mabounassif - 5 months ago 157
Python Question

# How to whiten matrix in PCA

I'm working with python and I've implemented the PCA using the following tutorial http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf

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?

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)

# 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.

Source (Stackoverflow)