# Statistical Identification of Uniformly Mutated Segments within Repeats

## Abstract

Given a long string of characters from a constant size (w.l.o.g. binary) alphabet we present an algorithm to determine whether its characters have been generated by a single i.i.d. random source. More specifically, consider all possible *k*-coin models for generating a binary string *S*, where each bit of *S* is generated via an independent toss of one of the *k* coins in the model. The choice of which coin to toss is decided by a random walk on the set of coins where the probability of a coin change is much lower than the probability of using the same coin repeatedly. We present a statistical test procedure which, for any given *S*, determines whether the *a posteriori* probability for *k* = 1 is higher than for any other *k* > 1. Our algorithm runs in time *O*(*l* ^{4} log *l*), where *l* is the length of *S*, through a dynamic programming approach which exploits the convexity of the *a posteriori* probability for *k*.

The problem we consider arises from two critical applications in analyzing long alignments between pairs of genomic sequences. A high alignment score between two DNA sequences usually indicates an evolutionary relationship, i.e. that the sequences have been generated as a result of one or more copy events followed by random point mutations. Such sequences may include functional regions (e.g. exons) as well as nonfunctional ones (e.g. introns). Functional regions with critical importance exhibit much lower mutation rates than non-functional DNA (or DNA

### Keywords

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