The Distribution of Short Word Match Counts between Markovian Sequences

Abstract

The D 2 statistic, which counts the number of word matches between two given sequences, has long been proposed as a measure of similarity for biological sequences. Much of the mathematically rigorous work carried out to date on the properties of the D 2 statistic has been restricted to the case of 'Bernoulli' sequences composed of identically and independently distributed letters. Here the properties of the distribution of this statistic for the biologically more realistic case of Markovian sequences is studied. The approach is novel in that Markovian dependency is defined for sequences with periodic boundary conditions, and this enables exact analytic formulae for the mean and variance to be derived. The formulae are confirmed using numerical simulations, and asymptotic approximations to the full distribution are tested.