8
$\begingroup$

I have a set of files consisting of randomly selected points from a dataset, each file belonging to a particular class. Each row in these files contains the coordinates in n-space of the point. I'd like to compare the distributions in n-space of each of these files - and am inspired by the K-S test for comparing histograms. From what I've read this method doesn't extend well to multivariate data. I had previously used PCA - but all of my variance collapsed into a single noisy dimension and clustering methods were useless.

My question - is there a reason I shouldn't just use an average of the K-S values across the histogram for each of the n-dimensions as a metric for the goodness of fit? Is there a better method for comparing these distributions?

$\endgroup$
3
$\begingroup$

ROOT supports Kolmogorov tests on higher dimensional histograms, and the notes (for the 2D version) suggest that there is a ambiguity--which they deal with by punting: calculate it both ways. I don't know if the code contains anymore details, but the comments sometimes have references to papers and the like.

There are some additional interesting comments in the notes to TH1::KolmogorovTest.

$\endgroup$
3
$\begingroup$

I'd calculate the mean $\overline x$ and the covariance matrix $C$ of the joint data set, and then do a K/S test on the univariate quantity $V(x):=(x-\overline x)^TC^{-1}(x-\overline x)$ evaluated on the parts. If the K/S test give a significant difference between the parts, there is one. If it gives no significant difference, the test is to be regarded as unconclusive.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.