normalize {pathwayPCA} | R Documentation

## 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 λ 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

[Package pathwayPCA version 1.6.3 Index]