Variable Length Markov Models Research Articles

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.

Bayesian classifiers for detecting HGT using fixed and variable order markov models of genomic signatures

Analyses of genomic signatures are gaining attention as they allow studies of species-specific relationships without involving alignments of homologous sequences. A naïve Bayesian classifier was built to discriminate between different bacterial compositions of short oligomers, also known as DNA words. The classifier has proven successful in identifying foreign genes in Neisseria meningitis. In this study we extend the classifier approach using either a fixed higher order Markov model (Mk) or a variable length Markov model (VLMk). We propose a simple algorithm to lock a variable length Markov model to a certain number of parameters and show that the use of Markov models greatly increases the flexibility and accuracy in prediction to that of a naïve model. We also test the integrity of classifiers in terms of false-negatives and give estimates of the minimal sizes of training data. We end the report by proposing a method to reject a false hypothesis of horizontal gene transfer. Software and Supplementary information available at www.cs.chalmers.se/~dalevi/genetic_sign_classifiers/.

Finding Short DNA Motifs Using Permuted Markov Models

Many short DNA motifs, such as transcription factor binding sites (TFBS) and splice sites, exhibit strong local as well as nonlocal dependence. We introduce permuted variable length Markov models (PVLMM) which could capture the potentially important dependencies among positions and apply them to the problem of detecting splice and TFB sites. They have been satisfactory from the viewpoint of prediction performance and also give ready biological interpretations of the sequence dependence observed. The issue of model selection is also studied.

Seq++: analyzing biological sequences with a range of Markov-related models

The seq++ package offers a reference set of programs and an extensible library to biologists and developers working on sequence statistics. Its generality arises from the ability to handle sequences described with any alphabet (nucleotides, amino acids, codons and others). seq++ enables sequence modelling with various types of Markov models, including variable length Markov models and the newly developed parsimonious Markov models, all of them potentially phased. Simulation modules are supplied for Monte Carlo methods. Hence, this toolbox allows the study of any biological process which can be described by a series of states taken from a finite set.

Algorithms for variable length Markov chain modeling

We present a general purpose implementation of variable length Markov models. Contrary to fixed order Markov models, these models are not restricted to a predefined uniform depth. Rather, by examining the training data, a model is constructed that fits higher order Markov dependencies where such contexts exist, while using lower order Markov dependencies elsewhere. As both theoretical and experimental results show, these models are capable of capturing rich signals from a modest amount of training data, without the use of hidden states. The source code is freely available at http://www.soe.ucsc.edu/~jill/src/

Markovian architectural bias of recurrent neural networks.

In this paper, we elaborate upon the claim that clustering in the recurrent layer of recurrent neural networks (RNNs) reflects meaningful information processing states even prior to training [1], [2]. By concentrating on activation clusters in RNNs, while not throwing away the continuous state space network dynamics, we extract predictive models that we call neural prediction machines (NPMs). When RNNs with sigmoid activation functions are initialized with small weights (a common technique in the RNN community), the clusters of recurrent activations emerging prior to training are indeed meaningful and correspond to Markov prediction contexts. In this case, the extracted NPMs correspond to a class of Markov models, called variable memory length Markov models (VLMMs). In order to appreciate how much information has really been induced during the training, the RNN performance should always be compared with that of VLMMs and NPMs extracted before training as the "null" base models. Our arguments are supported by experiments on a chaotic symbolic sequence and a context-free language with a deep recursive structure. Index Terms-Complex symbolic sequences, information latching problem, iterative function systems, Markov models, recurrent neural networks (RNNs).

Volatility Trading ia Temporal Pattern Recognition in Quantised Financial Time Series

We investigate the potential of the analysis of noisy non-stationary time series by quantising it into streams of discrete symbols and applying finite-memory symbolic predictors. Careful quantisation can reduce the noise in the time series to make model estimation more amenable. We apply the quantisation strategy in a realistic setting involving financial forecasting and trading. In particular, using historical data, we simulate the trading of straddles on the financial indexes DAX and FTSE 100 on a daily basis, based on predictions of the daily volatility differences in the underlying indexes. We propose a parametric, data-driven quantisation scheme which transforms temporal patterns in the series of daily volatility changes into grammatical and statistical patterns in the corresponding symbolic streams. As symbolic predictors operating on the quantised streams, we use the classical fixed-order Markov models, variable memory length Markov models and a novel variation of fractal-based predictors, introduced in its original form in Tin_ o and Dorffner [1]. The fractal-based predictors are designed to efficiently use deep memory. We compare the symbolic models with continuous techniques such as time-delay neural networks with continuous and categorical outputs, and GARCH models. Our experiments strongly suggest that the robust information reduction achieved by quantising the real-valued time series is highly beneficial. To deal with non-stationarity in financial daily time series, we propose two techniques that combine ‘sophisticated’ models fitted on the training data with a fixed set of simple-minded symbolic predictors not using older (and potentially misleading) data in the training set. Experimental results show that by quantising the volatility differences and then using symbolic predictive models, market makers can sometimes generate a statistically significant excess profit. We also mention some interesting observations regarding the memory structure in the series of daily volatility differences studied.

Adaptive context trees and text clustering

In the finite-alphabet context we propose four alternatives to fixed-order Markov models to estimate a conditional distribution. They consist in working with a large class of variable-length Markov models represented by context trees, and building an estimator of the conditional distribution with a risk of the same order as the risk of the best estimator for every model simultaneously, in a conditional Kullback-Leibler sense. Such estimators can be used to model complex objects like texts written in natural language and define a notion of similarity between them. This idea is illustrated by experimental results of unsupervised text clustering.

Learning Variable-Length Markov Models of Behavior

In recent years there has been an increased interest in the modeling and recognition of human activities involving highly structured and semantically rich behavior such as dance, aerobics, and sign language. A novel approach for automatically acquiring stochastic models of the high-level structure of an activity without the assumption of any prior knowledge is presented. The process involves temporal segmentation into plausible atomic behavior components and the use of variable-length Markov models for the efficient representation of behaviors. Experimental results that demonstrate the synthesis of realistic sample behaviors and the performance of models for long-term temporal prediction are presented.

Predicting the Future of Discrete Sequences from Fractal Representations of the Past

We propose a novel approach for building finite memory predictive models similar in spirit to variable memory length Markov models (VLMMs). The models are constructed by first transforming the n-block structure of the training sequence into a geometric structure of points in a unit hypercube, such that the longer is the common suffix shared by any two n-blocks, the closer lie their point representations. Such a transformation embodies a Markov assumption—n-blocks with long common suffixes are likely to produce similar continuations. Prediction contexts are found by detecting clusters in the geometric n-block representation of the training sequence via vector quantization. We compare our model with both the classical (fixed order) and variable memory length Markov models on five data sets with different memory and stochastic components. Fixed order Markov models (MMs) fail on three large data sets on which the advantage of allowing variable memory length can be exploited. On these data sets, our predictive models have a superior, or comparable performance to that of VLMMs, yet, their construction is fully automatic, which, is shown to be problematic in the case of VLMMs. On one data set, VLMMs are outperformed by the classical MMs. On this set, our models perform significantly better than MMs. On the remaining data set, classical MMs outperform the variable context length strategies.