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I have two questions:

Suppose we uncorrelate variables of a multivariate normal distribution using Cholesky transformation. Then:

  1. What is the relation between Mahalanobis distances before and after this transformation?

  2. Can I use this method to calculate the Mahalanobis distance between the means of two multivariate normal distributions with different means and different covariance matrices? In other word, can I calculate the Mahalanobis distance by uncorrelating each distribution and then using Euclidean distance?

Hossein Abedi
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  • There are two "versions" of Mahalanobis distance: 1) distance between two points of a cloud, 2) distance between a point and some cloud's centroid; you didn't refine what you mean. However, in both cases when the cloud is spherical with unit variances (i.e. the cov matrix is identity matrix) mahalanobis distance is equal to euclidean distance. – ttnphns May 07 '13 at 10:34
  • @ttnphns: For the first question, I mean the first version. But for the second question, Non of them! I want to measure the distance between centroids of two clouds. – Hossein Abedi May 07 '13 at 10:43
  • This is a particular case of (2). Mean1 is a point distanced from the cloud with Mean2 and Cov2 [and thus if Cov2 is identity, we're dealing with euclidean distance], while Mean2 is a point distanced from the cloud with Mean1 and Cov1 [and thus if Cov1 is identity, we're dealing with euclidean distance]. So, if both Cov1 and Cov2 is identity, mahalanobis distance b/w the centroids of the 2 clouds is the euclidean distance b/w them. – ttnphns May 07 '13 at 10:53
  • I have mentioned that the Cov1 and Cov2 are not equal. Do you think that using Cholesky transformation to make them (Cov1, Cov2) identity, allows us to use Euclidean distance? – Hossein Abedi May 07 '13 at 11:06
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    Yes, exactly. If $Cov1 \ne Cov2 \ne \bf I$ then "Mahalanobis distance" between two centroids is asymmetric (i.e. two different values, not one value). By the way, it would be nice if you explain or show right in your question (or leave a link) how exactly you do "Cholesky transformation" - because the term is a bit vague. – ttnphns May 07 '13 at 11:20
  • I thought, using this method I will have a metric for measuring the distance between the means of bivariate normal distributions. I've implemented this method using c++ and it seems that the distances obey metric conditions(just for 2000 bivariate normal distributions). Here is the link I have found: http://blogs.sas.com/content/iml/2012/02/08/use-the-cholesky-transformation-to-correlate-and-uncorrelate-variables/ – Hossein Abedi May 07 '13 at 11:25
  • 1) For a related discussion, see http://stats.stackexchange.com/questions/7912/mahalanobis-distance-between-two-bivariate-distributions-with-different-covarian?rq=1 2) I don't think you can use two different transformations and then make sense of the results. You might consider whether using the pooled covariance would meet your needs. – Rick May 07 '13 at 16:40

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