A Markov Chain Model of Population Growth with an Application in Epidemiology

This paper is concerned with a class of population growth processes in discrete time; the simple epidemic process is considered as a specific example. A Markov chain model is constructed and standard Markov methods are used to study the main biological concepts. A simple and explicit formula is obtained for the transient distribution of the population size. Then, the cost of the process is defined and the joint probability generating function of its components is derived. Finally, the results are extended to the case where the inter-transition periods are bounded i.i.d. random variables.     

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.     

Past states of continuous-time Markov models for ecological communities

Discrete-time Markov chains are often used to model communities of sessile organisms. The community is described by a set of discrete states, which may represent species or groups of species. Transitions between states are modelled using a stochastic matrix. A recent study showed how the time-reversal of such a Markov chain can be used to estimate the distribution of time since the last occurrence of some state of interest (such as empty space) at a point, given the current state of the point. However, if the underlying process operates in continuous time but is observed at regular intervals, this distribution describes the time since the last possible observation of the state of interest, rather than the time since its last occurrence. We show how to obtain the distribution of time since the last occurrence of a state of interest for a continuous-time homogeneous Markov chain. The expected time since the last occurrence of an initial state can be interpreted as a measure of the successional rank of a state. We show how to distinguish between different ways in which a state can have high successional rank. We apply our results to a marine subtidal community.     

Besetzungsmodelle

Starting with the classical occupancy with boxes, balls and probabilities for occupying boxes, we employ new models: a ball, which met a box, occupies this box with a probability depending on the size of box content, and - using several kinds of balls - by the composition of the content too. For that purpose several models are constructed with distinctive forms of the conditions. Dealing with these models - they are special MARKO vian chains - we pay attention mainly to the so called first passage time T_z for defined events Z; in particular for certain events Z we have to prove that T_z is a random variable, and have to set up the appropriate probability generating function.     

samc: an R package for connectivity modeling with spatial absorbing Markov chains

Quantifying landscape connectivity is fundamental to better understand and predict how populations respond to environmental change. Currently, popular methods to quantify landscape connectivity emphasize how landscape features provide resistance to movement. While many tools are available to quantify landscape resistance, these do not discern between two fundamentally different sources of resistance: movement behavior and mortality. To address this issue, we developed the samc R package that quantifies landscape connectivity using absorbing Markov chain theory. Within this mathematical framework, movements are represented as transient states in the Markov chain, while mortality is represented by transitions to absorbing states. Not only does this framework explicitly account for these different issues, it provides a probabilistic approach that can incorporate both short-term and long-term dynamics, as well as species distribution and abundance. The package includes functions to quantify life expectancy, long-term visitation rates, and various spatially and temporally explicit measures of mortality and movement at the local and landscape scales. These functions in samc have been optimized to find computationally practical solutions in landscapes comprised of > 2 × 10⁶ cells. Here, we illustrate the workflow of the samc package with publicly available movement and mortality data on the endangered Florida panther Puma concolor coryi. This analysis showed that movement and mortality are generally correlated except for locations near roads (areas of high mortality risk) that are within the dispersal range of source locations. This pattern would have been undetectable with current methods that quantify movement resistance. Overall, the samc package provides a means for implementing spatial absorbing Markov chains that can distinguish between movement behavior and mortality resulting in more reliable landscape connectivity measures.     

Bayesian analysis of discrete survival data with a hidden Markov chain

This paper considers the discrete survival data from a Bayesian point of view. A sequence of the baseline hazard functions, which plays an important role in the discrete hazard function, is modeled with a hidden Markov chain. It is explained how the resultant model is implemented via Markov chain Monte Carlo methods. The model is illustrated by an application of real data.     

Exact Markov chains versus diffusion theory for haploid random mating

Exact discrete Markov chains are applied to the Wright-Fisher model and the Moran model of haploid random mating. Selection and mutations are neglected. At each discrete value of time t there is a given number n of diploid monoecious organisms. The evolution of the population distribution is given in diffusion variables, to compare the two models of random mating with their common diffusion limit. Only the Moran model converges uniformly to the diffusion limit near the boundary. The Wright-Fisher model allows the population size to change with the generations. Diffusion theory tends to under-predict the loss of genetic information when a population enters a bottleneck.     

Bayesian Bootstrap Clones for Censored Markov Chains

In this article, we consider r observations from a non‐homogeneous censored Markov chain, with transition probability matrix P. For the product estimator of P proposed by Aalen and Johansen (1978) and Phelan (1988), we investigate the behavior of Bayesian bootstrap clones to approximate the sampling distribution of , and then construct approximate confidence interval. It is shown that the approximation based on the random‐weighted distribution is first‐order consistent. The performance of the Bayesian bootstrap clones (BBC) is also discussed by small sample simulation. Finally, we illustrate the BBC procedure in the application to the WHO malaria survey data (cf. Singer and Cohen 1970).     

Estimation of amino acid residue substitution rates at local spatial regions and application in protein function inference: a Bayesian Monte Carlo approach

The amino acid sequences of proteins provide rich information for inferring distant phylogenetic relationships and for predicting protein functions. Estimating the rate matrix of residue substitutions from amino acid sequences is also important because the rate matrix can be used to develop scoring matrices for sequence alignment. Here we use a continuous time Markov process to model the substitution rates of residues and develop a Bayesian Markov chain Monte Carlo method for rate estimation. We validate our method using simulated artificial protein sequences. Because different local regions such as binding surfaces and the protein interior core experience different selection pressures due to functional or stability constraints, we use our method to estimate the substitution rates of local regions. Our results show that the substitution rates are very different for residues in the buried core and residues on the solvent-exposed surfaces. In addition, the rest of the proteins on the binding surfaces also have very different substitution rates from residues. Based on these findings, we further develop a method for protein function prediction by surface matching using scoring matrices derived from estimated substitution rates for residues located on the binding surfaces. We show with examples that our method is effective in identifying functionally related proteins that have overall low sequence identity, a task known to be very challenging.     

应用马尔科夫链模型对草地螟发生程度的预测

草地螟Loxostege stictialis L.是我国北方农牧业生产上一种重要迁飞性、暴发性害虫,一旦暴发会给当地农牧生产造成严重危害.根据康保县1977-2008年1代草地螟幼虫发生程度的时间序列资料,应用马尔科夫链的转移概率预测法,构建了1～3阶转移概率矩阵,组建模型对该县2009-2011年1代草地螟发生程度进行了预测,结果与大田实际发生情况完全一致,准确率100％.对1980-2011年的历史资料进行回检,历史符合率89.9％,该方法可对草地螟进行长期预报,为草地螟长期预报提供了一种准确有效的方法,对草地螟发生程度的长期预报具有重要指导意义.     

A tutorial introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods

Recent Bayesian methods for the analysis of infectious disease outbreak data using stochastic epidemic models are reviewed. These methods rely on Markov chain Monte Carlo methods. Both temporal and non-temporal data are considered. The methods are illustrated with a number of examples featuring different models and datasets.     

MLE and Bayesian inference of age-dependent sensitivity and transition probability in periodic screening

This article extends previous probability models for periodic breast cancer screening examinations. The specific aim is to provide statistical inference for age dependence of sensitivity and the transition probability from the disease free to the preclinical state. The setting is a periodic screening program in which a cohort of initially asymptomatic women undergo a sequence of breast cancer screening exams. We use age as a covariate in the estimation of screening sensitivity and the transition probability simultaneously, both from a frequentist point of view and within a Bayesian framework. We apply our method to the Health Insurance Plan of Greater New York study of female breast cancer and give age-dependent sensitivity and transition probability density estimates. The inferential methodology we develop is also applicable when analyzing studies of modalities for early detection of other types of progressive chronic diseases.     

Use of Growth Indices in Markov Chain Models for Crop Yield Forecasting

An improved Markov chain model has been developed for forecasting of sugarcane yields in which growth indices of biometrical characters based on data from two stages simultaneously have been utilised. Comparisons were also made with the models in use viz. the regression model and the first order Markov chain model.