Sparse Markov Chains for Sequence Data

Jääskinen, Väinö; Xiong, Jie; Corander, Jukka; Koski, Timo

doi:10.1111/sjos.12053

Cited by 28 publications

(26 citation statements)

References 32 publications

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“…In contrast to approaches that incorporate temporal layers in methods for static network descriptions, we build our approach on describing the actual dynamics. We first formulate a generative model of discrete temporal processes based on arbitraryorder Markov chains with community structure [24][25][26][27]. Since our model generates event sequences, it does not aggregate data in time windows [13][14][15]17], and, other than the Markov model assumption, needs no a priori imposed timescales.…”

Section: Introductionmentioning

confidence: 99%

Modelling sequences and temporal networks with dynamic community structures

2017

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In evolving complex systems such as air traffic and social organisations, collective effects emerge from their many components’ dynamic interactions. While the dynamic interactions can be represented by temporal networks with nodes and links that change over time, they remain highly complex. It is therefore often necessary to use methods that extract the temporal networks’ large-scale dynamic community structure. However, such methods are subject to overfitting or suffer from effects of arbitrary, a priori-imposed timescales, which should instead be extracted from data. Here we simultaneously address both problems and develop a principled data-driven method that determines relevant timescales and identifies patterns of dynamics that take place on networks, as well as shape the networks themselves. We base our method on an arbitrary-order Markov chain model with community structure, and develop a nonparametric Bayesian inference framework that identifies the simplest such model that can explain temporal interaction data.

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Section: Introductionmentioning

confidence: 99%

Modelling sequences and temporal networks with dynamic community structures

2017

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“…Allowing the possibility of keeping only one parameter for the equal‐conditional probability classes affords both parsimony and flexibility, giving a better trade‐off of variance that arises from having too many parameters and bias from having a model that does not fit the data due to contexts that are too short. A VLMC is a special case of an SMM where in each equal‐conditional probability class γ l , the m ‐tuples have a common suffix (Jääskinen, Xiong, Koski, & Corander, ). Thus if

l_{γ_{j}}

denotes the length of the longest common suffix of strings of probability equivalence class γ j , then a VLMC has

l_{γ_{j}} > 0 \forall j .

Figure depicts the context tree of a VLMC with Σ = {0, 1} and m = 3.…”

Section: Extension Of Amc‐based Computation To Smmsmentioning

confidence: 99%

“…SMM (Xiong, Jääskinen, & Corander, ) have also been called minimal Markov models (García & González‐López, ), sparse Markov chains (Jääskinen et al, ), and partition Markov models (Fernández, García, & González‐López, ; García & González‐López, ).…”

Section: Extension Of Amc‐based Computation To Smmsmentioning

confidence: 99%

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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“…In a more general context, there are also sequences of several words, which are equivalent in that sense. See, for instance, [8,9]. For this kind of data, a model should retrieve and use the redundancy to improve the quality of the estimate.…”

Section: Introductionmentioning

confidence: 99%

Consistent Estimation of Partition Markov Models

Garćıa

González‐López

2017

Entropy

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Abstract:The Partition Markov Model characterizes the process by a partition L of the state space, where the elements in each part of L share the same transition probability to an arbitrary element in the alphabet. This model aims to answer the following questions: what is the minimal number of parameters needed to specify a Markov chain and how to estimate these parameters. In order to answer these questions, we build a consistent strategy for model selection which consist of: giving a size n realization of the process, finding a model within the Partition Markov class, with a minimal number of parts to represent the process law. From the strategy, we derive a measure that establishes a metric in the state space. In addition, we show that if the law of the process is Markovian, then, eventually, when n goes to infinity, L will be retrieved. We show an application to model internet navigation patterns.

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Sparse Markov Chains for Sequence Data

Cited by 28 publications

References 32 publications

Modelling sequences and temporal networks with dynamic community structures

Modelling sequences and temporal networks with dynamic community structures

Computation of exact probabilities associated with overlapping pattern occurrences

Consistent Estimation of Partition Markov Models

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