# Whitening transformation

A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix meaning that they are uncorrelated and all have variance 1.[1] The transformation is called "whitening" because it changes the input vector into a white noise vector.

Several other transformations are closely related to whitening: 1) the decorrelation transform removes only the correlations but leaves variances intact, 2) the standardization transform sets variances to 1 but leaves correlations intact, and 3) a coloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.[2]

## Definition

Suppose ${\displaystyle X}$ is a random (column) vector with non-singular covariance matrix ${\displaystyle M}$ and mean ${\displaystyle 0}$. Then the transformation ${\displaystyle Y=WX}$ with a whitening matrix ${\displaystyle W}$ satisfying the condition ${\displaystyle W^{T}W=M^{-1}}$ yields the whitened random vector ${\displaystyle Y}$ with unit diagonal covariance.

There are infinitely many possible whitening matrices ${\displaystyle W}$ that all satisfy the above condition. Commonly used choices are ${\displaystyle W=M^{-1/2}}$ (Mahalanobis or ZCA whitening), the Cholesky decomposition of ${\displaystyle M^{-1}}$ (Cholesky whitening), or the eigen-system of ${\displaystyle M}$ (PCA whitening).

Kessy et al. (2015) demonstrate that optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of ${\displaystyle X}$ and ${\displaystyle Y}$. For example, the unique optimal whitening transformation achieving maximal component-wise correlation between original ${\displaystyle X}$ and whitened ${\displaystyle Y}$ is produced by the whitening matrix ${\displaystyle W=P^{-1/2}V^{-1/2}}$ where ${\displaystyle P}$ is the correlation matrix and ${\displaystyle V}$ the variance matrix.

## Whitening a data matrix

Whitening a data matrix follows the same transformation as for random variables. An empirical whitening transform is obtained by estimating the covariance (e.g. by maximum likelihood) and subsequently constructing a corresponding estimated whitening matrix (e.g. by Cholesky decomposition).