Order Markov Model Research Articles

Weak convergence of some marked empirical processes: Application to testing heteroscedasticity

We present a procedure for testing the goodness-of-fit of the conditional variance function of a Markov model of order 1, under stationarity and ergodicity. The autoregressive parameter, the distribution of the noise and the stationary distribution of the observations are assumed to be unknown. Under the null hypothesis H 0 that the conditional variance function belongs to a class of parametric functions, we define an estimator {\ ilde{\ heta}}_{n} of 0 , the assumed true parameter, and we establish its consistency and asymptotic normality. We define a marked empirical process A n (), for which we state and prove a functional limit theorem under H 0 . The asymptotic behavior of this process is studied under fixed alternatives H 1 . Based on the process A n (), a chi-squared test is derived. Simulation experiments show that the test is powerful against some heteroscedastic time series models.

Genetic algorithms for the classification and prediction of precipitation occurrence

Using an approach similar to the biological processes of natural selection and evolution, the genetic algorithm (GA) is a nonconventional optimum search technique. Genetic algorithms have the ability to search large and complex decision spaces and handle nonconvexities. In this paper, the GA is applied for solving the optimum classification of rainy and non-rainy day occurrences based on vertical velocity, dewpoint depression, temperature and humidity data. The problem involves finding optimum classification based on known data, training the future prediction system and then making reliable predictions for rainfall occurrences which have significance in agricultural, transportation, water resources and tourism activities. Various statistical approaches require restrictive assumptions such as stationarity, homogeneity and normal probability distribution of the hydrological variables concerned. The GAs do not require any of these assumptions in their applications. The GA approach for the occurrence classifications and predictions is presented in steps and then the application of the methodology is shown for precipitation occurrence (non-occurrence) data. It has been shown that GAs give better results than classical approaches such as discriminant analysis. The application of the methodology is presented independently for the precipitation event occurrences and forecasting at the Lake Van station in eastern Turkey. Finally, the amounts of precipitation are predicted with a model similar to a third order Markov model whose parameters are estimated by the GA technique.

Determination of Bias in the Relative Abundance of Oligonucleotides in DNA Sequences

Different statistical measures of bias of oligonucleotide sequences in DNA sequences were compared, both by theoretical analysis and according to their abilities to predict the relative abundances of oligonucleotides in the genome of Escherichia coli. The expected frequency of an oligonucleotide calculated from a maximal order Markov model was shown to be a degenerate case of the expected frequency calculated from biases of all subwords arising when noncontiguous subwords exhibit no bias. Since (at least in E. coli) noncontiguous sequences exhibit significant bias, the total compositional bias approach is expected to represent biases in genomic sequences more faithfully than Markov approaches. In fact, the efficacy of statistics based on Markov analysis even at the highest order were inferior in predicting actual frequencies of oligonucleotides to methods that factored out biases of internal subwords with gaps. Using total compositional bias as a measure of relative abundance, tetranucleotide and hexanucleotide palindromes were found to be distributed differently from nonpalindromic sequences, with their means shifted somewhat towards underrepresentation. A subpopulation of palindromic hexanucleotides, however, was highly underrepresented, and this group consisted almost entirely of targets for Type II restriction enzymes found within strains of E. coli. Sites recognized by Type I endonucleases from related strains were not markedly biased, and with pentanucleotides, palindromic and nonpalindromic sequences had nearly identical distributions. The loss of restriction sites may be explained by the free transfer of plasmids encoding restriction enzymes and episodic selection for the presence of the enzymes.

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.

A note concerning the proper choice for Markov model order for daily precipitation in the humid tropics: a case study in Costa Rica

A note concerning the proper choice for Markov model order for daily precipitation in the humid tropics: a case study in Costa Rica

On the convergence of a clustering algorithm for protein-coding regions in microbial genomes.

As the number of fully sequenced prokaryotic genomes continues to grow rapidly, computational methods for reliably detecting protein-coding regions become even more important. Audic and Claverie (1998) Proc. Natl Acad. Sci. USA, 95, 10026-10031, have proposed a clustering algorithm for protein-coding regions in microbial genomes. The algorithm is based on three Markov models of order k associated with subsequences extracted from a given genome. The parameters of the three Markov models are recursively updated by the algorithm which, in simulations, always appear to converge to a unique stable partition of the genome. The partition corresponds to three kinds of regions: (1) coding on the direct strand, (2) coding on the complementary strand, (3) non-coding. Here we provide an explanation for the convergence of the algorithm by observing that it is essentially a form of the expectation maximization (EM) algorithm applied to the corresponding mixture model. We also provide a partial justification for the uniqueness of the partition based on identifiability. Other possible variations and improvements are briefly discussed.

Generalized Markov modeling for flat fading

This paper investigates the properties of a method for obtaining Markov models of unspecified order to be applied to narrow-band fading channels with additive white Gaussian noise. The models are obtained by applying the context tree pruning algorithm to experimental or simulated sequences. Fading environments are identified in which the extension from first-order to higher order models is justified. The paper presents, as examples, the evaluation of the covariance function and the packet error distribution.

Stochastic Modeling of the Van Lake Monthly Level Fluctuations in Turkey

Planning, design, construction and operation of lakeshore structures require information about the future likely extremes of the lake levels at a given risk percentage. Alternative future likely synthetic sequences can be numerically generated provided that the underlying generating mechanism of the lake level fluctuation phenomenon is identified. Simple linear and periodic nonlinear models are used for modeling the deterministic part in the lake level records. Linear trend is eliminated from the original lake level historic data by regression line technique. The nonlinear part needs two stages for its identification. First Fourier series is applied to model interannual periodicities in the lake level fluctuation series and then monthly standardization procedure is applied for seasonal periodic nonlinear component modeling. A second order Markov model is found suitable for the remaining stochastic parts. The application of the methodology is presented for the Lake Van monthly level data in eastern Turkey. Suitable models are identified and their parameters are estimated for trend, periodic and stochastic parts. Likely, synthetic lake levels are generated by the stochastic model and hence lake level extreme values are depicted for the next 2, 6, 12, 24, 60 and 120 months with risk calculations. Such risk calculations take into account the stochastic characteristics of the lake level fluctuations only. The deterministic parts as linear trends and periodicities are added to the stochastic extreme events for the actual simulation of the lake levels. The model presented in this paper is not for time prediction of future lake levels but rather for the simulation of possible equally likely extreme lake level value occurrences over any desired future period.

Markov model aided decoding for image transmission using soft-decision-feedback

Soft-decision-feedback MAP decoders are developed for joint source/channel decoding (JSCD) which uses the residual redundancy in two-dimensional sources. The source redundancy is described by a second order Markov model which is made available to the receiver for row-by-row decoding, wherein the output for one row is used to aid the decoding of the next row. Performance can be improved by generalizing so as to increase the vertical depth of the decoder. This is called sheet decoding, and entails generalizing trellis decoding of one-dimensional data to trellis decoding of two-dimensional data (2-D). The proposed soft-decision-feedback sheet decoder is based on the Bahl algorithm, and it is compared to a hard-decision-feedback sheet decoder which is based on the Viterbi algorithm. The method is applied to 3-bit DPCM picture transmission over a binary symmetric channel, and it is found that the soft-decision-feedback decoder with vertical depth V performs approximately as well as the hard-decision-feedback decoder with vertical depth V+1. Because the computational requirement of the decoders depends exponentially on the vertical depth, the soft-decision-feedbark decoder offers significant reduction in complexity. For standard monochrome Lena, at a channel bit error rate of 0.05, the V=1 and V=2 soft-decision-feedback decoder JSCD gains in RSNR are 5.0 and 6.3 dB, respectively.

A note concerning the proper choice for Markov model order for daily precipitation in the humid tropics: a case study in Costa Rica

The use of chain-dependent hydroclimatological models (sometimes referred to as ‘combined models’ or ‘two-part models’) in analysing daily precipitation requires that rainfall be modelled using both occurrence and intensity statistics. Markov processes in the context of precipitation climatology have been studied in such regions as monsoonal Asia, sub-Saharan Africa and South America. Many studies have indicated that the use of a first-order Markov model is often adequate when describing daily precipitation occurrences, particularly when working in temperate regions, but relatively little work has been done in the humid tropics regarding proper Markov model order, particularly in the western hemisphere. This research examines the occurrence characteristics of Costa Rican daily precipitation by comparing the Akaike and Bayesian information criteria (AIC and BIC) for three long-term meteorological stations. It is found that the most parsimonious models generally are those of first order (winter) or zero order (summer). Overall, the BIC yields less ambiguous results than the AIC, and thus, a higher level of model confidence is achieved when using the BIC as the model-order selection criteria. Copyright © 2000 Royal Meteorological Society

Directed Markov Point Processes as Limits of Partially Ordered Markov Models

In this paper, we consider spatial point processes and investigate members of a subclass of the Markov point processes, termed the directed Markov point processes (DMPPs), whose joint distribution can be written in closed form and, as a consequence, its parameters can be estimated directly. Furthermore, we show how the DMPPs can be simulated rapidly using a one-pass algorithm. A subclass of Markov random fields on a finite lattice, called partially ordered Markov models (POMMs), has analogous structure to that of DMPPs. In this paper, we show that DMPPs are the limits of auto-Poisson and auto-logistic POMMs. These and other results reveal a close link between inference and simulation for DMPPs and POMMs.

Stochastic modelling of daily rainfall in Nigeria: intra-annual variation of model parameters

A Markov model of order 1 may be used to describe the occurrence of wet and dry days in Nigeria. Such models feature two parameter sets; P 01 to characterise the probability of a wet day following a dry day and P 11 to characterise the probability of a wet day following a wet day. The model parameter sets, when estimated from historical records, are characterised by a distinctive seasonal behaviour. However, the comparison of this seasonal behaviour between rainfall stations is hampered by the noise reflecting the high variability of parameters on successive days. The first part of this article is concerned with methods for smoothing these inherently noisy parameter sets. Smoothing has been approached using Fourier series, averaging techniques, or a combination thereof. It has been found that different methods generally perform well with respect to estimation of the average number of wet events and the frequency duration curves of wet and dry events. Parameterisation of the P 01 parameter set is more successful than the P 11 in view of the relatively small number of wet events lasting two or more days. The second part of the article is concerned with describing the regional variation in smoothed parameter sets. There is a systematic variation in the P 01 parameter set as one moves northwards. In contrast, there is limited regional variation in the P 11 set. Although this regional variation in P 01 appears to be related to the gradual movement of the Inter Tropical Convergence Zone, the contrasting behaviour of the two parameter sets is difficult to explain on physical grounds.

Texture synthesis and pattern recognition for partially ordered Markov models

The uses of texture in image analysis are widespread, ranging from remotely sensed data to medical imaging to military applications. Image processing tasks that use texture characteristics include classification, region segmentation, and synthesis of data. While there are several approaches available for texture modeling, the research presented here is concerned with stochastic texture models. Stochastic approaches view a texture as the realization of a random field and are most useful when the texture appears noisy or when it lacks smooth geometric features. The model introduced in this paper is a subclass of Markov random fields (MRFs) called partially ordered Markov models (POMMs). Markov random fields are a class of stochastic models that incorporate spatial dependency between data points. One major disadvantage of MRFs is that, in general, an explicit form of the joint probability of the random variables describing the model is not obtainable. However, a popular subclass of MRFs, called Markov mesh models (MMMs), allows the explicit description of the joint probability in terms of spatially local conditional probabilities. We show how POMMs are a generalization of MMMs and demonstrate the versatility of POMMs to texture synthesis and pattern recognition in imaging. Specifically, we give a fast, one-pass algorithm for simulating textures using POMMs, and introduce examples of heterogeneous models that suggest potential applications for object recognition purposes. Then we address an inverse problem, where we present results from a series of statistical experiments designed to estimate parameters of stochastic texture models for both binary and gray value data. Although the applications in this paper focus on imaging, in their most general form, POMMs can be found in areas such as probabilistic expert systems, Bayesian hierarchical modeling, influence diagrams, and random graphs and networks.

Prediction in LMS-type adaptive algorithms for smoothly time varying environments

The aim of this correspondence is to improve the performance of the least mean square (LMS) and normalized-LMS (NLMS) adaptive algorithms in tracking of time-varying models. A new procedure for estimation of weight increments for including in the LMS-type adaptive algorithms is proposed. This procedure applies a simple smoothing on the increment of the estimated weights to estimate the speed of weights. The estimated speeds are then used to predict the weights for the next iteration. The efficiency of the algorithm is confirmed by simulation results. The algorithm has a very low order of arithmetic complexity. Moreover, this procedure could be combined with a wide class of adaptive filters (e.g., RLS, gradient lattice algorithm, etc.) to improve their behaviors. The proposed algorithm is obtained by simplifying a Kalman filter. To this end, a Markov model of second order is considered for the weight vector. This model shows that the estimation of parameter increments inferred from the predicted parameters improves the tracking performance.

Microstructural characterization of acrylonitrile/pentyl acrylate copolymers by one and 2-D NMR spectroscopy

ABSTRACT: Acrylonitrile/pentyl acrylate (A/P) copolymers of different monomer com-position were prepared by solution polymerization using benzoyl peroxide as initiator.Copolymer compositions were determined by elemental analysis and quantitative 13 C 1 H-NMR spectroscopy. The comonomer reactivity ratios, determined by both KelenTudos (KT) and nonlinear error in variables (EVM) methods are r A 5 0.75 and r p 5 0.45. 2-D heteronuclear correlation spectroscopy (HSQC) was used to simplify thecomplex 1 H spectra of A/P copolymers in terms of configurational and compositionalsequences. The microstructure was obtained in terms of the distribution of A- and P-centered triad sequences from 13 C 1 H-NMR spectra of the copolymers. The copolymer-ization mechanism was found to follow a first order Markov Model. © 1999 John Wiley S acrylonitrile; pentyl acrylate; copolymer; 2-D NMR;spectroscopy INTRODUCTION Homo and copolyacrylates are of great interest inmany industrial application because of the widerange of physical and chemical properties thatcan be covered by an appropriate choice of theside groups and copolymer systems. The differ-ences in the monomer sequence distributionwithin synthetic copolymers, result in differencesin the physical properties of the copolymers. Con-sequently, knowledge of the microstructure canbe crucial to understanding the macroscopic char-acteristics of polymers.

Image analysis with partially ordered markov models

Statistical approaches to image analysis, such as image restoration, segmentation, object classification, and reconstruction often require specification of a distributional model for the variability of the pixel intensities around the true image and a prior distributional model for the true image itself. Spatial dependence (i.e., nearby values tend to be more - or less - alike than those far apart) is often modeled by assuming a Markov random field (MRF) for the prior model and sometimes for the pixel-intensity model. When dealing with MRFs, there is typically an unwieldy normalizing constant that can cause inference to be either inefficient or computationally intensive. In this article, we propose a class of models that are a subset of the class of MRFs but whose members have probability distributions that can be written in closed form. This class, called the partially ordered Markov models (POMMs), contains as a special case the Markov mesh models (MMMs) and is seen to be an important subclass of graphical models used in the analysis of (Bayesian) networks. POMMs are used in experiments for both the forward problem of texture synthesis and the inverse problem of parameter estimation. Various images of textures are generated using POMMs and are seen not to exhibit any obvious directional patterns. Also, parameter estimates from maximum likelihood estimators are found using a real texture image, and the estimates are then used to generate a texture that is similar to the real data.

Optimal Order of Markov Models Applied to Bankchecks

The aim of this study is to show that the optimal order of Markov Model of cursive words can be rigorously stated in order to fit the structural properties of the observed data using Akaike information criterion. The method has been tested on French Postal check amounts up to order 4. An original structural representation of cursive words based on graphemes is used. The conditional probability to have a word model given an observed sequence of graphemes is computed independently of the length of the sequence. The recognition results obtained confirm the optimal order found using Akaike criterion.

Duration and recurrence of otitis media with effusion in children from birth to 3 years: prospective study using monthly otoscopy and tympanometry

Objective: To monitor the natural course of otitis media with effusion.Design: Prospective, longitudinal assessment of the state of the middle ear by otoscopy and tympanometry at monthly intervals from birth...

STOCHASTIC GENERATION OF ANNUAL AND MONTHLY EVAPORATION IN SAUDI ARABIA

Long records of hydrologic data as evaporation sequences are needed for proper design and operation of water resources projects. However, such records are usually short and therefore need to be generated by using stochastic procedures. The generated sequences, with the same statistical characteristics as those of the historical data, make it possible to analyze water resources systems more objectively and with optimum design. Stochastic generation of evaporation sequences are generally used in simulation of rainfall-runoff processes, and are used as an index in evapotranspiration sequences. In this study annual and monthly evaporation sequences were generated for ten selected stations in Saudi Arabia. For generating annual sequences a first order Markov model was used. Three autoregressive models were used for generating monthly sequences: the method of fragments, Thomas-Fiering model and the Two-Tier model. Performance of the models was evaluated by comparing the statistical parameters of the generated sequences with those of the historical data. Finally, the suitability of the models for Saudi Arabia was defined.

Over- and underrepresentation of short DNA words in herpesvirus genomes.

The relative abundance and rarity of DNA words have been recognized in previous biological studies to have implications for the regulation, repair, and evolutionary mechanisms of a genome. In this paper, we review several different measures of abundance and rarity of DNA words, including z-scores, representation ratios, and cross-ratios, that have appeared in the recent literature, and examine the concordance among them using the human cytomegalovirus genome sequence. We then rank all words of length k = 2, ..., 5 of seven herpesvirus genomes according to their abundance, as measured by one of the z-scores based upon a stationary Markov model of order k-2. Using a simple metric on the ranks of 2-words of the seven herpesvirus sequences, we construct an evolutionary tree. Several 3-words are observed to be consistently over- or underrepresented in all seven herpesviruses. Furthermore, clusters of some of the most over- and underrepresented 4- and 5-words in the genomes are identified with functional sites such as the origins of replication and regulatory signals of individual viruses.