Questions and Comments for Whitening and Coloring Transforms for Multivariate Gaussian Random Variables

A slecture by ECE student Maliha Hossain

Loosely based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.

This is the talk page for the sLecture notes on Whitening and Coloring Transforms for Multivariate Gaussian Random Variables. Please leave me a comment below if you have any questions, if you notice any errors or if you would like to discuss a topic further.

  • Tian's review: It's a great slecture, the explanation is in detail and the logic between the steps is quite clear. Also, the MATLAB example provides me with a certain level of intuition of how the whitening process works. This slecture greatly helps me to understand the transformation between different Gaussian distributions, in both mathematical sense and practical sense.
  • Thanks for your feedback Tian. I implemented some of your suggestions into the slecture. -MH

  • Tian's comment 1: As we can see, the whited data does not have exactly zero mean and an identity covariance matrix, there are some tiny errors, is there a way that we can characterize how good the whiting process is from those errors? I guess the error should be linear to the difference between the largest and smallest eigenvalues of the covariance matrix. This could be a further topic that readers want to explore.
  • Answer: I think the whitened data does have zero mean because we first center our samples about the origin by subtracting the sample mean. Thus, let {$ X_1,...,X_N $} be these centered samples with zero mean. Let {$ W_1,...,W_N $} be the corresponding whitened data set where $ W_i = $A$ X_i $, and A is a known matrix. Then,
$ \mathbf E[W] = \mathbf E[\mathbf AX] = \frac{1}{N} \sum_i^N \mathbf A X_i = \frac{\mathbf A}{N} \sum_i^N X_i = \mathbf{AE}[X] =\mathbf 0 $
So regardless of whether the mean is known or estimated, the sample average of the whitened data is 0.
The whitening transform is a one-to-one transformation so you can invert the transform to recover your original data exactly. If you don't know the parameters, you have to estimate them so the errors are really a result of the estimation techniques and not the whitening transform. -MH

  • Tian's comment 2: The real-world data that we process is usually not exactly multivariate Gaussian distribution, if the original data was not Gaussian distribution, or even very far away from a Gaussian distribution, can we still use this technique and how good the result would be? This could be an interesting question that readers could further explore.
  • Answer: This is a very good question. Yes you can whiten any data with this transform after you've estimated the mean and the covariance. I will mention this in the introduction. Thank you for bringing it to my attention. I can see how the way I've presented it can mislead readers into thinking this applies only in the Gaussian case. -MH

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