Package 'templateICAr'

Fitted (spatial) template ICA object from templateICA.

Set a threshold value for activation? A threshold value can be specified directly with u, or a z-score-like threshold in terms of standard deviations (the SD of values in the mean template) can be specified with z. Only one type of threshold can be used. Default: NULL (do not use a threshold). Either argument can also be a vector to test multiple thresholds at once, as long as type is not "!=" (to ensure the activation regions are successive subsets).

Significance level for hypothesis testing. Default: 0.01.

Type of region: ">" (default), "abs >", "<", or "!=". "abs >" tests for magnitude by taking the absolute value and then testing if they are greater than... .

If the input is a "tICA.[format]" model object, the type of multiple comparisons correction to use for p-values, or NULL for no correction. See help(p.adjust). Default: "BH" (Benjamini & Hochberg, i.e. the false discovery rate). Note that multiple comparisons will account for data locations, but not ICs.

If TRUE, display progress of algorithm. Default: FALSE.

Indices of ICs for which to identify activations. If NULL (default), use all ICs.

If TRUE identify significant deviations from the template mean, rather than significant areas of engagement. Default: FALSE.

a k x k 'matrix'

how many times to repeat mat

Matrix of Cholesky factorizations (upper triangular values) for all template sessions (nN*nM x nChol)

Pivot/reordering applied to FC matrices prior to Cholesky factorization

Number of samples to draw

Indices of diagonal upper triangular elements

Indices of off-diagonal upper triangular elements

A nLxnL matrix indexing the upper triangular elements

A numeric matrix, with each column being a centered timeseries. For fMRI data, X should be T timepoints by V brain locations.

Number of latent dimensions to estimate. If NULL (default), estimated using PESEL (Sobczyka et al. 2020).

Maximal number of principal components for automatic dimensionality selection with PESEL. Default: 100.

($V \times Q$ matrix) mean maps for each IC in template, where $Q$ is the number of ICs, $V=nvox$ is the number of data locations.

($V \times Q$ matrix) between-subject variance maps for each IC in template

NULL for spatial independence model, otherwise a list of objects of class "templateICA_mesh" containing the triangular mesh (see make_mesh) for each brain structure.

($V \times Q$ matrix) dimension-reduced fMRI data

(list) initial guess at parameter values: A ($QxQ$ mixing matrix), nu0_sq (residual variance from first level) and (for spatial model only) kappa (SPDE smoothness parameter for each IC map)

($Qx1$) diagonal elements of matrix proportional to residual variance.

For dimension reduction of the spatial template ICA model, which assumes that all IC's have the same smoothness parameter, $\kappa$

Maximum number of EM iterations. Default: 100.

Parallelize the computation? Default: FALSE. Can be the number of cores to use or TRUE, which will use the number available minus two. Not yet implemented for spatial template ICA.

Smallest proportion change between iterations. Default: 0.001.

Reduce the temporal dimension of the data using PCA? Default: TRUE for the spatial EM algorithm, and FALSE for the independent EM algorithm.

If TRUE, display progress of algorithm. Default: FALSE.

Empirical between-subject variance of covariance matrices (QxQ)

Empirical mean of covariance matrices (QxQ)

Empirical between-subject variance of covariance matrices (QxQ)

Empirical mean of covariance matrices (QxQ)

Vector of subject-level fMRI data in one of the following formats: CIFTI file paths, "xifti" objects, GIFTI file paths, "gifti" objects, NIFTI file paths, "nifti" objects, or $V \times T$ numeric matrices, where $V$ is the number of data locations and $T$ is the number of timepoints.

If BOLD2 is provided it must be in the same format as BOLD; BOLD will be the test data and BOLD2 will be the retest data. BOLD2 should be the same length as BOLD and have the same subjects in the same order. If BOLD2 is not provided, BOLD will be split in half; the first half will be the test data and the second half will be the retest data.

Group ICA maps in a format compatible with BOLD. Can also be a (vectorized) numeric matrix ($V \times Q$) no matter the format of BOLD. Its columns will be centered.

New: can also be a parcellation in CIFTI format (other formats to be implemented in the future). The parcellation should have the same locations and one column, with integer values indicating the parcel to which each location belongs to. Each parcel is modeled as a brain map; instead of the first step of dual regression, the medial timecourse of each parcel is used.

Required if BOLD are NIFTI file paths or "nifti" objects, and optional for other formats. For NIFTI data, this is a brain map formatted as a logical array of the same spatial dimensions as the fMRI data, with TRUE corresponding to in-mask voxels. For other data, this is a logical vector with the same length as the number of locations in GICA, with TRUE corresponding to in-mask locations.

Numeric indices of the group ICs to include in the template. If NULL, use all group ICs (default).

If inds is provided, the ICs not included will be removed after calculating dual regression, not before. This is because removing the ICs prior to dual regression would leave unmodelled signals in the data, which could bias the templates.

"local" (default), "global", or "none". Local scaling will divide each data location's time series by its estimated standard deviation. Global scaling will divide the entire data matrix by the mean image standard deviation (mean(sqrt(rowVars(BOLD)))).

Only applies if scale=="local" and BOLD represents surface data (CIFTI or GIFTI). To smooth the standard deviation estimates used for local scaling, provide the surface geometries along which to smooth as GIFTI geometry files or "surf" objects, as well as the smoothing FWHM (default: 2).

If scale_sm_FWHM==0, no smoothing of the local standard deviation estimates will be performed.

If scale_sm_FWHM>0 but scale_sm_surfL and scale_sm_surfR are not provided, the default inflated surfaces from the HCP will be used.

To create a "surf" object from data, see make_surf. The surfaces must be in the same resolution as the BOLD data.

The temporal resolution of the data, i.e. the time between volumes, in seconds. TR is required for detrending with hpf.

The frequency at which to apply a highpass filter to the data during pre-processing, in Hertz. Default: 0.01 Hertz. Set to 0 to disable the highpass filter.

The highpass filter serves to detrend the data, since low-frequency variance is associated with noise. Highpass filtering is accomplished by nuisance regression of discrete cosine transform (DCT) bases.

Note the TR argument is required for highpass filtering. If TR is not provided, hpf will be ignored.

Center BOLD across columns (each image)? This is equivalent to performing global signal regression. Default: FALSE.

Obtain dual regression estimates after denoising? Denoising is based on modeling and removing nuisance ICs. It may result in a cleaner estimate for smaller datasets, but it may be unnecessary (and time-consuming) for larger datasets.

Set Q2 to control denoising: use a positive integer to specify the number of nuisance ICs, NULL to have the number of nuisance ICs estimated by PESEL, or zero (default) to skip denoising.

If is.null(Q2), use Q2_max to specify the maximum number of nuisance ICs that should be estimated by PESEL. Q2_max must be less than $T * .75 - Q$ where $T$ is the minimum number of timepoints in each fMRI scan and $Q$ is the number of group ICs. If NULL (default), Q2_max will be set to $T * .50 - Q$, rounded.

Only applies if the entries of BOLD are CIFTI file paths. This is a character vector indicating which brain structure(s) to obtain: "left" (left cortical surface), "right" (right cortical surface) and/or "subcortical" (subcortical and cerebellar gray matter). Can also be "all" (obtain all three brain structures). Default: c("all").

Only applies if the entries of BOLD are CIFTI file paths. Resample the data upon reading it in? Default: NULL (no resampling).

Keep the DR estimates of S? If FALSE (default), do not save the DR estimates and only return the templates. If TRUE, the DR estimates of S are returned too. If a single file path, save the DR estimates as an RDS file at that location rather than returning them.

Keep the DR estimates of the FC cor(A)? If FALSE (default), do not save the DR estimates and only return the templates. If TRUE, the DR estimates of cor(A) and its Cholesky factor are returned too. If a single file path, save the DR estimates as an RDS file at that location rather than returning them.

Include the functional connectivity template? Default: TRUE.

Number of pivots to use in Cholesky-based FC template estimation. Set to zero to skip Cholesky-based FC template estimation. Default: 100.

Number of FC matrix samples to generate across all pivots. This should be a multiple of FC_nPivots.

Tolerance for variance of each data location. For each scan, locations which do not meet this threshold are masked out of the analysis. Default: 1e-6. Variance is calculated on the original data, before any normalization.

For computing the dual regression results for each subject: tolerance for number of locations masked out due to low variance or missing values. If more than this many locations are masked out, a subject is skipped without calculating dual regression. maskTol can be specified either as a proportion of the number of locations (between zero and one), or as a number of locations (integers greater than one). Default: .1, i.e. up to 10 percent of locations can be masked out.

If BOLD2 is provided, masks are calculated for both scans and then the intersection of the masks is used, for each subject.

For computing the variance decomposition across all subjects: tolerance for number of subjects masked out due to low variance or missing values at a given location. If more than this many subjects are masked out, the location's value will be NA in the templates. missingTol can be specified either as a proportion of the number of locations (between zero and one), or as a number of locations (integers greater than one). Default: .1, i.e. up to 10 percent of subjects can be masked out at a given location.

Parallelize the DR computations over subjects? Default: FALSE. Can be the number of cores to use or TRUE, which will use the number on the PC minus two. If the input data is in CIFTI format, the wb_path must also be provided.

Display progress updates? Default: TRUE.

The FC estimates from estimate_template.

Factor by which to adjust estimate of nu. Values < 1 will inflate the template variance to avoid an over-informative prior on FC.

the test/retest dual regression estimates, as an array with dimensions $M \times N \times (L \times V)$, where $M$ is the number of visits (2), $N$ is the number of subjects, $L$ is the number of IC networks, and $V$ is the number of data locations.

($L$ and $V$ are collapsed because they are treated equivalently in the context of calculating the variance decomposition and templates).

A length-two integer vector giving the dimensions $L$ and $V$ to reshape the result. Default: NULL (do not reshape the result).

INLA mesh

data

index of the data locations in the mesh

If TRUE, a formula based on the trace is used, otherwise the inverse correlation is used

The result of estimate_template

Use NULL (default) to just return the template objects directly. Otherwise, use a character vector of length 3 or 4 of file path(s) to save the output to: the mean template, the variance template, the variance decomposition, and the FC template if present, in that order. If one file name is provided, it will be appended with "_mean.[file_ext]" for the template mean map, "_var.[file_ext]" for the template variance map, "_varDecomp.rds" for the variance decomposition, and "_FC.rds" where [file_ext] will be "dscalar.nii" for CIFTI input, "nii" for NIFTI input, and "rds" for data input.

"non-negative" (default) or "unbiased"

vector of p AR parameters

number of time points in timeseries

Vector of file paths of CIFTI-format fMRI timeseries (*.dtseries.nii) for which to calculate group ICA

Use this argument if some subjects have more than one scan. This is a numeric or factor vector the same length as cifti_fnames. Scans for the same subject should have the same value. For example, if there are four subjects with two visits each, and the scans are ordered with the first subject's two scans first, then the second subject's two scans next, etc., then this argument should be rep(seq(4), each=2). If there are three subjects and four scans, with the last two scans belonging to the same subject, this argument should be c(1,2,3,3). If all subjects only have one scan, keep this argument as NULL (default).

Character vector indicating which brain structure(s) to obtain: "left" (left cortical surface), "right" (right cortical surface) and/or "subcortical" (subcortical and cerebellar gray matter). Can also be "all" (obtain all three brain structures). Default: c("all").

Maximum number of PCs to retain in initial subject-level PCA

Number of independent components to identify.

The maximum number of rows for the matrix on which group level PCA will be performed. This matrix is the result of temporal concatenation and contains length(cifti_fnames) * num_PCs rows.

Center BOLD columns, scale by the standard deviation, and detrend voxel timecourses? See norm_BOLD. Normalization is applied separately to each scan. Defaults: Center BOLD columns, scale by the global standard deviation, and apply a .01 Hz HPF if the TR is provided.

If TRUE, display progress updates

(Optional) If not specified, a xifti object will be returned but the GICA maps will not be written to a CIFTI file.

(Optional) File path to a surface GIFTI for the left cortex. If provided, will be part of xifti result object for visualization in R. Will also be used to perform any smoothing.

(Optional) File path to a surface GIFTI for the right cortex. If provided, will be part of xifti result object for visualization in R. Will also be used to perform any smoothing.

Smooth the CIFTI data prior to reading in each file? Default: TRUE. Use the following arguments to control the smoothing parameters.

See smooth_cifti. The defaults here are the same. Note that smooth_zeroes_as_NA will control both of the corresponding surface and volume arguments to smooth_cifti. These arguments only apply if smooth.

Inverse Wishart degrees of freedom parameter

Matrix dimension for IW distribution

Empirical mean of covariance matrices at element (i,j)

Empirical mean of covariance matrices at the ith diagonal element

Empirical mean of covariance matrices at the jth diagonal element

Inverse Wishart degrees of freedom parameter

Matrix dimension for IW distribution

Empirical mean of covariance matrices at element (i,j)

Value of hyperparameters

Data vector

SPDE G matrix

SPDE C matrix

Indices of data locations in the mesh

Object of class "surf". See make_surf and is.surf.

Subset of vertices to include in analysis, e.g. non-medial wall locations.

Subset of vertices to retain in mesh, e.g. non-medial wall locations. Must be a superset of inds_data.

Brain mask (matrix of 0 and 1 or TRUE and FALSE).

fMRI numeric data matrix ($V \times T$)

Center BOLD data across rows (each data location's time series) or columns (each time point's image)? Default: TRUE for row centering, and FALSE for column centering.

"local" (default), "global", or "none". Local scaling will divide each data location's time series by its estimated standard deviation. Global scaling will divide the entire data matrix by the mean image standard deviation (mean(sqrt(rowVars(BOLD)))).

Only applies if scale=="local" and BOLD represents CIFTI-format data. To smooth the standard deviation estimates used for local scaling, provide a "xifti" object with data locations in alignment with BOLD, as well as the smoothing FWHM (default: 2). If no "xifti" object is provided (default), do not smooth.

For local scaling with smoothing, the data must be unmasked to be mapped back to the surface. So if the data are masked, provide the mask here.

The temporal resolution of the data, i.e. the time between volumes, in seconds. TR is required for detrending with hpf.

The frequency at which to apply a highpass filter to the data during pre-processing, in Hertz. Default: 0.01 Hertz. Set to 0 to disable the highpass filter.

The highpass filter serves to detrend the data, since low-frequency variance is associated with noise. Highpass filtering is accomplished by nuisance regression of discrete cosine transform (DCT) bases.

Note the TR argument is required for highpass filtering. If TR is not provided, hpf will be ignored.

A square matrix to be orthonormalized.

The template from estimate_template.cifti

"mean", "sd", or "both" (default). By default the square root of the variance template is shown; another option is stat="var" to instead display the variance template directly.

"non-negative" (default) or "unbiased"

Additional arguments to view_xifti

The template from estimate_template.gifti

"mean", "sd", or "both" (default). By default the square root of the variance template is shown; another option is stat="var" to instead display the variance template directly.

"non-negative" (default) or "unbiased"

Additional arguments to view_xifti

The template from estimate_template.matrix

Additional arguments

The template from estimate_template.nifti

"mean" (default), "sd", or "var". ("sd" will show the square root of the variance template.)

Anatomical plane and which slice indices to show. Default: 9 axial slices.

"non-negative" (default) or "unbiased"

Additional arguments to oro.nifti::image

The activations from activations.cifti

"active" (default), "pvals", "pvals_adj", "tstats", or "vars".

Additional arguments to view_xifti

The result of templateICA with CIFTI data

"mean" (default), "se", or "both"

Additional arguments to view_xifti

The result of templateICA with NIFTI data

Additional arguments

The result of templateICA with NIFTI data

"mean" (default), "se"

Anatomical plane and which slice indices to show. Default: 9 axial slices.

Additional arguments

Estimated A matrix (T x Q)

Order of the AR model used to prewhiten the data at each location. If !aic (default), the order will be exactly ar_order. If aic, the order will be between zero and ar_order, as determined by the AIC.

The "template.cifti" object.

The new resampling resolution.

Give occasional updates? Default: FALSE.

A numerical matrix

if inverse=TRUE, compute inverse of square root

Object of class "template.cifti".

further arguments passed to or from other methods.

The template from estimate_template.cifti

Object of class "template.gifti".

further arguments passed to or from other methods.

The template from estimate_template.gifti

Object of class "template.matrix".

further arguments passed to or from other methods.

The template from estimate_template.cifti

Object of class "template.nifti".

further arguments passed to or from other methods.

The template from estimate_template.nifti

Object of class "tICA_act.cifti".

further arguments passed to or from other methods.

The activations from activations.cifti

Object of class "tICA_act.matrix".

further arguments passed to or from other methods.

The activations from activations

Object of class "tICA_act.nifti".

further arguments passed to or from other methods.

The activations from activations

Object of class "tICA.cifti".

further arguments passed to or from other methods.

The result of templateICA with CIFTI data

Object of class "tICA.matrix".

further arguments passed to or from other methods.

The template from estimate_template.cifti

Object of class "tICA.nifti".

further arguments passed to or from other methods.

The template from estimate_template.cifti

Vector of subject-level fMRI data in one of the following formats: CIFTI file paths, "xifti" objects, NIFTI file paths, "nifti" objects, or $V \times T$ numeric matrices, where $V$ is the number of data locations and $T$ is the number of timepoints.

If multiple BOLD data are provided, they will be independently centered, scaled, detrended (if applicable), and denoised (if applicable). Then they will be concatenated together followed by computing the initial dual regression estimate.

Template estimates in a format compatible with BOLD, from estimate_template.

Which calculation of the template variance to use: "non-negative" (default) or "unbiased". The unbiased template variance is based on the assumed mixed effects/ANOVA model, whereas the non-negative template variance adds to it to account for greater potential between-subjects variation. (The template mean is the same for either choice of tvar_method.)

"global", "local", or "none". Global scaling will divide the entire data matrix by the mean image standard deviation (mean(sqrt(rowVars(BOLD)))). Local scaling will divide each data location's time series by its estimated standard deviation. Default: "template", to use the same option used for estimation of the template.

Only applies if scale=="local" and BOLD represents CIFTI-format data. To smooth the standard deviation estimates used for local scaling, provide the surface geometries along which to smooth as GIFTI geometry files or "surf" objects, as well as the smoothing FWHM (default: "template" to use the same option used for estimation of the template).

If scale_sm_FWHM==0, no smoothing of the local standard deviation estimates will be performed.

If scale_sm_FWHM>0 but scale_sm_surfL and scale_sm_surfR are not provided, the default inflated surfaces from the HCP will be used.

To create a "surf" object from data, see make_surf. The surfaces must be in the same resolution as the BOLD data.

(Optional) Signals to regress from the data, given as a numeric matrix with the same number of rows as there are volumes in the BOLD data. If multiple BOLD sessions are provided, this argument can be a list to use different nuisance regressors for different sessions. Nuisance regression is performed as a first step, before centering, scaling, and denoising. An intercept column will automatically be added to nuisance. If NULL, no extra nuisance signals will be regressed from the data, but a nuisance regression will still be used if warranted by scrub or hpf.

(Optional) A numeric vector of integers indicating the indices of volumes to scrub from the BOLD data. (List the volumes to remove, not the ones to keep.) If multiple BOLD sessions are provided, this argument can be a list to remove different volumes for different sessions. Scrubbing is performed within nuisance regression by adding a spike regressor to the nuisance design matrix for each volume to scrub. If NULL, do not scrub.

These arguments control detrending. TR is the temporal resolution of the data, i.e. the time between volumes, in seconds; hpf is the frequency of the high-pass filter, in Hertz. Detrending is performed via nuisance regression of DCT bases. Default: "template" to use the values from the template. Be sure to set the correct TR if it's different for the new data compared to the data used in estimate_template.

Note that if multiple BOLD sessions are provided, their TR and hpf must be the same; both arguments accept only one value.

Center BOLD across columns (each image)? This is equivalent to performing global signal regression. Default: "template", to use the same option used for estimation of the template.

Denoise the BOLD data? Denoising is based on modeling and removing nuisance ICs. It may result in a cleaner estimate for smaller datasets, but it may be unnecessary (and time-consuming) for larger datasets.

Set Q2 to control denoising: use a positive integer to specify the number of nuisance ICs, NULL to have the number of nuisance ICs estimated by PESEL, or zero to skip denoising.

If is.null(Q2), use Q2_max to specify the maximum number of nuisance ICs that should be estimated by PESEL. Q2_max must be less than $T * .75 - Q$ where $T$ is the number of timepoints in BOLD and $Q$ is the number of group ICs. If NULL, Q2_max will be set to $T * .50 - Q$, rounded.

The defaults for both arguments is "template", to use the same option used for estimation of the template.

Only applies if the entries of BOLD are CIFTI file paths. This is a character vector indicating which brain structure(s) to obtain: "left" (left cortical surface), "right" (right cortical surface) and/or "subcortical" (subcortical and cerebellar gray matter). Can also be "all" (obtain all three brain structures). Default: "template" to use the same brainstructures present in the template).

Required if and only if the entries of BOLD are NIFTI file paths or "nifti" objects. This is a brain map formatted as a binary array of the same spatial dimensions as the fMRI data, with TRUE corresponding to in-mask voxels.

Subset of fMRI BOLD volumes to include in analysis, as a vector of numeric integers. If NULL (default), use all volumes. Subsetting is performed before any centering, scaling, detrending, and denoising. If multiple BOLD are provided, time_inds can be a list of the same length, with each element being a vector of numeric integers indicating the timepoints to include for each BOLD.

Tolerance for variance of each data location. For each scan, locations which do not meet this threshold are masked out of the analysis. Default: "template" to use the same brainstructures present in the template). Variance is calculated on the original data, before any normalization. Set to 0 to avoid removing locations due to low variance.

Should spatial modeling be performed? If NULL, assume spatial independence. Otherwise, provide meshes specifying the spatial priors assumed on each independent component. Each should represent a brain structure, between which spatial independence can be assumed.

If BOLD represents CIFTI-format data, spatial_model should give the left and right cortex surface geometries (whichever one(s) are being used) as "surf" objects or GIFTI surface geometry file paths. Spatial modeling is not yet available for the subcortex. This argument can also be TRUE, in which case spatial modeling will be performed with the surfaces included in the first entry of BOLD if it is a "xifti" object, or if those are not present available, the default inflated surfaces from ciftiTools.

If BOLD represents NIFTI-format data, spatial modeling is not yet available.

If BOLD is a numeric matrix, spatial_model should be a list of meshes (see make_mesh).

Only applies if BOLD represents CIFTI-format data. The target resolution for resampling (number of cortical surface vertices per hemisphere). For spatial modelling, a value less than 10000 is recommended for computational feasibility. If NULL (default), do not perform resampling.

Only applies if BOLD represents CIFTI-format data. Should medial wall (missing data) locations be removed from the mesh? If TRUE, faster computation but less accurate estimates at the boundary of wall.

Reduce the temporal dimension of the data using PCA? Default: TRUE. Skipping dimension reduction will slow the model estimation, but may result in more accurate results. Ignored for FC template ICA

Variational Bayes (VB) method for FC template ICA model: "VB1" (default) uses a conjugate Inverse-Wishart prior for the cor(A); "VB2" draws samples from p(cor(A)) to emulate the population distribution using a combination of Cholesky, SVD, and random pivoting. "none" Uses standard template ICA without FC prior

Maximum number of EM or VB iterations. Default: 100.

Minimum number of EM or VB iterations. Default: 3.

Smallest proportion change between iterations. Default: .001.

Starting value for kappa. Default: 0.2.

Parallelize the computation? Default: TRUE. Can be the number of cores to use or TRUE, which will use the number available minus two.

Prewhiten to account for residual autocorrelation? Default: FALSE. Only affects FC template ICA models.

If TRUE, display progress of algorithm

($V \times Q$ matrix) mean maps for each IC in template

($V \times Q$ matrix) between-subject variance maps for each IC in template

NULL for spatial independence model, otherwise a list of objects of class "templateICA_mesh" containing the triangular mesh (see make_mesh) for each brain structure.

($V \times Q$ matrix) dimension-reduced fMRI data

(list) current parameter estimates

$(Qx1)$ diagonal elements of residual covariance after dimension reduction

For dimension reduction

Vectorized template means

Sparse diagonal matrix of template standard deviations

The inverse of D times s0_vec

If TRUE, display progress of algorithm. Default: FALSE.

If TRUE. return the posterior mean and precision of the latent fields instead of the parameter estimates. Default: FALSE.

Which parameters to update. Either "all", "A" or "kappa".

For non-spatial model: updating nu0sq is recommended if dimension reduction was not performed, and is not recommended if it was.

Vector of upper triangular values

Are diagonal values included in x? Default: TRUE.

Inverse Wishart degrees of freedom parameter

Matrix dimension for IW distribution

Empirical between-subject variance of covariance matrices at element (i,j)

Empirical mean of covariance matrices at element (i,j)

Empirical mean of covariance matrices at the ith diagonal element

Empirical mean of covariance matrices at the jth diagonal element

Inverse Wishart degrees of freedom parameter

Matrix dimension for IW distribution

Empirical between-subject variances of CORRELATION matrix (upper triangle)

Empirical mean of CORRELATION matrix (upper triangle)

Penalty to assign if theoretical variance is smaller than empirical variance