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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:
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 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).
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