Talk:ECE662 Whitening and Coloring Transforms S14 MH - Rhea

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,

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

• Question

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