Variable Length Markov Chains Research Articles

Reference to Patients in Nurse Shift Handover Meetings: Exploring the Dynamics of Referring Expressions

ABSTRACT This paper investigates the dynamics of referring expressions in hospital nurse handover meetings when discussing patients. We apply the methods of the Variable Length Markov Chain (VLMC) and network analyses to model the use of referring expressions and evaluate relationships between them. The models reveal second-order dependencies emerging for metonymy and noun phrases. Specifically, metonymy shows a greater association with the beginning of a reference, particularly in the context of other metonymies. In contrast, noun phrases tend to be more strongly associated with later points in the reference. Further, we introduce the notion of referential typicality, which measures the conformity of sequences of referring expressions to anticipated patterns. We show, for example, that consecutive noun phrases fall outside the typical pattern, whereas metonymical sequences and sequences of pronouns are highly typical. The transitions from metonymy or nouns to pronouns also closely align with a highly typical pattern. Using a Generalized Additive Model (GAM), we then track the overall evolution of referential typicality throughout the duration of handover meetings, from their beginning to end. The study reveals a subtle increase in referential typicality towards the end of these sessions, indicating a trend towards more consistent referencing as the discourse unfolds.

Identifying Effective Connectivity between Stochastic Neurons with Variable-Length Memory Using a Transfer Entropy Rate Estimator.

Information theory explains how systems encode and transmit information. This article examines the neuronal system, which processes information via neurons that react to stimuli and transmit electrical signals. Specifically, we focus on transfer entropy to measure the flow of information between sequences and explore its use in determining effective neuronal connectivity. We analyze the causal relationships between two discrete time series, X:=Xt:t∈Z and Y:=Yt:t∈Z, which take values in binary alphabets. When the bivariate process (X,Y) is a jointly stationary ergodic variable-length Markov chain with memory no larger than k, we demonstrate that the null hypothesis of the test-no causal influence-requires a zero transfer entropy rate. The plug-in estimator for this function is identified with the test statistic of the log-likelihood ratios. Since under the null hypothesis, this estimator follows an asymptotic chi-squared distribution, it facilitates the calculation of p-values when applied to empirical data. The efficacy of the hypothesis test is illustrated with data simulated from a neuronal network model, characterized by stochastic neurons with variable-length memory. The test results identify biologically relevant information, validating the underlying theory and highlighting the applicability of the method in understanding effective connectivity between neurons.

Several Approaches for the Prediction of the Operating Modes of a Wind Turbine

Growing concern about climate change has intensified efforts to use renewable energy, with wind energy highlighted as a growing source. It is known that wind turbines are characterized by distinct operating modes that reflect production efficiency. In this work, we focus on the forecasting problem for univariate discrete-valued time series of operating modes. We define three prediction strategies to overcome the difficulties associated with missing data. These strategies are evaluated through experiments using five forecasting methods across two real-life datasets. Two of the forecasting methods have been introduced in the statistical literature as extensions of the well-known context algorithm: variable length Markov chains and Bayesian context tree. Additionally, we consider a Bayesian method based on conditional tensor factorization and two different smoothers from the classical tools for time series forecasting. After evaluating each pair prediction strategy/forecasting method in terms of prediction accuracy versus computational complexity, we provide guidance on the methods that are suitable for forecasting the time series of operating modes. The prediction results that we report demonstrate that high accuracy can be achieved with reduced computational resources.

A New Context Tree Inference Algorithm for Variable Length Markov Chain Model with Applications to Biological Sequence Analyses.

The statistical inference of high-order Markov chains (MCs) for biological sequences is vital for molecular sequence analyses but can be hindered by the high dimensionality of free parameters. In the seminal article by Bühlmann and Wyner, variable length Markov chain (VLMC) model was proposed to embed the full-order MC in a sparse structured context tree. In the key procedure of tree pruning of their proposed context algorithm, the word count-based statistic for each branch was defined and compared with a fixed cutoff threshold calculated from a common chi-square distribution to prune the branch of the context tree. In this study, we find that the word counts for each branch are highly intercorrelated, resulting in non-negligible effects on the distribution of the statistic of interest. We demonstrate that the inferred context tree based on the original context algorithm by Bühlmann and Wyner, which uses a fixed cutoff threshold based on a common chi-square distribution, can be systematically biased and error prone. We denote the original context algorithm as VLMC-Biased (VLMC-B). To solve this problem, we propose a new context tree inference algorithm using an adaptive tree-pruning scheme, termed VLMC-Consistent (VLMC-C). The VLMC-C is founded on the consistent branch-specific mixed chi-square distributions calculated based on asymptotic normal distribution of multiple word patterns. We validate our theoretical branch-specific asymptotic distribution using simulated data. We compare VLMC-C with VLMC-B on context tree inference using both simulated and real genome sequence data and demonstrate that VLMC-C outperforms VLMC-B for both context tree reconstruction accuracy and model compression capacity.

Some Notes on the Classification of Shift Spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts

The aim of this article is to find appropriate definitions for shifts of finite type and sofic shifts in a general context of symbolic dynamics. We start showing that the classical definitions of shifts of finite type and sofic shifts, as they are given in the context of finite-alphabet shift spaces on the one-dimensional monoid $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, do not fit for shift spaces over infinite alphabet or on other monoids. Therefore, by examining the core features in the classical definitions of shifts of finite type and sofic shifts, we propose general definitions that can be used in any context. The alternative definition given for shifts of finite type inspires the definition of a new class of shift spaces which intersects with the class of sofic shifts and includes shifts of finite type. This new class is named finitely defined shifts, and the non-finite-type shifts in it are named shifts of variable length. For the specific case of infinite-alphabet shifts on the lattice $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, shifts of variable length can be interpreted as the topological version of variable length Markov chains.

Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition

We consider a class of variable length Markov chains with a binary alphabet in which context tree is defined by adding finite trees with uniformly bounded height to the vertices of an infinite comb tree. Such type of Markov chain models the spike neuron patterns and also extends the class of persistent random walks. The main interest is the limiting properties of the empirical distribution of symbols from the alphabet. We obtain the strong law of large numbers, central limit theorem, and exact asymptotics for large and moderate deviations. The presence of an intrinsic renewal structure is the subject of discussion in the literature. Proofs are based on the construction of a renewals of the chain and the applying corresponding properties of the compound (or generalized) renewal processes.

Fast parallel construction of variable-length Markov chains

BackgroundAlignment-free methods are a popular approach for comparing biological sequences, including complete genomes. The methods range from probability distributions of sequence composition to first and higher-order Markov chains, where a k-th order Markov chain over DNA has 4^k formal parameters. To circumvent this exponential growth in parameters, variable-length Markov chains (VLMCs) have gained popularity for applications in molecular biology and other areas. VLMCs adapt the depth depending on sequence context and thus curtail excesses in the number of parameters. The scarcity of available fast, or even parallel software tools, prompted the development of a parallel implementation using lazy suffix trees and a hash-based alternative.ResultsAn extensive evaluation was performed on genomes ranging from 12Mbp to 22Gbp. Relevant learning parameters were chosen guided by the Bayesian Information Criterion (BIC) to avoid over-fitting. Our implementation greatly improves upon the state-of-the-art even in serial execution. It exhibits very good parallel scaling with speed-ups for long sequences close to the optimum indicated by Amdahl’s law of 3 for 4 threads and about 6 for 16 threads, respectively.ConclusionsOur parallel implementation released as open-source under the GPLv3 license provides a practically useful alternative to the state-of-the-art which allows the construction of VLMCs even for very large genomes significantly faster than previously possible. Additionally, our parameter selection based on BIC gives guidance to end-users comparing genomes.

Variable length Markov chain with exogenous covariates

Markov chains with variable length are useful stochastic models for data compression that avoid the curse of dimensionality faced by full Markov chains. In this article we introduce a variable length Markov chain whose transition probabilities depend not only on the state history but also on exogenous covariates through a generalized linear model. The goal of the proposed procedure is to estimate not only the context of the process, that is, the history of the process that is relevant for predicting the next state, but also the coefficients corresponding to the significant exogenous variables. The proposed method is consistent in the sense that the probability that the estimated context and the coefficients are equal to the true data generating mechanism tends to 1 as the sample size increases. Simulations suggest that, when covariates do contribute to the transition probabilities, the proposed procedure can recover both the tree structure and the regression parameters. It outperforms variable length Markov chains when covariates are present while yielding comparable results when covariates are absent. For models with fixed length, the accuracy of the proposed algorithm in recovering the true data generating mechanism is close to the methods available in the literature. The proposed methodology is used to predict the gains and losses of the Hang Seng index based on its own history and three large stock market indices.

Webpage Recommendation System Based on the Social Media Semantic Details of the Website

The web page recommendation is generated by using the navigational history from web server log files. Semantic Variable Length Markov Chain Model (SVLMC) is a web page recommendation system used to generate recommendation by combining a higher order Markov model with rich semantic data. The problem of state space complexity and time complexity in SVLMC was resolved by Semantic Variable Length confidence pruned Markov Chain Model (SVLCPMC) and Support vector machine based SVLCPMC (SSVLCPMC) meth-ods respectively. The recommendation accuracy was further improved by quickest change detection using Kullback-Leibler Divergence method. In this paper, socio semantic information is included with the similarity score which improves the recommendation accuracy. The social information from the social websites such as twitter is considered for web page recommendation. Initially number of web pages is collected and the similari-ty between web pages is computed by comparing their semantic information. The term frequency and inverse document frequency (tf-idf) is used to produce a composite weight, the most important terms in the web pages are extracted. Then the Pointwise Mutual Information (PMI) between the most important terms and the terms in the twitter dataset are calculated. The PMI metric measures the closeness between the twitter terms and the most important terms in the web pages. Then this measure is added with the similarity score matrix to provide the socio semantic search information for recommendation generation. The experimental results show that the pro-posed method has better performance in terms of prediction accuracy, precision, F1 measure, R measure and coverage.

Direction-of-change forecasting in commodity futures markets

This paper examines direction-of-change predictability in commodity futures markets using a variety of binary probabilistic techniques. As well as traditional techniques, we apply Variable Length Markov Chain (VLMC) analysis, an innovative technique popularised in computational biology when predicting DNA sequences (Bühlmann & Wyner, 1999). To the best of our knowledge, this is the first application of VLMC in finance. Our results show that both VLMC and technical analysis methods provide strong predictability of the direction-of-change of commodity returns, with annualised mean returns of approximately 8%, substantially higher than the passive long strategy. Our results suggest that a short-term learning effect is present in commodities market which can be exploited using innovative direction-of-change forecasting techniques.

Emergent Routines in Peer-Production

Current theories struggle to explain how participants in peer-production self-organize to produce high-quality knowledge in the absence of formal coordination mechanisms. The literature traditionally holds that norms, policies, and roles make coordination possible. However, peer-production is largely free from workflow constraints and most peer-production communities do not allocate or assign tasks. Yet, scholars have suggested that ordered work sequences can emerge in such settings. We refer to sequences of activities that emerge organically as components of “emergent routines.” The volunteer nature of peer-production, coupled with high degrees of turnover, makes learning and coordination difficult, calling into question the extent to which emergent routines could be ingrained in the community. The objective of this article is to characterize the work sequences that organically emerge in peer-production, as well as to understand the temporal dynamics of these emergent routine components. We center our empirical investigation on the peer-production of a set of 1,000 Wikipedia articles. Using a dataset of labeled wiki work, we employ Variable-Length Markov Chains (VLMC) to identify sequences of activities exhibiting structural dependence, cluster the sequences to identify components of emergent routines, and then track their prevalence over time. We find that work is organized according to several routine components, and that the prevalence of these components changes over time.

An Enhancement of Svm Based Semantically Enriched Variable Length Confidence Pruned Markov Chain Model Based Web Page Recommendation System

Semantic Variable Length Markov Chain Length Model (SVLMC) is a web page recommendation system which combined the fields of semantic web and web usage mining by the Markov transition probability matrix with rich semantic information extracted from web pages. Though it has high prediction accuracy, it has problem of high state space complexity. The high space complexity reduce the execution speed and reduce the performance of the system, which was resolved by Semantic Variable Length confidence pruned Markov Chain Model (SVLCPMC) model that provides high user satisfied recommendation and Confidence Pruned Markov Model (CPMM). The time consumption of CPMM was reduced by Support Vector Machine (SVM). But still the recommendation accuracy is still below the user satisfaction. So in this paper, quickest change detection using Kullback-Leibler Divergence method is introduced to improve the accuracy of recommendation generation by developing a scalable quickest change detection schemes that can be implemented recursively in a more complicated scenario of Markov model and it is included in the training data of SVM. Then the performance of web page recommendation is improved by ranking the web pages using page ranking technique. Thus the performance of web page recommendation generation system has been improved. The experiments are conducted to prove the effectiveness of the proposed work in terms of prediction accuracy, precision, recall, F1-measure, coverage and R measure.

Computation of exact probabilities associated with overlapping pattern occurrences

AbstractSearching 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

Recurrence of multidimensional persistent random walks. Fourier and series criteria

The recurrence and transience of persistent random walks built from variable length Markov chains are investigated. It turns out that these stochastic processes can be seen as Lévy walks for which the persistence times depend on some internal Markov chain: they admit Markov random walk skeletons. A recurrence versus transience dichotomy is highlighted. Assuming the positive recurrence of the driving chain, a sufficient Fourier criterion for the recurrence, close to the usual Chung–Fuchs one, is given and a series criterion is derived. The key tool is the Nagaev–Guivarc’h method. Finally, we focus on particular two-dimensional persistent random walks, including directionally reinforced random walks, for which necessary and sufficient Fourier and series criteria are obtained. Inspired by (Adv. Math. 208 (2007) 680–698), we produce a genuine counterexample to the conjecture of (Adv. Math. 117 (1996) 239–252). As for the one-dimensional case studied in (J. Theoret. Probab. 31 (2018) 232–243), it is easier for a persistent random walk than its skeleton to be recurrent. However, such examples are much more difficult to exhibit in the higher dimensional context. These results are based on a surprisingly novel – to our knowledge – upper bound for the Lévy concentration function associated with symmetric distributions.

Approximate group context tree

We study a variable length Markov chain model associated with a group of stationary processes that share the same context tree but each process has potentially different conditional probabilities. We propose a new model selection and estimation method which is computationally efficient. We develop oracle and adaptivity inequalities, as well as model selection properties, that hold under continuity of the transition probabilities and polynomial $\beta$-mixing. In particular, model misspecification is allowed. These results are applied to interesting families of processes. For Markov processes, we obtain uniform rate of convergence for the estimation error of transition probabilities as well as perfect model selection results. For chains of infinite order with complete connections, we obtain explicit uniform rates of convergence on the estimation of conditional probabilities, which have an explicit dependence on the processes’ continuity rates. Similar guarantees are also derived for renewal processes. Our results are shown to be applicable to discrete stochastic dynamic programming problems and to dynamic discrete choice models. We also apply our estimator to a linguistic study, based on recent work by Galves et al. [Ann. Appl. Stat. 6 (2012) 186–209], of the rhythmic differences between Brazilian and European Portuguese.

Probability and algorithmics: a focus on some recent developments

This article presents different recent theoretical results illustrating the interactions between probability and algorithmics. These contributions deal with various topics: cellular automata and calculability, variable length Markov chains and persistent random walks, perfect sampling via coupling from the past. All of them involve discrete dynamics on complex random structures.

Persistent Random Walks. I. Recurrence Versus Transience

We consider a walker on the line that at each step keeps the same direction with a probability which depends on the time already spent in the direction the walker is currently moving. These walks with memories of variable length can be seen as generalizations of directionally reinforced random walks introduced in Mauldin et al. (Adv Math 117(2):239-252, 1996). We give a complete and usable characterization of the recurrence or transience in terms of the probabilities to switch the direction and we formulate some laws of large numbers. The most fruitful situation emerges when the running times both have an infinite mean. In that case, these properties are related to the behaviour of some embedded random walk with an undefined drift so that these features depend on the asymptotics of the distribution tails related to the persistence times. In the other case, the criterion reduces to a null-drift condition. Finally, we deduce some criteria for a wider class of persistent random walks whose increments are encoded by a variable length Markov chain having-in full generality-no renewal pattern in such a way that their study does not reduce to a skeleton RW as for the original model.

On the Mathematical Consequences of Binning Spike Trains.

We initiate a mathematical analysis of hidden effects induced by binning spike trains of neurons. Assuming that the original spike train has been generated by a discrete Markov process, we show that binning generates a stochastic process that is no longer Markov but is instead a variable-length Markov chain (VLMC) with unbounded memory. We also show that the law of the binned raster is a Gibbs measure in the DLR (Dobrushin-Lanford-Ruelle) sense coined in mathematical statistical mechanics. This allows the derivation of several important consequences on statistical properties of binned spike trains. In particular, we introduce the DLR framework as a natural setting to mathematically formalize anticipation, that is, to tell "how good" our nervous system is at making predictions. In a probabilistic sense, this corresponds to condition a process by its future, and we discuss how binning may affect our conclusions on this ability. We finally comment on the possible consequences of binning in the detection of spurious phase transitions or in the detection of incorrect evidence of criticality.

Analyzing State Sequences with Probabilistic Suffix Trees: ThePSTRPackage

This article presents the PST R package for categorical sequence analysis with probabilistic suffix trees (PSTs), i.e., structures that store variable-length Markov chains (VLMCs). VLMCs allow to model high-order dependencies in categorical sequences with parsimonious models based on simple estimation procedures. The package is specifically adapted to the field of social sciences, as it allows for VLMC models to be learned from sets of individual sequences possibly containing missing values; in addition, the package is extended to account for case weights. This article describes how a VLMC model is learned from one or more categorical sequences and stored in a PST. The PST can then be used for sequence prediction, i.e., to assign a probability to whole observed or artificial sequences. This feature supports data mining applications such as the extraction of typical patterns and outliers. This article also introduces original visualization tools for both the model and the outcomes of sequence prediction. Other features such as functions for pattern mining and artificial sequence generation are described as well. The PST package also allows for the computation of probabilistic divergence between two models and the fitting of segmented VLMCs, where sub-models fitted to distinct strata of the learning sample are stored in a single PST.

Stochastic Processes With Random Contexts: A Characterization and Adaptive Estimators for the Transition Probabilities

This paper introduces the concept of random context representations for the transition probabilities of a finite-alphabet stochastic process. Processes with these representations generalize context tree processes (a.k.a. variable length Markov chains), and are proven to coincide with processes whose transition probabilities are almost surely continuous functions of the (infinite) past. This is similar to a classical result by Kalikow about continuous transition probabilities. Existence and uniqueness of a minimal random context representation are proven, and an estimator of the transition probabilities based on this representation is shown to have very good "pastwise adaptativity" properties. In particular, it achieves minimax performance, up to logarithmic factors, for binary renewal processes with bounded $2+\gamma$ moments.