Markov Approaches Research Articles

Modeling linkage disequilibrium in exact linkage computations: a comparison of first-order Markov approaches and the clustered-markers approach

Recent studies have shown that linkage disequilibrium (LD) between single-nucleotide polymorphism (SNP) markers is widespread. Assuming linkage equilibrium has been shown to cause false positives in linkage studies where parental genotypes are not available. Therefore, linkage analysis methods that can deal with LD are required to accurately analyze SNP marker data sets. We compared three approaches to deal with LD between markers: 1) The clustered-markers approach implemented in the computer program MERLIN; 2) The standard hidden Markov model (HMM) multipoint model augmented with a first-order Markov model for the allele frequencies of the founders, in which we considered both a Bayesian and a maximum-likelihood implementation of this approach; 3) The 'independent' SNPs approach, i.e., removing SNPs from the data set until the remaining SNPs have low levels of LD.We evaluated these approaches on the Illumina 6K SNP data set of affected sib-pairs of Problem 2. We found that the first-order Markov model was able to account for most of the strong LD in this data set. The difference between the Bayesian and maximum- likelihood implementation was small. An advantage of the first-order Markov model is that it does not require the user to specify parameters.

A binary decision diagram method for phased mission analysis of non-repairable systems

Phased mission analysis is carried out to predict the reliability of systems which undergo a series of phases, each with differing requirements for success, with the mission objective being achieved only on the successful completion of all phases. Many systems from a range of industries experience such missions. The methods used for phased mission analysis are dependent upon the repairability of the system during the phases. If the system is non-repairable, fault-tree-based methods offer an efficient solution. For repairable systems, Markov approaches can be used. This paper is concerned with the analysis of non-repairable systems. When the phased mission failure causes are represented using fault trees, it is shown that the binary decision diagram (BDD) method of analysis offers advantages in the solution process. A new way in which BDD models can be efficiently developed for phased mission analysis is proposed. The paper presents a methodology by which the phased mission models can be developed and analysed to produce the phase failure modes and the phase failure likelihoods.

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.

Quantum Brownian particle and memory effects.

The quantum Brownian particle, immersed in a heat bath, is described by a statistical operator whose evolution is ruled by a generalized master equation (GME). The heat bath's degrees of freedom are considered to be either white-noise or colored-noise correlated, while the GME is considered under either the Markov or non-Markov approaches. The comparisons between these considerations are fully developed, and their physical meaning is discussed.

A physiologically based continuous-time Markov approach to plant growth modelling in semi-arid woodlands

Abstract A model for simulating the combined effects of water, light and nutrients on tree, shrub and grass growth in a semi-arid woodland is described. Continuous-time Markov (CTM) assumptions are used to derive a plant growth model based on general resource availability. Physiological interpretation of states and transitions facilitates the use of the abstract continuous-time Markov approaches. The model treats resources as an interacting group, and provides an approach by which a single equation can be used across all growth forms. Simulation results for a range of leaf areas and diameters for trees, shrubs and grasses agree with field observations. The poor relationship between leaf area and diameter of plants observed in intact semi-arid woodlands is interpreted as a consequence of competitive interactions for resources.

Placing a Value on Human-Resources: A Markov Approach

Abstract A method of placing a monetary value on the human resources of an organization is explained. The purpose is to improve the quality of managerial decisions that concern people. Markov approaches, both stationary and time-varying, are used in combination with life analysis, a technique useful in industrial property valuation. This application illustrates a use of the modern theory of Markov chains utilizing stopping times.

A new approach for treating fluctuations in noise theory

A new method, introduced in a previous article, is shown to overcome certain physical difficulties encountered in noise problems when treated by more common approaches. In particular, an internal inconsistency in the original Langevin approach for treating Brownian motion is explicitly presented in which problems arise because the statistical treatment is carried out at a too macroscopic level. As another example, a more realistic expression for the retrogression of a fluctuation is obtained in which the cusp at t=0, characteristic of Markov approaches, is avoided. The method is extended to treating systems of subsystems, and the concept of internal noise is introduced to enable a simpler, more macroscopic noise analysis. The method is also applied to determine statistical properties of fluctuations in irreversible thermodynamcis. We hope that our discussion will further illustrate the origin, nature, and treatment of fluctuations in linear classical systems.