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Whitening of a Multivariate Gaussian Random Vector

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Whitening is the process of transforming a random vector with a known covariance matrix into a new random vector with a covariance matrix equal to the identity matrix. Accordingly, the elements of the new random vector have variance 1 and are uncorrelated. If the original random vector is a multidimensional Gaussian, then the elements of the new random vector have variance 1 and are statistically independent.

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In this Demonstration, samples of a zero mean of the three-dimensional random vector before whitening are generated using a moving-average process. The three-dimensional random vector is , where , with and statistically independent.

The covariance matrix for is then:

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Three-dimensional scatter plots of the samples of the correlated random vector and the corresponding whitened random vector are displayed in the graphic along with the whitening matrix as you vary the parameters of the covariance matrix for . This Demonstration highlights the result that the three-dimensional scatter plot before whitening is an ellipsoidal cloud while the three-dimensional scatter plot after whitening is a spherical cloud.

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Contributed by: Victor S. Frost (August 27)
(University of Kansas)
Open content licensed under CC BY-NC-SA

Details

Let be the covariance matrix of a random vector . Let be a diagonal matrix having the eigenvalues of as its diagonal elements. Define such that the columns of are the eigenvectors of the covariance matrix . Then the whitening matrix is . The whitened data is obtained from the samples of the correlated random vector using .

References

[1] S. S. Haykin, Adaptive Filter Theory, Englewood Cliffs, NJ: Prentice-Hall, 1986.

[2] K. S. Shanmugan and A. M. Breipohl, Random Signals: Detection, Estimation and Data Analysis, New York: Wiley, 1988.

[3] R. W. Picard. "Topic: Decorrelating and Then Whitening Data." (Aug 11, 2023) courses.media.mit.edu/2010fall/mas622j/whiten.pdf.

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