Getting help within R

Once qfasar has been installed, it can be attached and made available for use by typing the command library(qfasar).

To view a list of the contents of qfasar, type library(help=qfasar).

To view the documentation of a function, which may contain more technical information than is included in this vignette, type ?function_name.

Distribution list

To be notified when new versions of qfasar are available, e-mail a request to be added to the qfasar distribution list to jbromaghin@usgs.gov. Please include your name and affiliation in the request.

Prey signature data

The prey signature data frame contains information on prey types, unique sample identifiers, and fatty acid signature proportions or percentages. The formatting requirements for this file are:

• The data frame must be in row-major format, i.e., each row contains information for an individual prey animal.
• The first column must contain an alphanumeric designation of prey type. This will generally be a species of prey, or possibly subclasses of a species, e.g., large versus small or subadult versus adult.
• The second column must contain an alphanumeric identifier unique to each prey animal, i.e., a sample ID. The ID should not be a number only, so that R recognizes the ID as a factor. Alternatively, one could force R to treat a numeric ID as a factor when the data are read.
• The remaining columns must contain signature data. The data can be in the form of either proportions or percentages, as qfasar will internally convert the data to proportions for analysis.
• The data frame must have column names, such as “type”, “id”, name of fatty acid 1, name of fatty acid 2, etc. These can most easily be included by having the names in the first row of the data file and using the header=TRUE option when reading the data into R. Fatty acid names in the prey data frame must exactly match the names in the predator and fatty acid suite data frames (described below).
• NOTE: The data frame should contain data from all available fatty acids, not just a subset. The fatty acids to be used in an analysis are designated in the fatty acid suite data frame (described below).

As an example, the first few rows and columns of a prey signature data file used by Bromaghin et al. (2015), in comma separated variable (csv) format, follow. Note that extra spaces have been added between commas and the subsequent values to enhance readability here, but extra spaces should not be included in the data file.

Species,            ID,   c12.0,  c13.0,  Iso14,  c14.0,  c14.1w9,  c14.1w7,  c14.1w5
Atlantic argentine, E68,  0.17,   0.05,   0.02,   6.17,   0.21,     0.03,     0.07
Atlantic argentine, E69,  0.11,   0.05,   0.03,   6.37,   0.27,     0.03,     0.06
Atlantic argentine, E70,  0.14,   0.05,   0.03,   6.64,   0.26,     0.03,     0.06
Atlantic argentine, E71,  0.15,   0.05,   0.03,   7.16,   0.28,     0.02,     0.06
Atlantic argentine, E72,  0.13,   0.05,   0.02,   6.65,   0.29,     0.02,     0.05
Atlantic argentine, E73,  0.14,   0.04,   0.02,   5.68,   0.20,     0.02,     0.06
Atlantic argentine, E74,  0.16,   0.05,   0.02,   6.70,   0.25,     0.02,     0.06
Atlantic argentine, E75,  0.17,   0.04,   0.02,   6.32,   0.23,     0.03,     0.06
Atlantic argentine, E76,  0.11,   0.04,   0.02,   6.94,   0.27,     0.02,     0.06
Atlantic argentine, E77,  0.14,   0.05,   0.02,   6.97,   0.26,     0.03,     0.06
Butterfish,         E141, 0.30,   0.11,   0.03,   5.87,   0.09,     0.05,     0.05
Butterfish,         E142, 0.10,   0.10,   0.02,   4.93,   0.07,     0.03,     0.04

NOTE: the prey data file can contain columns with additional “tombstone” information such as sex, age, sampling date, sampling location, etc. If so, read the data file into a data frame and from that data frame create a second data frame in the required format. Including the sample ID in both data frames will allow the subsequent merging of qfasar results with the tombstone information.

Predator signature data

The predator signature data frame must be formatted exactly like the prey signature data frame (see Prey signature data), with each record being comprised of a predator type, a unique sample ID, and signature proportions.

The only difference between how prey and predator data are handled by qfasar is that the first column in the predator data frame is used to designate types of predators for which separate estimates of mean diet composition are desired. For example, Rode et al. (2014) estimated mean diet composition for four sex-age classes of polar bears, so the first column might contain the strings adult-male, adult-female, subadult-male, and subadult-female. If separate mean estimates are not desired for predator types, each row of the data frame should have an identical string in the first column, such as polar-bear.

NOTE: the predator data file can contain columns with additional “tombstone” information such as sex, age, sampling date, sampling location, etc. If so, read the data file into a data frame and from that data frame create a second data frame in the required format. Including the sample ID in both data frames will allow the subsequent merging of qfasar results with the tombstone information.

Fatty acid suite data

The fatty acid suite data frame contains fatty acid names, one or more sets of calibration coefficients, and definitions for one or more suites of fatty acids. The data frame must strictly meet the following formatting requirements.

• The first row of the data frame must contain header information, such as “fatty acid”, “CC 1”, “CC 2”, “Suite 1”, “Suite 2”, “Comment”. These can most easily be included by having the names in the first row of the data file and using the header=TRUE option when reading the data into R.
• The second row of the data frame must list “use_me” in the first column, a 1 in the column for the set of calibration coefficients to be used, a 1 in the column for the fatty acid suite to be used, and a 0 in all other columns.
• Starting with the third row, the first column must contain fatty acid names, which must exactly match the header names for the corresponding fatty acids in the prey and predator data frames (see Prey signature data).
• Starting with the third row, Columns 2 to k must contain calibration coefficients for each fatty acid. Multiple sets of calibration coefficients can be in the data frame. The set to be used must contain a 1 in Row 2 and any others must contain a 0 in Row 2.
• Starting with the third row, Columns k+1 to m must contain one or more definitions of fatty acid suites. Membership in a suite is defined by 0/1 indicators, with a 1 indicating membership. Definitions for multiple suites can be in the data frame. For example, two columns could contain indicators defining membership in the dietary and extended-dietary suites of fatty acids (Iverson et al. 2004). The suite to be used must contain a 1 in Row 2 and the others must contain a 0 in Row 2.
• An optional last column can contain comments, but care must be taken not to include commas in the comment field if the data are stored as csv files.

Use of the first row in the data frame to denote the set of calibration coefficients and fatty acid suite to be used in an analysis is intended to allow all available sets of calibration coefficients and all potential suites of fatty acids to be conveniently stored in the same file, while permitting fast and easy switching between them. It also allows all fatty acid data to be stored in a single file, which is more convenient and less confusing than having separate data files for each suite of fatty acids that one might want to use.

The first few rows of a csv fatty acid suite data file follow below as an example. This example has a single set of calibration coefficients (harbor_seal), definitions for two suites of fatty acids (dietary and ext_dietary), and comments in the last column. Note that extra spaces have been added between commas and the subsequent values to enhance readability here, but extra spaces should not be included in the data file.

fatty_acid, harbor_seal,  dietary,  ext_dietary,  comments
use_me,     1,            0,        1,            Values from Rosen and Tollit (2012) unless noted
c12.0,      0.863,        0,        0,  
c13.0,      0.516,        0,        0,  
Iso14,      0.806,        0,        0,  
c14.0,      0.588,        0,        1,  
c14.1w9,    0.823,        0,        0,  
c14.1w7,    2.853,        0,        0,  
c14.1w5,    12.137,       0,        0,  
Iso15,      0.538,        0,        0,  
Anti15,     0.785,        0,        0,  
c15.0,      0.673,        0,        0,  
c15.1w8,    0.942,        0,        0,  
c15.1w6,    6.372,        0,        0,  
Iso16,      0.812,        0,        0,  
c16.0,      0.479,        0,        1,  
c16.1w11,   1.898,        0,        0,  
c16.1w9,    3.506,        0,        0,  
c16.1w7,    2.377,        0,        1,  
c7Me16.0,   0.976,        0,        0,  
c16.1w5,    1.55,         0,        0,  
c16.2w6,    0.745,        1,        1,  
Iso17,      0.864,        0,        0,  
c16.2w4,    0.954,        1,        1,  
c16.3w6,    0.759,        1,        1,  
c17.0,      0.1,          0,        1,  
c16.3w4,    0.552,        1,        1,
c17.1,      1.618,        0,        0,
c16.3w1,    1,            1,        1,            From Nordstrom et al. (2008), BDL in Rosen & Tollit

Preparing fatty acid suite information

The function prep_fa() takes the fatty acid suite data frame as input and returns a list containing the following objects.

• cc - A numeric vector of the calibration coefficients to be used.
• use - A logical vector defining the fatty acid suite to be used.
• fa_names - A character vector of all fatty acid names.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

The following code snippet contains an example call to prep_fa().

# Process the fatty acid suite information.
fa_info <- prep_fa(df_fa = df_fa)
str(fa_info)

Preparing signature data

The function prep_sig() modifies raw fatty acid signatures in two important ways to prepare them for analysis, replacing invalid signature proportions and scaling signatures, each of which is described below.

• Modifying invalid signature proportions
• Scaling signatures

prep_sig() returns a list containing the following objects. Note that many of these objects are required inputs to other functions in qfasar.

• type - A character vector of the type of each signature.
• id - A character vector of the unique sample ID of each signature.
• n_types - The number of unique types.
• uniq_types - A character vector of the unique types, sorted alphanumerically.
• n_sig - The total number of signatures.
• type_ss - The number of signatures (sample size) for each unique type.
• loc - A vector or matrix giving the first and last locations of the signatures of each type, after being sorted by type and id.
• sig_rep - A vector or matrix of the original signatures, with any values missing or less than or equal to 0 replaced, in column-major format.
• n_fa_rep - The number of fatty acids in sig_rep.
• sig_scale - A vector or matrix of scaled signatures ready for analysis, sorted by type and id, in column-major format.
• n_fa_suite - The number of fatty acids in sig_scale.
• fa_suite - A character vector of the names of fatty acids in the suite to be used in the analysis.
• zero_rep_val - A constant associated with the method and value to be used to replace proportions that are missing or less than or equal to 0.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

# Prepare the signatures for analysis.
prey_info <- prep_sig(df_sig = df_prey,
                      fa_names = fa_info$fa_names,
                      use_fa = fa_info$use,
                      zero_rep = 90)
str(prey_info)
pred_info <- prep_sig(df_sig = df_pred,
                      fa_names = fa_info$fa_names,
                      use_fa = fa_info$use,
                      zero_rep = prey_info$zero_rep_val)
str(pred_info)

pred_info <- prep_sig(df_sig = df_pred,
                      fa_names = fa_info$fa_names,
                      use_fa = fa_info$use,
                      zero_rep = prey_info$zero_rep_val,
                      scale = 1)

# Compute the calibration coefficient for the augmented proportion.
new_cc <- cc_aug(sig_rep = prey_info$sig_rep,
                 sig_scale = prey_info$sig_scale,
                 cc_all = fa_info$cc,
                 use_fa = fa_info$use,
                 dist_meas = 1)

Chi-square distance power parameter

The chi-square distance involves a power transformation of signature proportions, with the power parameter being denoted gamma. The function comp_chi_gamma() implements the algorithm of Stewart et al. (2014) to find a suitable value of gamma.

The algorithm is initialized with inv_gamma equal to 1 and gamma is computed as 1/inv_gamma. The distances between all possible pairs of full signatures are computed (distances). For each pair of full signatures, the distances between all possible sub-signatures comprised of only two fatty acid proportions are computed (sub-distances). The proportion of sub-distances that exceed their corresponding distance is computed across all possible pairs of signatures. If that proportion is less than a user specified argument near_zero, the function returns with gamma equal to 1; otherwise, the function enters an iterative phase. At each iteration, inv_gamma is incremented by 1, gamma is computed as 1/inv_gamma, distances and sub-distances are recomputed, and the proportion of the sub-distances that exceed their corresponding distance is recomputed. The algorithm terminates when the proportion is less than the argument near_zero or the value of gamma is less than min_gamma.

The argument space must equal 1 or 2 (see Estimating individual diet and variance). If its value is 1, the calibration coefficients are used to map the signatures to the predator space prior to initializing the algorithm.

The following code snippet contains an example call to comp_chi_gamma().

# Estimate the chi-square distance gamma parameter.
gamma_est <- comp_chi_gamma(sigs = prey_info$sig_rep,
                            cc = fa_info$cc,
                            near_zero = 0.00001,
                            min_gamma = 0.05,
                            space = 1)
str(est_gamma)                            

comp_chi_gamma() returns a list containing the following objects:

• gamma - The estimated value of gamma.
• gamma_vec - A numeric vector containing the value of gamma at each step of the iteration.
• prop_vec - A numeric vector containing the proportion of all possible two-element signatures with distance exceeding that of the full signatures at each step of the iteration. This value is compared to the argument near_zero.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

In the above code block, note that gamma is estimated using the full signatures with any values equal to zero or missing replaced (prey_info$sig_rep) and all calibration coefficients. If using augmentation, one might initially think to call comp_chi_gamma() with augmented signatures (Scaling signatures). However, specification of a distance measure is required to obtain a calibration coefficient for the augmented proportion (Calibration coefficient for an augmented proportion). Consequently, if augmentation is used, gamma must be estimated using the full signatures. If traditional scaling is to be used (scale <- 1), one could estimate gamma either with the full signatures or with the scaled signatures, though we have no idea how similar the two estimates of gamma would be; with traditionally scaled signatures, one would need to restrict the calibration coefficients to those associated with fatty acids in the scaled signatures, i.e., my_ccs <- fa_info$cc[fa_info$use]. If one is not scaling signatures (scale <- 2), they will be scaled internally for the computation of gamma.

As the number of signatures in the library and/or the number of fatty acids in a signature increases, the number of possible pairs of signatures and the number of all possible two-proportion sub-signatures increases rapidly. Consequently, comp_chi_gamma() is likely to require long run times. However, it only needs to be run once for any particular library of signatures.

I have little experience with the chi-square distance and with different values of the gamma parameter. Stewart et al. (2014) state that the chi-square distance approaches the Aitchison distance as gamma decreases. However, as gamma decreases, the power-transformed signature components become increasingly similar in magnitude, a characteristic which seems counterintuitive for a distance measure, at least on the surface. Consequently, I provide no recommendations regarding gamma and provide this functionality for the sake of completeness and to facilitate future comparative research into the performance of QFASA with the chi-square distance and a range of values for gamma.

Estimating individual diet and variance

The diet composition of an individual predator is estimating by minimizing a measure of distance (Distance measures) between its observed signature and a modeled signature computed from the diet composition and mean prey-type signatures (Iverson et al. 2004). est_diet() uses the function Rsolnp::solnp() to minimize the distance (Ghalanos & Theussl 2014). The selection of a distance measure is controlled by the argument dist_meas:

• dist_meas == 1 yields the Aitchison distance measure.
• dist_meas == 2 yields the Kullback-Leibler measure.
• dist_meas == 3 yields the chi-square distance measure.

Diet estimation can occur in either the predator or prey space (Bromaghin et al. 2015; Estimation spaces), controlled by the argument space.

• space == 1 indicates estimation should occur in the predator space. This is the default value.
• space == 2 indicates estimation should occur in the prey space.

The variance matrix of an individual diet composition estimate is estimated through bootstrap sampling. The signatures of each prey type are indepedently sampled with replacement, with sample sizes equal to the prey-type sample sizes in the prey library, and amalgamated to construct a bootstrapped prey library. The bootstrapped prey library is then used to estimate the predator’s diet. This is repeated a specified number of times (argument ind_boot), resulting in replicated estimates of the diet of each predator. The empirical variance and covariance of the replicated prey-type estimates are used to construct a covariance matrix for each predator (Beck et al. 2007, Bromaghin et al. 2015). Note that the prey library is resampled independently for each predator.

If a particular analysis does not require the variances of the diet estimates for individual predators to be estimated, variance estimation can be disabled by setting the individual bootstrap sample size equal to zero (ind_boot <- 0).

If estimated, the variance matrices, one for each predator, are returned via the three-dimensional array var_ind. The predators index the third dimension. The following code snippet illustrates how to extract the variance information for an individual predator.

# Specify a predator.
this_pred <- 1

# Extract the variance matrix for this predator.
var_mat <- diet_est$var_ind[,,this_pred]
print(var_mat)

# Extract the standard errors for this predator.
se <- sqrt(diag(var_mat))
print(se)

Estimating mean diet and variance

Mean diet composition can be estimated for each type in the predator signature data (see Data frames). est_diet() provides three options for estimating mean diets, controlled by the argument mean_meth.

• mean_meth == 0 skips estimation of mean diet composition.
• mean_meth == 1 returns the empirical estimate of mean diet composition.
• mean_meth == 2 returns the “parameterized mean” estimate of mean diet composition.

The empirical mean estimator is simply the average of the estimated diets for individual predators within each predator type.

The “parameterized mean” estimator is an unpublished estimator that I have experimented with to a limited extent. This estimator reparameterizes the QFASA model with a single vector of diet proportions common to all predators and mean diet is estimated by minimizing the distance between the mean signature modeled from the mean diet proportions and each predator’s observed signature, summed over all predators. The parameterized mean estimator has not yet been thoroughly tested and its inclusion in qfasar is intended to facilitate planned research. Our limited work with the parameterized mean estimator to date suggests that it may perform well when predator signatures are homogeneous, but may be more sensitive to the presence of “outlier” signatures in the predator data than the empirical estimator.

est_diet() returns the estimated mean diet compositions in the object est_mean.

qfasar implements two methods of estimating the variance of mean diet composition estimates, the variance estimator of Beck et al. (2007) and a bootstrap estimator, controlled by the argument var_meth.

• var_meth == 0 skips variance estimation.
• var_meth == 1 yields the bootstrap estimator. This is the default value.
• var_meth == 2 yields the estimator of Beck et al. (2007), generalized to produce a variance matrix.

The bootstrap estimator draws independent samples of each prey type to form a bootstrap prey library, exactly as in Estimating individual diet and variance, and also a random sample of each predator type, with replacement and sample sizes equal to the observed sample sizes. Mean diet is estimated using the bootstrapped prey and predator signatures and the method indicated by the argument mean_meth. The argument mean_boot controls the number of times this process is replicated, and the replications are used to estimate the covariance matrix of the mean diet composition estimates for each predator type. Unpublished work indicates that the bootstrap estimator is more reliable than the Beck et al. (2007) estimator.

Note that the Beck estimator cannot be used in combination with the parameterized-mean estimator of mean diet composition.

est_diet() returns the estimated variances of the predator-type mean diet composition estimates in the object var_mean.

The following code snippet illustrates one method of extracting the diagonal of the variance matrix for all predator types, computing the standard errors, and combining mean diet estimates and associated standard errors into a single matrix.

# There must be a more elegant way to extract all the diagonals into a matrix,
# but the following works.

# Allocate memory for the standard errors.
stderr <- matrix(data = 0, nrow = prey_info$n_types, ncol = pred_info$n_types)
colnames(stderr) <- colnames(diet_est$est_mean)
rownames(stderr) <- rownames(diet_est$est_mean)

# Extract the diagonal of the variance matrix for each predator type and compute
# the standard errors. Note: I use li# for "loop index number".
for(li1 in 1:pred_info$n_types){
  stderr[,li1] <- sqrt(diag(diet_est$var_mean[,,li1]))
}

# Combine estimates of means and standard errors in one matrix.
est_comb <- rbind(diet_est$est_mean, stderr)
rownames(est_comb) <- c(paste("Mean", prey_info$uniq_types, sep = " - "),
                        paste("SE", prey_info$uniq_types, sep = " - "))

# Transpose the matrix (if desired?).
est_comb <- t(est_comb)

Adjusting for fat content

The function est_diet() estimates diet composition in terms of the fatty acid mass consumed (units [fatty acid mass]). Such estimates may be of direct ecological interest, especially for ecosystems in which lipids are particularly important. In other situations, one may wish to transform estimates to a different scale (Iverson et al. 2004). The qfasar function adj_diet_fat() performs such transformations.

adj_diet_fat() takes constants specifying the differential fat content of prey types, one or more estimates of diet composition, and potentially one or more variance matrices as input arguments. The diet composition and variance objects are intended to be the corresponding objects returned by the function est_diet(). Either estimates for individual predators or mean estimates for predator types may be passed to the function.

Note that the units of the fat-content constants (input argument prey_fat) control the units of the resulting transformed diet estimates. For example, if the units of prey_fat are [fatty acid mass]/[animal mass], the adjusted estimates are in terms of the relative mass of prey consumed (e.g., Bromaghin et al. 2013). Alternatively, if the units of prey_fat are [fatty acid mass]/[animal], the adjusted estimates are in terms of the relative numbers of prey animals consumed. It is the user’s responsibility to know the units associated with the fat content values and interpret the adjusted estimates correctly. In any case, the variance of the transformed diet estimates is estimated from the input arguments using the delta method (Seber 1982)

NOTE: values of the fat parameters are treated as known constants without variance.

The following code snippet illustrates an example call to adj_diet_fat().

# Estimate predator diets in terms of fatty acid mass consumed.
mass_est <- est_diet(pred_sigs = pred_info$sig_scale,
                     pred_uniq_types = pred_info$uniq_types,
                     pred_loc = pred_info$loc,
                     prey_sigs = prey_info$sig_scale,
                     prey_uniq_types = prey_info$uniq_types,
                     prey_loc = prey_info$loc,
                     cc = new_cc$cc,
                     space = my_space,
                     dist_meas = my_dist_meas,
                     gamma = my_gamma,
                     ind_boot = my_ind_boot,
                     mean_meth = my_mean_meth,
                     var_meth = my_var_meth,
                     mean_boot = my_mean_boot)


# Specify fat content, one value for each prey type and in the alphanumeric order of the
# prey-type names. Here, I generate random values purely for illustrative purposes,
# because I do not have any fat data handy.  With values, use the concatenate function:
# fat_vals <- c(val1, val2, ...).
fat_vals <- runif(n = length(prey_info$uniq_types), min = 0.25, max = 1.75)
names(fat_vals) <- prey_info$uniq_types


# Transform individual estimates of diet composition in terms of fatty acid mass
# to estimates in terms of some other desired scale. Note that the units of the
# fat constants determine the resulting scale. Ideally, the units would be
# (fatty acid mass)/(prey mass) or (fatty acid mass)/(prey animal).
trans_est_ind <- adj_diet_fat(prey_fat = fat_vals,
                              diet_est = mass_est$est_ind,
                              diet_var = mass_est$var_ind)
                              
# Transform mean estimates of diet composition in terms of fatty acid mass
# to estimates in terms of some other desired scale.                              
trans_est_mean <- adj_diet_fat(prey_fat = fat_vals,
                               diet_est = mass_est$est_mean,
                               diet_var = mass_est$var_mean)

str(trans_est_mean)

adj_diet_fat() returns a list containing the following objects:

• diet_est - A numeric vector or matrix of the estimated diet composition of individual predator(s) or predator type(s) in the units represented by the argument prey_fat.
• diet_var - A numeric matrix or array containing the estimated variance matrix for the estimated diet of individual predator(s) or predator type(s), in the units represented by prey_fat.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

Exploring signatures for structure

Prey and, perhaps to a lesser extent, predator signatures come with some pre-defined structure. A sample of signatures is required from each potential prey type, often species, so those a priori prey types represent structure within a prey library. Similarly, predator data may be compiled from some number of recognized classes, for example sex, age, or size classes. Such pre-defined structure helps frame an analysis in important ways, such as defining the prey types to which estimates of diet composition will pertain or classes of predators for which separate estimates of mean diet composition are desired.

However, one may reasonably wonder if additional structure exists within the signatures. For example, within a prey library structured by species, might there be clusters of individuals within one or more species that are more similar within clusters than between clusters? If so, there may be advantages to subdividing the prey library into a larger number of types.

qfasar implements an algorithm referred to as divisive magnetic clustering (dimac; Bromaghin et al. Under review) in the function dimac() to explore a collection of signatures for additional structure within defined types. The dimac algorithm was modified from the diana algorithm of the package cluster (Maechler et al. 2016). The algorithm is initialized with all signatures in one cluster. The first two “magnets”" are chosen as the two signatures having the greatest distance between them, using a distance measure of the user’s choice (Distance measures), and each signature is drawn to the nearest magnet to form clusters. The algorithm then enters an iterative phase. At each iteration, the cluster with the greatest average distance between its signatures and the mean signature is identified as the “active” cluster. The two signatures within the active cluster having the greatest distance between them are selected as new magnets. One of the two new magnets replaces the original magnet for the active cluster and the second starts the formation of an additional cluster. Each signature is drawn to the nearest magnet to form clusters, without regard for its cluster membership in the preceding iteration. Consequently, the algorithm is not simply bifurcating, but rather is much more dynamic and flexible. The iterations continue until each signature is in its own cluster.

dimac() is intended to be called after the fatty acid signatures have been prepared for analysis by calls to the functions prep_fa() and prep_sig(). The following code snippet illustrates a typical call to dimac().

# Explore prey signatures for sub-structure.
my_clust <- dimac(sigs = prey_info$sig_scale,
                  id = prey_info$id,
                  type = prey_info$uniq_types,
                  loc = prey_info$loc,
                  dist_meas = my_dist_meas)
str(my_clust)

dimac() returns a list containing the following elements:

• clust - A data frame denoting cluster assignments at each iteration of the algorithm.
• clust_dist - A numeric matrix of the summed distance within clusters at each iteration.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

Unfortunately, there is no objective method to determine the most appropriate number of clusters. It may seem tempting to use a large number of clusters in subsequent analyses to exploit fine scale structure in the data. Indeed, Meynier et al. (2010) used individual prey animals as prey types, so one might argue that having a large number of clusters is acceptable. One disadvantage of using a large number of prey types is that the speed of numerical optimization will be slow with large prey libraries. Also, partitioning a prey library into a large number of clusters can increase prey confounding between sub-types (Bromaghin et al. Under review). My recommendation is to use the results of dimac() conservatively and only partition prey types into multiple clusters if there is strong evidence for doing so. Examine the distance results in the object clust_dist returned by make_prey_part() and identify any large reductions in distance that are followed by a more gradual decrease in distance as the number of clusters increases further. An example of this pattern is provided by the prey type Gaspereau from the Scotian fish prey library (Budge et al. 2002; Bromaghin et al. 2016a).

Number_of_clusters <- 1:41
Distance <- c(209.8, 190.4, 107.6, 96.3, 88.8, 82.5, 76.5, 72.6, 64.2, 59.8,
              55.2, 48.3, 45.2, 42.1, 40.0, 33.9, 31.8, 29.5, 26.9, 25.3,
              23.0, 21.2, 19.1, 17.3, 16.2, 14.7, 14.8, 13.6, 11.1, 9.9,
              8.5, 7.3, 6.5, 5.6, 5.1, 4.1, 3.2, 2.4, 1.4, 0.6,
              0.0)
plot(x = Number_of_clusters, y = Distance, main="Scotian: Gaspereau")

Moving from one to two clusters provides little reduction in the summed distance within clusters, but the increase to three clusters is associated with a substantial drop in distance that is likely attributable to the discovery of some true structure within the signatures. As the number of clusters increases above three, distance decreases rather continuously to zero. I interpret this result as evidence that three clusters exist within the Gaspereau prey type and, in fact, the three clusters correspond well to ancillary information on the year and location samples were collected (Bromaghin et al. Under review).

Note that for diet estimation applications, partitioning a prey library into more clusters than the number of fatty acids used to estimate diet may result in estimates that are not unique. In such a case, estimates of diet composition need to be pooled into a smaller number of “reporting groups” (e.g., Bromaghin 2008; Meynier et al. 2010), which one would hope would largely eliminate any lack of uniqueness. See Bromaghin et al. (Under review) for a more detailed discussion of this issue.

# Specify the number of clusters for each prey type.
n_clust <- c(2, 1, 4, 3, 3, 3, 1, 3, 1, 2,
             2, 2, 3, 2, 3, 1, 1, 1, 3, 1,
             1, 1, 1, 1, 2, 2, 2, 1)

# Partition the prey library.
prey_part <- make_prey_part(sig = prey_info$sig_scale,
                            clust = my_clust$clust,
                            n_clust = n_clust)

# Specify the number of clusters for each prey type.
n_clust <- c(2, 1, 4, 3, 3, 3, 1, 3, 1, 2,
             2, 2, 3, 2, 3, 1, 1, 1, 3, 1,
             1, 1, 1, 1, 2, 2, 2, 1)

# Partition the prey library.
prey_part <- make_prey_part(sig = prey_info$sig_scale,
                            clust = my_clust$clust,
                            n_clust = n_clust)

# Estimate predator diets.
my_est <- est_diet(pred_sigs = pred_info$sig_scale,
                   pred_uniq_types = pred_info$uniq_types,
                   pred_loc = pred_info$loc,
                   prey_sigs = prey_part$sig_part,
                   prey_uniq_types = prey_part$uniq_types,
                   prey_loc = prey_part$loc,
                   cc = new_cc$cc,
                   space = my_space,
                   dist_meas = my_dist_meas,
                   gamma = my_gamma,
                   ind_boot = my_ind_boot,
                   mean_meth = my_mean_meth,
                   var_meth = my_var_meth,
                   mean_boot = my_mean_boot)
str(my_est)

# Pool estimates to the original prey types.
pool_est <- diet_pool(rep_grp = prey_part$pool_post,
                      est_ind = my_est$est_ind,
                      var_ind = my_est$var_ind,
                      est_mean = my_est$est_mean,
                      var_mean = my_est$var_mean)
str(pool_est)

rep_grp = matrix(c(1, 0, 0, 0, 0, 0, 0,
                   0, 1, 0, 0, 0, 0, 0,
                   0, 1, 0, 0, 0, 0, 0,
                   0, 0, 1, 0, 0, 0, 0,
                   0, 0, 0, 1, 0, 0, 0,
                   0, 0, 0, 1, 0, 0, 0,
                   0, 0, 0, 0, 1, 0, 0,
                   0, 0, 0, 0, 0, 1, 0,
                   0, 0, 0, 0, 0, 1, 0,
                   0, 0, 0, 0, 0, 0, 1),
                 nrow = 10, byrow = TRUE)

Leave-one-prey-out analysis

In any type of QFASA analysis, one might reasonably wonder how well the prey library might perform. For example, if the signatures of the individuals within each prey type are similar and quite different from the signatures of other prey types, QFASA might perform very well. Conversely, performance might suffer if some prey types are similar, which Bromaghin et al. (2016b) referred to as “prey confounding”, or if distinct clusters of similar prey occur within some prey types (see Exploring signatures for structure). Although such expectations seem reasonable, it is important to recognize that model performance is largely controlled by the complex interaction of a predator’s diet and characteristics of a prey library (Bromaghin et al. 2015, 2016a, 2016b). For example, performance can be poor if most prey types are distinct, but a predator’s diet is dominated by a subset of prey types that are confounded. The degree to which a predator diet is concentrated on distinct prey types has been referred to as “diet identifiability” (Bromaghin et al. 2016b). Of course, the accuracy of the calibration coefficients is also a critical determinant of model performance.

Despite the above cautions, it seems reasonable to investigate the performance of a prey library as long as one interprets the results with the cautions firmly in mind. There may be several ways this could be done. For example, Haynes et al. (2015) used what is commonly called “prey-on-prey” simulations to evaluate their prey library. The approach implemented in qfasar is a conceptually-similar leave-one-prey-out technique.

A leave-one-prey-out analysis in qfasar is conducted as follows. A single signature is temporarily removed from the prey library, the mean prey-type signatures are computed, the mixture of prey signatures that best approximates the removed signature is estimated as in diet estimation, after which the temporarily removed signature is returned to the library. This is done for all prey signatures in turn. The mean diet estimate is computed for the signatures of each prey type. This analysis performed on an ideal prey library would result in 100% of the diet estimate from each signature being attributed to the prey type to which the signature belongs. The degree of prey confounding within a library is reflected by the extent to which portions of the diet estimates are attributed to incorrect prey types.

In qfasar, a leave-one-prey-out analysis is conducted by a call to the function lopo(). The inputs lopo() requires are obtained by calls to the functions prep_sig() with the prey signature data, or possibly make_prey_part() if a prey library has been partitioned into clusters (see Exploring signatures for structure). The following code snippet contains an example call to lopo().

# Perform a leave-one-prey-out analysis with a partitioned library.
lib_perf <- lopo(sigs = prey_part$sig_part,
                 type = prey_part$type,
                 uniq_types = prey_part$uniq_types,
                 type_ss = prey_part$type_ss,
                 loc = prey_part$loc,
                 dist_meas = my_dist_meas,
                 gamma = my_gamma)
str(lib_perf)

lopo() returns a list containing the following elements:

• est - A square numerical matrix containing the mean distribution of estimates among all prey types, by prey-type (rows). Perfect estimation would result in a 1 in each element along the diagonal.
• mean_correct - The mean proportion correctly estimated across prey types, unweighted by prey-type sample sizes.
• total_correct - The proportion of all signatures correctly estimated, irrespective of prey type designations.
• n_conv - A integer vector containing the number of estimates that converged for each prey type.
• err_code - An integer error code(0 if no error is detected).
• err_message - A string containing a brief summary of the results.

Note:

• The statistics returned by lopo() are computed based on the estimates that successfully converge.
• Prey types represented by a single signature are skipped.
• Because of the numerical optimization involved in a leave-one-prey-out analysis, lopo() can take a few minutes to run with a large prey library.

# Perform a leave-one-prey-out analysis with the original library.
perf_orig <- lopo(sigs = prey_info$sig_scale,
                  type = prey_info$type,
                  uniq_types = prey_info$uniq_types,
                  type_ss = prey_info$type_ss,
                  loc = prey_info$loc,
                  dist_meas = my_dist_meas,
                  gamma = my_gamma)

# Partition the prey library.
prey_part <- make_prey_part(sig = prey_info$sig_scale,
                            clust = my_clust$clust,
                            n_clust = n_clust)

# Perform a leave-one-prey-out analysis with the partitioned library.
perf_part <- lopo(sigs = prey_part$sig_part,
                  type = prey_part$type,
                  uniq_types = prey_part$uniq_types,
                  type_ss = prey_part$type_ss,
                  loc = prey_part$loc,
                  dist_meas = my_dist_meas,
                  gamma = my_gamma)

# Pool results back to the original prey types.
pooled_results <- lopo_pool(est = perf_part$est,
                            n_conv = perf_part$n_conv,
                            type_ss = prey_part$type_ss,
                            pre = prey_part$pool_pre,
                            post = prey_part$pool_post)

# Compare correct allocations.
cbind(diag(perf_orig$est), diag(perf_part$est))

Signature & calibration coefficient consistency

Predator signatues are assumed to be a linear mixture of mean prey signatures (Iverson et al. 2004) and are modeled as such during diet estimation. Predator signature proportions should therefore be within the range of prey signature proportions and proportions outside the range of the prey proportions are indicative of a violation of one or both of the primary model assumptions, i.e., the prey library is incomplete or the calibration coefficients are inaccurate (Bromaghin et al. 2015, 2016a). Consequently, checking for predator proportions that are outside the range of mean prey proportions is an important diagnostic aid that may provide insights into the reliability of diet estimates.

qfasar uses the function pred_beyond_prey() to identify predator signature proportions that are outside the range of proportions observed among the individual and mean prey signatures. For purposes of diet estimation, proportions outside the range of the mean signatures are most important. However, pred_beyond_prey() also identifies predator proportions that are outside the range of the individual prey proportions for exploratory purposes.

pred_beyond_prey() is designed to be called with inputs returned by the function est_diet() (see Diet estimation). Although it is not conceptually necessary to estimate diets before performing this diagnostic check, doing so ensures that the predator and prey signatures have been transformed to the optimization space (Bromaghin et al. 2015) in which diets have been estimated. An example call to pred_beyond_prey() is contained in the following code snippet.

# Find predator proportions outside range of prey proportions.
beyond_prey <- pred_beyond_prey(pred_sigs = diet_est$pred_sigs,
                                prey_sigs = diet_est$prey_sigs,
                                mean_sigs = diet_est$mean_sigs)
str(beyond_prey)


# Compute percentage out of range by fatty acid, predator, and overall.
mean_by_fa <- apply(X = beyond_prey$beyond_mean, MARGIN = 1, FUN = mean)
mean_by_pred <- apply(X = beyond_prey$beyond_mean, MARGIN = 2, FUN = mean)
mean_all <- mean(beyond_prey$beyond_mean)

pred_beyond_prey() returns a list with the following objects:

• beyond_ind - A logical matrix with TRUE indicating that the corresponding predator proportion is outside the range of individual prey proportions.
• beyond_mean - A logical matrix with TRUE indicating that the corresponding predator proportion is outside the range of mean prey proportions.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

Although pred_beyond_prey() performs an important diagnostic check, how best to use the results may not always be obvious. Finding that all the predator proportions are within the range of the prey proportions is obviously desirable, but that may not be a realistic expectation when working with real data. A relatively small number of proportions being out of range may not be problematic. One might find that signs of problems are limited to a small number of predators or to one or two fatty acids across many predators. In such cases, a reasonable way forward may not be difficult to devise. However, as the signs of problems become increasingly prevalent, the best way to deal with them may not be clear. For example, Bromaghin et al. (2015) summarised an instance in which nearly half of all predator signature proportions were outside the range of the mean prey proportions. I can offer no general advice for how to proceed in such extreme cases.

Goodness of fit

In diet estimation (Diet estimation), a predator signature is modeled as a linear mixture of prey signatures and diet composition is estimated as the mixture proportions that minimize a measure of distance between observed and modeled predator signatures (Iverson et al. 2004). Consequently, one might expect that how well modeled signatures approximate observed signatures would be of interest, and potentially informative with respect to interpretation of the resulting diet estimates. While that seems likely to be true, methods to assess how well predator signatures are modeled have received almost no attention in the literature (but see Bromaghin et al. 2015).

A potentially informative byproduct of diet estimation is the value of the minimized distance measure associated with each estimated diet. The value will tend to be small when a modeled signature closely approximates an observed signature, and larger when a modeled signature approximates an observed signature less well. However, what value of the minimized distance measure might provide a warning flag for a potentially poor fit is an open question.

The function gof() represents a first attempt to address that question. The algorithm implemented in the function is based on the following logic. First, I assume that a predator consumes the mixture of prey specified by its estimated diet composition. Given that assumption, the expected value of the distance measure is, in a sense, fixed by the combination of mean prey signatures and the assumed diet composition (Bromaghin 2015). Large values of the distance measure are then most likely to occur when there is a high degree of random variation in a predator signature, which results only from the selection of individual prey within prey types if model assumptions are met. Within the framework of simulating predator signatures, variation is maximized when bootstrap sample sizes of the prey signatures used to construct a predator signature are minimized (Bromaghin 2015; Bromaghin et al. 2016b).

Implementing the above logic, gof() randomly samples a single signature from each prey type and weights those signatures by a predator’s estimated diet composition to construct a modeled signature. The modeled signature is then used to estimate diet. If the optimization function converges, the value of the minimized distance measure obtained with the modeled signature is compared to the value of the minimized distance measure obtained while estimating diet with the observed signature. This process is repeated a user-specified number of times (argument boot_gof) and the proportion of the simulated distance measure values that exceed the observed value of the distance measure is computed. gof() therefore constructs a statistic similar to a p-value, with small values being suggestive of predator signatures that may not have been closely approximated during diet estimation.

gof() is designed to be called with objects returned by est_diet() and prep_sig(), as well as associated constants; please refer to the function documentation for details. An example call to gof() is contained in the following code snippet.

# Check goodness-of-fit diagnostics.
gof_diag <- gof(prey_sigs = diet_est$prey_sigs,
                prey_loc = prey_info$loc,
                mean_sigs = diet_est$mean_sigs,
                diet_est = diet_est$est_ind,
                conv = diet_est$conv,
                obj_func = diet_est$obj_func,
                dist_meas = my_dist_meas,
                gamma = my_gamma,
                boot_gof = boot_gof)
str(gof_diag)
hist(gof_diag$p_val)

gof() returns a list containing the following objects:

• gof_ss - The number of diet estimates that converged for each predator, therefore producing a simulated value of the distance measure.
• p_val - The proportion of the simulated values of the distance measure that exceeded the value produced during diet estimation.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

NOTE: this algorithm implements an idea whose performance has not been explored. It has been included in qfasar to support future research on this topic. Also, like diet estimation itself, this function involves considerable bootstrap resampling and numerical optimization, which can result in very long run times.

Predator diet composition

A majority of simulations designed to investigate the performance of QFASA will probably involve some method of establishing predators with known diet compositions, simulating predator signatures based on those known diets (see Simulating predator signatures), and then evaluating the performance of diet estimators relative to the known “true diet”. Potential methods of establishing known diets, and their implementation in qfasar, are described in the following subsections:

• Realistic diets
• Random diets
• Diet grid

# Enter data.
diets <- matrix(c(0.065, 0.005, 0.051, 0.000, 0.873, 0.000, 0.006,
                  0.207, 0.014, 0.077, 0.000, 0.658, 0.000, 0.044),
                  ncol = 2)
rownames(diets) <- c("Bearded", "Beluga", "Bowhead", "Ribbon", "Ringed", "Spotted", "Walrus")
colnames(diets) <- c("AF", "AM")

# Check results.
print(diets)
colSums(diets)

# Generate some random diets.
diet_rand <- make_diet_rand(uniq_types = prey_info$uniq_types,
                            n_diet = 100000)
str(diet_rand)

# Generate a diet grid.
diet_grid <- make_diet_grid(uniq_types = prey_info$uniq_types,
                            inv_inc = 8)
str(diet_grid)

Simulating predator signatures

Investigations to study performance characteristics of QFASA may require predator signatures to be simulated so that thay have known properties, such as a known diet composition. Such simulated predator signatures have been called “pseudo-predators” (Iverson et al. 2004). The diet compositions of the simulated predators are then estimated, allowing the observed characteristics of the estimates to be compared with the known characteristics of the specified diet (e.g., Bromaghin et al. 2016a).

Predator signatures are simulated using a procedure that that closely parallels diet estimation. A diet composition must first be specified (see Predator diet composition). Conditioned on the specified diet, a bootstrap sample of signatures from each prey type is drawn (sampling with replacement) and the mean signature of each prey type is computed. A predator signature is constructed as a linear mixture of the mean prey signatures, with the diet composition as the mixture proportions.

An important question in simulating predator signatures is the number of signatures to bootstrap from each prey type. In the literature, the choice of bootstrap sample sizes has been made subjectively. As examples, Iverson et al. (2004) specified three levels of prey sample size (30, 60, and 90) and Bromaghin et al. (2015) used bootstrap sample sizes equal to the sample sizes in the prey library. Some authors apparently used subjectively-determined sample sizes but did not report the values (e.g., Thiemann et al. 2008; Wang et al. 2010; Haynes et al. 2015). However, Bromaghin et al. (2016b) found that bootstrap sample sizes strongly influence both the bias and variance of diet composition estimates. Consequently, the selection of bootstrap sample sizes is an important aspect of simulation design. Using sample sizes that are too small or too large may lead to an overly pessimistic or optimistic assessment, respectively, of model capabilities.

Bromaghin (2015) presented an objective method of establishing a bootstrap sample size for each prey type that produces simulated predator signatures having a realistic level of between-signature variation. That algorithm and the function implementing it are briefly described in a subsequent subsection (Bootstrap sample size algorithm).

qfasar implements the simulation of predator signatures in the function make_pred_sigs(). Predator signatures are first generated in the prey space and then transformed to their natural predator space. make_pred_sigs() is called using the following arguments:

• prey_sigs - A matrix of prey signatures in the prey space, intended to be the object sig_scale returned by a call to the function prep_sig(), or the object sig_part returned by a call to the function make_prey_part().
• prey_loc - A matrix giving the first and last locations of the signatures of each prey type within prey_sigs, intended to be the object loc returned by a call to one of the functions prep_fa() or make_prey_part().
• cc - A numeric vector containing the calibration coefficients, intended to be the object cc returned by a call to one of the functions prep_fa() or cc_aug().
• diet - A numeric vector specifying the predator diet composition as proportions.
• prey_ss - An integer vector specifying the bootstrap sample size to use for each prey type.
• n_pred - An integer specifying the number of predator signatures to generate.

make_pred_sigs() returns a list with the following components:

• sim_pred_sigs - A numeric matrix containing simulated predator signatures in the predator space.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string containing a brief summary of the results.

Please see the code block in the section Bootstrap sample size algorithm for an example call to make_pred_sigs().

# Find bootstrap sample sizes suitable for simulation work.
boot_ss <- find_boot_ss(pred_sigs = diet_est$pred_sigs,
                        pred_diets = diet_est$est_ind,
                        prey_sigs = diet_est$prey_sigs,
                        prey_loc = prey_info$loc,
                        n_pred_boot = 500)
str(boot_ss)
print(boot_ss$boot_ss)

# Simulate predator signatures.
my_sim_preds <- make_pred_sigs(prey_sigs = prey_info$sig_scale,
                               prey_loc = prey_info$loc,
                               cc = new_cc$cc,
                               diet = apply(X = diet_est$est_ind, MARGIN = 1,
                                            FUN = mean),
                               prey_ss = boot_ss$boot_ss,
                               n_pred = 1000)
str(my_sim_preds)                          

# Figure 1 of Bromaghin (2015).
plot(x = boot_algo$var_diet, y = boot_algo$var_sig, cex.lab=1.5,
     xlab="Variation in predator diets",
     ylab="Variation in predator signatures")
lines(x = boot_algo$var_diet, y = boot_algo$var_sig_smooth, lwd=3, col="red")


# Figure 2 of Bromaghin (2015).
iteration <- 1:length(boot_algo$mod_sig_var)
plot(x = iteration, y = boot_algo$mod_sig_var,
     pch = 16, cex.lab = 1.5, type = "b",
     xlab = "Iteration",
     ylab = "Modeled predator signature variance")
abline(boot_algo$var_target, 0)

Generating a ghost prey signature

One of the major assumptions of QFASA is that the prey library contains representatives of all prey types consumed by a predator. Bromaghin et al. (2016a) investigated the robustness of diet estimators to violations of this assumption. qfasar implements their method of constructing a signature for a prey type not in the library, termed a ghost prey, in the function make_ghost().

The ghost prey signature is constructed by maximizing the summed distance between the ghost prey signature and the mean prey signatures, while constraining the ghost signature proportions within reasonable bounds to ensure that the signature is somewhat realistic for the prey library. The definition of reasonable bounds is embodied in the argument ghost_err. ghost_err is a proportion greater than or equal to zero and less than 1 used to construct box constraints for each fatty acid in the signatures.

• lower bound equals (1 - ghost_err)*(smallest mean prey proportion)
• upper bound equals (1 + ghost_err)*(largest mean prey proportion)

The ghost prey signature is obtained by maximizing the summed distance between the signature and the mean prey signatures, constraining the signature to lie within the box constraints and sum to 1.

This method ensures that the ghost prey signature is somewhat distinct from the other prey types, but not so wildly different that it represents a completely different pattern from the other prey types. Although research into suitable values for ghost_err has not been conducted, it is probably advisable to use small to moderate values. Bromaghin et al. (2016) used a value of 0.25. As the value of ghost_err is increased, the resulting ghost prey signature will tend to become increasing different from any prey type in the library. If the value of ghost_err is too small, the ghost prey signature could potentially be similar to one or more prey types in the library, which is probably not what one would want for a simulation study.

The following code snippet contains an example call to make_ghost().

# Specify a level of error.
ghost_err <- 0.25

# Construct a ghost prey signature.
ghost <- make_ghost(prey_sigs = prey_info$sig_scale,
                    loc = prey_info$loc,
                    ghost_err = ghost_err,
                    dist_meas = 1,
                    gamma = NA)
str(ghost)

The function make_ghost() returns a list containing the following objects.

• sig - A numeric vector containing the ghost prey signature.
• dist - The summed distance between the ghost signature and the mean prey signatures.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.

The following code snippet illustrates how to simulate predator signatures incorporating a specified contribution of ghost prey to their diets.

## Generate predator signatures with a contribution from the ghost prey.

# Find bootstrap sample sizes suitable for simulation work.
# There is only one ghost signature, so its sample size should be 1.
boot_algo <- find_boot_ss(pred_sigs = diet_orig$pred_sigs,
                          pred_diets = diet_orig$est_ind,
                          prey_sigs = diet_orig$prey_sigs,
                          prey_loc = prey_info$loc,
                          n_pred_boot = 500)
ghost_ss <- c(boot_algo$boot_ss, 1)


# Append ghost signature to prey data.
ghost_sigs <- cbind(prey_info$sig_scale, ghost$sig)
colnames(ghost_sigs) <- c(colnames(prey_info$sig_scale), "ghost")

last_prey <- prey_info$loc[prey_info$n_types,2]+1
ghost_loc <- rbind(prey_info$loc, c(last_prey, last_prey))
rownames(ghost_loc) <- c(rownames(prey_info$loc), "ghost")
colnames(ghost_loc) <- c("first", "last")


# Generate a random diet for the non-ghost prey components.  These proportions
# could be established in whatever way makes sense in a particular application.
non_ghost <- make_diet_rand(uniq_types = prey_info$uniq_types, n_diet = 1)


# Specify ghost prey component and add to the non-ghost components.
ghost_prop <- 0.15
ghost_diet <- c(non_ghost$diet_rand*(1 - ghost_prop), ghost_prop)
print(ghost_diet)
sum(ghost_diet)


# Generate predator signatures.
ghost_pred_sigs <- make_pred_sigs(prey_sigs = ghost_sigs,
                                  prey_loc = ghost_loc,
                                  cc = new_cc$cc,
                                  diet = ghost_diet,
                                  prey_ss = ghost_ss,
                                  n_pred = 500)
str(ghost_pred_sigs)

Adding error to calibration coefficients

One of the major assumptions of QFASA is that the calibration coefficients are known perfectly. Bromaghin et al. (2016a) investigated the robustness of diet estimators to violations of this assumption. The function add_cc_err() uses the methods of Bromaghin et al. (2016a) to add error to a set of calibration coefficients.

The argument err_bound is used to compute box constraints for the calibration coefficients:

• lower bound equals (1 - err_bound)*cc_true
• upper bound equals (1 + err_bound)*cc_true

where the argument cc_true is the set of true calibration coefficients provided as input. A uniformly distributed random number is generated between the bounds for each calibration coefficient and the vector of coefficients is scaled so that their sum equals the sum of the true calibration coefficients. The mean relative absolute difference between the true and error-added calibration coefficients is computed as a measure of error for the entire vector (Bromaghin et al 2016a).

The influence of calibration coefficient error can be investigated by generating predator signatures with the true set of calibration coefficients and estimating diet with a set of coefficients incorporating error (Bromaghin et al. 2016a). An example call to add_cc_err() is contained in the following code block.

# Find bootstrap sample sizes suitable for simulation work.
boot_ss <- find_boot_ss(pred_sigs = diet_est$pred_sigs,
                        pred_diets = diet_est$est_ind,
                        prey_sigs = diet_est$prey_sigs,
                        prey_loc = prey_info$loc,
                        n_pred_boot = 500)
                        
                        
# Compute the mean diet to use in a simulation.
mean_diet <- apply(X = diet_est$est_ind, MARGIN = 1, FUN = mean)


# Simulate predator signatures using the true calibration coefficients.
my_sim_preds <- make_pred_sigs(prey_sigs = prey_info$sig_scale,
                               prey_loc = prey_info$loc,
                               cc = new_cc$cc,
                               diet = mean_diet,
                               prey_ss = boot_ss$boot_ss,
                               n_pred = 1000)


# Add error to the calibration coefficients.
cc_err <- add_cc_err(cc_true = new_cc$cc, err_bound = 0.25)
str(cc_err)
print(cc_err$err)


# Estimate diets of the simulated predators using the error-added calibration coefficients.
err_est <- est_diet(pred_sigs = pred_info$sig_scale,
                    pred_uniq_types = pred_info$uniq_types,
                    pred_loc = pred_info$loc,
                    prey_sigs = prey_info$sig_scale,
                    prey_uniq_types = prey_info$uniq_types,
                    prey_loc = prey_info$loc,
                    cc = cc_err$cc,
                    space = my_space,
                    dist_meas = my_dist_meas,
                    gamma = my_gamma,
                    ind_boot = my_ind_boot,
                    mean_meth = my_mean_meth,
                    var_meth = my_var_meth,
                    mean_boot = my_mean_boot)

The function add_cc_err() returns a list containing the following objects.

• cc - A numeric vector of calibration coefficients with error incorporated.
• err - The mean relative absolute error in the calibration coefficients.
• err_code - An integer error code (0 if no error is detected).
• err_message - A string contains a brief summary of the execution.