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PubMed2006

Analysing grouping of nucleotides in DNA sequences using lumped processes constructed from Markov chains.

The most commonly used models for analysing local dependencies in DNA sequences are (high-order) Markov chains. Incorporating knowledge relative to the possible grouping of the nucleotides enables to define dedicated sub-classes of Markov chains. The problem of formulating lumpability hypotheses for a Markov chain is therefore addressed. In the classical approach to lumpability, this problem can be formulated as the determination of an appropriate state space (smaller than the original state space) such that the lumped chain defined on this state space retains the Markov property. We propose a different perspective on lumpability where the state space is fixed and the partitioning of this state space is represented by a one-to-many probabilistic function within a two-level stochastic process. Three nested classes of lumped processes can be defined in this way as sub-classes of first-order Markov chains. These lumped processes enable parsimonious reparameterizations of Markov chains that help to reveal relevant partitions of the state space. Characterizations of the lumped processes on the original transition probability matrix are derived. Different model selection methods relying either on hypothesis testing or on penalized log-likelihood criteria are presented as well as extensions to lumped processes constructed from high-order Markov chains. The relevance of the proposed approach to lumpability is illustrated by the analysis of DNA sequences. In particular, the use of lumped processes enables to highlight differences between intronic sequences and gene untranslated region sequences.

3' Untranslated Regions↗

PubMed1994

Deriving non-homogeneous DNA Markov chain models by cluster analysis algorithm minimizing multiple alignment entropy.

Non-homogeneous Markov chain models can represent biologically important regions of DNA sequences. The statistical pattern that is described by these models is usually weak and was found primarily because of strong biological indications. The general method for extracting similar patterns is presented in the current paper. The algorithm incorporates cluster analysis, multiple alignment and entropy minimization. The method was first tested using the set of DNA sequences produced by Markov chain generators. It was shown that artificial gene sequences, which initially have been randomly set up along the multiple alignment panels, are aligned according to the hidden triplet phase. Then the method was applied to real protein-coding sequences and the resulting alignment clearly indicated the triplet phase and produced the parameters of the optimal 3-periodic non-homogeneous Markov chain model. These Markov models were already employed in the GeneMark gene prediction algorithm, which is used in genome sequencing projects. The algorithm can also handle the case in which the sequences to be aligned reveal different statistical patterns, such as Escherichia coli protein-coding sequences belonging to Class II and Class III. The algorithm accepts a random mix of sequences from different classes, and is able to separate them into two groups (clusters), align each cluster separately, and define a non-homogeneous Markov chain model for each sequence cluster.

Algorithms↗

PubMed2005

Modeling animals' behavioral response by Markov chain models for capture-recapture experiments.

A bivariate Markov chain approach that includes both enduring (long-term) and ephemeral (short-term) behavioral effects in models for capture-recapture experiments is proposed. The capture history of each animal is modeled as a Markov chain with a bivariate state space with states determined by the capture status (capture/noncapture) and marking status (marked/unmarked). In this framework, a conditional-likelihood method is used to estimate the population size and the transition probabilities. The classical behavioral model that assumes only an enduring behavioral effect is included as a special case of the bivariate Markovian model. Another special case that assumes only an ephemeral behavioral effect reduces to a univariate Markov chain based on capture/noncapture status. The model with the ephemeral behavioral effect is extended to incorporate time effects; in this model, in contrast to extensions of the classical behavioral model, all parameters are identifiable. A data set is analyzed to illustrate the use of the Markovian models in interpreting animals' behavioral response. Simulation results are reported to examine the performance of the estimators.

Animals↗

PubMed1993

Modeling of analog film-file radiographic retrievals. A Markov chain.

RATIONALE AND OBJECTIVES: Existing retrieval models for radiology film libraries have not incorporated the influence of previous retrievals. A Markov chain model for the retrieval rates from an analog film library is proposed as a means of considering this effect. METHODS: A Markov chain model was developed for the retrieval rates of an analog film library. The Markov chain model required identification of the states of the Markov chain, the required one-step transition probabilities between states, and the initial state probabilities. RESULTS AND CONCLUSIONS: The results from the Markov chain model compared favorably with the 30-day measurements (25,775 retrievals), but a large enough sample to determine a statistical confidence level was not considered.

Academic Medical Centers↗

PubMed1995

A scheme for constructing an irreducible Markov chain for pedigree data.

Estimating probabilities on a pedigree by dependent samples, namely realizations of a Markov chain, has been explored as an alternative method when exact computation is not feasible. If the transition kernel of the Markov chain is aperiodic and irreducible, convergence of the estimates to the true probabilities is guaranteed by the ergodic theorem. However, reducibility is a potential problem for genetic pedigree analysis unless the Markov chain is constructed appropriately. In the present paper, we propose a scheme for constructing an irreducible Markov chain for pedigree data. Transitions between communicating classes, which can be found explicitly, are made by using a Metropolis jumping kernel. The method has been demonstrated to be much more efficient than other currently existing methods.

ABO Blood-Group System↗

PubMed2004

A Markov chain approach to reconstruction of long haplotypes.

Haplotypes are important for association based gene mapping, but there are no practical laboratory methods for obtaining them directly from DNA samples. We propose simple Markov models for reconstruction of haplotypes for a given sample of multilocus genotypes. The models are aimed specifically for long marker maps, where linkage disequilibrium between markers may vary and be relatively weak. Such maps are ultimately used in chromosome or genome-wide association studies. Haplotype reconstruction with standard Markov chains is based on linkage disequilibrium (LD) between neighboring markers. Markov chains of higher order can capture LD in a neighborhood of a given size. We introduce a more flexible and robust model, MC-VL, which is based on a Markov chain of variable order. Experimental validation of the Markov chain methods on both a wide range of simulated data and real data shows that they clearly out perform previous methods on genetically long marker maps and are highly competitive with short maps, too. MC-VL performs well across different data sets and settings while avoiding the problem of manually choosing an appropriate order for the Markov chain, and it has low computational complexity.

Algorithms↗

PubMed1992

First and second moment of counts of words in random texts generated by Markov chains.

An exact expression for the variance of random frequency that a given word has in text generated by a Markov chain is presented. The result is applied to periodic Markov chains, which describe the protein-coding DNA sequences better than simple Markov chains. A new solution to the problem of word overlap is proposed. It was found that the expected frequency and overlapping properties determine most of the variance. The expectation and variance of counts for triplets are compared with experimental counts in Escherichia coli coding sequences.

Algorithms↗

PubMed2005

Markov chains: computing limit existence and approximations with DNA.

We present two algorithms to perform computations over Markov chains. The first one determines whether the sequence of powers of the transition matrix of a Markov chain converges or not to a limit matrix. If it does converge, the second algorithm enables us to estimate this limit. The combination of these algorithms allows the computation of a limit using DNA computing. In this sense, we have encoded the states and the transition probabilities using strands of DNA for generating paths of the Markov chain.

Algorithms↗

PubMed1993

Studies on the specificity of HIV protease: an application of Markov chain theory.

A sequence-coupled (Markov chain) model is proposed to predict the cleavage sites in proteins by proteases with extended specificity subsites. In addition to the probability of an amino acid occurring at each of these subsites as observed from a training set of oligopeptides known cleavable by HIV protease, the conditional probabilities as reflected by the neighbor-coupled effect along the subsite sequence are also taken into account. These conditional probabilities are derived from an expanded training set consisting of sufficiently large peptide sequences generated by the Monte Carlo sampling process. Very high accuracy was obtained in predicting protein cleavage sites by both HIV-1 and HIV-2 proteases. The new method provides a rapid and accurate means for analyzing the specificity of HIV protease, and hence can be used to help find effective inhibitors of HIV protease as potential drugs against AIDS. The principle of this method can also be used to study the specificity of any multisubsite enzyme.

Acquired Immunodeficiency Syndrome↗

PubMed1992

Approximating Markov chains.

A common framework of finite state approximating Markov chains is developed for discrete time deterministic and stochastic processes. Two types of approximating chains are introduced: (i) those based on stationary conditional probabilities (time averaging) and (ii) transient, based on the percentage of the Lebesgue measure of the image of cells intersecting any given cell. For general dynamical systems, stationary measures for both approximating chains converge weakly to stationary measures for the true process as partition width converges to 0. From governing equations, transient chains and resultant approximations of all n-time unit probabilities can be computed analytically, despite typically singular true-process stationary measures (no density function). Transition probabilities between cells account explicitly for correlation between successive time increments. For dynamical systems defined by uniformly convergent maps on a compact set (e.g., logistic, Henon maps), there also is weak continuity with a control parameter. Thus all moments are continuous with parameter change, across bifurcations and chaotic regimes. Approximate entropy is seen as the information-theoretic rate of entropy for approximating Markov chains and is suggested as a parameter for turbulence; a discontinuity in the Kolmogorov-Sinai entropy implies that in the physical world, some measure of coarse graining in a mixing parameter is required.

Journal Article↗

PubMed2007

Bayesian analysis of non-homogeneous Markov chains: application to mental health data.

In this paper we present a formal treatment of non-homogeneous Markov chains by introducing a hierarchical Bayesian framework. Our work is motivated by the analysis of correlated categorical data which arise in assessment of psychiatric treatment programs. In our development, we introduce a Markovian structure to describe the non-homogeneity of transition patterns. In doing so, we introduce a logistic regression set-up for Markov chains and incorporate covariates in our model. We present a Bayesian model using Markov chain Monte Carlo methods and develop inference procedures to address issues encountered in the analyses of data from psychiatric treatment programs. Our model and inference procedures are implemented to some real data from a psychiatric treatment study.

Adolescent↗

PubMed1998

Problems with determination of noncommunicating classes for Monte Carlo Markov chain applications in pedigree analysis.

Exact calculations for probabilities on complex pedigrees are computationally intensive and very often infeasible. Markov chain Monte Carlo methods are frequently used to approximate probabilities and likelihoods of interest. However, when a locus with more than two alleles is considered, the underlying Markov chain is not guaranteed to be irreducible and the results of such analyses are unreliable. A method for finding the noncommunicating classes of the Markov chain would be very useful in designing algorithms that can jump between these classes. In this paper, we will examine some existing work on this problem and point out its limitations. We will also comment on the difficulty of developing a useful algorithm.

Algorithms↗

PubMed1978

Quantitative nuclear image analysis: differentiation between normal, hyperplastic, and malignant appearing uterine glands in a paraffin section. IV. The use of Markov chain texture features in discriminant analysis.

Discriminant analysis was applied to Markov chain texture features and elementary features calculated from data from microscope photometry of nuclei in a paraffin section. The results from measurements on the nuclei of morphologically normal, atypical hyperplastic and carcinomatous uterine glands were used in discriminant analysis. With this technique it is possible to classify up to 88.1% of the nuclei correctly in one of the three groups of uterine glands. The discriminating power of several smaller subsets indicated that with more than 28 features there is hardly any increase in discriminating power. Discriminant analysis with a selection from elementary and Markov chain features provides objective criteria of assistance in histopathological diagnosis.

Cell Nucleus↗

PubMed2003

Bayes or bootstrap? A simulation study comparing the performance of Bayesian Markov chain Monte Carlo sampling and bootstrapping in assessing phylogenetic confidence.

Bayesian Markov chain Monte Carlo sampling has become increasingly popular in phylogenetics as a method for both estimating the maximum likelihood topology and for assessing nodal confidence. Despite the growing use of posterior probabilities, the relationship between the Bayesian measure of confidence and the most commonly used confidence measure in phylogenetics, the nonparametric bootstrap proportion, is poorly understood. We used computer simulation to investigate the behavior of three phylogenetic confidence methods: Bayesian posterior probabilities calculated via Markov chain Monte Carlo sampling (BMCMC-PP), maximum likelihood bootstrap proportion (ML-BP), and maximum parsimony bootstrap proportion (MP-BP). We simulated the evolution of DNA sequence on 17-taxon topologies under 18 evolutionary scenarios and examined the performance of these methods in assigning confidence to correct monophyletic and incorrect monophyletic groups, and we examined the effects of increasing character number on support value. BMCMC-PP and ML-BP were often strongly correlated with one another but could provide substantially different estimates of support on short internodes. In contrast, BMCMC-PP correlated poorly with MP-BP across most of the simulation conditions that we examined. For a given threshold value, more correct monophyletic groups were supported by BMCMC-PP than by either ML-BP or MP-BP. When threshold values were chosen that fixed the rate of accepting incorrect monophyletic relationship as true at 5%, all three methods recovered most of the correct relationships on the simulated topologies, although BMCMC-PP and ML-BP performed better than MP-BP. BMCMC-PP was usually a less biased predictor of phylogenetic accuracy than either bootstrapping method. BMCMC-PP provided high support values for correct topological bipartitions with fewer characters than was needed for nonparametric bootstrap.

Bayes Theorem↗

PubMed2004

Assessing convergence of Markov chain Monte Carlo simulations in hierarchical Bayesian models for population pharmacokinetics.

Advances in computer hardware and the associated computer-intensive algorithms made feasible by these advances [like Markov chain Monte Carlo (MCMC) data analysis techniques] have made possible the application of hierarchical full Bayesian methods in analyzing pharmacokinetic and pharmacodynamic (PK-PD) data sets that are multivariate in nature. Pharmacokinetic data analysis in particular has been one area that has seized upon this technology to refine estimates of drug parameters from sparse data gathered in a large, highly variable population of patients. A drawback in this type of analysis is that it is difficult to quantitatively assess convergence of the Markov chains to a target distribution, and thus, it is sometimes difficult to assess the reliability of estimates gained from this procedure. Another complicating factor is that, although the application of MCMC methods to population PK-PD problems has been facilitated by new software designed for the PK-PD domain (specifically PKBUGS), experts in PK-PD may not have the necessary experience with MCMC methods to detect and understand problems with model convergence. The objective of this work is to provide an example of a set of diagnostics useful to investigators, by analyzing in detail three convergence criteria (namely the Raftery and Lewis, Geweke, and Heidelberger and Welch methods) on a simulated problem and with a rule of thumb of 10,000 chain elements in the Markov chain. We used two publicly available software packages to assess convergence of MCMC parameter estimates; the first performs Bayesian parameter estimation (PKBUGS/WinBUGS), and the second is focused on posterior analysis of estimates (BOA). The main message that seems to emerge is that accurately estimating confidence regions for the parameters of interest is more demanding than estimating the parameter means. Together, these tools provide numerical means by which an investigator can establish confidence in convergence and thus in the estimated parameters derived from hierarchical full Bayesian pharmacokinetic data analysis.

Algorithms↗

PubMed1987

Mono- through hexanucleotide composition of the Escherichia coli genome: a Markov chain analysis.

Several statistical methods were tested for accuracy in predicting observed frequencies of di- through hexanucleotides in 74,444 bp of E. coli DNA. A Markov chain was most accurate overall, whereas other methods, including a random model based on mononucleotide frequencies, were very inaccurate. When ranked highest to lowest abundance, the observed frequencies of oligonucleotides up to six bases in length in E. coli DNA were highly asymmetric. All ordered abundance plots had a wide linear range containing the majority of the oligomers which deviated sharply at the high and low ends of the curves. In general, values predicted by a Markov chain closely followed the overall shape of the ordered abundance curves. A simple equation was derived by which the frequency of any nucleotide longer than four bases in the E. coli genome (or any genome) can be relatively accurately estimated from the nested set of component tri- and tetranucleotides by serial application of a 3rd order Markov chain. The equation yielded a mean ratio of 1.03 +/- 0.94 for the observed-to-expected frequencies of the 4,096 hexanucleotides. Hence, the method is a relatively accurate but not perfect predictor of the length in nucleotides between hexanucleotide sites. Higher accuracy can be achieved using a 4th order Markov chain and larger data sets. The high asymmetry in oligonucleotide abundance means that in the E. coli genome of 4.2 X 10(6) bp many relatively short sequences of 7-9 bp are very rare or absent.

Base Composition↗

PubMed2001

Markov chain Monte Carlo linkage analysis of complex quantitative phenotypes.

We report a Markov chain Monte Carlo analysis of the five simulated quantitative traits in Genetic Analysis Workshop 12 using the Loki software. Our objectives were to determine the efficacy of the Markov chain Monte Carlo method and to test a new scoring technique. Our initial blind analysis, on replicate 42 (the "best replicate") successfully detected four out of the five disease loci and found no false positives. A power analysis shows that the software could usually detect 4 of the 10 trait/gene combinations at an empirical point-wise p-value of 1.5 x 10(-4).

Chromosome Mapping↗

PubMed2003

Exact test of Hardy-Weinberg equilibrium by Markov chain Monte Carlo.

The assumption of Hardy-Weinberg equilibrium (HWE) among alleles is of fundamental importance in genetic studies. There are numerous testing methods for it using genotype counts data. The exact test is used when the sample size is not large enough for asymptotic approximations. There are several numerical methods to carry out this test, such as complete enumeration, Monte Carlo and Markov chain Monte Carlo simulations. Complete enumeration is impractical in many applications, especially when the table counts are large. The Monte Carlo method is simple to use but still difficult when the table counts become large. The Markov chain Monte Carlo method, by sampling a sub-table each time, is suitable for this latter situation. Based on switches among a few (no more than four) cells, the existing Markov chain samplers are highly dependent and inefficient for large tables. Here we consider a new Markov chain sampling, in which a sub-table of user-specified size is updated at each iteration. The resulting chain is less dependent, and the sampling is flexible and efficient. The conventional test for HWE is based on a few test statistics, such as the likelihood and the chi-squared statistic. To expand the family of test statistics, we consider a class of divergence measures for the departure of HWE. Examples are given as illustrations.

Alleles↗