Drifting Markov Models with Polynomial Drift and Applications to DNA Sequences

Abstract

In this article, we introduce the drifting Markov models (DMMs) which are inhomogeneous Markov models designed for modeling the heterogeneities of sequences (in our case DNA or protein sequences) in a more flexible way than homogeneous Markov chains or even hidden Markov models (HMMs). We focus here on the polynomial drift: the transition matrix varies in a polynomial way. To show the reliability of our models on DNA, we exhibit high similarities between the probability distributions of nucleotides obtained by our models and the frequencies of these nucleotides computed by using a sliding window. In a further step, these DMMs can be used as the states of an HMM: on each of its segments, the observed process can be modeled by a drifting Markov model. Search of rare words in DNA sequences remains possible with DMMs and according to the fits provided, DMMs turn out to be a powerful tool for this purpose. The software is available on request from the author. It will soon be integrated on seq++ library (http://stat.genopole.cnrs.fr/seqpp/).

Reliability and Survival Analysis for Drifting Markov Models: Modeling and Estimation

“…The proof is straightforward and we omit it. In a most general case, it is given in Lemma 2 2 Note that one could consider a k order linear drifting Markov chain, k ∈ N * , by allowing a dependence of order k in Equation (1) and letting Π 0 et Π 1 be the transition probabilities of Markov chains of order k. Note also that the linear drifting can be generalized to polynomial drifting of a certain degree d. See [37] for more details on these points.…”

“…The purpose of this section is twofold: first, continuing the direction developed in [37], we will consider different types of data for which the estimators of the characteristics of a drifting Markov chain will be derived. Second, we will estimate the associated reliability indicators.…”

“…As in [37], we minimize a quadratic form of the different parameters which is the sum of "prediction errors". For each sequence h, at each position t in the sequence, knowing X t−1 = u, we want Π t n (u, v) to be the nearest possible to 1 if X t = v or the nearest possible to 0 if X t = v. We minimize the sum of squares of the errors…”

“…A possible solution is to consider a non-homogeneity that is "smooth", controlled, of a known shape. An example of this type in a Markov framework consists in the so called drifting Markov chains, introduced in [37]. For these processes, the Markov transition matrix is a linear (polynomial) function/mixture of two (several) Markov transition matrices.…”

Moments of the Count of a Regular Expression in a Heterogeneous Random Sequence

“…Most of this research mainly focus on homogeneous models because this assumption usually allows to obtain simpler and computationally more efficient formulas. Heterogenous models are nevertheless often encountered, either directly as continuous process [9,49] or, more often, as discrete process though hidden Markov models -HMMs - [42,15,47]. In the context of HMMs the forward/backward algorithm [4,18] allows computing efficiently the posterior distribution of the hidden states given the observations which is a heterogenous Markov model.…”