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Results for “Markov Chain”

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PubMed1999

Markov chain models and estimation of absolute progression rates: application to cataract progression in diabetic adults.

BACKGROUND: We present a case study in the use of Markov chain models of disease progression, with exponential regression to model the effects of covariates. METHODS: An exponential regression model was developed for a three-state Markov chain to model progression of cataracts in diabetic patients, with a view to estimation of absolute progression rates. Two methods of estimation were applied, a non-linear least squares approximation, and Markov Chain Monte Carlo (MCMC). RESULTS: Both methods gave estimated transition rates which can readily be transformed to absolute progression probabilities. Agreement was reasonable for most but not all of the parameters. CONCLUSIONS: The MCMC estimates had more conservative variance estimates.

Aged↗
PubMed1999

Bayesian phylogenetic inference via Markov chain Monte Carlo methods.

We derive a Markov chain to sample from the posterior distribution for a phylogenetic tree given sequence information from the corresponding set of organisms, a stochastic model for these data, and a prior distribution on the space of trees. A transformation of the tree into a canonical cophenetic matrix form suggests a simple and effective proposal distribution for selecting candidate trees close to the current tree in the chain. We illustrate the algorithm with restriction site data on 9 plant species, then extend to DNA sequences from 32 species of fish. The algorithm mixes well in both examples from random starting trees, generating reproducible estimates and credible sets for the path of evolution.

Algorithms↗
PubMed2006

Bayesian analysis of correlated evolution of discrete characters by reversible-jump Markov chain Monte Carlo.

We describe a Bayesian method for investigating correlated evolution of discrete binary traits on phylogenetic trees. The method fits a continuous-time Markov model to a pair of traits, seeking the best fitting models that describe their joint evolution on a phylogeny. We employ the methodology of reversible-jump (RJ) Markov chain Monte Carlo to search among the large number of possible models, some of which conform to independent evolution of the two traits, others to correlated evolution. The RJ Markov chain visits these models in proportion to their posterior probabilities, thereby directly estimating the support for the hypothesis of correlated evolution. In addition, the RJ Markov chain simultaneously estimates the posterior distributions of the rate parameters of the model of trait evolution. These posterior distributions can be used to test among alternative evolutionary scenarios to explain the observed data. All results are integrated over a sample of phylogenetic trees to account for phylogenetic uncertainty. We implement the method in a program called RJ Discrete and illustrate it by analyzing the question of whether mating system and advertisement of estrus by females have coevolved in the Old World monkeys and great apes.

Animals↗
PubMed1998

Evaluation by Markov chain models of a non-randomised breast cancer screening programme in women aged under 50 years in Sweden.

STUDY OBJECTIVE: To apply Markov chain models that have previously been used on data in randomised trials of breast cancer screening to data from an uncontrolled service screening programme; to compare results with those from a randomised trial. DESIGN: A service screening programme in Uppsala county, Sweden, inviting 25,660 women aged 39-49 to mammographic screening every 20 months, and the Swedish Two-County Trial inviting 19,844 women aged 40-49 to two yearly screening, compared with 15,604 women of the same age in an unscreened control group. Data were analysed using Markov chain models and quasi-likelihood estimation procedures. MAIN RESULTS: The results with respect to parameters of disease progression were very similar between the two studies. Use of estimated progression rates to predict the effect on mortality ranged from a 10% to a 17% reduction in breast cancer mortality in the Two-County Study and predicted a 15% reduction in the Uppsala programme. These compare well with the observed mortality reduction of 13% in the Two-County Trial. CONCLUSIONS: The screening in the Uppsala programme is likely to have a similar effect to that observed in the Two-County Trial. It is feasible to evaluate non-randomised service screening programmes using Markov chain models.

Adult↗
PubMed2004

Improved spike-sorting by modeling firing statistics and burst-dependent spike amplitude attenuation: a Markov chain Monte Carlo approach.

Spike-sorting techniques attempt to classify a series of noisy electrical waveforms according to the identity of the neurons that generated them. Existing techniques perform this classification ignoring several properties of actual neurons that can ultimately improve classification performance. In this study, we propose a more realistic spike train generation model. It incorporates both a description of "nontrivial" (i.e., non-Poisson) neuronal discharge statistics and a description of spike waveform dynamics (e.g., the events amplitude decays for short interspike intervals). We show that this spike train generation model is analogous to a one-dimensional Potts spin-glass model. We can therefore tailor to our particular case the computational methods that have been developed in fields where Potts models are extensively used, including statistical physics and image restoration. These methods are based on the construction of a Markov chain in the space of model parameters and spike train configurations, where a configuration is defined by specifying a neuron of origin for each spike. This Markov chain is built such that its unique stationary density is the posterior density of model parameters and configurations given the observed data. A Monte Carlo simulation of the Markov chain is then used to estimate the posterior density. We illustrate the way to build the transition matrix of the Markov chain with a simple, but realistic, model for data generation. We use simulated data to illustrate the performance of the method and to show that this approach can easily cope with neurons firing doublets of spikes and/or generating spikes with highly dynamic waveforms. The method cannot automatically find the "correct" number of neurons in the data. User input is required for this important problem and we illustrate how this can be done. We finally discuss further developments of the method.

Action Potentials↗
PubMed2006

Numerical characterization of DNA sequences based on the k-step Markov chain transition probability.

A DNA sequence can be regarded as a discrete-time Markov chain. Based on k-step transition probabilities, we construct a series of 4 x 4 k-step transition matrices to characterize the DNA primary sequences. According to the properties of Markov chains, we obtain distributions of A, T, C and G, and analyze the changes among them from yesterday to tomorrow. We can calculate the probabilities of nucleotide triples of DNA primary sequences. Finally, we introduce a correlation of this kind of transition matrices and consider it as an invariant to analyze the similarities/dissimilarities of DNA sequences.

Adenosine↗
PubMed2003

Finding starting points for Markov chain Monte Carlo analysis of genetic data from large and complex pedigrees.

Genetic data from founder populations are advantageous for studies of complex traits that are often plagued by the problem of genetic heterogeneity. However, the desire to analyze large and complex pedigrees that often arise from such populations, coupled with the need to handle many linked and highly polymorphic loci simultaneously, poses challenges to current standard approaches. A viable alternative to solving such problems is via Markov chain Monte Carlo (MCMC) procedures, where a Markov chain, defined on the state space of a latent variable (e.g., genotypic configuration or inheritance vector), is constructed. However, finding starting points for the Markov chains is a difficult problem when the pedigree is not single-locus peelable; methods proposed in the literature have not yielded completely satisfactory solutions. We propose a generalization of the heated Gibbs sampler with relaxed penetrances (HGRP) of Lin et al., ([1993] IMA J. Math. Appl. Med. Biol. 10:1-17) to search for starting points. HGRP guarantees that a starting point will be found if there is no error in the data, but the chain usually needs to be run for a long time if the pedigree is extremely large and complex. By introducing a forcing step, the current algorithm substantially reduces the state space, and hence effectively speeds up the process of finding a starting point. Our algorithm also has a built-in preprocessing procedure for Mendelian error detection. The algorithm has been applied to both simulated and real data on two large and complex Hutterite pedigrees under many settings, and good results are obtained. The algorithm has been implemented in a user-friendly package called START.

Algorithms↗
PubMedDate not supplied

[Markov chain analysis of the movement of geriatric pensioners].

The movements of geriatric patients through the eight health zones of Canton Vaud (Switzerland) are analyzed. The procedure for modelling the patients' moves between various states with a Markov chain is detailed and the results presented. Markov chain analysis may constitute a predicting model. Movement prediction may then assist in delineating various health areas. For the time being, the limits of health areas are usually decided upon following conventional administrative patterns without any regard to reality. Markov chain analysis could be of assistance in rationalizing this process.

Aged↗
PubMed1987

A multistate Markov chain model for evaluating a sterilization policy.

"A multistate Markov chain model corresponding to varying fertility and mortality rates at different levels of surviving children of a couple was developed. Asymptotic probabilities of having a fixed number of children have been worked out." The implied geographical focus is on India.

Asia↗
PubMed1988

The use of Markov chain models in studying the evolution of the proteins.

In a recent paper Giulio & Caldararo (1987) used a Markov chain model to study the evolution of proteins. Unfortunately, their use of a first-order Markov chain model at the amino acid level is incorrect. The model has to be applied at the codon level [Jorre & Curnow (1975a)] followed by amalgamation of the codon states corresponding to each amino acid and of the three codons specifying termination. The model is correctly applied in this paper. The results obtained do not differ substantially from those obtained by Giulio & Caldararo (1987). The interpretation of the results as supporting the neutralist view of protein evolution is criticized.

Amino Acids↗
PubMed1996

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

Journal Article↗
PubMed1977

Polydisperse particulate solids mixing and segregation: nonstationary Markov chains.

The feasibility of analyzing interparticulate translocations within agitated beds of polydisperse particulate solids in terms of Markov chains was investigated. A binary mixture of spherical particles subjected to vertical sine wave vibration is shown to behave in accordance with a nonstationary. Markov chain having singly stochastic transition matrixes. The transition probability elements of such matrixes, calculated from a knowledge of the initial and final occupancy vectors, agree with those estimated using tracer particles. With appropriate restrictions, the former method, based on the solution of simultaneous equations, permits a quantitative evaluation of particle mobilities throughout the bed.

Chemistry, Pharmaceutical↗
PubMed2003

Evaluation of a stochastic Markov-chain model for the development of forelimb injuries in Thoroughbred racehorses.

OBJECTIVE: To evaluate a Markov-chain model for the development of forelimb injuries in Thoroughbreds and to use the model to determine effects of reducing sprint distance on incidence of metacarpal condylar fracture (CDY) and severe suspensory apparatus injury (SSAI). SAMPLE POPULATION: Weekly exercise and injury data for 122 Thoroughbreds during racing or training. PROCEDURE: Weekly data were used to construct a Markov-chain model with 5 states (uninjured [UNINJ], palpable suspensory apparatus injury [PSAII, SSAI, CDY, and lost to follow-up [LOST]). Transition probabilities between UNINJ and PSAI were estimated as a function of weekly sprint distance by use of linear regression analysis. The model was used to predict distributions of annual CDY and SSAI incidences in southern California racehorses and was validated by using CDY incidence reported by racetrack practitioners. The model was modified by reducing the number of sprint distances that were > 6 furlongs (> 1.20 km) by 20%, and CDY and SSAI incidences were compared with those generated by the baseline model. RESULTS: The model accurately fit development of injuries in the sample population but overestimated development of injuries in the southern California racehorse population. Development of and recovery from PSAI were correlated with distance run at high speeds. Reducing by 20% the number of sprints run at distances > 6 furlongs significantly reduced modeled annual CDY and SSAI incidence by 9%. CONCLUSIONS AND CLINICAL RELEVANCE: Reducing the number of sprints at distances > 6 furlongs, particularly among horses with PSAI, reduces risk of CDY and SSAI.

Animals↗
PubMed1999

Evaluation of screening for nasopharyngeal carcinoma: trial design using Markov chain models.

In this paper, we develop a Markov chain model to estimate parameters pertaining to the natural history of nasopharyngeal carcinoma (NPC). The model is of progression from no disease to Epstein-Barr virus (EBV) infection, preclinical screen-detectable tumour and clinical tumour. We derive tentative estimates of the parameters of the model, based on limited published data, to assess the efficacy of serum screening in conjunction with clinical assessment (indirect mirror examination for NPC), for example the average duration of the preclinical screen-detectable phase is estimated as 3.1 years. We further apply these parameters to a hypothetical screening trial in the Hong Kong population to assess the efficacy of serum screening with clinical assessment by different combinations of screening regime. Results suggest: (1) there is no substantial difference between 3-yearly and 6-yearly serum screening; and (2) within the same serum screening regime annual and 3-yearly clinical assessment can prevent 33% and 28% of deaths from NPC respectively. Prediction of deaths and surrogate end points can be used to estimate the required sample size and duration for designing a randomized trial of screening for NPC. Based on these findings and power projections, we suggest a design for a randomized trial in a high incidence area such as Hong Kong.

Disease Progression↗
PubMed2001

Bayesian selection of continuous-time Markov chain evolutionary models.

We develop a reversible jump Markov chain Monte Carlo approach to estimating the posterior distribution of phylogenies based on aligned DNA/RNA sequences under several hierarchical evolutionary models. Using a proper, yet nontruncated and uninformative prior, we demonstrate the advantages of the Bayesian approach to hypothesis testing and estimation in phylogenetics by comparing different models for the infinitesimal rates of change among nucleotides, for the number of rate classes, and for the relationships among branch lengths. We compare the relative probabilities of these models and the appropriateness of a molecular clock using Bayes factors. Our most general model, first proposed by Tamura and Nei, parameterizes the infinitesimal change probabilities among nucleotides (A, G, C, T/U) into six parameters, consisting of three parameters for the nucleotide stationary distribution, two rate parameters for nucleotide transitions, and another parameter for nucleotide transversions. Nested models include the Hasegawa, Kishino, and Yano model with equal transition rates and the Kimura model with a uniform stationary distribution and equal transition rates. To illustrate our methods, we examine simulated data, 16S rRNA sequences from 15 contemporary eubacteria, halobacteria, eocytes, and eukaryotes, 9 primates, and the entire HIV genome of 11 isolates. We find that the Kimura model is too restrictive, that the Hasegawa, Kishino, and Yano model can be rejected for some data sets, that there is evidence for more than one rate class and a molecular clock among similar taxa, and that a molecular clock can be rejected for more distantly related taxa.

Animals↗
PubMed1985

The N of 1 Markov chain design as a means of studying the stages of psychotherapy.

Research on the process or, more specifically, the stages of psychotherapy has been difficult to carry out and can lead to spurious results. An initial step in furthering knowledge of the stages of successful therapy would be the use of an N of 1 design with Markov chain analyses. This design would allow one to incorporate the uniqueness of each therapy dyad, the interactional nature of the psychotherapeutic process, and a statistical test of the changes in the interactional behavior over time that is not often available in nonbehavioral single case designs. The purpose of this paper is to familiarize the reader with N of 1 designs and Markov chain methodologies and to propose how they could be combined in an attempt to study the stages of successful therapy. A hypothetical study is presented to exemplify the operation and value of the proposed N of 1 Markov chain design.

Dominance-Subordination↗
PubMed2005

Searching on patch networks using correlated random walks: space usage and optimal foraging predictions using Markov chain models.

We describe a novel representation of a discrete correlated random walk as the transition matrix of a Markov chain with the displacements as the states. Such a representation makes it possible to utilize results from the theory of absorbing Markov chains, to make biologically interesting predictions without having to resort to Monte Carlo simulations. Our motivation for constructing such a representation is to explore the relationship between the movement strategy of an animal searching for resources upon a network of patches, and its consequent utilization of space and foraging success. As an illustrative case study, we have determined the optimal movement strategy and the consequent usage of space for a central place forager utilizing a continuous movement space which is discretized as a hexagonal lattice. The optimal movement strategy determines the size of the optimal home range. In this example, the animal uses mnemokinesis, which is a sinuosity regulating mechanism, to return it to the central place. The movement strategy thus refers to the choice of the intrinsic path sinuosity and the strength of the mnemokinetic mechanism. Although the movement space has been discretized as a regular lattice in this example, the method can be readily applied to naturally compartmentalized movement spaces, such as forest canopy networks. This paper is thus an attempt at incorporating results from the theory of random walk-based animal movements into Foraging Theory.

Animals↗
PubMed2005

The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain.

Localized Ca(2+) elevations known as Ca(2+) puffs and sparks are cellular signals that arise from the cooperative activity of clusters of inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca(2+) release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of these intracellular Ca(2+)-regulated Ca(2+) channels are coupled via a mathematical representation of Ca(2+) microdomain, simulated Ca(2+) release sites may exhibit the phenomenon of "stochastic Ca(2+) excitability" where the inositol 1,4,5-trisphosphate receptors (IP(3)Rs) or ryanodine receptors (RyRs) open and close in a concerted fashion. Interestingly, under some conditions simulated puffs and sparks can be observed even when the single-channel model used does not include slow Ca(2+) inactivation or, indeed, any long-lived closed/refractory state [V. Nguyen, R. Mathias, G. Smith, Stochastic automata network descriptor for Markov chain models of instantaneously-coupled intracellular Ca(2+) channels, Bull. Math. Biol. 67 (2005) 393-432]. In this case, termination of the localized Ca(2+) elevation occurs when all of the intracellular channels at a release site simultaneously close through a process referred to as stochastic attrition [M. Stern, Theory of excitation-contraction coupling in cardiac muscle, Biophys. J. 63 (1992) 497-517]. In this paper, we investigate the statistical properties of stochastic attrition viewed as an absorption time on a terminating Markov chain that represents a Ca(2+) release site composed of N two-state channels that are activated by Ca(2+). Assuming that the local [Ca(2+)] experienced by a channel depends only on the number of open channels at the Ca(2+) release site (i.e., instantaneous mean-field coupling [ibid.], we derive the probability distribution function for the time until stochastic attrition occurs and present an analytical formula for the expectation of this random variable. We explore how the contribution of stochastic attrition to the termination of Ca(2+) puffs and sparks depends on the number of channels at a release site, the source amplitude of the channels (i.e., the strength of the coupling), the background [Ca(2+)], channel kinetics, and the cooperactivity of Ca(2+) binding. Because we explicitly model the Ca(2+) regulation of the intracellular channels, our results differ markedly from (and in fact generalize) preliminary analyses that assume the intracellular Ca(2+) channels are uncoupled and consequently independent.

Animals↗