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

The function sgccak() is called by sgcca() and does not have to be used by the user. sgccak() enables the computation of SGCCA block components, outer weight vectors, etc., for each block and each dimension.

sgccak(A, C, c1 = rep(1, length(A)), scheme = "centroid",
  tol = .Machine$double.eps, init = "svd", bias = TRUE,
  verbose = TRUE)

Arguments

A	A list that contains the $J$ blocks of variables from which block components are constructed. It could be eiher the original matrices ($X_1, X_2, ..., X_J$) or the residual matrices ($X_{h1}, X_{h2}, ..., X_{hJ}$).
C	A design matrix that describes the relationships between blocks.
c1	A $1 * J$ vector that contains the value of c1 applied to each block. The L1 bound on a[[j]] is $$ \\|a_{j}\\|_{\ell_1} \leq c_1[j] \sqrt{p_j}.$$ with $p_j$ the number of variables of $X_j$ and with c1[j] between 0 and 1 (larger L1 bound corresponds to less penalization).
scheme	Either "horst", "factorial" or "centroid" (default: centroid).
tol	Stopping value for convergence.
init	Mode of initialization of the SGCCA algorithm. Either by Singular Value Decompostion ("svd") or random ("random") (default: "svd").
bias	Logical value for biaised ($1/n$) or unbiaised ($1/(n-1)$) estimator of the var/cov.
verbose	Reports progress while computing, if verbose = TRUE (default: TRUE).

Value

A list with the following elements:

A $n * J$ matrix of SGCCA block components.

A list of $J$ elements. Each element contains the outer weight vector of each block.

crit

The values of the objective function at each iteration of the iterative procedure.

converg

Speed of convergence of the alogrithm to reach the tolerance.

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).

scheme

The scheme chosen by the user (user specified).

A	A list that contains the \(J\) blocks of variables from which block components are constructed. It could be eiher the original matrices (\(X_1, X_2, ..., X_J\)) or the residual matrices (\(X_{h1}, X_{h2}, ..., X_{hJ}\)).
C	A design matrix that describes the relationships between blocks.
c1	A \(1 * J\) vector that contains the value of c1 applied to each block. The L1 bound on a[[j]] is $$ \\|a_{j}\\|_{\ell_1} \leq c_1[j] \sqrt{p_j}.$$ with \(p_j\) the number of variables of \(X_j\) and with c1[j] between 0 and 1 (larger L1 bound corresponds to less penalization).
scheme	Either "horst", "factorial" or "centroid" (default: centroid).
tol	Stopping value for convergence.
init	Mode of initialization of the SGCCA algorithm. Either by Singular Value Decompostion ("svd") or random ("random") (default: "svd").
bias	Logical value for biaised (\(1/n\)) or unbiaised (\(1/(n-1)\)) estimator of the var/cov.
verbose	Reports progress while computing, if verbose = TRUE (default: TRUE).