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 each 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,
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 ${\displaystyle \Sigma }$ and mean ${\displaystyle 0}$. Then the transformation ${\displaystyle Y=WX}$ with a whitening matrix ${\displaystyle W}$ satisfying the condition ${\displaystyle W^{\mathrm {T} }W=\Sigma ^{-1}}$ yields the whitened random vector ${\displaystyle Y}$ with unit diagonal covariance.

If ${\displaystyle X}$ has non-zero mean ${\displaystyle \mu }$, then whitening can be performed by ${\displaystyle Y=W(X-\mu )}$.

There are infinitely many possible whitening matrices ${\displaystyle W}$ that all satisfy the above condition. Commonly used choices are ${\displaystyle W=\Sigma ^{-1/2}}$ (Mahalanobis or ZCA whitening), ${\displaystyle W=L^{T}}$ where ${\displaystyle L}$ is the Cholesky decomposition of ${\displaystyle \Sigma ^{-1}}$ (Cholesky whitening),^[3] or the eigen-system of ${\displaystyle \Sigma }$ (PCA whitening).^[4]

Optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of ${\displaystyle X}$ and ${\displaystyle Y}$.^[3] 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 diagonal 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).

High-dimensional whitening

This modality is a generalization of the pre-whitening procedure extended to more general spaces where ${\displaystyle X}$ is usually assumed to be a random function or other random objects in a Hilbert space ${\displaystyle H}$. One of the main issues of extending whitening to infinite dimensions is that the covariance operator has an unbounded inverse in ${\displaystyle H}$. Nevertheless, if one assumes that Picard condition holds for ${\displaystyle X}$ in the range space of the covariance operator, whitening becomes possible.^[5] A whitening operator can be then defined from the factorization of the Moore–Penrose inverse of the covariance operator, which has effective mapping on Karhunen–Loève type expansions of ${\displaystyle X}$. The advantage of these whitening transformations is that they can be optimized according to the underlying topological properties of the data, thus producing more robust whitening representations. High-dimensional features of the data can be exploited through kernel regressors or basis function systems.^[6]

R implementation

An implementation of several whitening procedures in R, including ZCA-whitening and PCA whitening but also CCA whitening, is available in the "whitening" R package ^[7] published on CRAN. The R package "pfica"^[8] allows the computation of high-dimensional whitening representations using basis function systems (B-splines, Fourier basis, etc.).

See also

• Decorrelation
• Principal component analysis
• Weighted least squares
• Canonical correlation
• Mahalanobis distance (is Euclidean after W. transformation).