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. The transformation is called "whitening" because it changes the input vector into a white noise vector.
Several other transformations are closely related to whitening:
- the decorrelation transform removes only the correlations but leaves variances intact,
- the standardization transform sets variances to 1 but leaves correlations intact,
- a coloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.
Suppose is a random (column) vector with non-singular covariance matrix and mean . Then the transformation with a whitening matrix satisfying the condition yields the whitened random vector with unit diagonal covariance.
There are infinitely many possible whitening matrices that all satisfy the above condition. Commonly used choices are (Mahalanobis or ZCA whitening), the Cholesky decomposition of (Cholesky whitening), or the eigen-system of (PCA whitening).
Optimal whitening transforms can be singled out by investigating the cross-covariance and cross-correlation of and . For example, the unique optimal whitening transformation achieving maximal component-wise correlation between original and whitened is produced by the whitening matrix where is the correlation matrix and 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).
Example of ZCA whitening
An example of Python implementation of ZCA whitening 
import numpy as np def zca_whitening_matrix(X): """ Function to compute ZCA whitening matrix (aka Mahalanobis whitening). INPUT: X: [M x N] matrix. Rows: Variables Columns: Observations OUTPUT: ZCAMatrix: [M x M] matrix """ # Covariance matrix [column-wise variables]: Sigma = (X-mu)' * (X-mu) / N sigma = np.cov(X, rowvar=True) # [M x M] # Singular Value Decomposition. X = U * np.diag(S) * V U,S,V = np.linalg.svd(sigma) # U: [M x M] eigenvectors of sigma. # S: [M x 1] eigenvalues of sigma. # V: [M x M] transpose of U # Whitening constant: prevents division by zero epsilon = 1e-5 # ZCA Whitening matrix: U * Lambda * U' ZCAMatrix = np.dot(U, np.dot(np.diag(1.0/np.sqrt(S + epsilon)), U.T)) # [M x M] return ZCAMatrix
- Koivunen, A.C.; Kostinski, A.B. (1999). "The Feasibility of Data Whitening to Improve Performance of Weather Radar". American Meteorological Society. doi:10.1175/1520-0450(1999)038<0741:TFODWT>2.0.CO;2.
- Hossain, Miliha. "Whitening and Coloring Transforms for Multivariate Gaussian Random Variables". Project Rhea. Retrieved 21 March 2016.
- Friedman, J. (1987). "Exploratory Projection Pursuit". ISSN 0162-1459. JSTOR 2289161.
- Agnan Kessy, Alex Lewin & Korbinian Strimmer (2018) Optimal Whitening and Decorrelation, The American Statistician, DOI: 10.1080/00031305.2016.1277159
- The ZCA whitening transformation. Appendix A of Learning Multiple Layers of Features from Tiny Images by A. Krizhevsky.