Probabilistic and Statistical Properties of Words: An Overview

Abstract

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-overlap as well as due to overlap between words. The results can be used to derive approximate, and conservative, confidence intervals for tests.

On the number of word occurrences in a semi-Markov sequence of letters

“…This result generalizes the well known formula for the probability of a word occurrence in a Markovian sequence of letters, see [5].…”

“…Reinert et al [5] have called N (n) count of word w. In this section we shall derive IE(N (n)) and Var (N (n)) under certain conditions.…”

Enhancing the Symbolic Aggregate Approximation Method Using Updated Lookup Tables

“…So these data sets are very diverse. We also tested these datasets using the Euclidean distance, because this distance is widely used in the time series data mining community [7,12].…”

Importance Sampling of Word Patterns in DNA and Protein Sequences

“…Statistical significance of over-representation of these word patterns provides valuable clues to biologists. Consequently, much work has been done on the use of asymptotic limiting distributions to approximate these pvalues (Prum et al, 1995;Reinert et al, 2000;Régnier, 2000;Robin et al, 2002;Huang et al, 2004;Leung et al, 2005;Mitrophanov and Borodovsky, 2006;Pape et al, 2008). However, the approximations may not be accurate for short words or for words consisting of repeats and most theoretical approximations work only in specific settings.…”