Feature selection given non-normal variables?

Question

I'm trying to reduce noise (improve separability) among groups in a data set with 26 numerical variables and 10.000 samples. Each sample is a chemical profile, with each variable indicating the quantity of some chemical constituent. Most of the variables are far from normal.

My question is: how can I test whether samples are from separate multivariate distributions?

My intuition is to bootstrap the means for each variable within groups and then perform a permuted MANOVA test (or Hotelling's T-square) to simulate a null distribution. I know this works in the univariate case, as I've done so on microarray data, but I'm rather confused about whether or not I'm still violating assumptions.

The idea is to select samples I'm confident (at some alpha) are from different distributions, then use them as the basis of a clustering function. My data is highly collinear, which may indicate that certain samples have a common origin, and therefore the assigning of novel samples to known clusters would be informative.

Thanks a lot.

Answer 1

As a preprocessing, you may want to consider the usual Box-Cox/Power transformation of your variables: http://en.wikipedia.org/wiki/Power_transform#Box-Cox_transformation This will bring them closer to "gaussianity". I assume you don't know in advance how many groups/clusters are present ? My preference would be to estimate different densities (e.g., mixtures of Gaussians) and choose the best performing based on an in sample criterion (AIC etc.) or using cross validation. Then the Gaussian centroids are the bases for your clusters. Trouble is cluster assignment is probabilistic which may not suit your needs... Good luck !