_tilde2_[cluster_i] = Zt_[cluster_i].transpose() * ((y_tilde_[cluster_i].array() / sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array()).matrix());
					}
				}
				else {
					TriangularSolveGivenCholesky<T_chol, T_mat, vec_t, vec_t>(chol_facts_[cluster_i], Zty_[cluster_i], y_tilde_[cluster_i], false);
					if (also_calculate_ytilde2) {
						vec_t ytilde_aux;
						TriangularSolveGivenCholesky<T_chol, T_mat, vec_t, vec_t>(chol_facts_[cluster_i], y_tilde_[cluster_i], ytilde_aux, true);
						y_tilde2_[cluster_i] = Zt_[cluster_i].transpose() * ytilde_aux;
					}
				}
			}
		}//end CalcYtilde

		/*!
		* \brief Calculate y^T*Psi^-1*y
		* \param[out] yTPsiInvy y^T*Psi^-1*y
		* \param all_clusters If true, then y^T*Psi^-1*y is calculated for all clusters / data and cluster_ind is ignored
		* \param cluster_ind Cluster index
		* \param CalcYAux_already_done If true, it is assumed that y_aux_=Psi^-1y_ has already been calculated (only relevant for not use_woodbury_identity_)
		* \param CalcYtilde_already_done If true, it is assumed that y_tilde = L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z), has already been calculated (only relevant for use_woodbury_identity_)
		*/
		void CalcYTPsiIInvY(double& yTPsiInvy,
			bool all_clusters,
			data_size_t cluster_ind,
			bool CalcYAux_already_done,
			bool CalcYtilde_already_done) {
			CHECK(gauss_likelihood_);
			yTPsiInvy = 0;
			std::vector<data_size_t> clusters_iterate;
			if (all_clusters) {
				clusters_iterate = unique_clusters_;
			}
			else {
				clusters_iterate = std::vector<data_size_t>(1);
				clusters_iterate[0] = cluster_ind;
			}
			for (const auto& cluster_i : clusters_iterate) {
				if (y_.find(cluster_i) == y_.end()) {
					Log::REFatal("Response variable data (y_) for random effects model has not been set. Call 'SetY' first.");
				}
				if (!covariance_matrix_has_been_factorized_) {
					Log::REFatal("Factorisation of covariance matrix has not been done. Call 'CalcCovFactor' first.");
				}
				if (gp_approx_ == "vecchia") {
					if (CalcYAux_already_done) {
						yTPsiInvy += (y_[cluster_i].transpose() * y_aux_[cluster_i])(0, 0);
					}
					else {
						vec_t y_aux_sqrt = B_[cluster_i][0] * y_[cluster_i];
						yTPsiInvy += (y_aux_sqrt.transpose() * D_inv_[cluster_i][0] * y_aux_sqrt)(0, 0);
					}
				}//end gp_approx_ == "vecchia"
				else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering" || gp_approx_ == "full_scale_vecchia") {
					if (!CalcYAux_already_done) {
						CalcYAux(1., false);
					}
					yTPsiInvy += (y_[cluster_i].transpose() * y_aux_[cluster_i])(0, 0);
				}//end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
				else if (use_woodbury_identity_) {
					if (matrix_inversion_method_ == "cholesky") {
						if (!CalcYtilde_already_done) {
							CalcYtilde(false);//y_tilde = L^-1 * Z^T * y, L = chol(Sigma^-1 + Z^T * Z)
						}
						else if ((int)y_tilde_[cluster_i].size() != cum_num_rand_eff_[cluster_i][num_comps_total_]) {
							Log::REFatal("y_tilde = L^-1 * Z^T * y has not the correct number of data points. Call 'CalcYtilde' first.");
						}
						yTPsiInvy += (y_[cluster_i].transpose() * y_[cluster_i])(0, 0) - (y_tilde_[cluster_i].transpose() * y_tilde_[cluster_i])(0, 0);

					}//end cholesky
					else if (matrix_inversion_method_ == "iterative") {
						if (!CalcYAux_already_done) {
							CalcYAux(1., false);
						}
						yTPsiInvy += (y_[cluster_i].transpose() * y_aux_[cluster_i])(0, 0);
					}//end iterative
					else {
						Log::REFatal("Matrix inversion method '%s' is not supported.", matrix_inversion_method_.c_str());
					}
				}//end use_woodbury_identity_
				else {//not use_woodbury_identity_
					if (CalcYAux_already_done) {
						yTPsiInvy += (y_[cluster_i].transpose() * y_aux_[cluster_i])(0, 0);
					}
					else {
						vec_t y_aux_sqrt;
						TriangularSolveGivenCholesky<T_chol, T_mat, vec_t, vec_t>(chol_facts_[cluster_i], y_[cluster_i], y_aux_sqrt, false);
						yTPsiInvy += (y_aux_sqrt.transpose() * y_aux_sqrt)(0, 0);
					}
				}//end not use_woodbury_identity_
			}//end not gp_approx_ == "vecchia"
		}//end CalcYTPsiIInvY

		/*!
		* \brief Update linear fixed-effect coefficients using generalized least squares (GLS)
		*/
		void UpdateCoefGLS() {
			CHECK(gauss_likelihood_);
			vec_t y_aux(num_data_);
			GetYAux(y_aux);
			den_mat_t XT_psi_inv_X;
			CalcXTPsiInvX(X_, XT_psi_inv_X);
			beta_ = XT_psi_inv_X.llt().solve(X_.transpose() * y_aux);
		}

		/*!
		* \brief Calculate the Fisher information for covariance parameters. Note: you need to call CalcCovFactor first
		* \param cov_pars_in Covariance parameters
		* \param[out] FI Fisher information
		* \param transf_scale If true, the derivative is taken on the transformed scale (= nugget factored out and on log-scale) otherwise on the original scale.
		*			Transformed scale is used for estimation, the original scale for standard errors.
		* \param include_error_var If true, the error variance parameter (=nugget effect) is also included, otherwise not
		* \param use_saved_psi_inv If false, the inverse covariance matrix Psi^-1 is calculated, otherwise a saved version is used.
		*						   For iterative methods for grouped random effects, if false, P^(-1) z_i is calculated, otherwise a saved version is used
		*/
		void CalcFisherInformation(const vec_t& cov_pars_in,
			den_mat_t& FI,
			bool transf_scale,
			bool include_error_var,
			bool use_saved_psi_inv) {
			vec_t cov_pars;
			MaybeKeepVarianceConstant(cov_pars_in, cov_pars);
			CHECK(gauss_likelihood_);
			if (include_error_var) {
				FI = den_mat_t(num_cov_par_, num_cov_par_);
			}
			else {
				FI = den_mat_t(num_cov_par_ - 1, num_cov_par_ - 1);
			}
			FI.setZero();
			int first_cov_par = include_error_var ? 1 : 0;
			if (use_stochastic_trace_for_Fisher_information_Vecchia_) {
				if (saved_rand_vec_fisher_info_.size() == 0) {
					for (const auto& cluster_i : unique_clusters_) {
						saved_rand_vec_fisher_info_[cluster_i] = false;
					}
				}
			}
			if (gp_approx_ == "vecchia") {
				CalcFisherInformation_Vecchia(FI, transf_scale, include_error_var, first_cov_par);
			}//end gp_approx_ == "vecchia"
			else if (gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering") {
				CalcFisherInformation_FITC_FSA(cov_pars, FI, transf_scale, include_error_var, first_cov_par);
			}//end gp_approx_ == "fitc" || gp_approx_ == "full_scale_tapering"
			else if (gp_approx_ == "full_scale_vecchia") {
				Log::REFatal("'CalcFisherInformation': this is not implemented for the approximation '%s' ", gp_approx_.c_str());
			}
			else if (use_woodbury_identity_) {
				CalcFisherInformation_Only_Grouped_REs_Woodbury(cov_pars, FI, transf_scale, include_error_var, use_saved_psi_inv, first_cov_par);
			}//end use_woodbury_identity_
			else {//gp_approx_ == "none" and not use_woodbury_identity_
				for (const auto& cluster_i : unique_clusters_) {
					T_mat psi_inv;
					if (use_saved_psi_inv) {
						psi_inv = psi_inv_[cluster_i];
					}
					else {
						CalcPsiInv(psi_inv, cluster_i, false);
					}
					if (!transf_scale) {
						psi_inv /= cov_pars[0];//psi_inv has been calculated with a transformed parametrization, so we need to divide everything by cov_pars[0] to obtain the covariance matrix
					}
					//Calculate Psi^-1 * derivative(Psi)
					std::vector<T_mat> psi_inv_deriv_psi(num_cov_par_ - 1);
					int deriv_par_nb = 0;
					for (int j = 0; j < num_comps_total_; ++j) {//there is currently no possibility to loop over the parameters directly
						for (int jpar = 0; jpar < re_comps_[cluster_i][0][j]->num_cov_par_; ++jpar) {
							psi_inv_deriv_psi[deriv_par_nb] = psi_inv * *(re_comps_[cluster_i][0][j]->GetZSigmaZtGrad(jpar, transf_scale, cov_pars[0]));
							deriv_par_nb++;
						}
					}
					//Calculate Fisher information
					if (include_error_var) {
						//First calculate terms for nugget effect / noise variance parameter
						if (transf_scale) {//Optimization is done on transformed scale (error variance factored out and log-scale)
							//The derivative for the nugget variance on the transformed scale is the original covariance matrix Psi, i.e. psi_inv_grad_psi_sigma2 is the identity matrix.
							FI(0, 0) += num_data_per_cluster_[cluster_i] / 2.;
							for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
								FI(0, par_nb + 1) += psi_inv_deriv_psi[par_nb].diagonal().sum() / 2.;
							}
						}
						else {//Original scale for asymptotic covariance matrix
							//The derivative for the nugget variance is the identity matrix, i.e. psi_inv_grad_psi_sigma2 = psi_inv.
							FI(0, 0) += ((double)(psi_inv.cwiseProduct(psi_inv)).sum()) / 2.;
							for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
								FI(0, par_nb + 1) += ((double)(psi_inv.cwiseProduct(psi_inv_deriv_psi[par_nb])).sum()) / 2.;
							}
						}
					}
					//Remaining covariance parameters
					for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
						T_mat psi_inv_grad_psi_par_nb_T = psi_inv_deriv_psi[par_nb].transpose();
						FI(par_nb + first_cov_par, par_nb + first_cov_par) += ((double)(psi_inv_grad_psi_par_nb_T.cwiseProduct(psi_inv_deriv_psi[par_nb])).sum()) / 2.;
						for (int par_nb_cross = par_nb + 1; par_nb_cross < num_cov_par_ - 1; ++par_nb_cross) {
							FI(par_nb + first_cov_par, par_nb_cross + first_cov_par) += ((double)(psi_inv_grad_psi_par_nb_T.cwiseProduct(psi_inv_deriv_psi[par_nb_cross])).sum()) / 2.;
						}
						psi_inv_deriv_psi[par_nb].resize(0, 0);//not needed anymore
						psi_inv_grad_psi_par_nb_T.resize(0, 0);
					}
				}//end loop over clusterI
			}//end gp_approx_ == "none" and not use_woodbury_identity_
			FI.triangularView<Eigen::StrictlyLower>() = FI.triangularView<Eigen::StrictlyUpper>().transpose();
			//for (int i = 0; i < std::min((int)FI.rows(),4); ++i) {//For debugging only
			//    for (int j = i; j < std::min((int)FI.cols(),4); ++j) {
			//	    Log::REInfo("FI(%d,%d) %g", i, j, FI(i, j));
			//    }
			//}
		}//end CalcFisherInformation

		void CalcFisherInformation_Vecchia(den_mat_t& FI,
			bool transf_scale,
			bool include_error_var,
			int first_cov_par) {
			CHECK(gauss_likelihood_);
			CHECK(gp_approx_ == "vecchia");
			for (const auto& cluster_i : unique_clusters_) {
				//Note: if transf_scale==false, then all matrices and derivatives have been calculated on the original scale for the Vecchia approximation, that is why there is no disinction for 'transf_scale'
				if (use_stochastic_trace_for_Fisher_information_Vecchia_) {
					// Using Hutchinson's trace estimator
					sp_mat_t D(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
					D.setIdentity();
					D.diagonal().array() = D_inv_[cluster_i][0].diagonal().array().pow(-1);
					// Sample vectors
					if (!saved_rand_vec_fisher_info_[cluster_i]) {
						rand_vec_fisher_info_[cluster_i].resize(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
						GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vec_fisher_info_[cluster_i]);
						if (reuse_rand_vec_trace_) {//Use same random vectors for each iteration && cluster_i == end(unique_cluster) Tim
							saved_rand_vec_fisher_info_[cluster_i] = true;
						}
					}
					den_mat_t BT_inv_rand_vec;
					TriangularSolve<sp_mat_t, den_mat_t, den_mat_t>(B_[cluster_i][0], rand_vec_fisher_info_[cluster_i], BT_inv_rand_vec, true);
					den_mat_t D_BT_inv_rand_vec = D * BT_inv_rand_vec;
					den_mat_t Bi_D_BT_inv_rand_vec;
					TriangularSolve<sp_mat_t, den_mat_t, den_mat_t>(B_[cluster_i][0], D_BT_inv_rand_vec, Bi_D_BT_inv_rand_vec, false);//Bi_D_BT_inv_rand_vec = B^-1 * D * B^-T * rand_vec
					D_BT_inv_rand_vec.resize(0, 0);
					for (int par_nb = 1; par_nb < num_cov_par_; ++par_nb) {
						den_mat_t minus_dB_Bi_D_BT_inv_rand_vec = -B_grad_[cluster_i][0][par_nb - 1] * Bi_D_BT_inv_rand_vec + D_grad_[cluster_i][0][par_nb - 1] * BT_inv_rand_vec;//minus_dB_Bi_D_BT_inv_rand_vec = -dBk * B^-1 * D * B^-T * rand_vec + dDk * B^-T * rand_vec
						sigma_inv_sigma_grad_rand_vec_[par_nb] = (B_[cluster_i][0].transpose() * (D_inv_[cluster_i][0] * minus_dB_Bi_D_BT_inv_rand_vec)) - (B_grad_[cluster_i][0][par_nb - 1]).transpose() * BT_inv_rand_vec;
					}
					Bi_D_BT_inv_rand_vec.resize(0, 0);
					BT_inv_rand_vec.resize(0, 0);
					if (include_error_var && !transf_scale) {
						//The derivative for the nugget variance is the identity matrix on the orginal scale, i.e. psi_inv_grad_psi_sigma2 = psi_inv
						sigma_inv_sigma_grad_rand_vec_[0] = B_[cluster_i][0].transpose() * (D_inv_[cluster_i][0] * (B_[cluster_i][0] * rand_vec_fisher_info_[cluster_i]));
					}
					//Calculate Fisher information
					sp_mat_t D_inv_B_grad_B_inv, B_grad_B_inv_D;
					if (include_error_var) {
						//First calculate terms for nugget effect / noise variance parameter
						if (transf_scale) {//Optimization is done on transformed scale (in particular, log-scale)
							//The derivative for the nugget variance on the log scale is the original covariance matrix Psi, i.e. psi_inv_grad_psi_sigma2 is the identity matrix.
							FI(0, 0) += num_data_per_cluster_[cluster_i] / 2.;
							for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
								FI(0, par_nb + 1) += (double)((D_inv_[cluster_i][0].diagonal().array() * D_grad_[cluster_i][0][par_nb].diagonal().array()).sum()) / 2.;
							}
							//Remaining covariance parameters
							for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
								for (int par_nb_cross = par_nb; par_nb_cross < num_cov_par_ - 1; ++par_nb_cross) {
									FI(par_nb + first_cov_par, par_nb_cross + first_cov_par) += (sigma_inv_sigma_grad_rand_vec_[par_nb + 1]).cwiseProduct(sigma_inv_sigma_grad_rand_vec_[par_nb_cross + 1]).colwise().sum().mean() / 2.;
								}
							}
						}
						else {//Original scale for asymptotic covariance matrix
							for (int par_nb = 0; par_nb < num_cov_par_; ++par_nb) {
								for (int par_nb_cross = par_nb; par_nb_cross < num_cov_par_; ++par_nb_cross) {
									FI(par_nb, par_nb_cross) += (sigma_inv_sigma_grad_rand_vec_[par_nb]).cwiseProduct(sigma_inv_sigma_grad_rand_vec_[par_nb_cross]).colwise().sum().mean() / 2.;
								}
							}
						}
					}//end include_error_var
					else {//!include_error_var
						for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
							for (int par_nb_cross = par_nb; par_nb_cross < num_cov_par_ - 1; ++par_nb_cross) {
								FI(par_nb, par_nb_cross) += (sigma_inv_sigma_grad_rand_vec_[par_nb + 1]).cwiseProduct(sigma_inv_sigma_grad_rand_vec_[par_nb_cross + 1]).colwise().sum().mean() / 2.;
							}
						}
					}
				}//end use_stochastic_trace_for_Fisher_information_Vecchia_
				else {//!use_stochastic_trace_for_Fisher_information_Vecchia_
					//Calculate auxiliary matrices for use below
					sp_mat_t Identity(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
					Identity.setIdentity();
					sp_mat_t B_inv;
					TriangularSolve<sp_mat_t, sp_mat_t, sp_mat_t>(B_[cluster_i][0], Identity, B_inv, false);//No noticeable difference in (n=500, nn=100/30) compared to using eigen_sp_Lower_sp_RHS_cs_solve()
					//eigen_sp_Lower_sp_RHS_cs_solve(B_[cluster_i][0], Identity, B_inv, true);
					sp_mat_t D(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
					D.setIdentity();
					D.diagonal().array() = D_inv_[cluster_i][0].diagonal().array().pow(-1);
					sp_mat_t D_inv_2 = sp_mat_t(num_data_per_cluster_[cluster_i], num_data_per_cluster_[cluster_i]);
					D_inv_2.setIdentity();
					D_inv_2.diagonal().array() = D_inv_[cluster_i][0].diagonal().array().pow(2);
					//Calculate derivative(B) * B^-1
					std::vector<sp_mat_t> B_grad_B_inv(num_cov_par_ - 1);
					for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
						B_grad_B_inv[par_nb] = B_grad_[cluster_i][0][par_nb] * B_inv;
					}
					//Calculate Fisher information
					sp_mat_t D_inv_B_grad_B_inv, B_grad_B_inv_D;
					if (include_error_var) {
						//First calculate terms for nugget effect / noise variance parameter
						if (transf_scale) {//Optimization is done on transformed scale (in particular, log-scale)
							//The derivative for the nugget variance on the log scale is the original covariance matrix Psi, i.e. psi_inv_grad_psi_sigma2 is the identity matrix.
							FI(0, 0) += num_data_per_cluster_[cluster_i] / 2.;
							for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
								FI(0, par_nb + 1) += (double)((D_inv_[cluster_i][0].diagonal().array() * D_grad_[cluster_i][0][par_nb].diagonal().array()).sum()) / 2.;
							}
						}
						else {//Original scale for asymptotic covariance matrix
							int ind_grad_nugget = num_cov_par_ - 1;
							D_inv_B_grad_B_inv = D_inv_[cluster_i][0] * B_grad_[cluster_i][0][ind_grad_nugget] * B_inv;
							B_grad_B_inv_D = B_grad_[cluster_i][0][ind_grad_nugget] * B_inv * D;
							double diag = (double)((D_inv_2.diagonal().array() * D_grad_[cluster_i][0][ind_grad_nugget].diagonal().array() * D_grad_[cluster_i][0][ind_grad_nugget].diagonal().array()).sum());
							FI(0, 0) += ((double)(B_grad_B_inv_D.cwiseProduct(D_inv_B_grad_B_inv)).sum() + diag / 2.);
							for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
								B_grad_B_inv_D = B_grad_B_inv[par_nb] * D;
								diag = (double)((D_inv_2.diagonal().array() * D_grad_[cluster_i][0][ind_grad_nugget].diagonal().array() * D_grad_[cluster_i][0][par_nb].diagonal().array()).sum());
								FI(0, par_nb + 1) += ((double)(B_grad_B_inv_D.cwiseProduct(D_inv_B_grad_B_inv)).sum() + diag / 2.);
							}
						}
					}//end include_error_var
					//Remaining covariance parameters
					for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
						D_inv_B_grad_B_inv = D_inv_[cluster_i][0] * B_grad_B_inv[par_nb];
						for (int par_nb_cross = par_nb; par_nb_cross < num_cov_par_ - 1; ++par_nb_cross) {
							B_grad_B_inv_D = B_grad_B_inv[par_nb_cross] * D;
							double diag = (double)((D_inv_2.diagonal().array() * D_grad_[cluster_i][0][par_nb].diagonal().array() * D_grad_[cluster_i][0][par_nb_cross].diagonal().array()).sum());
							FI(par_nb + first_cov_par, par_nb_cross + first_cov_par) += ((double)(B_grad_B_inv_D.cwiseProduct(D_inv_B_grad_B_inv)).sum() + diag / 2.);
						}
					}
				}//end !use_stochastic_trace_for_Fisher_information_Vecchia_
			}//end loop over cluster_i
		}//end CalcFisherInformation_Vecchia

		void CalcFisherInformation_FITC_FSA(const vec_t& cov_pars,
			den_mat_t& FI,
			bool transf_scale,
			bool include_error_var,
			int first_cov_par) {
			CHECK(gauss_likelihood_);
			for (const auto& cluster_i : unique_clusters_) {
				// Hutchinson's Trace estimator
				// Sample vectors
				if (!saved_rand_vec_fisher_info_[cluster_i]) {
					rand_vec_fisher_info_[cluster_i].resize(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
					GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vec_fisher_info_[cluster_i]);
					if (reuse_rand_vec_trace_) {//Use same random vectors for each iteration && cluster_i == end(unique_cluster) Tim
						saved_rand_vec_fisher_info_[cluster_i] = true;
					}
				}
				std::shared_ptr<T_mat> sigma_resid;
				den_mat_t sigma_inv_rand_vec_nugget;
				int deriv_par_nb = 0;
				for (int j = 0; j < num_comps_total_; ++j) {
					if (gp_approx_ == "full_scale_tapering" && matrix_inversion_method_ == "iterative") {
						re_comps_resid_[cluster_i][0][j]->CalcSigma();
						// Subtract predictive process covariance
						re_comps_resid_[cluster_i][0][j]->SubtractPredProcFromSigmaForResidInFullScale(chol_ip_cross_cov_[cluster_i][0], true);
						// Apply Taper
						re_comps_resid_[cluster_i][0][j]->ApplyTaper();
						if (gauss_likelihood_) {
							re_comps_resid_[cluster_i][0][j]->AddConstantToDiagonalSigma(1.);//add nugget effect variance
						}
						sigma_resid = re_comps_resid_[cluster_i][0][j]->GetZSigmaZt();
					}
					const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][j]->GetSigmaPtr();
					//den_mat_t sigma_ip_inv_sigma_cross_cov = chol_fact_sigma_ip_[cluster_i][0].solve((*cross_cov).transpose());
					den_mat_t sigma_ip_inv_sigma_cross_cov;
					GPBoost::solve_linear_sys(chol_fact_sigma_ip_[cluster_i][0], (*cross_cov).transpose(), sigma_ip_inv_sigma_cross_cov, GPU_use_);
					int num_par_comp = re_comps_ip_[cluster_i][0][j]->num_cov_par_;
					// Inverse of Sigma residual times cross covariance
					den_mat_t Sigma_inv_cross_cov;
					den_mat_t Sigma_inv_rand_vec;
					if (matrix_inversion_method_ == "cholesky" && gp_approx_ == "full_scale_tapering") {
						Sigma_inv_cross_cov = chol_fact_resid_[cluster_i].solve(*cross_cov);
						Sigma_inv_rand_vec = chol_fact_resid_[cluster_i].solve(rand_vec_fisher_info_[cluster_i]);
					}
					for (int jpar = 0; jpar < num_par_comp; ++jpar) {
						// Derivative of Components
						std::shared_ptr<den_mat_t> cross_cov_grad = re_comps_cross_cov_[cluster_i][0][j]->GetZSigmaZtGrad(jpar, transf_scale, cov_pars[0]);
						den_mat_t sigma_ip_stable_grad = *(re_comps_ip_[cluster_i][0][j]->GetZSigmaZtGrad(jpar, transf_scale, cov_pars[0]));
						den_mat_t sigma_ip_grad_inv_sigma_cross_cov = sigma_ip_stable_grad * sigma_ip_inv_sigma_cross_cov;
						if (gp_approx_ == "full_scale_tapering") {
							// Initialize Residual Process
							re_comps_resid_[cluster_i][0][j]->CalcSigma();
							std::shared_ptr<T_mat> sigma_resid_grad = re_comps_resid_[cluster_i][0][j]->GetZSigmaZtGrad(jpar, transf_scale, cov_pars[0]);
							// Subtract gradient of predictive process covariance
							SubtractProdFromMat<T_mat>(*sigma_resid_grad, -sigma_ip_inv_sigma_cross_cov, sigma_ip_grad_inv_sigma_cross_cov, true);
							SubtractProdFromMat<T_mat>(*sigma_resid_grad, (*cross_cov_grad).transpose(), sigma_ip_inv_sigma_cross_cov, false);
							SubtractProdFromMat<T_mat>(*sigma_resid_grad, sigma_ip_inv_sigma_cross_cov, (*cross_cov_grad).transpose(), false);
							// Apply taper
							re_comps_resid_[cluster_i][0][j]->ApplyTaper(*(re_comps_resid_[cluster_i][0][j]->dist_), *sigma_resid_grad);
							// Inverse times Gradient times Random vectors
							// Gradient times Random vectors
							den_mat_t sigma_resid_grad_rand_vec(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
							sigma_resid_grad_rand_vec.setZero();
#pragma omp parallel for schedule(static)   
							for (int i = 0; i < num_rand_vec_trace_; ++i) {
								sigma_resid_grad_rand_vec.col(i) += (*sigma_resid_grad) * rand_vec_fisher_info_[cluster_i].col(i);
							}
							sigma_resid_grad_rand_vec += sigma_ip_inv_sigma_cross_cov.transpose() * ((*cross_cov_grad).transpose() * rand_vec_fisher_info_[cluster_i])
								+ (*cross_cov_grad) * (sigma_ip_inv_sigma_cross_cov * rand_vec_fisher_info_[cluster_i])
								- sigma_ip_inv_sigma_cross_cov.transpose() * (sigma_ip_grad_inv_sigma_cross_cov * rand_vec_fisher_info_[cluster_i]);
							// Inverse times Gradient times Random vectors
							den_mat_t sigma_inv_sigma_grad_rand_vec_interim(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
							if (matrix_inversion_method_ == "cholesky") {
								den_mat_t Sigma_inv_Grad_rand_vec = chol_fact_resid_[cluster_i].solve(sigma_resid_grad_rand_vec);
								sigma_inv_sigma_grad_rand_vec_[deriv_par_nb] = Sigma_inv_Grad_rand_vec - Sigma_inv_cross_cov * (chol_fact_sigma_woodbury_[cluster_i].solve((*cross_cov).transpose() * Sigma_inv_Grad_rand_vec));
							}
							else if (matrix_inversion_method_ == "iterative") {
								if (cg_preconditioner_type_ == "fitc") {
									const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov_preconditioner, chol_ip_cross_cov_[cluster_i][0], sigma_resid_grad_rand_vec, sigma_inv_sigma_grad_rand_vec_interim, NaN_found,
										num_data_per_cluster_[cluster_i], num_rand_vec_trace_, cg_max_num_it_tridiag_, cg_delta_conv_, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}
								else {
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov, chol_ip_cross_cov_[cluster_i][0], sigma_resid_grad_rand_vec, sigma_inv_sigma_grad_rand_vec_interim, NaN_found,
										num_data_per_cluster_[cluster_i], num_rand_vec_trace_, cg_max_num_it_tridiag_, cg_delta_conv_, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}
								sigma_inv_sigma_grad_rand_vec_[deriv_par_nb] = sigma_inv_sigma_grad_rand_vec_interim;
							}
							// Gradient times Inverse times Random vectors
							// Inverse times Random vectors
							den_mat_t sigma_inv_rand_vec(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
							if (matrix_inversion_method_ == "cholesky") {
								sigma_inv_rand_vec = Sigma_inv_rand_vec - Sigma_inv_cross_cov * (chol_fact_sigma_woodbury_[cluster_i].solve((*cross_cov).transpose() * Sigma_inv_rand_vec));
								sigma_inv_rand_vec_nugget = sigma_inv_rand_vec;
							}
							else if (matrix_inversion_method_ == "iterative") {
								if (cg_preconditioner_type_ == "fitc") {
									const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov_preconditioner, chol_ip_cross_cov_[cluster_i][0], rand_vec_fisher_info_[cluster_i], sigma_inv_rand_vec, NaN_found,
										num_data_per_cluster_[cluster_i], num_rand_vec_trace_, cg_max_num_it_tridiag_, cg_delta_conv_, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}
								else {
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov, chol_ip_cross_cov_[cluster_i][0], rand_vec_fisher_info_[cluster_i], sigma_inv_rand_vec, NaN_found,
										num_data_per_cluster_[cluster_i], num_rand_vec_trace_, cg_max_num_it_tridiag_, cg_delta_conv_, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}
								sigma_inv_rand_vec_nugget = sigma_inv_rand_vec;
							}
							// Gradient times Inverse times Random vectors
							sigma_grad_sigma_inv_rand_vec_[deriv_par_nb] = (*sigma_resid_grad) * sigma_inv_rand_vec;
							sigma_grad_sigma_inv_rand_vec_[deriv_par_nb] += sigma_ip_inv_sigma_cross_cov.transpose() * ((*cross_cov_grad).transpose() * sigma_inv_rand_vec)
								+ (*cross_cov_grad) * (sigma_ip_inv_sigma_cross_cov * sigma_inv_rand_vec)
								- sigma_ip_inv_sigma_cross_cov.transpose() * (sigma_ip_grad_inv_sigma_cross_cov * sigma_inv_rand_vec);
						}
						else if (gp_approx_ == "fitc") {
							den_mat_t sigma_ip_stable_grad_nugget = *(re_comps_ip_[cluster_i][0][j]->GetZSigmaZtGrad(jpar, transf_scale, 1.));
							vec_t FITC_Diag_grad = vec_t::Zero(num_data_per_cluster_[cluster_i]);
							if (re_comps_cross_cov_[cluster_i][0][0]->VarianceOnDiagonal()) {
								FITC_Diag_grad.array() += sigma_ip_stable_grad_nugget.coeffRef(0, 0);
							}
							else {
								CHECK(num_data_per_cluster_[cluster_i] == re_comps_cross_cov_[cluster_i][0][j]->GetNumUniqueREs());
#pragma omp parallel for schedule(static)
								for (int ii = 0; ii < re_comps_cross_cov_[cluster_i][0][j]->GetNumUniqueREs(); ++ii) {
									FITC_Diag_grad[ii] += re_comps_cross_cov_[cluster_i][0][j]->GetZSigmaZtGradDiagonal_ii(ii, jpar, transf_scale, 1.);//uses coords of all data points to calculate Sigma[ii,ii]
								}
							}
#pragma omp parallel for schedule(static)
							for (int ii = 0; ii < num_data_per_cluster_[cluster_i]; ++ii) {
								FITC_Diag_grad[ii] -= 2 * sigma_ip_inv_sigma_cross_cov.col(ii).dot((*cross_cov_grad).row(ii))
									- sigma_ip_inv_sigma_cross_cov.col(ii).dot(sigma_ip_grad_inv_sigma_cross_cov.col(ii));
							}
							// Inverse times Gradient times Random vectors
							// Gradient times Random vectors
							den_mat_t sigma_resid_grad_rand_vec = FITC_Diag_grad.asDiagonal() * rand_vec_fisher_info_[cluster_i];
							sigma_resid_grad_rand_vec += sigma_ip_inv_sigma_cross_cov.transpose() * ((*cross_cov_grad).transpose() * rand_vec_fisher_info_[cluster_i])
								+ (*cross_cov_grad) * (sigma_ip_inv_sigma_cross_cov * rand_vec_fisher_info_[cluster_i])
								- sigma_ip_inv_sigma_cross_cov.transpose() * (sigma_ip_grad_inv_sigma_cross_cov * rand_vec_fisher_info_[cluster_i]);
							// Inverse times Gradient times Random vectors
							den_mat_t FITC_Diag_inv_Grad_rand_vec = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * sigma_resid_grad_rand_vec;
							den_mat_t FITC_Diag_inv_cross_cov = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * (*cross_cov);
							sigma_inv_sigma_grad_rand_vec_[deriv_par_nb] = FITC_Diag_inv_Grad_rand_vec - FITC_Diag_inv_cross_cov * (chol_fact_sigma_woodbury_[cluster_i].solve((*cross_cov).transpose() * FITC_Diag_inv_Grad_rand_vec));
							// Gradient times Inverse times Random vectors
							// Inverse times Random vectors
							den_mat_t FITC_Diag_inv_rand_vec = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * rand_vec_fisher_info_[cluster_i];
							den_mat_t sigma_inv_rand_vec = FITC_Diag_inv_rand_vec - FITC_Diag_inv_cross_cov * (chol_fact_sigma_woodbury_[cluster_i].solve((*cross_cov).transpose() * FITC_Diag_inv_rand_vec));
							sigma_inv_rand_vec_nugget = sigma_inv_rand_vec;
							// Gradient times Inverse times Random vectors
							sigma_grad_sigma_inv_rand_vec_[deriv_par_nb] = FITC_Diag_grad.asDiagonal() * sigma_inv_rand_vec;
							sigma_grad_sigma_inv_rand_vec_[deriv_par_nb] += sigma_ip_inv_sigma_cross_cov.transpose() * ((*cross_cov_grad).transpose() * sigma_inv_rand_vec)
								+ (*cross_cov_grad) * (sigma_ip_inv_sigma_cross_cov * sigma_inv_rand_vec)
								- sigma_ip_inv_sigma_cross_cov.transpose() * (sigma_ip_grad_inv_sigma_cross_cov * sigma_inv_rand_vec);
						}
						deriv_par_nb += 1;
					}
				}
				if (include_error_var) {
					//First calculate terms for nugget effect / noise variance parameter
					if (transf_scale) {//Optimization is done on transformed scale (error variance factored out and log-scale)
						//The derivative for the nugget variance on the transformed scale is the original covariance matrix Psi, i.e. psi_inv_grad_psi_sigma2 is the identity matrix.
						FI(0, 0) += num_data_per_cluster_[cluster_i] / 2.;
						for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
							FI(0, par_nb + 1) += (rand_vec_fisher_info_[cluster_i]).cwiseProduct(sigma_inv_sigma_grad_rand_vec_[par_nb]).colwise().sum().mean() / 2.;
						}
					}
					else {//Original scale for asymptotic covariance matrix
						//The derivative for the nugget variance is the identity matrix, i.e. psi_inv_grad_psi_sigma2 = psi_inv.
						FI(0, 0) += (sigma_inv_rand_vec_nugget).cwiseProduct(sigma_inv_rand_vec_nugget).colwise().sum().mean() / 2.;
						for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
							FI(0, par_nb + 1) += (sigma_inv_rand_vec_nugget).cwiseProduct(sigma_inv_sigma_grad_rand_vec_[par_nb]).colwise().sum().mean() / 2.;
						}
					}
				}
				//Remaining covariance parameters
				for (int par_nb = 0; par_nb < num_cov_par_ - 1; ++par_nb) {
					for (int par_nb_cross = par_nb; par_nb_cross < num_cov_par_ - 1; ++par_nb_cross) {
						FI(par_nb + first_cov_par, par_nb_cross + first_cov_par) += (sigma_grad_sigma_inv_rand_vec_[par_nb]).cwiseProduct(sigma_inv_sigma_grad_rand_vec_[par_nb_cross]).colwise().sum().mean() / 2.;
					}
				}
				if (!transf_scale) {
					FI /= (cov_pars[0] * cov_pars[0]);
				}
			}//end loop over cluster_i
		}//end CalcFisherInformation_FITC_FSA

		void CalcFisherInformation_Only_Grouped_REs_Woodbury(const vec_t& cov_pars,
			den_mat_t& FI,
			bool transf_scale,
			bool include_error_var,
			bool use_saved_psi_inv,
			int first_cov_par) {
			CHECK(gauss_likelihood_);
			CHECK(use_woodbury_identity_);
			for (const auto& cluster_i : unique_clusters_) {
				if (matrix_inversion_method_ == "cholesky") {
					//Notation used below: M = Sigma^-1 + ZtZ, Sigma = cov(b) b=latent random effects, L=chol(M) i.e. M=LLt, MInv = M^-1 = L^-TL^-1
					if (!use_saved_psi_inv) {
						LInvZtZj_[cluster_i] = std::vector<T_mat>(num_comps_total_);
						if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ == ZtZj_ and L_inv are diagonal  
							LInvZtZj_[cluster_i][0] = ZtZ_[cluster_i];
							LInvZtZj_[cluster_i][0].diagonal().array() /= sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array();
						}
						else {
							for (int j = 0; j < num_comps_total_; ++j) {
								if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
									TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), P_ZtZj_[cluster_i][j], LInvZtZj_[cluster_i][j], false);
								}
								else {
									TriangularSolve<T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i].CholFactMatrix(), ZtZj_[cluster_i][j], LInvZtZj_[cluster_i][j], false);
								}
							}
						}
					}
					if (include_error_var) {
						if (transf_scale) {//Optimization is done on transformed scale (error variance factored out and log-scale)
							//The derivative for the nugget variance on the transformed scale is the original covariance matrix Psi, i.e. psi_inv_grad_psi_sigma2 is the identity matrix.
							FI(0, 0) += num_data_per_cluster_[cluster_i] / 2.;
							for (int j = 0; j < num_comps_total_; ++j) {
								double trace_PsiInvGradPsi = Zj_square_sum_[cluster_i][j] - LInvZtZj_[cluster_i][j].squaredNorm();
								FI(0, j + 1) += trace_PsiInvGradPsi * cov_pars[j + 1] / 2.;
							}
						}//end transf_scale
						else {//not transf_scale
							T_mat MInv_ZtZ;//=(Sigma_inv + ZtZ)^-1 * ZtZ
							if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ == ZtZj_ and L_inv are diagonal 
								MInv_ZtZ = T_mat(ZtZ_[cluster_i].rows(), ZtZ_[cluster_i].cols());
								MInv_ZtZ.setIdentity();//initialize
								MInv_ZtZ.diagonal().array() = ZtZ_[cluster_i].diagonal().array() / (sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().square());
							}
							else {
								SolveGivenCholesky<T_chol, T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i], ZtZ_[cluster_i], MInv_ZtZ);

							}
							T_mat MInv_ZtZ_t = MInv_ZtZ.transpose();
							FI(0, 0) += (num_data_per_cluster_[cluster_i] - 2. * MInv_ZtZ.diagonal().sum() + (double)(MInv_ZtZ.cwiseProduct(MInv_ZtZ_t)).sum()) / (cov_pars[0] * cov_pars[0] * 2.);
							for (int j = 0; j < num_comps_total_; ++j) {
								T_mat ZjZ_MInv_ZtZ_t = MInv_ZtZ_t * ZtZj_[cluster_i][j];
								double trace_PsiInvGradPsi;
								if (num_comps_total_ > 1) {
									T_mat MInv_ZtZj;
									SolveGivenCholesky<T_chol, T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i], ZtZj_[cluster_i][j], MInv_ZtZj);
									trace_PsiInvGradPsi = Zj_square_sum_[cluster_i][j] - 2. * (double)(LInvZtZj_[cluster_i][j].squaredNorm()) +
										(double)(ZjZ_MInv_ZtZ_t.cwiseProduct(MInv_ZtZj)).sum();
								}
								else {
									trace_PsiInvGradPsi = Zj_square_sum_[cluster_i][j] - 2. * (double)(LInvZtZj_[cluster_i][j].squaredNorm()) +
										(double)(ZjZ_MInv_ZtZ_t.cwiseProduct(MInv_ZtZ)).sum();
								}
								FI(0, j + 1) += trace_PsiInvGradPsi / (cov_pars[0] * cov_pars[0] * 2.);
							}
						}//end not transf_scale
					}//end include_error_var
					//Remaining covariance parameters
					int counter = 0;
					for (int j = 0; j < num_comps_total_; ++j) {
						sp_mat_t* Z_j = nullptr;
						if (!linear_kernel_use_woodbury_identity_) {
							Z_j = re_comps_[cluster_i][0][j]->GetZ();
						}					
						for (int k = j; k < num_comps_total_; ++k) {
							// if used for Fisher scoring, this is repeatedly done -> save quantities that do not change over iterations
							if (!Zjt_Zk_saved_) {
								if (linear_kernel_use_woodbury_identity_) {
									CHECK(num_comps_total_ == 1);
									Zjt_Zk_[cluster_i].push_back(ZtZ_[cluster_i]);
								}
								else {
									CHECK(Z_j != nullptr);
									sp_mat_t* Z_k = re_comps_[cluster_i][0][k]->GetZ();
									Zjt_Zk_[cluster_i].push_back((T_mat)((*Z_j).transpose() * (*Z_k)));
								}
								Zjt_Zk_squaredNorm_[cluster_i].push_back(Zjt_Zk_[cluster_i][counter].squaredNorm());
							}
							T_mat LInvZtZj_t_LInvZtZk = LInvZtZj_[cluster_i][j].transpose() * LInvZtZj_[cluster_i][k];
							double FI_jk = Zjt_Zk_squaredNorm_[cluster_i][counter] + LInvZtZj_t_LInvZtZk.squaredNorm() -
								2. * (double)(Zjt_Zk_[cluster_i][counter].cwiseProduct(LInvZtZj_t_LInvZtZk)).sum();
							if (transf_scale) {
								FI_jk *= cov_pars[j + 1] * cov_pars[k + 1];
							}
							else {
								FI_jk /= cov_pars[0] * cov_pars[0];
							}
							FI(j + first_cov_par, k + first_cov_par) += FI_jk / 2.;
							counter++;
						}//end loop k
					}//end loop j
				}// end cholesky
				else if (matrix_inversion_method_ == "iterative") {
					// Sample vectors
					if (!saved_rand_vec_fisher_info_[cluster_i]) {
						rand_vec_fisher_info_[cluster_i].resize(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
						GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vec_fisher_info_[cluster_i]);
						if (reuse_rand_vec_trace_) {
							saved_rand_vec_fisher_info_[cluster_i] = true;
						}
					}
					//Z^T z_i
					den_mat_t Zt_RV(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						Zt_RV.col(i) = Zt_[cluster_i] * rand_vec_fisher_info_[cluster_i].col(i);
					}
					//(Sigma^(-1) + Z^T Z)^(-1) Z^T z_i
					den_mat_t MInv_Zt_RV(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
					CGRandomEffectsMat(SigmaI_plus_ZtZ_rm_[cluster_i], Zt_RV, MInv_Zt_RV, NaN_found,
						cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_,
						cg_max_num_it_tridiag_, cg_delta_conv_, cg_preconditioner_type_,
						L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i]);
					if (NaN_found) {
						Log::REFatal("CalcFisherInformation_Only_Grouped_REs_Woodbury: Nan or Inf value generated in the conjugate gradient method ");
					}
					//Z (Sigma^(-1) + Z^T Z)^(-1) Z^T z_i
					den_mat_t Z_MInv_Zt_RV(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
					for (int i = 0; i < num_rand_vec_trace_; ++i) {
						Z_MInv_Zt_RV.col(i) = Zt_[cluster_i].transpose() * MInv_Zt_RV.col(i);
					}
					std::vector<den_mat_t> Zj_Zjt_RV_minus_Z_MInv_Zt_Zj_Zjt_RV(num_comps_total_, den_mat_t(num_data_per_cluster_[cluster_i], num_rand_vec_trace_));
					std::vector<den_mat_t> Zj_Zjt_RV_minus_Zj_Zjt_Z_MInv_Zt_RV(num_comps_total_, den_mat_t(num_data_per_cluster_[cluster_i], num_rand_vec_trace_));
					for (int j = 0; j < num_comps_total_; ++j) {
						sp_mat_t Zj_Zjt;
						if (linear_kernel_use_woodbury_identity_) {
							CHECK(num_comps_total_ == 1);
							Zj_Zjt = (Zt_[cluster_i].transpose()) * Zt_[cluster_i];
						}
						else {
							sp_mat_t* Zj = re_comps_[cluster_i][0][j]->GetZ();
							Zj_Zjt = ((*Zj) * (*Zj).transpose());
						}
						//Z_j Z_j^T z_i
						den_mat_t Zj_Zjt_RV(num_data_per_cluster_[cluster_i], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
						for (int i = 0; i < num_rand_vec_trace_; ++i) {
							Zj_Zjt_RV.col(i) = Zj_Zjt * rand_vec_fisher_info_[cluster_i].col(i);
						}
						//Z^T Z_j Z_j^T z_i
						den_mat_t Zt_Zj_Zjt_RV(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
#pragma omp parallel for schedule(static)   
						for (int i = 0; i < num_rand_vec_trace_; ++i) {
							Zt_Zj_Zjt_RV.col(i) = Zt_[cluster_i] * Zj_Zjt_RV.col(i);
						}
						//(Sigma^(-1) + Z^T Z)^(-1) Z^T Z_j Z_j^T z_i
						den_mat_t MInv_Zt_Zj_Zjt_RV(cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_);
						CGRandomEffectsMat(SigmaI_plus_ZtZ_rm_[cluster_i], Zt_Zj_Zjt_RV, MInv_Zt_Zj_Zjt_RV, NaN_found,
							cum_num_rand_eff_[cluster_i][num_comps_total_], num_rand_vec_trace_,
							cg_max_num_it_tridiag_, cg_delta_conv_, cg_preconditioner_type_,
							L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i]);
						if (NaN_found) {
							Log::REFatal("CalcFisherInformation_Only_Grouped_REs_Woodbury: Nan or Inf value generated in the conjugate gradient method ");
						}
						//Z_j Z_j^T - Z (Sigma^(-1) + Z^T Z)^(-1) Z_j Z_j^T z_i
#pragma omp parallel for schedule(static)   
						for (int i = 0; i < num_rand_vec_trace_; ++i) {
							Zj_Zjt_RV_minus_Z_MInv_Zt_Zj_Zjt_RV[j].col(i) = Zj_Zjt_RV.col(i) - Zt_[cluster_i].transpose() * MInv_Zt_Zj_Zjt_RV.col(i);
						}
						//Z_j Z_j^T Z (Sigma^(-1) + Z^T Z)^(-1) Z^T z_i
#pragma omp parallel for schedule(static)   
						for (int i = 0; i < num_rand_vec_trace_; ++i) {
							Zj_Zjt_RV_minus_Zj_Zjt_Z_MInv_Zt_RV[j].col(i) = Zj_Zjt_RV.col(i) - Zj_Zjt * Z_MInv_Zt_RV.col(i);
						}
					}
					if (include_error_var) {
						if (transf_scale) {
							FI(0, 0) += num_data_per_cluster_[cluster_i] / 2.;
							for (int j = 0; j < num_comps_total_; ++j) {
								double trace_PsiInvGradPsi = ((rand_vec_fisher_info_[cluster_i].cwiseProduct(Zj_Zjt_RV_minus_Z_MInv_Zt_Zj_Zjt_RV[j])).colwise().sum()).mean();
								FI(0, j + 1) += trace_PsiInvGradPsi * cov_pars[j + 1] / 2.;
							}
						}//end transf_scale
						else {//not transf_scale
							den_mat_t RV_minus_Z_MInv_Zt_RV = rand_vec_fisher_info_[cluster_i] - Z_MInv_Zt_RV;
							FI(0, 0) += ((RV_minus_Z_MInv_Zt_RV.cwiseProduct(RV_minus_Z_MInv_Zt_RV)).colwise().sum()).mean() / (cov_pars[0] * cov_pars[0] * 2.);
							for (int j = 0; j < num_comps_total_; ++j) {
								FI(0, j + 1) += ((RV_minus_Z_MInv_Zt_RV.cwiseProduct(Zj_Zjt_RV_minus_Z_MInv_Zt_Zj_Zjt_RV[j])).colwise().sum()).mean() / (cov_pars[0] * cov_pars[0] * 2.);
							}
						}//end not transf_scale
					}//end include_error_var
					//Remaining covariance parameters
					for (int j = 0; j < num_comps_total_; ++j) {
						for (int k = j; k < num_comps_total_; ++k) {
							double FI_jk = (((Zj_Zjt_RV_minus_Zj_Zjt_Z_MInv_Zt_RV[j]).cwiseProduct(Zj_Zjt_RV_minus_Z_MInv_Zt_Zj_Zjt_RV[k])).colwise().sum()).mean();
							if (transf_scale) {
								FI_jk *= cov_pars[j + 1] * cov_pars[k + 1];
							}
							else {
								FI_jk /= cov_pars[0] * cov_pars[0];
							}
							FI(j + first_cov_par, k + first_cov_par) += FI_jk / 2.;
						}//end loop k
					}//end loop j
				}//end iterative
				else {
					Log::REFatal("Matrix inversion method '%s' is not supported ", matrix_inversion_method_.c_str());
				}
			}//end loop over clusters
			if (transf_scale) {
				Zjt_Zk_saved_ = true;
			}
			else {
				Zjt_Zk_saved_ = false;
			}
		}//end CalcFisherInformation_Only_Grouped_REs_Woodbury			

		/*!
		* \brief Calculate the standard deviations for the MLE of the covariance parameters as the diagonal of the inverse Fisher information (on the orignal scale and not the transformed scale used in the optimization, for "gaussian" likelihood only)
		* \param cov_pars MLE of covariance parameters
		* \param[out] std_dev Standard deviations
		*/
		void CalcStdDevCovPar(const vec_t& cov_pars,
			vec_t& std_dev) {
			CHECK(gauss_likelihood_);
			SetCovParsComps(cov_pars);
			CalcCovFactor(false, cov_pars[0]);
			if (gp_approx_ == "vecchia") {
				std::vector<int> estimate_cov_par_index_temp = estimate_cov_par_index_;
				estimate_cov_par_index_ = std::vector<int>(num_cov_par_, 1);
				CalcGradientVecchia(false, cov_pars[0], true);
				estimate_cov_par_index_ = estimate_cov_par_index_temp;
			}
			den_mat_t FI;
			CalcFisherInformation(cov_pars, FI, false, true, false);
			std_dev = FI.inverse().diagonal().array().sqrt().matrix();
		}

		/*!
		* \brief Calculate standard deviations for the MLE of the regression coefficients as the diagonal of the inverse Fisher information (for "gaussian" likelihood only)
		* \param cov_pars MLE of covariance parameters
		* \param X Covariate data for linear fixed-effect
		* \param[out] std_dev Standard deviations
		*/
		void CalcStdDevCoef(const vec_t& cov_pars,
			const den_mat_t& X,
			vec_t& std_dev) {
			CHECK(gauss_likelihood_);
			if ((int)std_dev.size() >= num_data_) {
				Log::REWarning("Sample size too small to calculate standard deviations for coefficients");
				for (int i = 0; i < (int)std_dev.size(); ++i) {
					std_dev[i] = std::numeric_limits<double>::quiet_NaN();
				}
			}
			else {
				SetCovParsComps(cov_pars);
				CalcCovFactor(true, 1.);
				den_mat_t FI((int)X.cols(), (int)X.cols());
				CalcXTPsiInvX(X, FI);
				FI /= cov_pars[0];
				std_dev = FI.inverse().diagonal().array().sqrt().matrix();
			}
		}

		/*!
		* \brief Calculate standard deviations for the MLE of the regression coefficients as the square root of diagonal of a numerically approximated inverse Hessian
		* \param num_covariates Number of covariates / coefficients
		* \param beta Regression coefficients
		* \param cov_pars Covariance parameters
		* \param fixed_effects Externally provided fixed effects component of location parameter
		* \param[out] std_dev_beta Standard deviations
		*/
		void CalcStdDevCoefNonGaussian(int num_covariates,
			const vec_t& beta,
			const vec_t& cov_pars,
			const double* fixed_effects,
			vec_t& std_dev_beta) {
			den_mat_t H(num_covariates, num_covariates);// Aproximate Hessian calculated as the Jacobian of the gradient
			const double mach_eps = std::numeric_limits<double>::epsilon();
			vec_t delta_step = beta * std::pow(mach_eps, 1.0 / 3.0);// based on https://math.stackexchange.com/questions/1039428/finite-difference-method
			vec_t fixed_effects_vec, beta_change1, beta_change2, grad_beta_change1, grad_beta_change2;
			vec_t unused_dummy;
			for (int i = 0; i < num_covariates; ++i) {
				// Beta plus / minus delta
				beta_change1 = beta;
				beta_change2 = beta;
				beta_change1[i] += delta_step[i];
				beta_change2[i] -= delta_step[i];
				// Gradient vector at beta plus / minus delta
				UpdateFixedEffects(beta_change1, fixed_effects, fixed_effects_vec);
				CalcCovFactorOrModeAndNegLL(cov_pars, fixed_effects_vec.data());
				CalcGradPars(cov_pars, 1., false, true, unused_dummy, grad_beta_change1, false, false, fixed_effects_vec.data(), true);
				UpdateFixedEffects(beta_change2, fixed_effects, fixed_effects_vec);
				CalcCovFactorOrModeAndNegLL(cov_pars, fixed_effects_vec.data());
				CalcGradPars(cov_pars, 1., false, true, unused_dummy, grad_beta_change2, false, false, fixed_effects_vec.data(), true);
				// Approximate gradient of gradient
				H.row(i) = (grad_beta_change1 - grad_beta_change2) / (2. * delta_step[i]);
			}
			den_mat_t Hsym = (H + H.transpose()) / 2.;
			// (Very) approximate standard deviations as square root of diagonal of inverse Hessian
			std_dev_beta = Hsym.inverse().diagonal().array().sqrt().matrix();
		}//end CalcStdDevCoefNonGaussian

		/*!
		* \brief Calculate numerically approximated Hessian for the covariance and auxiliary parameters
		* \param cov_aux_pars Covariance and auxiliary (if present) parameters
		* \param include_error_var If true, the error variance parameter (=nugget effect) is also included, otherwise not
		* \param fixed_effects Externally provided fixed effects component of location parameter
		* \param[out] Hessian matrix obtained as numerical Jacobian of the gradient
		*/
		void CalcHessianCovParAuxPars(const vec_t& cov_aux_pars,
			bool include_error_var,
			const double* fixed_effects,
			den_mat_t& Hessian) {
			if (estimate_aux_pars_) {
				CHECK(cov_aux_pars.size() == num_cov_par_ + NumAuxPars());
			}
			else {
				CHECK(cov_aux_pars.size() == num_cov_par_);
			}
			int length_grad = num_cov_par_;
			int offset = 0;
			if (gauss_likelihood_ && !include_error_var) {
				length_grad -= 1;
				offset = 1;
			}
			if (estimate_aux_pars_) {
				length_grad += NumAuxPars();
			}
			den_mat_t H(length_grad, length_grad);// Aproximate Hessian calculated as the Jacobian of the gradient
			vec_t log_pars = cov_aux_pars.array().log().matrix();
			const double h_eps = std::pow(std::numeric_limits<double>::epsilon(), 1.0 / 3.0);
			vec_t delta_step = log_pars * h_eps;// based on https://math.stackexchange.com/questions/1039428/finite-difference-method
			for (int i = 0; i < (int)delta_step.size(); ++i) {//avoid problems when log_pars is close to 0
				if (delta_step[i] < h_eps) {
					delta_step[i] = h_eps;
				}
			}
			vec_t pars_change1, pars_change2, grad_change1, grad_change2;
			vec_t unused_dummy;
			for (int i = 0; i < length_grad; ++i) {
				// cov_aux_pars plus / minus delta
				pars_change1 = cov_aux_pars;
				pars_change2 = cov_aux_pars;
				pars_change1[i + offset] *= std::exp(delta_step[i + offset]);//gradient is taken on log-scale
				pars_change2[i + offset] *= std::exp(-delta_step[i + offset]);
				if (estimate_aux_pars_) {
					SetAuxPars(pars_change1.data() + num_cov_par_);
				}
				CalcCovFactorOrModeAndNegLL(pars_change1.segment(0, num_cov_par_), fixed_effects);
				CalcGradPars(pars_change1.segment(0, num_cov_par_), 1., true, false, grad_change1, unused_dummy, include_error_var, false, fixed_effects, false);
				if (estimate_aux_pars_) {
					SetAuxPars(pars_change2.data() + num_cov_par_);
				}
				CalcCovFactorOrModeAndNegLL(pars_change2.segment(0, num_cov_par_), fixed_effects);
				CalcGradPars(pars_change2.segment(0, num_cov_par_), 1., true, false, grad_change2, unused_dummy, include_error_var, false, fixed_effects, false);
				// Approximate gradient of gradient
				H.row(i) = (grad_change1 - grad_change2) / (2. * delta_step[i + offset]);
			}
			Hessian = (H + H.transpose()) / 2.;
		}//end CalcHessianCovParAuxPars

		/*!
		 * \brief Prepare for prediction: set respone variable data, factorize covariance matrix and calculate Psi^{-1}y_obs or calculate Laplace approximation (if required)
		* \param cov_pars Covariance parameters of components
		* \param coef Coefficients for linear covariates
		* \param y_obs Response variable for observed data
		* \param calc_cov_factor If true, the covariance matrix of the observed data is factorized otherwise a previously done factorization is used
		* \param fixed_effects Fixed effects component of location parameter for observed data (only used for non-Gaussian likelihoods)
		* \param predict_training_data_random_effects If true, the goal is to predict training data random effects
		* \param sample_prior If true, prior samples are generated
		 */
		void SetYCalcCovCalcYAuxForPred(const vec_t& cov_pars,
			const vec_t& coef,
			const double* y_obs,
			bool calc_cov_factor,
			const double* fixed_effects,
			bool predict_training_data_random_effects,
			bool sample_prior) {
			const double* fixed_effects_ptr = fixed_effects;
			vec_t fixed_effects_vec;
			// Set response data and fixed effects
			if (gauss_likelihood_) {
				if (has_covariates_ || fixed_effects != nullptr) {
					vec_t resid;
					if (y_obs != nullptr) {
						resid = Eigen::Map<const vec_t>(y_obs, num_data_);
					}
					else {
						resid = y_vec_;
					}
					if (has_covariates_) {
						resid -= X_ * coef;
					}
					//add external fixed effects to linear predictor
					if (fixed_effects != nullptr) {
#pragma omp parallel for schedule(static)
						for (data_size_t i = 0; i < num_data_; ++i) {
							resid[i] -= fixed_effects[i];
						}
					}
					SetY(resid.data());
				}//end if has_covariates_
				else {//no covariates
					if (y_obs != nullptr) {
						SetY(y_obs);
					}
				}//end no covariates
			}//end if gauss_likelihood_
			else {//if not gauss_likelihood_
				if (has_covariates_) {
					fixed_effects_vec = vec_t(num_data_ * num_sets_re_);
					for (int igp = 0; igp < num_sets_re_; ++igp) {
						fixed_effects_vec.segment(num_data_ * igp, num_data_) = X_ * (coef.segment(num_covariates_ * igp, num_covariates_));
					}
					//add external fixed effects to linear predictor
					if (fixed_effects != nullptr) {
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_data_ * num_sets_re_; ++i) {
							fixed_effects_vec[i] += fixed_effects[i];
						}
					}
					fixed_effects_ptr = fixed_effects_vec.data();
				}
				if (y_obs != nullptr) {
					SetY(y_obs);
				}
			}//end if not gauss_likelihood_
			//TODO (low prio): the factorization needs to be done only for the GP realizations / clusters for which predictions are made (currently it is done for all)
			SetCovParsComps(cov_pars);
			if (!(gp_approx_ == "vecchia" && gauss_likelihood_) || predict_training_data_random_effects || (sample_prior && gp_approx_ == "vecchia")) {
				// no need to call CalcCovFactor here for the Vecchia approximation for Gaussian data, this is done in the prediction steps below, 
				//	but when predicting training data random effects, this is required
				if (calc_cov_factor) {
					if (ShouldRedetermineNearestNeighborsVecchiaInducingPointsFITC(true)) {
						RedetermineNearestNeighborsVecchiaInducingPointsFITC(true);//called if gp_approx_ == "vecchia" or  gp_approx_ == "full_scale_vecchia" and neighbors are selected based on correlations and not distances or gp_approx_ == "fitc" with ard kernel
					}
					CalcCovFactor(true, 1.);
					if (!gauss_likelihood_) {
						//We reset the initial modes to 0. This is done to avoid that different calls to the prediction function lead to (very small) differences
						//	as the mode is calculated from different starting values.
						//	If one is willing to accept these (very) small differences, one could disable this with the advantage of having faster predictions
						//	as the mode does not need to be found anew.
						for (const auto& cluster_i : unique_clusters_) {
							likelihood_[cluster_i]->InitializeModeAvec();
						}
						CalcModePostRandEffCalcMLL(fixed_effects_ptr, false);
					}//end not gauss_likelihood_
				}//end if calc_cov_factor
				if (gauss_likelihood_) {
					if (optimizer_cov_pars_ == "lbfgs_not_profile_out_nugget" || optimizer_cov_pars_ == "lbfgs") {
						CalcSigmaComps();
					}
					CalcYAux(1., false);//note: in some cases a call to CalcYAux() could be avoided (e.g. no covariates and not GPBoost algorithm)...
				}
			}//end not (gp_approx_ == "vecchia" && gauss_likelihood_)
		}// end SetYCalcCovCalcYAuxForPred

		/*!
		 * \brief Calculate predictions (conditional mean and covariance matrix) for one cluster
		 * \param cluster_i Cluster index for which prediction are made
		 * \param num_data_pred Total number of prediction locations (over all clusters)
		 * \param num_data_per_cluster_pred Keys: Labels of independent realizations of REs/GPs, values: number of prediction locations per independent realization
		 * \param data_indices_per_cluster_pred Keys: labels of independent clusters, values: vectors with indices for data points that belong to the every cluster
		 * \param re_group_levels_pred Group levels for the grouped random effects (re_group_levels_pred[j] contains the levels for RE number j)
		 * \param re_group_rand_coef_data_pred Random coefficient data for grouped REs
		 * \param gp_coords_mat_pred Coordinates for prediction locations
		 * \param gp_rand_coef_data_pred Random coefficient data for GPs
		 * \param predict_cov_mat If true, the predictive/conditional covariance matrix is calculated (default=false) (predict_var and predict_cov_mat cannot be both true)
		 * \param predict_var If true, the predictive/conditional variances are calculated (default=false) (predict_var and predict_cov_mat cannot be both true)
		 * \param predict_response If true, the response variable (label) is predicted, otherwise the latent random effects
		 * \param sample_posterior If true, posterior samples are generated
		 * \param sample_prior If true, prior samples are generated
		 * \param num_post_samples Number of posterior samples
		 * \param num_prior_samples Number of prior samples
		 * \param[out] mean_pred_id Predictive mean
		 * \param[out] cov_mat_pred_id Predictive covariance matrix
		 * \param[out] var_pred_id Predictive variances
		 * \param[out] post_samples_id Sample from posterior without the mean, the mean is added at then end
		 * \param[out] prior_samples_id Sample from prior 
		 */
		void CalcPred(data_size_t cluster_i,
			int num_data_pred,
			std::map<data_size_t, int>& num_data_per_cluster_pred,
			std::map<data_size_t, std::vector<int>>& data_indices_per_cluster_pred,
			const std::vector<std::vector<re_group_t>>& re_group_levels_pred,
			const double* re_group_rand_coef_data_pred,
			const den_mat_t& gp_coords_mat_pred,
			const double* gp_rand_coef_data_pred,
			bool predict_cov_mat,
			bool predict_var,
			bool predict_response,
			bool sample_posterior,
			bool sample_prior,
			int num_post_samples,
			int num_prior_samples,
			vec_t& mean_pred_id,
			T_mat& cov_mat_pred_id,
			vec_t& var_pred_id,
			den_mat_t& post_samples_id,
			den_mat_t& prior_samples_id,
			const sp_mat_t& Bpo,
			sp_mat_t& Bp,
			const vec_t& Dp,
			bool VecchiaCondObsOnly) {
			CHECK(gp_approx_ != "fitc" && gp_approx_ != "full_scale_tapering" &&
				gp_approx_ != "full_scale_vecchia" && !(gp_approx_ == "vecchia" && !grouped_RE_and_vecchia_GP_));
			int num_REs_obs, num_REs_pred;
			if (only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
				num_REs_pred = (int)re_group_levels_pred[0].size();
				num_REs_obs = re_comps_[cluster_i][0][0]->GetNumUniqueREs();
			}
			else if (only_one_GP_calculations_on_RE_scale_) {
				num_REs_pred = (int)gp_coords_mat_pred.rows();
				num_REs_obs = re_comps_[cluster_i][0][0]->GetNumUniqueREs();
			}
			else {
				num_REs_pred = num_data_per_cluster_pred[cluster_i];
				num_REs_obs = num_data_per_cluster_[cluster_i];
			}
			bool cov_mat_only_for_simulation = false;
			if (sample_posterior) {
				if (!only_one_grouped_RE_calculations_on_RE_scale_ && !only_one_grouped_RE_calculations_on_RE_scale_for_prediction_ &&
					!use_woodbury_identity_ && !grouped_RE_and_vecchia_GP_) {
					if (!predict_cov_mat) {
						cov_mat_only_for_simulation = true;
					}
					predict_cov_mat = true;//currently only classical case without an approximation (except tapering) supported
				}
			}
			if (predict_var) {
				if (gauss_likelihood_ && predict_response) {
					var_pred_id = vec_t::Ones(num_REs_pred);//nugget effect
				}
				else {
					var_pred_id = vec_t::Zero(num_REs_pred);
				}
			}
			if (predict_cov_mat) {
				cov_mat_pred_id = T_mat(num_REs_pred, num_REs_pred);
				if (gauss_likelihood_ && predict_response) {
					cov_mat_pred_id.setIdentity();//nugget effect
				}
				else {
					cov_mat_pred_id.setZero();
				}
			}
			T_mat cross_cov;//Cross-covariance between prediction and observation points
			sp_mat_t Ztilde;//Matrix which relates existing random effects to prediction samples (used only if use_woodbury_identity_ and not only_one_grouped_RE_calculations_on_RE_scale_)
			sp_mat_t Sigma;//Covariance matrix of random effects (used only if use_woodbury_identity_ || grouped_RE_and_vecchia_GP_ and not only_one_grouped_RE_calculations_on_RE_scale_)
			std::vector<data_size_t> random_effects_indices_of_pred;//Indices that indicate to which training data random effect every prediction point is related. -1 means to none in the training data
			//Calculate (cross-)covariance matrix
			int cn = 0;//component number counter
			bool dont_add_but_overwrite = true;
			if (only_one_grouped_RE_calculations_on_RE_scale_ || only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
				if (sample_posterior) {
					Log::REFatal("Posterior sampling is not implemented for single-level grouped random effects ");
				}
				std::shared_ptr<RECompGroup<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGroup<T_mat>>(re_comps_[cluster_i][0][0]);
				if (predict_cov_mat) {
					re_comp->AddPredCovMatrices(re_group_levels_pred[0], cross_cov, cov_mat_pred_id,
						true, predict_cov_mat, true, true, nullptr);
				}
				random_effects_indices_of_pred = std::vector<data_size_t>(re_group_levels_pred[0].size());
				re_comp->RandomEffectsIndicesPred(re_group_levels_pred[0], random_effects_indices_of_pred.data());//Note re_group_levels_pred[0] contains only data for cluster_i; see above
				if (predict_var) {
					if (gauss_likelihood_) {
						re_comp->AddPredUncondVarNewGroups(var_pred_id.data(), num_REs_pred, nullptr, re_group_levels_pred[0]);
					}
					else {
						re_comp->AddPredUncondVar(var_pred_id.data(), num_REs_pred, nullptr);
					}
				}
			}
			else if (use_woodbury_identity_ || grouped_RE_and_vecchia_GP_) {
				if (sample_posterior) {
					Log::REFatal("Posterior sampling is not implemented for multi-level grouped random effects ");
				}
				Ztilde = sp_mat_t(num_data_per_cluster_pred[cluster_i], cum_num_rand_eff_[cluster_i][num_re_group_total_]);
				if (linear_kernel_use_woodbury_identity_) {
					Ztilde = gp_coords_mat_pred.sparseView();
					std::shared_ptr<RECompGP<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][ind_intercept_gp_]);
					if (predict_cov_mat) {
						T_mat cov_mat_pred_id_aux;
						ConvertTo_T_mat_FromDense<T_mat>(re_comp->GetMarginalVariance() * gp_coords_mat_pred * gp_coords_mat_pred.transpose(), cov_mat_pred_id_aux);
						cov_mat_pred_id += cov_mat_pred_id_aux;
					}
					if (predict_var) {
						if (matrix_inversion_method_ == "cholesky") {
							var_pred_id += re_comp->GetMarginalVariance() * gp_coords_mat_pred.rowwise().squaredNorm();
						}
						//else {//"iterative"
						//	//nothing to do as there are no "new groups"			
						//}
					}
				}//end linear_kernel_use_woodbury_identity_
				else {//!linear_kernel_use_woodbury_identity_
					bool has_ztilde = false;
					std::vector<Triplet_t> triplets(num_data_per_cluster_pred[cluster_i] * num_re_group_total_);
					for (int j = 0; j < num_group_variables_; ++j) {
						if (!drop_intercept_group_rand_effect_[j]) {
							std::shared_ptr<RECompGroup<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGroup<T_mat>>(re_comps_[cluster_i][0][cn]);
							std::vector<re_group_t> group_data_pred;
							for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
								group_data_pred.push_back(re_group_levels_pred[j][id]);
							}
							re_comp->CalcInsertZtilde(group_data_pred, nullptr, cum_num_rand_eff_[cluster_i][cn], cn, triplets, has_ztilde);
							if (predict_cov_mat) {
								re_comp->AddPredCovMatrices(group_data_pred, cross_cov, cov_mat_pred_id,
									false, true, false, false, nullptr);//Note: cross_cov is not used, only unconditional predictive covariance is added
							}
							if (predict_var) {
								if (matrix_inversion_method_ == "iterative" || grouped_RE_and_vecchia_GP_) {
									re_comp->AddPredUncondVarNewGroups(var_pred_id.data(), num_REs_pred, nullptr, group_data_pred);
								}
								else if (matrix_inversion_method_ == "cholesky") {
									//re_comp->AddPredUncondVarNewGroups(var_pred_id.data(), num_REs_pred, nullptr, group_data_pred);//if VERSION 2 is used in PredictLaplaceApproxGroupedRE()
									re_comp->AddPredUncondVar(var_pred_id.data(), num_REs_pred, nullptr);
								}
							}
							cn += 1;
						}
					}
					if (num_re_group_rand_coef_ > 0) {//Random coefficient grouped random effects
						for (int j = 0; j < num_re_group_rand_coef_; ++j) {
							std::shared_ptr<RECompGroup<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGroup<T_mat>>(re_comps_[cluster_i][0][cn]);
							std::vector<re_group_t> group_data_pred;
							std::vector<double> rand_coef_data;
							for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
								rand_coef_data.push_back(re_group_rand_coef_data_pred[j * num_data_pred + id]);
								group_data_pred.push_back(re_group_levels_pred[ind_effect_group_rand_coef_[j] - 1][id]);//subtract 1 since counting starts at one for this index
							}
							re_comp->CalcInsertZtilde(group_data_pred, rand_coef_data.data(), cum_num_rand_eff_[cluster_i][cn], cn, triplets, has_ztilde);
							if (predict_cov_mat) {
								re_comp->AddPredCovMatrices(group_data_pred, cross_cov, cov_mat_pred_id,
									false, true, false, false, rand_coef_data.data());
							}
							if (predict_var) {
								re_comp->AddPredUncondVar(var_pred_id.data(), num_REs_pred, rand_coef_data.data());
							}
							cn += 1;
						}
					}
					if (has_ztilde) {
						Ztilde.setFromTriplets(triplets.begin(), triplets.end());
					}
				}//end !linear_kernel_use_woodbury_identity_
				CalcSigmaOrInverseUseWoodbury(Sigma, cluster_i, false);
			}//end use_woodbury_identity_ || grouped_RE_and_vecchia_GP_
			else {//!only_one_grouped_RE_calculations_on_RE_scale_ && !use_woodbury_identity_ && ! grouped_RE_and_vecchia_GP_
				// classical case without any approximation (except tapering)
				if (num_re_group_ > 0) {//Grouped random effects
					for (int j = 0; j < num_group_variables_; ++j) {
						if (!drop_intercept_group_rand_effect_[j]) {
							std::shared_ptr<RECompGroup<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGroup<T_mat>>(re_comps_[cluster_i][0][cn]);
							std::vector<re_group_t> group_data;
							for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
								group_data.push_back(re_group_levels_pred[j][id]);
							}
							re_comp->AddPredCovMatrices(group_data, cross_cov, cov_mat_pred_id,
								true, predict_cov_mat, dont_add_but_overwrite, false, nullptr);
							dont_add_but_overwrite = false;
							if (predict_var) {
								re_comp->AddPredUncondVar(var_pred_id.data(), num_REs_pred, nullptr);
							}
							cn += 1;
						}
					}
				}//end grouped random effects
				if (num_re_group_rand_coef_ > 0) { //Random coefficient grouped random effects
					for (int j = 0; j < num_re_group_rand_coef_; ++j) {
						std::shared_ptr<RECompGroup<T_mat>> re_comp = std::dynamic_pointer_cast<RECompGroup<T_mat>>(re_comps_[cluster_i][0][cn]);
						std::vector<re_group_t> group_data;
						std::vector<double> rand_coef_data;
						for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
							rand_coef_data.push_back(re_group_rand_coef_data_pred[j * num_data_pred + id]);
							group_data.push_back(re_group_levels_pred[ind_effect_group_rand_coef_[j] - 1][id]);//subtract 1 since counting starts at one for this index
						}
						re_comp->AddPredCovMatrices(group_data, cross_cov, cov_mat_pred_id,
							true, predict_cov_mat, false, false, rand_coef_data.data());
						if (predict_var) {
							re_comp->AddPredUncondVar(var_pred_id.data(), num_REs_pred, rand_coef_data.data());
						}
						cn += 1;
					}
				}//end random coefficient grouped random effects
				//Gaussian process
				if (num_gp_ > 0 && !grouped_RE_and_vecchia_GP_) {
					std::shared_ptr<RECompGP<T_mat>> re_comp_base = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][cn]);
					T_mat cross_dist; // unused dummy variable
					re_comp_base->AddPredCovMatrices(re_comp_base->coords_, gp_coords_mat_pred, cross_cov,
						cov_mat_pred_id, true, predict_cov_mat, dont_add_but_overwrite, nullptr,
						false, cross_dist);
					dont_add_but_overwrite = false;
					if (predict_var) {
						if (!(re_comp_base->VarianceOnDiagonal())) {
#pragma omp parallel for schedule(static)
							for (int ii = 0; ii < (int)gp_coords_mat_pred.rows(); ++ii) {
								vec_t coord_pred_i = gp_coords_mat_pred.row(ii);
								var_pred_id[ii] += re_comp_base->CalculateCovarianceOneEntry(coord_pred_i, coord_pred_i);
							}
						}
						else {
							re_comp_base->AddPredUncondVar(var_pred_id.data(), num_REs_pred, nullptr);
						}
					}
					cn += 1;
					if (num_gp_rand_coef_ > 0) {
						std::shared_ptr<RECompGP<T_mat>> re_comp;
						//Random coefficient Gaussian processes
						for (int j = 0; j < num_gp_rand_coef_; ++j) {
							re_comp = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_[cluster_i][0][cn]);
							std::vector<double> rand_coef_data;
							for (const auto& id : data_indices_per_cluster_pred[cluster_i]) {
								rand_coef_data.push_back(gp_rand_coef_data_pred[j * num_data_pred + id]);
							}
							re_comp->AddPredCovMatrices(re_comp_base->coords_, gp_coords_mat_pred, cross_cov,
								cov_mat_pred_id, true, predict_cov_mat, false, rand_coef_data.data(),
								false, cross_dist);
							if (predict_var) {
								if (!(re_comp->VarianceOnDiagonal())) {
									Log::REFatal("Predictive variances are currently not yet implemented for this covariance and random GP coefficients ");
								}
								re_comp->AddPredUncondVar(var_pred_id.data(), num_REs_pred, rand_coef_data.data());
							}
							cn += 1;
						}
					}
				}// end Gaussian process
			}//end calculate cross-covariances for !only_one_grouped_RE_calculations_on_RE_scale_ && !use_woodbury_identity_ && !grouped_RE_and_vecchia_GP_
			// Calculate predictive means and covariances
			if (gauss_likelihood_) {//Gaussian data
				if (only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
					vec_t Zt_y_aux(num_REs_obs);
					CalcZtVGivenIndices(num_data_per_cluster_[cluster_i], num_REs_obs,
						re_comps_[cluster_i][0][cn]->random_effects_indices_of_data_.data(), y_aux_[cluster_i].data(), Zt_y_aux.data(), true);
					mean_pred_id = vec_t::Zero(random_effects_indices_of_pred.size());
					double sigma2 = re_comps_[cluster_i][0][0]->cov_pars_[0];
#pragma omp parallel for schedule(static)
					for (int i = 0; i < (int)random_effects_indices_of_pred.size(); ++i) {
						if (random_effects_indices_of_pred[i] >= 0) {
							mean_pred_id[i] = sigma2 * Zt_y_aux[random_effects_indices_of_pred[i]];
						}
					}
				}//end only_one_grouped_RE_calculations_on_RE_scale_for_prediction_
				else if (use_woodbury_identity_) {
					vec_t v_aux = Zt_[cluster_i] * y_aux_[cluster_i];
					vec_t v_aux2 = Sigma * v_aux;
					mean_pred_id = Ztilde * v_aux2;
				}//end use_woodbury_identity_
				else {
					mean_pred_id = cross_cov * y_aux_[cluster_i];
				}
				if (predict_cov_mat && only_one_grouped_RE_calculations_on_RE_scale_for_prediction_) {
					sp_mat_t* Z = re_comps_[cluster_i][0][0]->GetZ();
					T_mat cross_cov_temp = cross_cov;
					cross_cov = cross_cov_temp * (*Z).transpose();
					cross_cov_temp.resize(0, 0);
				}
				if (predict_cov_mat) {
					if (use_woodbury_identity_) {
						if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
							T_mat ZtM_aux = (T_mat)(Zt_[cluster_i] * cross_cov.transpose());
							ZtM_aux = sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().inverse().matrix().asDiagonal() * ZtM_aux;
							cov_mat_pred_id -= (T_mat)(cross_cov * cross_cov.transpose());
							cov_mat_pred_id += (T_mat)(ZtM_aux.transpose() * ZtM_aux);
						}
						else {
							if (matrix_inversion_method_ == "iterative") {
								den_mat_t pred_cov_global = den_mat_t::Zero(num_REs_pred, num_REs_pred);
								vec_t SigmaI_diag_sqrt = Sigma.diagonal().cwiseInverse().cwiseSqrt();
								if (!cg_generator_seeded_) {
									cg_generator_ = RNG_t(seed_rand_vec_trace_);
									cg_generator_seeded_ = true;
								}
								int num_threads;
#ifdef _OPENMP
								num_threads = omp_get_max_threads();
#else
								num_threads = 1;
#endif
								std::uniform_int_distribution<> unif(0, 2147483646);
								std::vector<RNG_t> parallel_rngs;
								for (int ig = 0; ig < num_threads; ++ig) {
									int seed_local = unif(cg_generator_);
									parallel_rngs.push_back(RNG_t(seed_local));
								}
#pragma omp parallel
								{
									int thread_nb;
#ifdef _OPENMP
									thread_nb = omp_get_thread_num();
#else
									thread_nb = 0;
#endif
									RNG_t rng_local = parallel_rngs[thread_nb];
									den_mat_t pred_cov_private = den_mat_t::Zero(num_REs_pred, num_REs_pred);
#pragma omp for
									for (int i = 0; i < nsim_var_pred_; ++i) {
										//z_i ~ N(0,I)
										std::normal_distribution<double> ndist(0.0, 1.0);
										vec_t rand_vec_pred_I_1(cum_num_rand_eff_[cluster_i][num_comps_total_]), rand_vec_pred_I_2(num_data_per_cluster_[cluster_i]);
										for (int j = 0; j < cum_num_rand_eff_[cluster_i][num_comps_total_]; j++) {
											rand_vec_pred_I_1(j) = ndist(rng_local);
										}
										for (int j = 0; j < num_data_per_cluster_[cluster_i]; j++) {
											rand_vec_pred_I_2(j) = ndist(rng_local);
										}
										//z_i ~ N(0,(Sigma^(-1) + Z^T Z))
										vec_t rand_vec_pred_SigmaI_plus_ZtZ = SigmaI_diag_sqrt.asDiagonal() * rand_vec_pred_I_1 + Zt_[cluster_i] * rand_vec_pred_I_2;
										vec_t rand_vec_pred_SigmaI_plus_ZtZ_inv(cum_num_rand_eff_[cluster_i][num_comps_total_]);
										//z_i ~ N(0,(Sigma^(-1) + Z^T Z)^(-1))
										int num_cg_steps_dummy;
										CGRandomEffectsVec(SigmaI_plus_ZtZ_rm_[cluster_i], rand_vec_pred_SigmaI_plus_ZtZ, rand_vec_pred_SigmaI_plus_ZtZ_inv, NaN_found, cg_max_num_it_, cg_delta_conv_pred_, true, THRESHOLD_ZERO_RHS_CG_,
											true, cg_preconditioner_type_, L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i], SigmaI_plus_ZtZ_inv_diag_[cluster_i], num_cg_steps_dummy
											//cum_num_rand_eff_[cluster_i], num_comps_total_, P_SSOR_D1_inv_[cluster_i], P_SSOR_D2_inv_[cluster_i], P_SSOR_B_rm_[cluster_i]
										);
										if (NaN_found) {
											Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
										}
										//z_i ~ N(0, Z_p (Sigma^(-1) + Z^T Z)^(-1) Z_p^T)
										vec_t rand_vec_pred = Ztilde * rand_vec_pred_SigmaI_plus_ZtZ_inv;
										pred_cov_private += rand_vec_pred * rand_vec_pred.transpose();
									} //end for loop
#pragma omp critical
									{
										pred_cov_global += pred_cov_private;
									}
								} // end #pragma omp parallel
								pred_cov_global /= nsim_var_pred_;
								T_mat pred_cov_T_mat;
								ConvertTo_T_mat_FromDense<T_mat>(pred_cov_global, pred_cov_T_mat);
								cov_mat_pred_id -= (T_mat)(Ztilde * Sigma * Ztilde.transpose()); //TODO: create and call AddPredCovMatrices only for new groups and remove this line.
								cov_mat_pred_id += pred_cov_T_mat;
							} //end iterative
							else { //begin cholesky
								T_mat M_aux;
								TriangularSolveGivenCholesky<T_chol, T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i], ZtZ_[cluster_i], M_aux, false);
								sp_mat_t ZtildeSigma = Ztilde * Sigma;
								T_mat M_aux2 = M_aux * ZtildeSigma.transpose();
								M_aux.resize(0, 0);
								cov_mat_pred_id -= (T_mat)(ZtildeSigma * ZtZ_[cluster_i] * ZtildeSigma.transpose());
								cov_mat_pred_id += (T_mat)(M_aux2.transpose() * M_aux2);
							} //end cholesky
						}
					}
					else {
						T_mat M_aux;
						TriangularSolveGivenCholesky<T_chol, T_mat, T_mat, T_mat>(chol_facts_[cluster_i], cross_cov.transpose(), M_aux, false);
						cov_mat_pred_id -= (T_mat)(M_aux.transpose() * M_aux);
					}
				}//end predict_cov_mat
				if (predict_var) {
					if (use_woodbury_identity_) {
						if (num_re_group_total_ == 1 && num_comps_total_ == 1) {//only one random effect -> ZtZ_ is diagonal
							vec_t SigmaI_plus_ZtZ_inv = sqrt_diag_SigmaI_plus_ZtZ_[cluster_i].array().square().inverse().matrix();
#pragma omp parallel for schedule(static)
							for (int i = 0; i < (int)random_effects_indices_of_pred.size(); ++i) {
								if (random_effects_indices_of_pred[i] >= 0) {
									var_pred_id[i] += SigmaI_plus_ZtZ_inv[random_effects_indices_of_pred[i]];
								}
							}
						}
						else {//more than one grouped RE component
							if (matrix_inversion_method_ == "iterative") {
								vec_t pred_var_global = vec_t::Zero(num_REs_pred);
								//Variance reduction
								sp_mat_rm_t Ztilde_P_sqrt_invt_rm;
								vec_t varred_global, c_cov, c_var;
								if (cg_preconditioner_type_ == "incomplete_cholesky" || cg_preconditioner_type_ == "ssor") {
									varred_global = vec_t::Zero(num_REs_pred);
									c_cov = vec_t::Zero(num_REs_pred);
									c_var = vec_t::Zero(num_REs_pred);
									//Calculate P^(-0.5) explicitly
									sp_mat_rm_t Identity_rm(cum_num_rand_eff_[cluster_i][num_comps_total_], cum_num_rand_eff_[cluster_i][num_comps_total_]);
									Identity_rm.setIdentity();
									sp_mat_rm_t P_sqrt_invt_rm;
									if (cg_preconditioner_type_ == "incomplete_cholesky") {
										TriangularSolve<sp_mat_rm_t, sp_mat_rm_t, sp_mat_rm_t>(L_SigmaI_plus_ZtZ_rm_[cluster_i], Identity_rm, P_sqrt_invt_rm, true);
									}
									else {
										TriangularSolve<sp_mat_rm_t, sp_mat_rm_t, sp_mat_rm_t>(P_SSOR_L_D_sqrt_inv_rm_[cluster_i], Identity_rm, P_sqrt_invt_rm, true);
									}
									//Z_po P^(-T/2)
									Ztilde_P_sqrt_invt_rm = Ztilde * P_sqrt_invt_rm;
								}
								if (!cg_generator_seeded_) {
									cg_generator_ = RNG_t(seed_rand_vec_trace_);
									cg_generator_seeded_ = true;
								}
								int num_threads;
#ifdef _OPENMP
								num_threads = omp_get_max_threads();
#else
								num_threads = 1;
#endif
								std::uniform_int_distribution<> unif(0, 2147483646);
								std::vector<RNG_t> parallel_rngs;
								for (int ig = 0; ig < num_threads; ++ig) {
									int seed_local = unif(cg_generator_);
									parallel_rngs.push_back(RNG_t(seed_local));
								}
#pragma omp parallel
								{
									int thread_nb;
#ifdef _OPENMP
									thread_nb = omp_get_thread_num();
#else
									thread_nb = 0;
#endif
									RNG_t rng_local = parallel_rngs[thread_nb];
									vec_t pred_var_private = vec_t::Zero(num_REs_pred);
									vec_t varred_private;
									vec_t c_cov_private;
									vec_t c_var_private;
									//Variance reduction
									if (cg_preconditioner_type_ == "incomplete_cholesky" || cg_preconditioner_type_ == "ssor") {
										varred_private = vec_t::Zero(num_REs_pred);
										c_cov_private = vec_t::Zero(num_REs_pred);
										c_var_private = vec_t::Zero(num_REs_pred);
									}
#pragma omp for
									for (int i = 0; i < nsim_var_pred_; ++i) {
										//RV - Rademacher
										std::uniform_real_distribution<double> udist(0.0, 1.0);
										vec_t rand_vec_init(num_REs_pred);
										double u;
										for (int j = 0; j < num_REs_pred; j++) {
											u = udist(rng_local);
											if (u > 0.5) {
												rand_vec_init(j) = 1.;
											}
											else {
												rand_vec_init(j) = -1.;
											}
										}
										//Z_po^T RV
										vec_t Z_tilde_t_RV = Ztilde.transpose() * rand_vec_init;
										//Part 2: (Sigma^(-1) + Z^T Z)^(-1) Z_po^T RV
										vec_t MInv_Ztilde_t_RV(cum_num_rand_eff_[cluster_i][num_comps_total_]);
										int num_cg_steps_dummy;
										CGRandomEffectsVec(SigmaI_plus_ZtZ_rm_[cluster_i], Z_tilde_t_RV, MInv_Ztilde_t_RV, NaN_found, cg_max_num_it_, cg_delta_conv_pred_, true, THRESHOLD_ZERO_RHS_CG_,
											true, cg_preconditioner_type_, L_SigmaI_plus_ZtZ_rm_[cluster_i], P_SSOR_L_D_sqrt_inv_rm_[cluster_i], SigmaI_plus_ZtZ_inv_diag_[cluster_i], num_cg_steps_dummy
											//cum_num_rand_eff_[cluster_i], num_comps_total_, P_SSOR_D1_inv_[cluster_i], P_SSOR_D2_inv_[cluster_i], P_SSOR_B_rm_[cluster_i]
										);
										if (NaN_found) {
											Log::REFatal("There was Nan or Inf value generated in the Conjugate Gradient Method!");
										}
										//Part 2: Z_po (Sigma^(-1) + Z^T Z)^(-1) Z_po^T RV
										vec_t rand_vec_final = Ztilde * MInv_Ztilde_t_RV;
										pred_var_private += rand_vec_final.cwiseProduct(rand_vec_init);
										//Variance reduction
										if (cg_preconditioner_type_ == "incomplete_cholesky" || cg_preconditioner_type_ == "ssor") {
											//Stochastic: Z_po P^(-0.5T) P^(-0.5) Z_po^T RV
											vec_t P_sqrt_inv_Ztilde_t_RV = Ztilde_P_sqrt_invt_rm.transpose() * rand_vec_init;
											vec_t rand_vec_varred = Ztilde_P_sqrt_invt_rm * P_sqrt_inv_Ztilde_t_RV;
											varred_private += rand_vec_varred.cwiseProduct(rand_vec_init);
											c_cov_private += varred_private.cwiseProduct(pred_var_private);
											c_var_private += varred_private.cwiseProduct(varred_private);
										}

									} //end for loop
#pragma omp critical
									{
										pred_var_global += pred_var_private;
										//Variance reduction
										if (cg_preconditioner_type_ == "incomplete_cholesky" || cg_preconditioner_type_ == "ssor") {
											varred_global += varred_private;
											c_cov += c_cov_private;
											c_var += c_var_private;
										}
									}
								} // end #pragma omp parallel
								pred_var_global /= nsim_var_pred_;
								var_pred_id += pred_var_global;
								//Variance reduction
								if (cg_preconditioner_type_ == "incomplete_cholesky" || cg_preconditioner_type_ == "ssor") {
									varred_global /= nsim_var_pred_;
									c_cov /= nsim_var_pred_;
									c_var /= nsim_var_pred_;
									//Deterministic: diag(Z_po P^(-0.5T) P^(-0.5) Z_po^T)
									vec_t varred_determ = Ztilde_P_sqrt_invt_rm.cwiseProduct(Ztilde_P_sqrt_invt_rm) * vec_t::Ones(cum_num_rand_eff_[cluster_i][num_comps_total_]);
									//optimal c
									c_cov -= varred_global.cwiseProduct(pred_var_global);
									c_var -= varred_global.cwiseProduct(varred_global);
									vec_t c_opt = c_cov.array() / c_var.array();
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < c_opt.size(); ++i) {
										if (c_var.coeffRef(i) == 0) {
											c_opt[i] = 1;
										}
									}
									var_pred_id += c_opt.cwiseProduct(varred_determ - varred_global);
								}
							} //end iterative
							else { //begin cholesky
								T_mat M_aux;
								TriangularSolveGivenCholesky<T_chol, T_mat, sp_mat_t, T_mat>(chol_facts_[cluster_i], ZtZ_[cluster_i], M_aux, false);
								sp_mat_t ZtildeSigma = Ztilde * Sigma;
								T_mat M_aux2 = M_aux * ZtildeSigma.transpose();
								M_aux.resize(0, 0);
								sp_mat_t SigmaZtilde_ZtZ = ZtildeSigma * ZtZ_[cluster_i];
								sp_mat_t M_aux3 = ZtildeSigma.cwiseProduct(SigmaZtilde_ZtZ);
								M_aux2 = M_aux2.cwiseProduct(M_aux2);
#pragma omp parallel for schedule(static)
								for (int i = 0; i < num_REs_pred; ++i) {
									var_pred_id[i] -= M_aux3.row(i).sum() - M_aux2.col(i).sum();
								}
							}
						}
					}//end use_woodbury_identity_
					else {//not use_woodbury_identity_
						T_mat M_aux2;
						TriangularSolveGivenCholesky<T_chol, T_mat, T_mat, T_mat>(chol_facts_[cluster_i], cross_cov.transpose(), M_aux2, false);
						M_aux2 = M_aux2.cwiseProduct(M_aux2);
#pragma omp parallel for schedule(static)
						for (int i = 0; i < num_REs_pred; ++i) {
							var_pred_id[i] -= M_aux2.col(i).sum();
						}
					}//end not use_woodbury_identity_
				}//end predict_var
			}//end gauss_likelihood_
			if (!gauss_likelihood_) {//not gauss_likelihood_
				const double* fixed_effects_cluster_i_ptr = nullptr;
				// Note that fixed_effects_cluster_i_ptr is not used since calc_mode == false
				// The mode has been calculated already before in the Predict() function above
				if ((use_woodbury_identity_ && !only_one_grouped_RE_calculations_on_RE_scale_) || grouped_RE_and_vecchia_GP_) {
					likelihood_[cluster_i]->PredictLaplaceApproxGroupedRE(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, SigmaI_[cluster_i], grouped_RE_and_vecchia_GP_, B_[cluster_i][0], D_inv_[cluster_i][0], Bpo, Bp, Dp, VecchiaCondObsOnly, Ztilde, Sigma,
						mean_pred_id, cov_mat_pred_id, var_pred_id,
						predict_cov_mat, predict_var, false);
				}
				else if (only_one_grouped_RE_calculations_on_RE_scale_) {
					likelihood_[cluster_i]->PredictLaplaceApproxOnlyOneGroupedRECalculationsOnREScale(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr,
						re_comps_[cluster_i][0][0]->cov_pars_[0],
						random_effects_indices_of_pred.data(), (data_size_t)random_effects_indices_of_pred.size(), cross_cov,
						mean_pred_id, cov_mat_pred_id, var_pred_id,
						predict_cov_mat, predict_var, false);
				}
				else {
					likelihood_[cluster_i]->PredictLaplaceApproxStable(y_[cluster_i].data(), y_int_[cluster_i].data(),
						fixed_effects_cluster_i_ptr, ZSigmaZt_[cluster_i], cross_cov,
						mean_pred_id, cov_mat_pred_id, var_pred_id,
						predict_cov_mat, predict_var, false);
				}
			}//end not gauss_likelihood_
			if (sample_posterior) {
				if (!only_one_grouped_RE_calculations_on_RE_scale_ && !only_one_grouped_RE_calculations_on_RE_scale_for_prediction_ &&
					!use_woodbury_identity_ && !grouped_RE_and_vecchia_GP_) {
					CHECK(predict_cov_mat);
					T_chol chol_fact_cov_mat;
					chol_fact_cov_mat.compute(cov_mat_pred_id);
					if (cov_mat_only_for_simulation) {
						cov_mat_pred_id.resize(0, 0);
					}
					post_samples_id = den_mat_t(num_REs_pred, num_post_samples);					
					GenRandVecNormalParallel(seed_rng_, cg_generator_counter_, post_samples_id);
					post_samples_id = chol_fact_cov_mat.CholFactMatrix().template triangularView<Eigen::Lower>() * post_samples_id;
					if (CholeskyHasPermutation<T_chol>(chol_fact_cov_mat)) {
						ApplyPermutationCholeskyFactor<den_mat_t, T_chol>(chol_fact_cov_mat, post_samples_id, post_samples_id, /*transpose=*/true);
					}
				}
			}//end sample_posterior
			if (sample_prior) {
				if (!only_one_grouped_RE_calculations_on_RE_scale_ && !only_one_grouped_RE_calculations_on_RE_scale_for_prediction_ &&
					!use_woodbury_identity_ && !grouped_RE_and_vecchia_GP_) {
					prior_samples_id = den_mat_t(num_REs_obs, num_prior_samples);
					GenRandVecNormalParallel(seed_rng_, cg_generator_counter_, prior_samples_id);
					prior_samples_id = chol_facts_[cluster_i].CholFactMatrix().template triangularView<Eigen::Lower>() * prior_samples_id;
					if (CholeskyHasPermutation<T_chol>(chol_facts_[cluster_i])) {
						ApplyPermutationCholeskyFactor<den_mat_t, T_chol>(chol_facts_[cluster_i], prior_samples_id, prior_samples_id, /*transpose=*/true);
					}
				}
			}//end sample_prior
		}//end CalcPred

		void SetVecchiaPredType(const char* vecchia_pred_type) {
			vecchia_pred_type_ = std::string(vecchia_pred_type);
			if (gauss_likelihood_) {
				if (SUPPORTED_VECCHIA_PRED_TYPES_GAUSS_.find(vecchia_pred_type_) == SUPPORTED_VECCHIA_PRED_TYPES_GAUSS_.end()) {
					Log::REFatal("Prediction type '%s' is not supported for the Veccia approximation ", vecchia_pred_type_.c_str());
				}
			}
			else {
				if (SUPPORTED_VECCHIA_PRED_TYPES_NONGAUSS_.find(vecchia_pred_type_) == SUPPORTED_VECCHIA_PRED_TYPES_NONGAUSS_.end()) {
					Log::REFatal("Prediction type '%s' is not supported for the Veccia approximation for non-Gaussian likelihoods ", vecchia_pred_type_.c_str());
				}
				if (vecchia_pred_type_ == "order_obs_first_cond_obs_only") {
					vecchia_pred_type_ = "latent_order_obs_first_cond_obs_only";
				}
				if (vecchia_pred_type_ == "order_obs_first_cond_all") {
					vecchia_pred_type_ = "latent_order_obs_first_cond_all";
				}
			}
			vecchia_pred_type_has_been_set_ = true;
		}//end SetVecchiaPredType

		/*!
		* \brief Calculate predictions (conditional mean and covariance matrix) using the PP/FSA approximation
		* \param cluster_i Cluster index for which prediction are made
		* \param num_data_per_cluster_pred Keys: Labels of independent realizations of REs/GPs, values: number of prediction locations per independent realization
		* \param num_data_per_cluster Keys: Labels of independent realizations of REs/GPs, values: number of observed locations per independent realization
		* \param gp_coords_mat_pred Coordinates for prediction locations
		* \param calc_pred_cov If true, the covariance matrix is also calculated
		* \param calc_pred_var If true, predictive variances are also calculated
		* \param predict_response If true, the response variable (label) is predicted, otherwise the latent random effects
		* \param[out] pred_mean Predictive mean (only for Gaussian likelihoods)
		* \param[out] pred_cov Predictive covariance matrix (only for Gaussian likelihoods)
		* \param[out] pred_var Predictive variances (only for Gaussian likelihoods)
		* \param nsim_var_pred Number of random vectors
		* \param cg_delta_conv_pred Tolerance level for L2 norm of residuals for checking convergence in conjugate gradient algorithm when being used for prediction
		*/
		void CalcPredFITC_FSA(data_size_t cluster_i,
			const den_mat_t& gp_coords_mat_pred,
			bool calc_pred_cov,
			bool calc_pred_var,
			bool predict_response,
			vec_t& pred_mean,
			T_mat& pred_cov,
			vec_t& pred_var,
			int nsim_var_pred,
			const double cg_delta_conv_pred) {
			int num_REs_obs = re_comps_cross_cov_[cluster_i][0][0]->GetNumUniqueREs();
			int num_REs_pred = (int)gp_coords_mat_pred.rows();
			// Initialization of Components C_pm & C_pn & C_pp
			den_mat_t cross_cov_pred_ip, chol_ip_cross_cov_ip_pred;
			T_mat sigma_resid_pred_obs, cross_dist_resid, sigma_resid_pred, cross_dist_resid_pred;
			T_mat cov_mat_pred_obs, cov_mat_pred; // unused dummy variables
			std::shared_ptr<T_mat> sigma_resid;
			sp_mat_t fitc_resid_pred_obs;//FITC residual correction for entries for which the prediction and training coordinates are the same
			bool has_fitc_correction = false;
			if (num_comps_total_ > 1) {
				Log::REFatal("CalcPredFITC_FSA is not implemented when num_comps_total_ > 1");
			}
			// Construct components
			const den_mat_t* cross_cov = re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr();
			den_mat_t sigma_ip_stable = *(re_comps_ip_[cluster_i][0][0]->GetZSigmaZt());
			// Cross-covariance between predictions and inducing points C_pm
			den_mat_t cov_mat_pred_id, cross_dist; // unused dummy variables
			std::shared_ptr<RECompGP<den_mat_t>> re_comp_cross_cov_cluster_i_pred_ip = re_comps_cross_cov_[cluster_i][0][0];
			re_comp_cross_cov_cluster_i_pred_ip->AddPredCovMatrices(re_comp_cross_cov_cluster_i_pred_ip->coords_ind_point_, gp_coords_mat_pred, cross_cov_pred_ip,
				cov_mat_pred_id, true, false, true, nullptr, false, cross_dist);
			// Construct residual part for cross covariance between prediction and training locations
			if (gp_approx_ == "full_scale_tapering") {
				// Cross-covariance between predictions and observations C_pn (tapered)
				std::shared_ptr<RECompGP<T_mat>> re_comps_resid_po_cluster_i = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_resid_[cluster_i][0][0]);
				re_comps_resid_po_cluster_i->AddPredCovMatrices(re_comps_resid_po_cluster_i->coords_, gp_coords_mat_pred, sigma_resid_pred_obs,
					cov_mat_pred_obs, true, false, true, nullptr, true, cross_dist_resid);
				//den_mat_t sigma_ip_inv_cross_cov_T = chol_fact_sigma_ip_[cluster_i][0].solve((*cross_cov).transpose());// Calculate Cm_inv * C_mn part of predictive process
				den_mat_t sigma_ip_inv_cross_cov_T;
				GPBoost::solve_linear_sys(chol_fact_sigma_ip_[cluster_i][0], (*cross_cov).transpose(), sigma_ip_inv_cross_cov_T, GPU_use_);
				SubtractProdFromNonSqMat<T_mat>(sigma_resid_pred_obs, cross_cov_pred_ip.transpose(), sigma_ip_inv_cross_cov_T);// Subtract predictive process (prediction) covariance
				re_comps_resid_po_cluster_i->ApplyTaper(cross_dist_resid, sigma_resid_pred_obs);// Apply taper
			}//end gp_approx_ == "full_scale_tapering"
			else if (gp_approx_ == "fitc") {
				// check whether "FITC diagonal" correction needs to be added for duplicate coordinates
				std::vector<Triplet_t> triplets;
				double sigma2 = sigma_ip_stable.coeffRef(0, 0);
				std::vector<double> coords_sum(num_REs_obs), coords_pred_sum(num_REs_pred);
#pragma omp parallel for schedule(static)
				for (int i = 0; i < num_REs_obs; ++i) {
					coords_sum[i] = (re_comp_cross_cov_cluster_i_pred_ip->coords_)(i, Eigen::all).sum();
				}
#pragma omp parallel for schedule(static)
				for (int i = 0; i < num_REs_pred; ++i) {
					coords_pred_sum[i] = gp_coords_mat_pred(i, Eigen::all).sum();
				}
				den_mat_t sigma_ip_inv_cross_cov_T;
#pragma omp parallel for schedule(static)
				for (int ii = 0; ii < num_REs_pred; ++ii) {
					double Sigma_ii;
					if (re_comps_cross_cov_[cluster_i][0][0]->VarianceOnDiagonal()) {
						Sigma_ii = sigma2;
					}
					else {
						vec_t coord_pred_i = gp_coords_mat_pred.row(ii);
						Sigma_ii = re_comps_cross_cov_[cluster_i][0][0]->CalculateCovarianceOneEntry(coord_pred_i, coord_pred_i);
					}
					for (int jj = 0; jj < num_REs_obs; ++jj) {
						if (TwoNumbersAreEqual<double>(coords_pred_sum[ii], coords_sum[jj])) {
							bool are_the_same = true;
							for (int ic = 0; ic < (int)gp_coords_mat_pred.cols(); ++ic) {//loop over coordinates
								are_the_same = are_the_same && TwoNumbersAreEqual<double>(gp_coords_mat_pred.coeff(ii, ic), (re_comp_cross_cov_cluster_i_pred_ip->coords_).coeff(jj, ic));
							}
							if (are_the_same) {
#pragma omp critical
								{
									if (!has_fitc_correction) {
										has_fitc_correction = true;
										sigma_ip_inv_cross_cov_T = chol_fact_sigma_ip_[cluster_i][0].solve((*cross_cov).transpose());
										GPBoost::solve_linear_sys(chol_fact_sigma_ip_[cluster_i][0], (*cross_cov).transpose(), sigma_ip_inv_cross_cov_T, GPU_use_);
									}
								}
								double fitc_corr_ij = Sigma_ii - (cross_cov_pred_ip.row(ii)).dot(sigma_ip_inv_cross_cov_T.col(jj));
#pragma omp critical
								triplets.push_back(Triplet_t(ii, jj, fitc_corr_ij));
							}
						}
					}
				}
				if (has_fitc_correction) {
					fitc_resid_pred_obs = sp_mat_t(num_REs_pred, num_REs_obs);
					fitc_resid_pred_obs.setFromTriplets(triplets.begin(), triplets.end());
				}
			}//end gp_approx_ == "fitc"
			// Calculating predictive mean for gauss_likelihood_
			if (gauss_likelihood_) {
				if (gp_approx_ == "full_scale_tapering") {
					if (matrix_inversion_method_ == "cholesky") {
						pred_mean = cross_cov_pred_ip * (chol_fact_sigma_woodbury_[cluster_i].solve((*cross_cov).transpose() * (chol_fact_resid_[cluster_i].solve(y_[cluster_i]))));
					}
					else {
						pred_mean = cross_cov_pred_ip * (chol_fact_sigma_ip_[cluster_i][0].solve((*cross_cov).transpose() * y_aux_[cluster_i]));
					}
					pred_mean += sigma_resid_pred_obs * y_aux_[cluster_i];
				}
				else if (gp_approx_ == "fitc") {
					pred_mean = cross_cov_pred_ip * (chol_fact_sigma_woodbury_[cluster_i].solve((*cross_cov).transpose() * (fitc_resid_diag_[cluster_i].cwiseInverse().cwiseProduct(y_[cluster_i]))));
					if (has_fitc_correction) {
						pred_mean += fitc_resid_pred_obs * y_aux_[cluster_i];
					}
				}//end if gp_approx_ == "fitc"
			}//end gauss_likelihood_
			// Calculating predicitve covariances and variances
			if (calc_pred_cov || calc_pred_var) {
				// Initialize vector and matrices (and add nugget for gaussian case)
				if (calc_pred_var) {
					if (gauss_likelihood_ && predict_response) {
						pred_var = vec_t::Ones(num_REs_pred);
					}
					else {
						pred_var = vec_t::Zero(num_REs_pred);
					}
				}
				if (calc_pred_cov) {
					if (num_REs_pred > 10000) {
						Log::REInfo("The computational complexity and the storage of the predictive covariance martix heavily depend on the number of prediction location. "
							"Therefore, if this number is large we recommend only computing the predictive variances ");
					}
					pred_cov = T_mat(num_REs_pred, num_REs_pred);
					if (gauss_likelihood_ && predict_response) {
						pred_cov.setIdentity();
					}
					else {
						pred_cov.setZero();
					}
				}
				// Calculate quantities needed below
				bool calc_diag_resid_var_pred = (gp_approx_ == "fitc") ||
					(gauss_likelihood_ && gp_approx_ == "full_scale_tapering" && calc_pred_var && matrix_inversion_method_ != "iterative" && calc_pred_cov_var_FSA_cholesky_ != "exact");
				if (calc_pred_cov || calc_diag_resid_var_pred) {
					//TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_ip_[cluster_i][0], cross_cov_pred_ip.transpose(), chol_ip_cross_cov_ip_pred, false);
					GPBoost::solve_lower_triangular(chol_fact_sigma_ip_[cluster_i][0], cross_cov_pred_ip.transpose(), chol_ip_cross_cov_ip_pred, GPU_use_);
				}
				vec_t resid_diag_pred;
				if (calc_diag_resid_var_pred) {
					resid_diag_pred = vec_t::Zero(num_REs_pred);
					if (re_comps_cross_cov_[cluster_i][0][0]->VarianceOnDiagonal()) {
						resid_diag_pred.array() += sigma_ip_stable.coeffRef(0, 0);
					}
					else {
#pragma omp parallel for schedule(static)
						for (int ii = 0; ii < num_REs_pred; ++ii) {
							vec_t coord_pred_i = gp_coords_mat_pred.row(ii);
							resid_diag_pred[ii] += re_comps_cross_cov_[cluster_i][0][0]->CalculateCovarianceOneEntry(coord_pred_i, coord_pred_i);
						}
					}
#pragma omp parallel for schedule(static)
					for (int ii = 0; ii < num_REs_pred; ++ii) {
						resid_diag_pred[ii] -= chol_ip_cross_cov_ip_pred.col(ii).array().square().sum();
					}
				}
				// Add unconditional variances
				if (calc_pred_var) {
					if (gp_approx_ == "full_scale_tapering" && (!gauss_likelihood_ || (matrix_inversion_method_ == "iterative" || calc_pred_cov_var_FSA_cholesky_ == "exact"))) {
						re_comp_cross_cov_cluster_i_pred_ip->AddPredUncondVar(pred_var.data(), num_REs_pred, nullptr);
					}
					else {
						CHECK(calc_diag_resid_var_pred);
						pred_var += resid_diag_pred;
					}
				}// end add unconditional variances
				// Add unconditional covariances
				if (calc_pred_cov) {
					T_mat PP_Part;
					if (gp_approx_ == "fitc") {
						CHECK(calc_diag_resid_var_pred);
						pred_cov += resid_diag_pred.asDiagonal();
					}
					else if (gp_approx_ == "full_scale_tapering") {
						if (matrix_inversion_method_ == "iterative") {
							Log::REFatal("Predictive covariance matrices are currently not implemented for the '%s' approximation with matrix_inversion_method = 'iterative' and "
								"for the 'fitc' approximation when having multiple observations at the same location ", gp_approx_.c_str());
						}
						if (!gauss_likelihood_ || matrix_inversion_method_ == "iterative" || calc_pred_cov_var_FSA_cholesky_ == "exact") {
							ConvertTo_T_mat_FromDense<T_mat>(chol_ip_cross_cov_ip_pred.transpose() * chol_ip_cross_cov_ip_pred, PP_Part);
							pred_cov += PP_Part;
						}
						std::shared_ptr<RECompGP<T_mat>> re_comps_resid_pp_cluster_i = std::dynamic_pointer_cast<RECompGP<T_mat>>(re_comps_resid_[cluster_i][0][0]);
						re_comps_resid_pp_cluster_i->AddPredCovMatrices(gp_coords_mat_pred, gp_coords_mat_pred, sigma_resid_pred,
							cov_mat_pred, true, false, true, nullptr, true, cross_dist_resid_pred);
						SubtractInnerProdFromMat<T_mat>(sigma_resid_pred, chol_ip_cross_cov_ip_pred, false);// Subtract predictive process (predict) 
						re_comps_resid_pp_cluster_i->ApplyTaper(cross_dist_resid_pred, sigma_resid_pred);// Apply taper
						pred_cov += sigma_resid_pred;
					}
				}// end add unconditional covariances
				// Calculate remaining part of predictive covariance
				if (gauss_likelihood_) {
					if (gp_approx_ == "fitc") {
						den_mat_t Maux_rhs = cross_cov_pred_ip.transpose();
						sp_mat_t resid_obs_inv_resid_pred_obs_t;
						if (has_fitc_correction) {
							resid_obs_inv_resid_pred_obs_t = fitc_resid_diag_[cluster_i].cwiseInverse().asDiagonal() * (fitc_resid_pred_obs.transpose());
							Maux_rhs -= (*cross_cov).transpose() * resid_obs_inv_resid_pred_obs_t;
						}
						den_mat_t woodburry_part_sqrt;
						TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_woodbury_[cluster_i], Maux_rhs, woodburry_part_sqrt, false);
						if (calc_pred_cov) {
							T_mat Maux;
							ConvertTo_T_mat_FromDense<T_mat>(woodburry_part_sqrt.transpose() * woodburry_part_sqrt, Maux);
							pred_cov += Maux;
							if (has_fitc_correction) {
								den_mat_t diag_correction = fitc_resid_pred_obs * resid_obs_inv_resid_pred_obs_t;
								T_mat diag_correction_T_mat;
								ConvertTo_T_mat_FromDense<T_mat>(diag_correction, diag_correction_T_mat);
								pred_cov -= diag_correction_T_mat;
							}
						}//end calc_pred_cov
						if (calc_pred_var) {
#pragma omp parallel for schedule(static)
							for (int i = 0; i < num_REs_pred; ++i) {
								pred_var[i] += woodburry_part_sqrt.col(i).array().square().sum();
							}
							if (has_fitc_correction) {
#pragma omp parallel for schedule(static)
								for (int i = 0; i < num_REs_pred; ++i) {
									pred_var[i] -= fitc_resid_pred_obs.row(i).dot(resid_obs_inv_resid_pred_obs_t.col(i));
								}
							}
						}//end calc_pred_var
					}//end gp_approx_ == "fitc"
					else if (gp_approx_ == "full_scale_tapering") {
						if (matrix_inversion_method_ == "cholesky") {
							if (calc_pred_cov_var_FSA_cholesky_ == "stochastic_stable") {
								den_mat_t resid_obs_inv_cross_cov;
								SolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_fact_resid_[cluster_i], (*cross_cov), resid_obs_inv_cross_cov);
								den_mat_t Maux_rhs = cross_cov_pred_ip.transpose() - (sigma_resid_pred_obs * resid_obs_inv_cross_cov).transpose();
								den_mat_t woodburry_part_sqrt;
								TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_woodbury_[cluster_i], Maux_rhs, woodburry_part_sqrt, false);
								// Use stochastic estimate of sigma_resid_pred_obs * resid_obs_inv * sigma_resid_pred_obs.transpose()
								cg_generator_ = RNG_t(seed_rand_vec_trace_);
								den_mat_t rand_vecs(nsim_var_pred, num_REs_obs);
								GenRandVecNormalParallel(seed_rand_vec_trace_, cg_generator_counter_, rand_vecs);
								den_mat_t sample_resid_cov;
								vec_t sample_resid_var;
								if (calc_pred_cov) {
									sample_resid_cov = den_mat_t(num_REs_pred, num_REs_pred);
									sample_resid_cov.setZero();
								}
								if (calc_pred_var) {
									sample_resid_var = vec_t::Zero(num_REs_pred);
								}
#pragma omp parallel for schedule(static)   
								for (int i = 0; i < nsim_var_pred; ++i) {
									vec_t rand_vec_i = rand_vecs.row(i);
									TriangularSolveGivenCholesky<T_chol, T_mat, vec_t, vec_t>(chol_fact_resid_[cluster_i], rand_vec_i, rand_vec_i, true);
									vec_t rand_vec_resid = sigma_resid_pred_obs * rand_vec_i;
									if (calc_pred_cov) {
										den_mat_t sample_resid_cov_private = rand_vec_resid * rand_vec_resid.transpose();
#pragma omp critical
										{
											sample_resid_cov += sample_resid_cov_private;
										}
									}
									if (calc_pred_var) {
										vec_t pred_resid_var_private = rand_vec_resid.cwiseProduct(rand_vec_resid);
#pragma omp critical
										{
											sample_resid_var += pred_resid_var_private;
										}
									}
								}
								if (calc_pred_cov) {
									sample_resid_cov /= nsim_var_pred;
								}
								if (calc_pred_var) {
									sample_resid_var /= nsim_var_pred;

									// Alterative version for predictive variances using aproach of Bekas et al. (2007)
									// Note: if negative predictive variances are an issue (with the above approach or the approach of Bekas et al.),
									//			then this approach of Bekas et al. could be modified such that sigma_resid_pred is included as well, 
									//			i.e., estimating the diagonal of (sigma_resid_pred - sigma_resid_pred_obs * sigma_resid_obs^-1 * sigma_resid_pred_obs^T)
									//			instead of the diagonal of sigma_resid_pred_obs * sigma_resid_obs^-1 * sigma_resid_pred_obs^T
//									sample_resid_var = vec_t::Zero(num_REs_pred);
//									den_mat_t rand_vec_s(nsim_var_pred, num_REs_pred);
//									GenRandVecRademacher(cg_generator_, rand_vec_s);//note (26.09.2025): 'GenRandVecRademacher' has been replaced by 'GenRandVecRademacherParallel'
//#pragma omp parallel for schedule(static)
//									for (int i = 0; i < nsim_var_pred; ++i) {
//										vec_t rand_vec_i_0 = rand_vec_s.row(i);
//										vec_t rand_vec_i_2 = chol_fact_resid_[cluster_i].solve(sigma_resid_pred_obs.transpose() * (rand_vec_i_0));
//										vec_t pred_resid_var_private = rand_vec_i_0.cwiseProduct(sigma_resid_pred_obs * rand_vec_i_2);
//#pragma omp critical
//										{
//											sample_resid_var += pred_resid_var_private;
//										}
//									}
//									sample_resid_var /= nsim_var_pred;
								}
								// end stochastic estimate of sigma_resid_pred_obs * resid_obs_inv * sigma_resid_pred_obs.transpose()
								if (calc_pred_cov) {
									T_mat Maux;
									ConvertTo_T_mat_FromDense<T_mat>(woodburry_part_sqrt.transpose() * woodburry_part_sqrt, Maux);
									pred_cov += Maux;
									ConvertTo_T_mat_FromDense<T_mat>(sample_resid_cov, Maux);
									pred_cov -= Maux;
								}//end calc_pred_cov
								if (calc_pred_var) {
#pragma omp parallel for schedule(static)
									for (int i = 0; i < num_REs_pred; ++i) {
										pred_var[i] += woodburry_part_sqrt.col(i).array().square().sum();
									}
									pred_var -= sample_resid_var;
								}//end calc_pred_var
							}//end calc_pred_cov_var_FSA_cholesky_ == "stochatic"
							else if (calc_pred_cov_var_FSA_cholesky_ == "exact_stable") {
								//The following is too slow and requires too much memory
								den_mat_t resid_obs_inv_sqrt_cross_cov;
								TriangularSolveGivenCholesky<T_chol, T_mat, den_mat_t, den_mat_t>(chol_fact_resid_[cluster_i], (*cross_cov), resid_obs_inv_sqrt_cross_cov, false);
								T_mat resid_obs_inv_sqrt_resid_pred_obs_t;
								TriangularSolveGivenCholesky<T_chol, T_mat, T_mat, T_mat>(chol_fact_resid_[cluster_i], sigma_resid_pred_obs.transpose(), resid_obs_inv_sqrt_resid_pred_obs_t, false);
								den_mat_t Maux_rhs = cross_cov_pred_ip.transpose() - resid_obs_inv_sqrt_cross_cov.transpose() * resid_obs_inv_sqrt_resid_pred_obs_t;
								den_mat_t woodburry_part_sqrt;
								TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_woodbury_[cluster_i], Maux_rhs, woodburry_part_sqrt, false);
								if (calc_pred_cov) {
									T_mat Maux;
									ConvertTo_T_mat_FromDense<T_mat>(woodburry_part_sqrt.transpose() * woodburry_part_sqrt, Maux);
									pred_cov += Maux;
									pred_cov -= (T_mat)(resid_obs_inv_sqrt_resid_pred_obs_t.transpose() * resid_obs_inv_sqrt_resid_pred_obs_t);
								}//end calc_pred_cov
								if (calc_pred_var) {
#pragma omp parallel for schedule(static)
									for (int i = 0; i < num_REs_pred; ++i) {
										pred_var[i] += woodburry_part_sqrt.col(i).array().square().sum();
										pred_var[i] -= resid_obs_inv_sqrt_resid_pred_obs_t.col(i).squaredNorm();
									}
								}//end calc_pred_var
							}//end (calc_pred_cov_var_FSA_cholesky_ == "exact_stable"
							else if (calc_pred_cov_var_FSA_cholesky_ == "exact") {
								if (calc_pred_cov) {
									Log::REFatal("Predictive covariance matrices are currently not implemented for gp_approx = 'full_scale_tapering' and "
										"for the 'fitc' approximation when having multiple observations at the same location. Use gp_approx = 'full_scale_tapering_pred_var_stochastic_stable' instead");
									den_mat_t sigma_obs_pred_dense = (*cross_cov) * chol_fact_sigma_ip_[cluster_i][0].solve(cross_cov_pred_ip.transpose());
									sigma_obs_pred_dense += sigma_resid_pred_obs.transpose();
									den_mat_t sigma_resid_inv_sigma_obs_pred = chol_fact_resid_[cluster_i].solve(sigma_obs_pred_dense);
									den_mat_t sigma_resid_inv_sigma_obs_pred_cross_cov_pred_ip = sigma_resid_inv_sigma_obs_pred * cross_cov_pred_ip;
									T_mat cross_cov_part, woodbury_Part;
									ConvertTo_T_mat_FromDense<T_mat>(sigma_obs_pred_dense.transpose() * sigma_resid_inv_sigma_obs_pred, cross_cov_part);
									pred_cov -= cross_cov_part;
									//Note: there is something wrong in the code here (08.11.2024)
									ConvertTo_T_mat_FromDense<T_mat>(sigma_resid_inv_sigma_obs_pred_cross_cov_pred_ip * chol_fact_resid_[cluster_i].solve(sigma_resid_inv_sigma_obs_pred_cross_cov_pred_ip.transpose()), woodbury_Part);
									pred_cov += woodbury_Part;
								}//end calc_pred_cov
								if (calc_pred_var) {
									den_mat_t sigma_resid_inv_cross_cov = chol_fact_resid_[cluster_i].solve((*cross_cov));// sigma_resid^-1 * cross_cov
									den_mat_t sigma_ip_inv_cross_cov_pred_T = chol_fact_sigma_ip_[cluster_i][0].solve(cross_cov_pred_ip.transpose());// sigma_ip^-1 * cross_cov_pred^T
									den_mat_t auto_cross_cov = ((*cross_cov).transpose() * sigma_resid_inv_cross_cov) * sigma_ip_inv_cross_cov_pred_T;// cross_cov^T * sigma_resid^-1 * cross_cov * sigma_ip^-1 * cross_cov_pred
									den_mat_t sigma_resid_pred_obs_sigma_resid_inv_cross_cov(num_REs_pred, (*cross_cov).cols());// Sigma_resid_pred * sigma_resid^-1 * cross_cov
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < sigma_resid_pred_obs_sigma_resid_inv_cross_cov.cols(); ++i) {
										sigma_resid_pred_obs_sigma_resid_inv_cross_cov.col(i) = sigma_resid_pred_obs * sigma_resid_inv_cross_cov.col(i);
									}
#pragma omp parallel for schedule(static)
									for (int i = 0; i < num_REs_pred; ++i) {
										pred_var[i] -= 2 * sigma_ip_inv_cross_cov_pred_T.col(i).dot(sigma_resid_pred_obs_sigma_resid_inv_cross_cov.transpose().col(i))
											+ auto_cross_cov.col(i).dot(sigma_ip_inv_cross_cov_pred_T.col(i));
									}
									vec_t sigma_resid_inv_sigma_resid_pred_col;
									T_mat* R_ptr = &sigma_resid_pred_obs;
									for (int i = 0; i < num_REs_pred; ++i) {
										TriangularSolveGivenCholesky<T_chol, T_mat, vec_t, vec_t>(chol_fact_resid_[cluster_i], ((vec_t)(R_ptr->row(i))).transpose(), sigma_resid_inv_sigma_resid_pred_col, false);
										pred_var[i] -= sigma_resid_inv_sigma_resid_pred_col.array().square().sum();
									}
									// Woodburry matrix part
									den_mat_t Woodburry_fact_sigma_resid_inv_cross_cov;
									TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(chol_fact_sigma_woodbury_[cluster_i], sigma_resid_inv_cross_cov.transpose(), Woodburry_fact_sigma_resid_inv_cross_cov, false);
									den_mat_t auto_cross_cov_pred = (Woodburry_fact_sigma_resid_inv_cross_cov * (*cross_cov)) * sigma_ip_inv_cross_cov_pred_T;
									den_mat_t sigma_resid_pred_obs_Woodburry_fact(num_REs_pred, (*cross_cov).cols());
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < sigma_resid_pred_obs_Woodburry_fact.cols(); ++i) {
										sigma_resid_pred_obs_Woodburry_fact.col(i) = sigma_resid_pred_obs * Woodburry_fact_sigma_resid_inv_cross_cov.transpose().col(i);
									}
#pragma omp parallel for schedule(static)
									for (int i = 0; i < num_REs_pred; ++i) {
										pred_var[i] += 2 * auto_cross_cov_pred.col(i).dot(sigma_resid_pred_obs_Woodburry_fact.transpose().col(i))
											+ auto_cross_cov_pred.col(i).array().square().sum()
											+ sigma_resid_pred_obs_Woodburry_fact.transpose().col(i).array().square().sum();
									}
								}//end calc_pred_var
							}//end calc_pred_cov_var_FSA_cholesky_ == "exact"
							else {
								Log::REFatal("calc_pred_cov_var_FSA_cholesky_ = %s is not supported", calc_pred_cov_var_FSA_cholesky_.c_str());
							}
						}//end matrix_inversion_method_ == "cholesky"
						else if (matrix_inversion_method_ == "iterative") {
							sigma_resid = re_comps_resid_[cluster_i][0][0]->GetZSigmaZt();// Residual matrix
							if (calc_pred_cov) {
								// Whole cross-covariance as dense matrix 
								den_mat_t sigma_obs_pred_dense = (*cross_cov) * chol_fact_sigma_ip_[cluster_i][0].solve(cross_cov_pred_ip.transpose());
								sigma_obs_pred_dense += sigma_resid_pred_obs.transpose();
								den_mat_t sigma_inv_sigma_obs_pred;
								if (cg_preconditioner_type_ == "fitc") {
									const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov_preconditioner, chol_ip_cross_cov_[cluster_i][0], sigma_obs_pred_dense, sigma_inv_sigma_obs_pred, NaN_found,
										num_REs_obs, num_REs_pred, cg_max_num_it_tridiag_, cg_delta_conv_pred, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}
								else {
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov, chol_ip_cross_cov_[cluster_i][0], sigma_obs_pred_dense, sigma_inv_sigma_obs_pred, NaN_found,
										num_REs_obs, num_REs_pred, cg_max_num_it_tridiag_, cg_delta_conv_pred, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}

								T_mat cross_cov_part;
								ConvertTo_T_mat_FromDense<T_mat>(sigma_obs_pred_dense.transpose() * sigma_inv_sigma_obs_pred, cross_cov_part);
								pred_cov -= cross_cov_part;
							} // end calc_pred_cov 
							// Calculate remaining part of predictive variances
							if (calc_pred_var) {
								// Stochastic Diagonal
								vec_t stoch_part_pred_var(num_REs_pred);
								stoch_part_pred_var.setZero();
								// Precondtioner
								vec_t diag_P_stoch;
								vec_t diag_P;
								// To compute optimal c for variance reduction
								vec_t c_var, c_p_z, c_p;
								// Exact Diagonal (Preconditioner)
								if (cg_preconditioner_type_ == "fitc") {
									diag_P_stoch.resize(num_REs_pred);
									diag_P_stoch.setZero();
									diag_P.resize(num_REs_pred);
									c_var.resize(num_REs_pred);
									c_var.setZero();
									c_p_z.resize(num_REs_pred);
									c_p_z.setZero();
									c_p.resize(num_REs_pred);
									c_p.setZero();
									T_mat sigma_resid_pred_obs_pred_var = sigma_resid_pred_obs * (diagonal_approx_inv_preconditioner_[cluster_i].cwiseSqrt()).asDiagonal();
									T_mat* R_ptr_2 = &sigma_resid_pred_obs_pred_var;
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < num_REs_pred; ++i) {
										diag_P[i] = ((vec_t)(R_ptr_2->row(i))).array().square().sum();
									}
								}

								int num_threads;
#ifdef _OPENMP
								num_threads = omp_get_max_threads();
#else
								num_threads = 1;
#endif
								std::uniform_int_distribution<> unif(0, 2147483646);
								std::vector<RNG_t> parallel_rngs;
								for (int ig = 0; ig < num_threads; ++ig) {
									int seed_local = unif(cg_generator_);
									parallel_rngs.push_back(RNG_t(seed_local));
								}
#pragma omp parallel
								{
#pragma omp for nowait
									for (int i = 0; i < nsim_var_pred; ++i) {
										//z_i ~ N(0,I)
										int thread_nb;
#ifdef _OPENMP
										thread_nb = omp_get_thread_num();
#else
										thread_nb = 0;
#endif
										// Sample vector
										std::uniform_real_distribution<double> udist(0.0, 1.0);
										vec_t rand_vec_probe_init(num_REs_pred);
										for (int j = 0; j < num_REs_pred; j++) {
											// Map uniform [0,1) to Rademacher -1 or 1
											rand_vec_probe_init(j) = (udist(parallel_rngs[thread_nb]) < 0.5) ? -1.0 : 1.0;
										}
										// sigma_resid_pred^T * rand_vec_probe_init
										vec_t rand_vec_probe_pred = sigma_resid_pred_obs.transpose() * rand_vec_probe_init;

										// sigma_resid^-1 * rand_vec_probe_pred
										den_mat_t sigma_resid_inv_pv(num_REs_obs, 1);
										CGFSA_RESID<T_mat>(*sigma_resid, rand_vec_probe_pred.matrix(), sigma_resid_inv_pv, NaN_found, num_REs_obs, 1,
											cg_max_num_it_tridiag_, cg_delta_conv_pred,
											cg_preconditioner_type_, diagonal_approx_inv_preconditioner_[cluster_i]);
										// sigma_resid_pred * sigma_resid_inv_pv
										den_mat_t rand_vec_probe_final = sigma_resid_pred_obs * sigma_resid_inv_pv;

										vec_t sample_sigma = (rand_vec_probe_final.col(0)).cwiseProduct(rand_vec_probe_init);
										// Stochastic Diagonal (Preconditioner)
										if (cg_preconditioner_type_ == "fitc") {
											vec_t preconditioner_rand_vec_probe = diagonal_approx_inv_preconditioner_[cluster_i].asDiagonal() * rand_vec_probe_pred;
											vec_t rand_vec_probe_cv = sigma_resid_pred_obs * preconditioner_rand_vec_probe;
											vec_t sample_P = rand_vec_probe_cv.cwiseProduct(rand_vec_probe_init);
#pragma omp critical
											{
												diag_P_stoch += sample_P;
												c_var.array() += (sample_P - diag_P).array().square();
												c_p_z += (sample_P - diag_P).cwiseProduct(sample_sigma);
												c_p += (sample_P - diag_P);
											}
										}
#pragma omp critical
										{
											stoch_part_pred_var += sample_sigma;
										}
									}
								}
								stoch_part_pred_var /= nsim_var_pred;
								if (cg_preconditioner_type_ == "fitc") {
									diag_P_stoch /= nsim_var_pred;
									c_var /= nsim_var_pred;
									c_p_z /= nsim_var_pred;
									c_p /= nsim_var_pred;
									vec_t c_cov = c_p_z - stoch_part_pred_var.cwiseProduct(c_p);
									// Optimal c
									vec_t c_opt = c_cov.array() / c_var.array();
									// Correction if c_opt_i = inf
#pragma omp parallel for schedule(static)   
									for (int i = 0; i < c_opt.size(); ++i) {
										if (c_var.coeffRef(i) == 0) {
											c_opt[i] = 1;
										}
									}
									stoch_part_pred_var += c_opt.cwiseProduct(diag_P - diag_P_stoch);
								}
								pred_var -= stoch_part_pred_var;
								// CG: sigma_resid^-1 * cross_cov
								den_mat_t sigma_resid_inv_cross_cov(num_REs_obs, (*cross_cov).cols());
								CGFSA_RESID<T_mat>(*sigma_resid, *cross_cov, sigma_resid_inv_cross_cov, NaN_found, num_REs_obs, (int)(*cross_cov).cols(),
									cg_max_num_it_tridiag_, cg_delta_conv_pred,
									cg_preconditioner_type_, diagonal_approx_inv_preconditioner_[cluster_i]);
								// CG: sigma^-1 * cross_cov
								den_mat_t sigma_inv_cross_cov(num_REs_obs, (*cross_cov).cols());
								if (cg_preconditioner_type_ == "fitc") {
									const den_mat_t* cross_cov_preconditioner = re_comps_cross_cov_preconditioner_[cluster_i][0][0]->GetSigmaPtr();
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid,*cross_cov_preconditioner, chol_ip_cross_cov_[cluster_i][0], *cross_cov, sigma_inv_cross_cov, NaN_found,
										num_REs_obs, (int)(*cross_cov).cols(), cg_max_num_it_tridiag_, cg_delta_conv_pred, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}
								else {
									CGFSA_MULTI_RHS<T_mat>(*sigma_resid, *cross_cov, chol_ip_cross_cov_[cluster_i][0], *cross_cov, sigma_inv_cross_cov, NaN_found,
										num_REs_obs, (int)(*cross_cov).cols(), cg_max_num_it_tridiag_, cg_delta_conv_pred, cg_preconditioner_type_,
										chol_fact_woodbury_preconditioner_[cluster_i], diagonal_approx_inv_preconditioner_[cluster_i]);
								}

								// sigma_ip^-1 * cross_cov_pred
								//den_mat_t sigma_ip_inv_cross_cov_pred_T = chol_fact_sigma_ip_[cluster_i][0].solve(cross_cov_pred_ip.transpose());
								den_mat_t sigma_ip_inv_cross_cov_pred_T;
								GPBoost::solve_linear_sys(chol_fact_sigma_ip_[cluster_i][0], cross_cov_pred_ip.transpose(), sigma_ip_inv_cross_cov_pred_T, GPU_use_);
								// cross_cov^T * sigma^-1 * cross_cov
								den_mat_t auto_cross_cov = (*cross_cov).transpose() * sigma_inv_cross_cov;
								// cross_cov^T * sigma^-1 * cross_cov * sigma_ip^-1 * cross_cov_pred
								den_mat_t auto_cross_cov_sigma_ip_inv_cross_cov_pred = auto_cross_cov * sigma_ip_inv_cross_cov_pred_T;
								// sigma_resid_pred * sigma^-1 * cross_cov
								den_mat_t sigma_resid_pred_obs_sigma_inv_cross_cov(num_REs_pred, (*cross_cov).cols());
#pragma omp parallel for schedule(static)   
								for (int i = 0; i < sigma_resid_pred_obs_sigma_inv_cross_cov.cols(); ++i) {
									sigma_resid_pred_obs_sigma_inv_cross_cov.col(i) = sigma_resid_pred_obs * sigma_inv_cross_cov.col(i);
								}
								// cross_cov^T * sigma_resid^-1 * cross_cov
								den_mat_t cross_cov_sigma_resid_inv_cross_cov = (*cross_cov).transpose() * sigma_resid_inv_cross_cov;
								// Ensure symmetry
								cross_cov_sigma_resid_inv_cross_cov = (cross_cov_sigma_resid_inv_cross_cov + cross_cov_sigma_resid_inv_cross_cov.transpose()) / 2;
								// Woodburry factor
								chol_den_mat_t Woodburry_fact_chol;
								Woodburry_fact_chol.compute(sigma_ip_stable + cross_cov_sigma_resid_inv_cross_cov);
								den_mat_t Woodburry_fact;
								TriangularSolveGivenCholesky<chol_den_mat_t, den_mat_t, den_mat_t, den_mat_t>(Woodburry_fact_chol, sigma_resid_inv_cross_cov.transpose(), Woodburry_fact, false);
								den_mat_t sigma_resid_pred_obs_WF(num_REs_pred, (*cross_cov).cols());
#pragma omp parallel for schedule(static)   
								for (int i = 0; i < sigma_resid_pred_obs_WF.cols(); ++i) {
									sigma_resid_pred_obs_WF.col(i) = sigma_resid_pred_obs * Woodburry_fact.transpose().col(i);
								}
#pragma omp parallel for schedule(static)
								for (int i = 0; i < num_REs_pred; ++i) {
									pred_var[i] -= sigma_ip_inv_cross_cov_pred_T.col(i).dot(auto_cross_cov_sigma_ip_inv_cross_cov_pred.col(i))
										+ 2 * sigma_ip_inv_cross_cov_pred_T.col(i).dot(sigma_resid_pred_obs_sigma_inv_cross_cov.transpose().col(i))
										- sigma_resid_pred_obs_WF.transpose().col(i).array().square().sum();
								}
								if ((pred_var.array() < 0.0).any()) {
									Log::REWarning("There are negative estimates for variances. Use more sample vectors to reduce the variability of the stochastic estimate.");
								}
							}//end calc_pred_var 
						}// end matrix_inversion_method_ == "iterative"
					}//end gp_approx_ == "full_scale_tapering")
				}//if gauss_likelihood_
			}//end calc_pred_cov || calc_pred_var
			if (!gauss_likelihood_ && gp_approx_ == "fitc") {
				const double* fixed_effects_cluster_i_ptr = nullptr;
				likelihood_[cluster_i]->PredictLaplaceApproxFITC(y_[cluster_i].data(),
					y_int_[cluster_i].data(),
					fixed_effects_cluster_i_ptr,
					re_comps_ip_[cluster_i][0][0]->GetZSigmaZt(),
					chol_fact_sigma_ip_[cluster_i][0],
					re_comps_cross_cov_[cluster_i][0][0]->GetSigmaPtr(),
					fitc_resid_diag_[cluster_i],
					cross_cov_pred_ip,
					has_fitc_correction,
					fitc_resid_pred_obs,
					pred_mean,
					pred_cov,
					pred_var,
					calc_pred_cov,
					calc_pred_var,
					false,
					GPU_use_);
			}//end !gauss_likelihood_
		}//end CalcPredFITC_FSA

		friend class REModel;

	};

}  // end namespace GPBoost

#endif   // GPB_RE_MODEL_TEMPLATE_H_
