rgccak: Internal function for computing the RGCCA parameters (RGCCA block components, outer weight vectors, etc.).

Description

The function rgccak() is called by rgcca() and does not have to be used by the user. The function rgccak() computes the RGCCA block components, outer weight vectors, etc., for each block and each dimension. Depending on the dimensionality of each block \(X_j , j = 1, ..., J\), the primal (when \(n > p_j\)) or the dual (when \(n < p_j\)) algorithm is used (see Tenenhaus et al. 2015)

Usage

rgccak(A, C, tau = "optimal", scheme = "centroid", scale = FALSE,
  verbose = FALSE, init = "svd", bias = TRUE, tol = 1e-08)

Arguments

A list that contains the \(J\) blocks of variables. Either the blocks (\(X_1, X_2, ..., X_J\)) or the residual matrices (\(X_{h1}, X_{h2}, ..., X_{hJ}\)).

A design matrix that describes the relationships between blocks. (Default: complete design).

tau

A \(1 * J\) vector that contains the values of the shrinkage parameters \(\tau_j\), \( j=1, ..., J\). (Default: \(\tau_j = 1\), \( j=1, ..., J\)). If tau = "optimal" the shrinkage intensity paramaters are estimated using the Schafer and Strimmer (2005) analytical formula.

scheme

The value is "horst", "factorial", "centroid" or any diffentiable convex scheme function g designed by the user (default: "centroid").

scale

if scale = TRUE, each block is standardized to zero means and unit variances (default: TRUE).

verbose

Will report progress while computing if verbose = TRUE (default: TRUE).

init

The mode of initialization to use in the RGCCA algorithm. The alternatives are either by Singular Value Decompostion or random (default : "svd").

bias

A logical value for either a biaised or unbiaised estimator of the var/cov.

tol

Stopping value for convergence.

Value

A \(n * J\) matrix of RGCCA outer components

A \(n * J\) matrix of RGCCA inner components

A list of outer weight vectors

crit

The values of the objective function to be optimized in each iteration of the iterative procedure.

AVE

Indicators of model quality based on the Average Variance Explained (AVE): AVE(for one block), AVE(outer model), AVE(inner model).

A design matrix that describes the relationships between blocks (user specified).

tau

\(1 * J\) vector containing the value for the tau penalties applied to each of the \(J\) blocks of data (user specified)

scheme

The scheme chosen by the user (user specified).

References

Tenenhaus M., Tenenhaus A. and Groenen PJF (2017), Regularized generalized canonical correlation analysis: A framework for sequential multiblock component methods, Psychometrika, in press

Tenenhaus A., Philippe C., & Frouin V. (2015). Kernel Generalized Canonical Correlation Analysis. Computational Statistics and Data Analysis, 90, 114-131.

Tenenhaus A. and Tenenhaus M., (2011), Regularized Generalized Canonical Correlation Analysis, Psychometrika, Vol. 76, Nr 2, pp 257-284.

Schafer J. and Strimmer K., (2005), A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Statist. Appl. Genet. Mol. Biol. 4:32.