How to whiten matrix in PCA

Question

I'm working with Python and I've implemented the PCA using this tutorial.

Everything works great, I got the Covariance I did a successful transform, brought it make to the original dimensions not problem.

But how do I perform whitening? I tried dividing the eigenvectors by the eigenvalues:

S, V = numpy.linalg.eig(cov)
V = V / S[:, numpy.newaxis]

and used V to transform the data but this led to weird data values. Could someone please shred some light on this?

Answer 1

Here's a numpy implementation of some Matlab code for matrix whitening I got from here.

import numpy as np

def whiten(X,fudge=1E-18):

   # the matrix X should be observations-by-components

   # get the covariance matrix
   Xcov = np.dot(X.T,X)

   # eigenvalue decomposition of the covariance matrix
   d, V = np.linalg.eigh(Xcov)

   # a fudge factor can be used so that eigenvectors associated with
   # small eigenvalues do not get overamplified.
   D = np.diag(1. / np.sqrt(d+fudge))

   # whitening matrix
   W = np.dot(np.dot(V, D), V.T)

   # multiply by the whitening matrix
   X_white = np.dot(X, W)

   return X_white, W

You can also whiten a matrix using SVD:

def svd_whiten(X):

    U, s, Vt = np.linalg.svd(X, full_matrices=False)

    # U and Vt are the singular matrices, and s contains the singular values.
    # Since the rows of both U and Vt are orthonormal vectors, then U * Vt
    # will be white
    X_white = np.dot(U, Vt)

    return X_white

The second way is a bit slower, but probably more numerically stable.