Optimal Whitening and Decorrelation |
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Authors: | Agnan Kessy Alex Lewin Korbinian Strimmer |
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Institution: | 1. Statistics Section, Department of Mathematics, Imperial College London, South Kensington Campus, London, United Kingdom;2. Department of Mathematics, Brunel University London, Kingstone Lane, Uxbridge, United Kingdom;3. Epidemiology and Biostatistics, School of Public Health, Imperial College London, Norfolk Place, London, United Kingdom |
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Abstract: | Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example, based on principal component analysis (PCA), Cholesky matrix decomposition, and zero-phase component analysis (ZCA), among others. Here, we provide an overview of the underlying theory and discuss five natural whitening procedures. Subsequently, we demonstrate that investigating the cross-covariance and the cross-correlation matrix between sphered and original variables allows to break the rotational invariance and to identify optimal whitening transformations. As a result we recommend two particular approaches: ZCA-cor whitening to produce sphered variables that are maximally similar to the original variables, and PCA-cor whitening to obtain sphered variables that maximally compress the original variables. |
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Keywords: | CAR score CAT score Cholesky decomposition Decorrelation Principal components analysis Whitening ZCA-Mahalanobis transformation |
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