normalize: Normalize and reconstruct the eigenvalues of a data matrix...

In gabrielodom/pathwayPCA: Integrative Pathway Analysis with Modern PCA Methodology and Gene Selection

Normalize and reconstruct the eigenvalues of a data matrix for supervised PCA

Description

Normalize the columns of a project matrix. For each eigenvector, swap the signs of the vector elements if the first entry is negative. See "Details" for more information.

Usage

normalize(B, d)

Arguments

B	A projection matrix: often the matrix of the left singular vectors given by the Singular Value Decomposition of a data matrix or Grammian.
d	The number of columns of B to normalize.

Details

This function is designed to reconstruct the original first d left singular vectors of a data matrix from the first d eigenvectors of the Grammian of that data matrix. Basically, after the data matrix has been centred, the left singular vectors of that data matrix and the left singular vectors of the Grammian of that data matrix are equal up to a sign. This function reverses that sign so that the two sets of singular vectors are equal.

Consider the internal workings of the aespca function. This "sign flipping" changes the eigenvectors of xtx into the left singular vectors of scale(X, , center = TRUE, scale = TRUE). Instead of calculating the Grammian, regularising it (by adding some small \lambda value to the diagonal), taking the SVD of the regularized Grammian, and extracting the first d eigenvectors, why don't we just extract the first d singular vectors directly from the scaled data matrix itself? The regularisation effect only inflates the singular- or eigen-values anyway, so it has no effect on the singular vectors in any way. Moreover, the aespca function does not even call for the eigen-values at all, so this whole process is supurfluous. The only wrinkle is adapting the lars.lsa and aespca functions to only operate on the data matrix.

Furthermore, the lars function can take in the full data, instead of just a Grammian. As an enhancement, we should either update our copy of the lars function in lars.lsa, or make a call to the exported lars function. ENHANCEMENT.

Value

A matrix of the eigenvectors or left singular vectors in B transformed to be the left singular values of the original data matrix.

See Also

aespca; lars.lsa; AESPCA_pVals

Examples

  # DO NOT CALL THIS FUNCTION DIRECTLY.
  # Use AESPCA_pVals() instead

gabrielodom/pathwayPCA documentation built on July 10, 2023, 3:32 a.m.