Variable Length Markov Chains Research Articles

Optimal segmentation using tree models

Sequence data are abundant in application areas such as computational biology, environmental sciences, and telecommunication. Many real-life sequences have a strong segmental structure, with segments of different complexities. In this paper we study the description of sequence segments using variable length Markov chains (VLMCs), also known as tree models. We discover the segment boundaries of a sequence and at the same time we obtain a VLMC for each segment. Such a context tree contains the probability distribution vectors that capture the essential features of the corresponding segment. We use the Bayesian information criterion (BIC) and the Krichevsky-Trofimov probability (KT) to select the number of segments of a sequence. On DNA data the method selects segments that closely correspond to the annotated regions of the genes.

Association Mapping via Regularized Regression Analysis of Single-Nucleotide–Polymorphism Haplotypes in Variable-Sized Sliding Windows

Large-scale haplotype association analysis, especially at the whole-genome level, is still a very challenging task without an optimal solution. In this study, we propose a new approach for haplotype association analysis that is based on a variable-sized sliding-window framework and employs regularized regression analysis to tackle the problem of multiple degrees of freedom in the haplotype test. Our method can handle a large number of haplotypes in association analyses more efficiently and effectively than do currently available approaches. We implement a procedure in which the maximum size of a sliding window is determined by local haplotype diversity and sample size, an attractive feature for large-scale haplotype analyses, such as a whole-genome scan, in which linkage disequilibrium patterns are expected to vary widely. We compare the performance of our method with that of three other methods--a test based on a single-nucleotide polymorphism, a cladistic analysis of haplotypes, and variable-length Markov chains--with use of both simulated and experimental data. By analyzing data sets simulated under different disease models, we demonstrate that our method consistently outperforms the other three methods, especially when the region under study has high haplotype diversity. Built on the regression analysis framework, our method can incorporate other risk-factor information into haplotype-based association analysis, which is becoming an increasingly necessary step for studying common disorders to which both genetic and environmental risk factors contribute.

Evaluating Variable-Length Markov Chain Models for Analysis of User Web Navigation Sessions

Markov models have been widely used to represent and analyze user Web navigation data. In previous work, we have proposed a method to dynamically extend the order of a Markov chain model and a complimentary method for assessing the predictive power of such a variable-length Markov chain. Herein, we review these two methods and propose a novel method for measuring the ability of a variable-length Markov model to summarize user Web navigation sessions up to a given length. Although the summarization ability of a model is important to enable the identification of user navigation patterns, the ability to make predictions is important in order to foresee the next link choice of a user after following a given trail so as, for example, to personalize a Web site. We present an extensive experimental evaluation providing strong evidence that prediction accuracy increases linearly with summarization ability

Markov approximation and consistent estimation of unbounded probabilistic suffix trees

We consider infinite order chains whose transition probabilities depend on a finite suffix of the past. These suffixes are of variable length and the set of the lengths of all suffix is unbounded. We assume that the probability transitions for each of these suffixes are continuous with exponential decay rate. For these chains, we prove the weak consistency of a modification of Rissanen's algorithm Context which estimates the length of the suffix needed to predict the next symbol, given a finite sample. This generalizes to the unbounded case the original result proved for variable length Markov chains in the seminal paper Rissanen (1983). Our basic tool is the canonical Markov approximation which enables to approximate the chain of infinite order by a sequence of variable length Markov chains of increasing order. Our proof is constructive and we present an explicit decreasing upper bound for the probability of wrong estimation of the length of the current suffix.

Testing the Predictive Power of Variable History Web Usage

We present two methods for testing the predictive power of a variable length Markov chain induced from a collection of user web navigation sessions. The collection of sessions is split into a training and a test set. The first method uses a χ2 statistical test to measure the significance of the distance between the distribution of the probabilities assigned to the test trails by a Markov model build from the full collection of sessions and a model built from the training set. The statistical test measures the ability of the model to generalise its predictions to the unseen sessions from the test set. The second method evaluates the model ability to predict the last page of a navigation session based on the preceding pages viewed by recording the mean absolute error of the rank of the last occurring page among the predictions provided by the model. Experimental results conducted on both real and random data sets are reported and the results show that in most cases a second-order model is able to capture sufficient history to predict the next link choice with high accuracy.

Multilocus Association Mapping Using Variable-Length Markov Chains

I propose a new method for association-based gene mapping that makes powerful use of multilocus data, is computationally efficient, and is straightforward to apply over large genomic regions. The approach is based on the fitting of variable-length Markov chain models, which automatically adapt to the degree of linkage disequilibrium (LD) between markers to create a parsimonious model for the LD structure. Edges of the fitted graph are tested for association with trait status. This approach can be thought of as haplotype testing with sophisticated windowing that accounts for extent of LD to reduce degrees of freedom and number of tests while maximizing information. I present analyses of two published data sets that show that this approach can have better power than single-marker tests or sliding-window haplotypic tests.

A generalization of the PST algorithm: modeling the sparse nature of protein sequences

A central problem in genomics is to determine the function of a protein using the information contained in its amino acid sequence. Variable length Markov chains (VLMC) are a promising class of models that can effectively classify proteins into families and they can be estimated in linear time and space. We introduce a new algorithm, called Sparse Probabilistic Suffix Trees (SPST), that identifies equivalence between the contexts of a VLMC. We show that, in many cases, the identification of these equivalence can improve the classification rate of the classical Probabilistic Suffix Trees (PST) algorithm. We also show that better classification can be achieved by identifying representative fingerprints in the amino acid chains, and this variation in the SPST algorithm is called F-SPST.

A New Order Estimator for Fixed and Variable Length Markov Models with Applications to DNA Sequence Similarity

Recently Peres and Shields discovered a new method for estimating the order of a stationary fixed order Markov chain. They showed that the estimator is consistent by proving a threshold result. While this threshold is valid asymptotically in the limit, it is not very useful for DNA sequence analysis where data sizes are moderate. In this paper we give a novel interpretation of the Peres-Shields estimator as a sharp transition phenomenon. This yields a precise and powerful estimator that quickly identifies the core dependencies in data. We show that it compares favorably to other estimators, especially in the presence of variable dependencies. Motivated by this last point, we extend the Peres-Shields estimator to Variable Length Markov Chains. We compare it to a well-established estimator and show that it is superior in terms of the predictive likelihood. We give an application to the problem of detecting DNA sequence similarity in plasmids.

Variable Length Markov Chains: Methodology, Computing, and Software

This article presents a tutorial and new, publicly available computational tools for variable length Markov chains (VLMC). VLMCs are Markov chains with the additional attractive structure that their memories depend on a variable number of lagged values, depending on what the actual past (the lagged values) looks like. They build a very flexible class of tree-structured models for categorical time series. Fitting VLMCs from data is a nontrivial computational task. We provide an efficient implementation of the so-called context algorithm which requires only O(n log(n)) operations. The implementation, which is publicly available, includes additional important new features and options: diagnostics, goodness of fit, simulation and bootstrap, residuals, and tuning the context algorithm. Our tutorial is presented with a version in R which is available from the Comprehensive R Archive Network (CRAN). The exposition is self-contained, gives rigorous and partly new mathematical descriptions, and is illustrated by analyzing a DNA sequence from the Epstein-Barr virus.

Estimation of General Stationary Processes by Variable Length Markov Chains

Abstract.We develop new results about a sieve methodology for the estimation of minimal state spaces and probability laws in the class of stationary processes defined on finite categorical spaces. Using a sieve approximation with variable length Markov chains of increasing order, we show that an adapted version of the Context algorithm yields asymptotically correct estimates for the minimal state space and for the underlying probability distribution. As a side product, the method of sieves yields a nice graphical tree representation for the potentially infinite dimensional minimal state space of the data generating process, which is very useful for exploration of the memory.

Sieve Bootstrap With Variable-Length Markov Chains for Stationary Categorical Time Series

A bootstrap for stationary categorical time series based on the method of sieves is studied here. The data-generating process is approximated by the so-called variable-length Markov chain (VLMC), a flexible class of Markov models that allows for parsimonious structure. Then the resampling is given by simulating from the fitted model. It is shown that for a whole class of stationary categorical time series that is more general than VLMC, the VLMC sieve has faster rate of convergence for variance estimation than the more general block bootstrap. Results are illustrated from a theoretical and empirical perspective. For the latter, a real data application about (in-) homogeneity classification of a DNA strand is also presented. Finally, the VLMC sieve scheme enjoys an implementational advantage of using the plug-in rule for bootstrapping a statistical procedure, which generally is not the case for the block method.

Model Selection for Variable Length Markov Chains and Tuning the Context Algorithm

We consider the model selection problem in the class of stationary variable length Markov chains (VLMC) on a finite space. The processes in this class are still Markovian of high order, but with memory of variable length. Various aims in selecting a VLMC can be formalized with different non-equivalent risks, such as final prediction error or expected Kullback-Leibler information. We consider the asymptotic behavior of different risk functions and show how they can be generally estimated with the same resampling strategy. Such estimated risks then yield new model selection criteria. In particular, we obtain a data-driven tuning of Rissanen's tree structured context algorithm which is a computationally feasible procedure for selection and estimation of a VLMC.

Dynamic adaptive partitioning for nonlinear time series

SUMMARY We propose a dynamic adaptive partitioning scheme for nonparametric analysis of stationary nonlinear time series. It yields estimates of the whole probability distribution of the underlying process. We use information from past values to construct adaptive partitioning in a dynamic fashion which is then different from the more common static schemes in the regression set-up. The idea of dynamic partitioning is new. We make it constructive by an approach based on quantisation of the data and adaptively modelling partition cells with a parsimonious Markov chain. The methodology is formulated in terms of a new model class, the so-called quantised variable length Markov chains. It is a new extension of finite-valued variable length Markov chains to processes with values in Rd. We discuss estimation, explore asymptotic properties of the new method and give some numerical results which reflect the finite sample behaviour.

Variable length Markov chains

We study estimation in the class of stationary variable length Markov chains (VLMC) on a finite space. The processes in this class are still Markovian of high order, but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high-order Markov chains. From an algorithmic view, the VLMC model class has attracted interest in information theory and machine learning, but statistical properties have not yet been explored. Provided that good estimation is available, the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest. The latter is exemplified with some DNA data. A version of the tree-structured context algorithm, proposed by Rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of VLMCs. This remains true even when the underlying model increases in dimensionality. Fur-thermore, consistent estimation of minimal state spaces and mixing properties of fitted models are given. We also propose a new bootstrap scheme based on fitted VLMCs. We show its validity for quite general stationary categorical time series and for a broad range of statistical procedures.