Testing statistical hypothesis on Random Trees

In this paper we address the problem of identifying differences between populations of trees. Besides the theoretical relevance of this problem, we are interested in testing if trees characterizing protein sequences from different families constitute samples of significantly different distributions. In this context, trees are obtained by modelling protein sequences as Variable Length Markov Chains (VLMC), estimating the relevant motifs that are sufficient to predict the next amino acid in the sequence. We assign to each protein family an underlying VLMC model, which induces a distribution on the space of all trees. Our goal is to test if two (or more) populations of trees comes from different distributions. Our approach is based on a hypothesis test proposed recently by Balding et al (2004) (BFFS–test), which involves a Kolmogorov type statistics that roughly speaking, maximizes the difference between the expected distance structure that characterize the samples of the populations. A naive approach to calculate effectively the test statistic is quite difficult, since it is based on a supremo defined over the space of all trees, which grows exponentially fast. We show how to transform this problem into a max-flow over a network which can be solved using a Ford Fulkerson algorithm in polynomial time on the maximal number of nodes of the random tree. We also describe conditions that imply the characterization of the measure by the marginal distributions of each node (occupancy node probabilities) of the random tree, which validate the use of the BFFS–test for measure discrimination. We study the performance of the test via simulations on Galton-Watson processes. We apply the test to 10 randomly chosen samples of protein families from the Pfam database, and show that the underlying distributions over the set of trees are significantly different, confirming the coherence of the selected families. This is a novel mathematical approach to validate if automatic clustering results are indeed coherent families.