Poisson approximation for search of rare words in DNA sequences

TL;DR

This paper introduces a new psi-mixing method for accurately approximating the distribution of rare word occurrences in DNA sequences modeled by Markov chains, improving tail-bound estimates over traditional methods.

Contribution

The paper develops a local error bound approach using psi-mixing, providing more precise thresholds for over- or under-represented words in biological sequences compared to Chen-Stein methods.

Findings

Psi-mixing method yields better tail-bound accuracy.

New thresholds for word over- or under-representation.

Software PANOW implements the method.

Abstract

Using recent results on the occurrence times of a string of symbols in a stochastic process with mixing properties, we present a new method for the search of rare words in biological sequences generally modelled by a Markov chain. We obtain a bound on the error between the distribution of the number of occurrences of a word in a sequence (under a Markov model) and its Poisson approximation. A global bound is already given by a Chen-Stein method. Our approach, the psi-mixing method, gives local bounds. Since we only need the error in the tails of distribution, the global uniform bound of Chen-Stein is too large and it is a better way to consider local bounds. We search for two thresholds on the number of occurrences from which we can regard the studied word as an over-represented or an under-represented one. A biological role is suggested for these over- or under-represented words. Our method gives such thresholds for a panel of words much broader than the Chen-Stein method. Comparing the methods, we observe a better accuracy for the psi-mixing method for the bound of the tails of distribution. We also present the software PANOW (available at http://stat.genopole.cnrs.fr/software/panowdir/) dedicated to the computation of the error term and the thresholds for a studied word.