In the following, an overview is given on statistical and probabilistic properties of words, as occurring in the analysis of biological sequences. Counts of occurrence, counts of clumps, and renewal counts are distinguished, and exact distributions as well as normal approximations, Poisson process approximations, and compound Poisson approximations are derived. Here, a sequence is modelled as a stationary ergodic Markov chain; a test for determining the appropriate order of the Markov chain is described. The convergence results take the error made by estimating the Markovian transition probabilities into account. The main tools involved are moment generating functions, martingales, Stein’s method, and the Chen-Stein method. Similar results are given for occurrences of multiple patterns, and, as an example, the problem of unique recoverability of a sequence from SBH chip data is discussed. Special emphasis lies on disentangling the complicated dependence structure between word occurrences, due to self-over...

Probabilistic and Statistical Properties of Words: An Overview

Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecu- lar sequence analysis. In this paper, we review the ways in which Stein’s method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration.We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein’s method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods.

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Compound Poisson approximation: a user's guide

Consider a sequence of Bernoulli trials with success probability p, and let N,,k denote the number of success runs of length k ?_i 2 among the first n trials. The Stein-Chen method is employed to obtain a total variation upper bound for the rate of convergence of N,,k to a Poisson random variable under the standard condition

Poisson approximations for runs and patterns of rare events

Consider a stationary Markov chain {X j } j=1 n with state space consisting of the ξ-letter alphabet set Λ = {a 1 , a 2 ,...,a ξ }. We study the variables M=M(n,k) and N=N(n,k), defined, respectively, as the number of overlapping and non-overlapping occurrences of a fixed periodic k-letter word, and use the Stein-Chen method to obtain compound Poisson approximations for their distribution.

Compound Poisson approximations for word patterns under Markovian hypotheses

How do you distinguish a cat from a dog by their DNA? Did Shakespeare really write all of his plays? Pattern matching techniques can offer answers to these questions and to many others, from molecular biology, to telecommunications, to classifying Twitter content. This book for researchers and graduate students demonstrates the probabilistic approach to pattern matching, which predicts the performance of pattern matching algorithms with very high precision using analytic combinatorics and analytic information theory. Part I compiles known results of pattern matching problems via analytic methods. Part II focuses on applications to various data structures on words, such as digital trees, suffix trees, string complexity and string-based data compression. The authors use results and techniques from Part I and also introduce new methodology such as the Mellin transform and analytic depoissonization. More than 100 end-of-chapter problems help the reader to make the link between theory and practice.

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Analytic Pattern Matching

A sequence of independent experiments is performed, each one producing a letter from a given alphabet. We study the number of overlapping appearances of a given pattern of letters and we prove that, under quite general conditions, the number of overlapping appearances of long patterns is approximately distributed according to a Polya-Aeppli distribution.

O. Chrysaphinou

Papers

A limit theorem on the number of overlapping appearances of a pattern in a sequence of independent trials