[Show abstract][Hide abstract] ABSTRACT: We consider a model for an epidemic in a population that occupies
geographically distinct locations. The disease is spread within subpopulations
by contacts between infective and susceptible individuals, and is spread
between subpopulations by the migration of infected individuals. We show how
susceptible individuals can act collectively to limit the spread of disease
during the initial phase of an epidemic, by specifying the distribution that
minimises the growth rate of the epidemic when the infectives are migrating so
as to maximise the growth rate. We also give an explicit strategy that
minimises the basic reproduction number, which is also shown be optimal in
terms of the probability of extinction and total size of the epidemic.
Preview · Article · Aug 2015 · Journal of Theoretical Biology
[Show abstract][Hide abstract] ABSTRACT: We study a variant of Hanski's incidence function model that accounts for the
evolution over time of landscape characteristics which affect the persistence
of local populations. In particular, we allow the probability of local
extinction to evolve according to a Markov chain. This covers the widely
studied case where patches are classified as being either suitable or
unsuitable for occupancy. Threshold conditions for persistence of the
population are obtained using an approximating deterministic model that is
realized in the limit as the number of patches becomes large.
[Show abstract][Hide abstract] ABSTRACT: Hanski’s incidence function model is one of the most widely used metapopulation models in ecology. It models the presence/absence of a species at spatially distinct habitat patches as a discrete-time Markov chain whose transition probabilities are determined by the physical landscape. In this analysis, the limiting behaviour of the model is studied as the number of patches increases and the size of the patches decreases. Two different limiting cases are identified depending on whether or not the metapopulation is initially near extinction. Basic properties of the limiting models are derived.
No preview · Article · Jun 2014 · Journal of Applied Probability
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the relationship between certain stochastic and
deterministic versions of Hanski's incidence function model and the spatially
realistic Levins model. We show that the stochastic version can be well
approximated in a certain sense by the deterministic version when the number of
habitat patches is large, provided that the presence or absence of individuals
in a given patch is influenced by a large number of other patches. Explicit
bounds on the deviation between the stochastic and deterministic models are
Preview · Article · Jan 2014 · Journal of Mathematical Biology
[Show abstract][Hide abstract] ABSTRACT: We develop a stochastic metapopulation model that accounts for spatial structure as well as within patch dynamics. Using a deterministic approximation derived from a functional law of large numbers, we develop conditions for extinction and persistence of the metapopulation in terms of the birth, death and migration parameters. Interestingly, we observe the Allee effect in a metapopulation comprising two patches of greatly different sizes, despite there being decreasing patch specific per-capita birth rates. We show that the Allee effect is due to way the migration rates depend on the population density of the patches.
No preview · Article · Nov 2013 · Mathematical biosciences
[Show abstract][Hide abstract] ABSTRACT: We construct a stochastic patch occupancy metapopulation model that incorporates variation in habitat quality and an Allee-like effect. Using some basic results from stochastic ordering, we investigate the effect of habitat degradation on the persistence of the metapopulation. In particular, we show that for a metapopulation with Allee-like effect habitat degradation can cause a dramatic decrease in the level of persistence while in the absence of an Allee-like effect this decrease is more gradual.
No preview · Article · Jan 2013 · Ecological Modelling
[Show abstract][Hide abstract] ABSTRACT: We consider a Markov chain model similar to the stochastic logistic model except that it allows for variation amongst individuals in the population. We prove that as the population size grows, the process can be approximated by a deterministic process. The equilibrium points of the limiting process and their stability are determined. Applications to modelling epidemics and metapopulations are discussed.
No preview · Article · Oct 2012 · Mathematical biosciences
[Show abstract][Hide abstract] ABSTRACT: Metapopulation models have been used to better understand the conditions necessary for the persistence of the metapopulation. In this paper, we study a stochastic patch occupancy model that incorporates variation in quality and connectivity of the habitat patches. Two important assumptions are imposed in our analysis. Firstly, the distance between patches has a special form. This amounts to assuming that migrating individuals follow certain pathways. Secondly, the area of the habitat patches is assumed to scale with the number of patches in the metapopulation. Under these assumptions, a deterministic limit is obtained as the number of patches goes to infinity. Using the deterministic limiting process, a condition for persistence of the metapopulation is derived.
[Show abstract][Hide abstract] ABSTRACT: A discrete-time SIS model is presented that allows individuals in the
population to vary in terms of their susceptibility to infection and their rate
of recovery. This model is a generalisation of the metapopulation model
presented in McVinish and Pollett (2010). The main result of the paper is a
central limit theorem showing that fluctuations in the proportion of infected
individuals around the limiting proportion converges to a Gaussian random
variable when appropriately rescaled. In contrast to the case where there is no
variation amongst individuals, the limiting Gaussian distribution has a nonzero
No preview · Article · Jun 2012 · Journal of Applied Probability
[Show abstract][Hide abstract] ABSTRACT: Since its introduction in the early 90’s, the idea of using importance sampling (IS) with Markov chain Monte Carlo (MCMC) has found many applications. This paper examines problems associated with its application to repeated evaluation of related posterior distributions with a particular focus on Bayesian model validation. We demonstrate that, in certain applications, the curse of dimensionality can be reduced by a simple modification of IS. In addition to providing new theoretical insight into the behaviour of the IS approximation in a wide class of models, our result facilitates the implementation of computationally intensive Bayesian model checks. We illustrate the simplicity, computational savings and potential inferential advantages of the proposed approach through two substantive case studies, notably computation of Bayesian p-values for linear regression models and simulation-based model checking. Supplementary materials including appendices and the R code for Section 3.1.2 are available online.
Full-text · Article · Jan 2012 · Journal of Computational and Graphical Statistics
[Show abstract][Hide abstract] ABSTRACT: In the Bayesian community, an ongoing imperative is to develop efficient
algorithms. An appealing approach is to form a hybrid algorithm by combining
ideas from competing existing techniques. This paper addresses issues in
designing hybrid methods by considering selected case studies: the delayed
rejection algorithm, the pinball sampler, the Metropolis adjusted Langevin
algorithm, and the population Monte Carlo algorithm. We observe that even if
each component of a hybrid algorithm has individual strengths, they may not
contribute equally or even positively when they are combined. Moreover, even if
the statistical efficiency is improved, from a practical perspective there are
technical issues to be considered such as applicability and computational
workload. In order to optimize performance of the algorithm in real time, these
issues should be taken into account.
[Show abstract][Hide abstract] ABSTRACT: To model a fault that can be caused by more than one source, a mixture of conditional Gaussian transitions is proposed. The conditional means are modelled by recurrent neural networks. An expectation-maximization (EM) algorithm is used to estimate model parameters. By grouping known types of faults it is possible to form a bank of different fault models.
[Show abstract][Hide abstract] ABSTRACT: The population Monte Carlo algorithm is an iterative importance sampling scheme for solving static problems. We examine the
population Monte Carlo algorithm in a simplified setting, a single step of the general algorithm, and study a fundamental
problem that occurs in applying importance sampling to high-dimensional problem. The precision of the computed estimate from
the simplified setting is measured by the asymptotic variance of estimate under conditions on the importance function. We
demonstrate the exponential growth of the asymptotic variance with the dimension and show that the optimal covariance matrix
for the importance function can be estimated in special cases.
KeywordsAsymptotic variance of estimate–Central limit theorem–Importance sampling–Markov chain Monte Carlo–Population Monte Carlo
No preview · Article · Jun 2011 · Methodology And Computing In Applied Probability
[Show abstract][Hide abstract] ABSTRACT: Stochastic patch occupancy models (SPOMs) are a class of discrete time Markov chains used to model the presence/absence of a population in a collection of habitat patches. This class of model is popular with ecologists due to its ability to incorporate important factors of the habitat patch network such as connectivity and distance between patches as well as heterogeneity in patch characteristics. We present an asymptotic examination of a simple type of SPOM called the mainland-island model. In this model a single patch called the mainland is connected to a large number of smaller patches called islands and each island is only connected to the mainland. We discuss the limiting behaviour of the SPOM as the number of islands increases and the size of the islands decrease relative to the mainland. We demonstrate that a variety of limiting behaviours is possible depending on the scaling of the island size and on the heterogeneity of habitat quality.
Preview · Article · May 2011 · Journal of Mathematical Biology
[Show abstract][Hide abstract] ABSTRACT: We propose a model for the presence/absence of a population in a collection of habitat patches. This model assumes that colonisation and extinction of the patches occur as distinct phases. Importantly, the local extinction probabilities are allowed to vary between patches. This permits an investigation of the effect of habitat degradation on the persistence of the population. The limiting behaviour of the model is examined as the number of habitat patches increases to ∞. This is done in the case where the number of patches and the initial number of occupied patches increase at the same rate, and for the case where the initial number of occupied patches remains fixed.
Preview · Article · Dec 2010 · Advances in Applied Probability
[Show abstract][Hide abstract] ABSTRACT: A new approximate Bayesian computation (ABC) algorithm is proposed specifically designed for models involving quantile distributions.
The proposed algorithm compares favourably with two other ABC algorithms when applied to examples involving quantile distributions.
KeywordsApproximate Bayesian computation-Likelihood-free inference-Markov chain Monte Carlo-Quantile distributions-Quantile regression
No preview · Article · Nov 2010 · Statistics and Computing
[Show abstract][Hide abstract] ABSTRACT: Harmful algal blooms (HABs) are a worldwide problem that have been increasing in frequency and extent over the past several decades. HABs severely damage aquatic ecosystems by destroying benthic habitat, reducing invertebrate and fish populations, and affecting larger species such as dugong that rely on seagrasses for food. Few statistical models for predicting HAB occurrences have been developed, and in common with most predictive models in ecology, those that have been developed do not fully account for uncertainties in parameters and model structure. This makes management decisions based on these predictions more risky than might be supposed. We used a probit time series model and Bayesian model averaging (BMA) to predict occurrences of blooms of Lyngbya majuscula, a toxic cyanophyte, in Deception Bay, Queensland, Australia. We found a suite of useful predictors for HAB occurrence, with temperature figuring prominently in models with the majority of posterior support, and a model consisting of the single covariate, average monthly minimum temperature, showed by far the greatest posterior support. A comparison of alternative model averaging strategies was made with one strategy using the full posterior distribution and a simpler approach that utilized the majority of the posterior distribution for predictions but with vastly fewer models. Both BMA approaches showed excellent predictive performance with little difference in their predictive capacity. Applications of BMA are still rare in ecology, particularly in management settings. This study demonstrates the power of BMA as an important management tool that is capable of high predictive performance while fully accounting for both parameter and model uncertainty.
Full-text · Article · Oct 2009 · Ecological Applications
[Show abstract][Hide abstract] ABSTRACT: We consider the consistency of the Bayes factor in goodness of fit testing for a parametric family of densities against a non-parametric alternative. Sufficient conditions for consistency of the Bayes factor are determined and demonstrated with priors using certain mixtures of triangular densities.
Full-text · Article · Jun 2009 · Scandinavian Journal of Statistics
[Show abstract][Hide abstract] ABSTRACT: The sensitivity to the specification of the prior in a hidden Markov model describing homogeneous segments of DNA sequences is considered. An intron from the chimpanzee α-fetoprotein gene, which plays an important role in embryonic development in mammals, is analysed. Three main aims are considered: (i) to assess the sensitivity to prior specification in Bayesian hidden Markov models for DNA sequence segmentation; (ii) to examine the impact of replacing the standard Dirichlet prior with a mixture Dirichlet prior; and (iii) to propose and illustrate a more comprehensive approach to sensitivity analysis, using importance sampling. It is obtained that (i) the posterior estimates obtained under a Bayesian hidden Markov model are indeed sensitive to the specification of the prior distributions; (ii) compared with the standard Dirichlet prior, the mixture Dirichlet prior is more flexible, less sensitive to the choice of hyperparameters and less constraining in the analysis, thus improving posterior estimates; and (iii) importance sampling was computationally feasible, fast and effective in allowing a richer sensitivity analysis.
Full-text · Article · Mar 2009 · Computational Statistics & Data Analysis
[Show abstract][Hide abstract] ABSTRACT: In areas such as biology, geology and meteorology data often occur as angles. This paper examines the use of Bayesian mixtures of triangular distributions for the semiparametric analysis of circular data. Applications to density estimation, goodness-of-fit testing and semiparametric regression are demonstrated.
Preview · Article · Jun 2008 · Computational Statistics & Data Analysis