白化变换

原文地址：http://www.cnblogs.com/kevinGaoblog/archive/2012/06/20/2556335.html

The whitening transformation is a decorrelation method which transforms a set of random variables having the covariance matrix Σ into a set of new random variables whose covariance is aI, where a is a constant and I is the identity matrix. The new random variables are uncorrelated and all have variance 1. The method is called "whitening" because it transforms the input matrix to the form of white noise, which by definition is uncorrelated and has uniform variance. It differs from decorrelation in that the variances are made to be equal, rather than merely making the covariances zero. That is, where decorrelation results in a diagonal covariance matrix, whitening produces a scalar multiple of the identity matrix.

Definition

Define X to be a random vector with covariance matrix Σ and mean 0. The matrix Σ can be written as the outer product of X and X^T

Define Σ^1/2 as

Define the new random vector Y = Σ^-1/2X. The covariance of Y is

Thus, Y is a white random vector.
Based on the fact that the covariance matrix is always positive semi-definite, Σ^1/2 can be derived using eigenvalue decomposition:

Where the matrix Λ^1/2 is a diagonal matrix with each element being square root of the corresponding element in Λ. To show that this equation follows from the prior one, multiply by the transpose: