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Computation of identity by descent probabilities conditional on DNA markers via a Monte Carlo Markov Chain method.

The accurate estimation of the probability of identity by descent (IBD) at loci or genome positions of interest is paramount to the genetic study of quantitative and disease resistance traits. We present a Monte Carlo Markov Chain method to compute IBD probabilities between individuals conditional on DNA markers and on pedigree information. The IBDs can be obtained in a completely general pedigree at any genome position of interest, and all marker and pedigree information available is used. The method can be split into two steps at each iteration. First, phases are sampled using current genotypic configurations of relatives and second, crossover events are simulated conditional on phases. Internal track is kept of all founder origins and crossovers such that the IBD probabilities averaged over replicates are rapidly obtained. We illustrate the method with some examples. First, we show that all pedigree information should be used to obtain line origin probabilities in F2 crosses. Second, the distribution of genetic relationships between half and full sibs is analysed in both simulated data and in real data from an F2 cross in pigs.

Journal Article↗
PubMed2002

A DNA sequence evolution analysis generalized by simulation and the markov chain monte carlo method implicates strand slippage in a majority of insertions and deletions.

To study the mechanisms for local evolutionary changes in DNA sequences involving slippage-type insertions and deletions, an alignment approach is explored that can consider the posterior probabilities of alignment models. Various patterns of insertion and deletion that can link the ancestor and descendant sequences are proposed and evaluated by simulation and compared by the Markov chain Monte Carlo (MCMC) method. Analyses of pseudogenes reveal that the introduction of the parameters that control the probability of slippage-type events markedly augments the probability of the observed sequence evolution, arguing that a cryptic involvement of slippage occurrences is manifested as insertions and deletions of short nucleotide segments. Strikingly, approximately 80% of insertions in human pseudogenes and approximately 50% of insertions in murids pseudogenes are likely to be caused by the slippage-mediated process, as represented by BC in ABCD --> ABCBCD. We suggest that, in both human and murids, even very short repetitive motifs, such as CAGCAG, CACACA, and CCCC, have approximately 10- to 15-fold susceptibility to insertions and deletions, compared to nonrepetitive sequences. Our protocol, namely, indel-MCMC, thus seems to be a reasonable approach for statistical analyses of the early phase of microsatellite evolution.

Base Sequence↗
PubMed2004

Markov-chain Monte Carlo approach to the design of multilayer thin-film optical coatings.

We reconsider the problem of locating the globally optimal solution of a multilayer-optical-coating design problem, within some predetermined space of parameters, with the aim of obtaining a robust technique that requires a minimum of user intervention. The approach we adopt centers on exploring the space of the parameters of interest by using a Markov-chain Monte Carlo sampling algorithm. This technique enables one to locate the global optimum automatically with high confidence and without the need for a good starting design. It also allows the trivial inclusion of prior constraints on the variables and provides a natural means for investigating the robustness of the optimal solution.

Journal Article↗
PubMed1998

A stochastic approximation algorithm with Markov chain Monte-carlo method for incomplete data estimation problems.

We propose a general procedure for solving incomplete data estimation problems. The procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model, the nonlinear structural measurement error model, and the random effects model. The procedure is based on the general principle of stochastic approximation and the Markov chain Monte-Carlo method. Applying the theory on adaptive algorithms, we derive conditions under which the proposed procedure converges. Simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model.

Journal Article↗
PubMed2003

Correlations between parameters in risk models: estimation and propagation of uncertainty by Markov Chain Monte Carlo.

Monte Carlo simulation has become the accepted method for propagating parameter uncertainty through risk models. It is widely appreciated, however, that correlations between input variables must be taken into account if models are to deliver correct assessments of uncertainty in risk. Various two-stage methods have been proposed that first estimate a correlation structure and then generate Monte Carlo simulations, which incorporate this structure while leaving marginal distributions of parameters unchanged. Here we propose a one-stage alternative, in which the correlation structure is estimated from the data directly by Bayesian Markov Chain Monte Carlo methods. Samples from the posterior distribution of the outputs then correctly reflect the correlation between parameters, given the data and the model. Besides its computational simplicity, this approach utilizes the available evidence from a wide variety of structures, including incomplete data and correlated and uncorrelated repeat observations. The major advantage of a Bayesian approach is that, rather than assuming the correlation structure is fixed and known, it captures the joint uncertainty induced by the data in all parameters, including variances and covariances, and correctly propagates this through the decision or risk model. These features are illustrated with examples on emissions of dioxin congeners from solid waste incinerators.

Journal Article↗
PubMed2003

Cool walking: a new Markov chain Monte Carlo sampling method.

Effective relaxation processes for difficult systems like proteins or spin glasses require special simulation techniques that permit barrier crossing to ensure ergodic sampling. Numerous adaptations of the venerable Metropolis Monte Carlo (MMC) algorithm have been proposed to improve its sampling efficiency, including various hybrid Monte Carlo (HMC) schemes, and methods designed specifically for overcoming quasi-ergodicity problems such as Jump Walking (J-Walking), Smart Walking (S-Walking), Smart Darting, and Parallel Tempering. We present an alternative to these approaches that we call Cool Walking, or C-Walking. In C-Walking two Markov chains are propagated in tandem, one at a high (ergodic) temperature and the other at a low temperature. Nonlocal trial moves for the low temperature walker are generated by first sampling from the high-temperature distribution, then performing a statistical quenching process on the sampled configuration to generate a C-Walking jump move. C-Walking needs only one high-temperature walker, satisfies detailed balance, and offers the important practical advantage that the high and low-temperature walkers can be run in tandem with minimal degradation of sampling due to the presence of correlations. To make the C-Walking approach more suitable to real problems we decrease the required number of cooling steps by attempting to jump at intermediate temperatures during cooling. We further reduce the number of cooling steps by utilizing "windows" of states when jumping, which improves acceptance ratios and lowers the average number of cooling steps. We present C-Walking results with comparisons to J-Walking, S-Walking, Smart Darting, and Parallel Tempering on a one-dimensional rugged potential energy surface in which the exact normalized probability distribution is known. C-Walking shows superior sampling as judged by two ergodic measures.

Journal Article↗
PubMed1987

The effect of codon usage on the oligonucleotide composition of the E. coli genome and identification of over- and underrepresented sequences by Markov chain analysis.

As shown in the accompanying paper (5), the oligonucleotide composition of the E. coli genome is highly asymmetric for sequences up to 6 bp in length when ranked from highest to lowest abundance. We show here that this largely reflects codon usage because heavily used codons were found in the highly abundant oligomers whereas rarely used codons, with some exceptions, occurred in sequences in low abundance. Furthermore, linear regression analysis revealed a strong correlation between the frequencies of each trinucleotide and its usage as a codon. Dinucleotides are also not randomly distributed across each codon position and the dinucleotide composition of genes that are transcribed but not translated (rRNA and tRNA genes) was highly related to that seen in genes encoding polypeptides. However, 45 tetra-, 8 penta-, and 6 hexanucleotides were significantly over- or underabundant by Markov chain analysis and could not be accounted for by codon usage. Of these underrepresented sequences, many were palindromes, including the Dam methylation site.

Base Sequence↗
PubMed2004

Algorithms for variable length Markov chain modeling.

UNLABELLED: 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. AVAILABILITY: The source code is freely available at http://www.soe.ucsc.edu/~jill/src/

Algorithms↗
PubMed2006

Numerical solutions for patterns statistics on Markov chains.

We propose here a review of the methods available to compute pattern statistics on text generated by a Markov source. Theoretical, but also numerical aspects are detailed for a wide range of techniques (exact, Gaussian, large deviations, binomial and compound Poisson). The SPatt package (Statistics for Pattern, free software available at http://stat.genopole.cnrs.fr/spatt) implementing all these methods is then used to compare all these approaches in terms of computational time and reliability in the most complete pattern statistics benchmark available at the present time.

Algorithms↗