Distributions of pattern statistics in sparse Markov models

Martin, Donald E. K.

doi:10.1007/s10463-019-00714-6

Cited by 4 publications

(6 citation statements)

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“…Here, the improvement of Markov chain‐based techniques through the methods of Martin & Noé, and Martin () for reducing the number of Markov chain states was highlighted. The extension of the computation of distributions of statistics of overlapping patterns to sequences satisfying a SMM given in Martin () was presented as well. The SMM model affords enhanced modeling capabilities, allowing the grouping of parameters for conditional probabilities given contexts when it is warranted.…”

Section: Discussionmentioning

confidence: 99%

“…How then would one use an AMC to compute distributions of pattern statistics in SMM? What is outlined here is from Martin (). While knowing the last m observations at each time point means that there is sufficient information to compute the conditional probabilities of an SMM, using the same state space as for m th‐order Markov chains would ignore the possibility that equivalent m ‐tuples could be combined, resulting in a smaller state space.…”

Section: Extension Of Amc‐based Computation To Smmsmentioning

confidence: 99%

“…The computation of the distribution of pattern statistic Z then proceeds through the matrix updates of Equation (8) as before. Conditions for equivalent strings are given in Theorem 3 of Martin (), and are listed below. As the algorithm for setting up the state space of the AMC for the SMM case is very similar to the case of an m th‐order Markovian sequence (the main difference being the possible combination of m ‐tuples in the SMM case), the reader is referred to Martin (, ) for more details and examples of the state space's setup. Strings q and q ′ of length m are equivalent if and only if: aThey lie in the same equal‐conditional probability class γ j and have m ‐tuple suffixes in the same equal‐conditional probability class after concatenating arbitrary strings of lengths

1, \dots, m - l_{γ_{j}} - 1;

bThey have the same updates to statistic Z on the occurrence of all strings of Comp q ∪ Comp q ′ . Strings q and q ′ of length greater than m are equivalent if and only if:

q_{∣ q ∣ - m + 1}^{∣ q ∣} \sim {(q^{'})}_{∣ q' ∣ - m + 1}^{∣ q' ∣},

They have the same updates to statistic Z on the occurrence of all strings of Comp q ∪ Comp q ′ . …”

Section: Extension Of Amc‐based Computation To Smmsmentioning

confidence: 99%

“…The many combinations of prefixes of words representing overlapping spaced seed hits and positions that may or may not be covered can render the number of states of an AMC to be extremely large, and the computation to be much more difficult than for sensitivity. Thus, there are relatively few studies where seed coverage is considered (see Benson & Mak, ; Martin, , ; Martin & Noé, ; Noé, and Noé & Martin, ). The setup of ϒ to compute the coverage distribution for a spaced seed of length k > m is discussed next, following Martin & Noé, , and Martin (, ).…”

Section: Application To Spaced Seed Coveragementioning

confidence: 99%

“…Thus, there are relatively few studies where seed coverage is considered (see Benson & Mak, ; Martin, , ; Martin & Noé, ; Noé, and Noé & Martin, ). The setup of ϒ to compute the coverage distribution for a spaced seed of length k > m is discussed next, following Martin & Noé, , and Martin (, ). The binary sequence X is assumed to be stationary.…”

Section: Application To Spaced Seed Coveragementioning

confidence: 99%

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Computation of exact probabilities associated with overlapping pattern occurrences

Martin

2019

WIREs Computational Stats

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Searching for patterns in data is important because it can lead to the discovery of sequence segments that play a functional role. The complexity of pattern statistics that are used in data analysis and the need of the sampling distribution of those statistics for inference renders efficient computation methods as paramount. This article gives an overview of the main methods used to compute distributions of statistics of overlapping pattern occurrences, specifically, generating functions, correlation functions, the Goulden-Jackson cluster method, recursive equations, and Markov chain embedding. The underlying data sequence will be assumed to be higher-order Markovian, which includes sparse Markov models and variable length Markov chains as special cases. Also considered will be recent developments for extending the computational capabilities of the Markov chain-based method through an algorithm for minimizing the size of the chain's state space, as well as improved data modeling capabilities through sparse Markov models. An application to compute a distribution used as a test statistic in sequence alignment will serve to illustrate the usefulness of the methodology. This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Pattern Recognition Data: Types and Structure > Categorical Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms K E Y W O R D S Auxiliary Markov chain, distribution of a pattern statistic, Markovian sequences, sparse Markov models, VLMC

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Section: Discussionmentioning

confidence: 99%

Section: Extension Of Amc‐based Computation To Smmsmentioning

confidence: 99%

1, \dots, m - l_{γ_{j}} - 1;

bThey have the same updates to statistic Z on the occurrence of all strings of Comp q ∪ Comp q ′ . Strings q and q ′ of length greater than m are equivalent if and only if: