# Comparing Copula VAR Models

## Overview

The dcvar package provides three copula VAR models:

1. DC-VAR (`dcvar()`): Time-varying correlation via a random walk on the Fisher-z scale. Best for smooth, gradual changes.

2. HMM Copula (`dcvar_hmm()`): Discrete regime-switching with K hidden states. Best for abrupt changes between distinct coupling regimes.

3. Constant Copula (`dcvar_constant()`): Single time-invariant correlation. Serves as a null/baseline model.

## Mathematical Background

For normal-margin fits, all three models share a common decomposition of the bivariate likelihood into marginal and copula components. The package also supports exponential, gamma, and skew-normal margins for these three models, but the copula structure below is the common part.

### VAR(1) Marginals

Each variable follows a VAR(1) process with innovations:

\[y_t = \mu + \Phi \, (y_{t-1} - \mu) + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \text{diag}(\sigma_{\varepsilon}^2))\]

The normal marginal densities provide the per-variable likelihoods in this special case. For non-normal margins, dcvar uses the same VAR recursion and Gaussian copula but swaps in the corresponding marginal likelihood and transform.

### Gaussian Copula

The dependence between variables is captured by a Gaussian copula with time-varying correlation \(\rho_t\). The log-likelihood contribution of the copula at time \(t\) is:

\[\log c(u_{1t}, u_{2t}; \rho_t) = -\frac{1}{2}\log(1 - \rho_t^2) + \frac{\rho_t^2 \, (z_{1t}^2 + z_{2t}^2) - 2\rho_t \, z_{1t} z_{2t}}{2(1 - \rho_t^2)}\]

where \(z_{it} = \Phi^{-1}(u_{it})\) and \(u_{it} = \Phi(\varepsilon_{it} / \sigma_{\varepsilon,i})\).

### How the Models Differ

- DC-VAR: The correlation evolves as a random walk on the Fisher-z scale: \(z_t = z_{t-1} + \omega_t\) with \(\omega_t \sim N(0, \sigma_\omega^2)\), and \(\rho_t = \tanh(z_t)\).

- HMM Copula: The correlation is state-dependent: \(\rho_t = \rho_{s_t}\), where \(s_t \in \{1, \ldots, K\}\) follows a Markov chain with transition matrix \(A\). State-specific correlations use an ordered constraint (\(\rho_1 < \rho_2 < \ldots\)) to prevent label switching.

- Constant Copula: \(\rho_t = \rho\) for all \(t\). A special case of both the DC-VAR (\(\sigma_\omega = 0\)) and HMM (\(K = 1\)).

## Fitting All Three Models

    library(dcvar)

    # Simulate data with a step-function trajectory
    sim <- simulate_dcvar(
      n_time = 200,
      rho_trajectory = rho_step(200, rho_before = 0.7, rho_after = 0.3),
      seed = 42
    )

    # Fit all three models on the same data
    fit_dc  <- dcvar(sim$Y_df, vars = c("y1", "y2"), seed = 1)
    fit_hmm <- dcvar_hmm(sim$Y_df, vars = c("y1", "y2"), K = 2, seed = 2)
    fit_con <- dcvar_constant(sim$Y_df, vars = c("y1", "y2"), seed = 3)

## LOO-CV Comparison

    dcvar_compare(dcvar = fit_dc, hmm = fit_hmm, constant = fit_con)

## Extracting Tidy Summaries

All fit objects support `as.data.frame()` for exporting tidy parameter summaries. The three core time-series fits shown here also support `fitted()`/`predict()` for one-step-ahead values; SEM and multilevel fits do not currently implement those methods:

    # Full parameter summary as a data frame
    param_df <- as.data.frame(fit_dc)
    head(param_df)

    # Predictions with marginal intervals
    pred_df <- predict(fit_hmm)
    head(pred_df)

## HMM-Specific Outputs

The HMM model provides additional outputs related to state inference:

    # State posteriors, Viterbi path, transition matrix
    states <- hmm_states(fit_hmm)

    # State-specific rho values
    states$rho_state

    # Transition matrix
    states$A

    # Visualise state posteriors
    plot(fit_hmm, type = "states")

    # Transition matrix heatmap
    plot(fit_hmm, type = "transition")

## Clinical Interpretation

`interpret_rho_trajectory()` provides a model-aware narrative:

    interpret_rho_trajectory(fit_dc)
    interpret_rho_trajectory(fit_hmm)
    interpret_rho_trajectory(fit_con)

## When to Use Which Model

- Smooth dynamics (therapy effects, developmental trends): Use DC-VAR
- Abrupt regime switches (treatment onset, relapse): Use HMM Copula
- No time variation expected: Use Constant Copula as baseline
- Uncertain: Fit all three and compare with LOO-CV