Multiple pattern matching: a Markov chain approach

Lladser, Manuel E.; Betterton, Meredith D.; Knight, Rob

doi:10.1007/s00285-007-0109-3

Cited by 27 publications

(32 citation statements)

References 82 publications

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“…(16) (16) is undefined and the constant C should be computed as −P(z 1 )/Q (z 1 ), where P(z) and Q(z) are the respective numerator and denominator of S γ (z) in expression (15). Once again, the analytic combinatorics estimates are accurate.…”

Section: Substitutions and Deletionsmentioning

confidence: 99%

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Analytic Combinatorics for Computing Seeding Probabilities

Filion

2018

Algorithms

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Seeding heuristics are the most widely used strategies to speed up sequence alignment in bioinformatics. Such strategies are most successful if they are calibrated, so that the speed-versus-accuracy trade-off can be properly tuned. In the widely used case of read mapping, it has been so far impossible to predict the success rate of competing seeding strategies for lack of a theoretical framework. Here, we present an approach to estimate such quantities based on the theory of analytic combinatorics. The strategy is to specify a combinatorial construction of reads where the seeding heuristic fails, translate this specification into a generating function using formal rules, and finally extract the probabilities of interest from the singularities of the generating function. The generating function can also be used to set up a simple recurrence to compute the probabilities with greater precision. We use this approach to construct simple estimators of the success rate of the seeding heuristic under different types of sequencing errors, and we show that the estimates are accurate in practical situations. More generally, this work shows novel strategies based on analytic combinatorics to compute probabilities of interest in bioinformatics.

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Section: Substitutions and Deletionsmentioning

confidence: 99%

“…For instance, see [15] for a thorough review of finite automata, their application to pattern matching in biological sequences and the use of generating functions to compute certain probabilities of occurrence. In [16], Fu and Koutras study the distribution of runs in Bernoulli trials using Markov chain embeddings.…”

Section: Related Workmentioning

confidence: 99%

Analytic Combinatorics for Computing Seeding Probabilities

Filion

2018

Algorithms

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“…See also Lladser et al [16]. Approximations using the normal distribution and the Poisson distribution are in Schbath [31], Nicodème et al [21], Reinert et al [28] and Roquain and Schbath [30].…”

Section: P Pudlomentioning

confidence: 99%

Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

Pudlo

2010

ESAIM: PS

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Abstract.To establish lists of words with unexpected frequencies in long sequences, for instance in a molecular biology context, one needs to quantify the exceptionality of families of word frequencies in random sequences. To this aim, we study large deviation probabilities of multidimensional word counts for Markov and hidden Markov models. More specifically, we compute local Edgeworth expansions of arbitrary degrees for multivariate partial sums of lattice valued functionals of finite Markov chains. This yields sharp approximations of the associated large deviation probabilities. We also provide detailed simulations. These exhibit in particular previously unreported periodic oscillations, for which we provide theoretical explanations.Mathematics Subject Classification. 60J10, 60F10, 60J55, 92D20, 60F05.

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“…In [17] the embedding of X into a deterministic finite automaton (DFA) G = (V, A, f, q, T ) that recognizes a regular pattern L is defined as the infinite sequence of V -valued random variables…”

Section: Introductionmentioning

confidence: 99%

Markovian embeddings of general random strings

Lladser

2008

2008 Proceedings of the Fifth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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Let A be a finite set and X a sequence of A-valued random variables. We do not assume any particular correlation structure between these random variables; in particular, X may be a non-Markovian sequence. An adapted embedding of X is a sequence of the form R(X1), R(X1, X2), R(X1, X2, X3), etc where R is a transformation defined over finite length sequences. In this extended abstract we characterize a wide class of adapted embeddings of X that result in a first-order homogeneous Markov chain. We show that any transformation R has a unique coarsest refinement R in this class such that R (X1), R (X1, X2), R (X1, X2, X3), etc is Markovian. (By refinement we mean that R (u) = R (v) implies R(u) = R(v), and by coarsest refinement we mean that R is a deterministic function of any other refinement of R in our class of transformations.) We propose a specific embedding that we denote as R X which is particularly amenable for analyzing the occurrence of patterns described by regular expressions in X. A toy example of a non-Markovian sequence of 0's and 1's is analyzed thoroughly: discrete asymptotic distributions are established for the number of occurrences of a certain regular pattern in X1, ..., Xn as n → ∞ whereas a Gaussian asymptotic distribution is shown to apply for another regular pattern.

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Multiple pattern matching: a Markov chain approach

Cited by 27 publications

References 82 publications

Analytic Combinatorics for Computing Seeding Probabilities

Analytic Combinatorics for Computing Seeding Probabilities

Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

Markovian embeddings of general random strings

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