Decorrelation in Statistics: The Mahalanobis Transformation

An image can be compressed if, and only if, its pixels are correlated. This is mentioned many times in Chapter 4, as well as the fact that any image compression method results in decorrelating the pixels. As a result, any transformation that decorrelates values can be the basis for an image compression method. Several such transformations are described in Section 4.4. The topic of this document is the little-known Mahalanobis transformation, used by statisticians to decorrelate two random variables. The discussion starts by defining the square root of a matrix and the concept of a centring matrix. This is followed by a review of the concepts of mean, variance, and covariance, in order to derive relations that are needed later. The transformation and its inverse are then introduced and Mathematica and Matlab codes for it are provided. A proof is included, to show that this transformation really decorrelates the two variables. Matrix Concepts. This short section shows (1) how a matrix A can be raised to any rational power r/s and (2) the definition of the centring matrix H. The spectral decomposition (also known as the Jordan decomposition) theorem claims that any symmetric matrix A can be written as