Stephen P. Brooks

Optimization using simulated annealing

Much work has been published on the theoretical aspects of simulated annealing. This paper provides a brief overview of this theory and provides an introduction to the practical aspects of function optimization using this approach. Different implementations of the general simulated annealing algorithm are discussed, and two examples are used to illustrate the behaviour of the algorithm in low dimensions. A third example illustrates a hybrid approach, combining simulated annealing with traditional techniques.

Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK

Foot-and-mouth disease (FMD) in the UK provides an ideal opportunity to explore optimal control measures for an infectious disease. The presence of fine-scale spatio-temporal data for the 2001 epidemic has allowed the development of epidemiological models that are more accurate than those generally created for other epidemics and provide the opportunity to explore a variety of alternative control measures. Vaccination was not used during the 2001 epidemic; however, the recent DEFRA (Department for Environment Food and Rural Affairs) contingency plan details how reactive vaccination would be considered in future. Here, using the data from the 2001 epidemic, we consider the optimal deployment of limited vaccination capacity in a complex heterogeneous environment. We use a model of FMD spread to investigate the optimal deployment of reactive ring vaccination of cattle constrained by logistical resources. The predicted optimal ring size is highly dependent upon logistical constraints but is more robust to epidemiological parameters. Other ways of targeting reactive vaccination can significantly reduce the epidemic size; in particular, ignoring the order in which infections are reported and vaccinating those farms closest to any previously reported case can substantially reduce the epidemic. This strategy has the advantage that it rapidly targets new foci of infection and that determining an optimal ring size is unnecessary.

Accuracy of models for the 2001 foot-and-mouth epidemic

Since 2001 models of the spread of foot-and-mouth disease, supported by the data from the UK epidemic, have been expounded as some of the best examples of problem-driven epidemic models. These claims are generally based on a comparison between model results and epidemic data at fairly coarse spatio-temporal resolution. Here, we focus on a comparison between model and data at the individual farm level, assessing the potential of the model to predict the infectious status of farms in both the short and long terms. Although the accuracy with which the model predicts farms reporting infection is between 5 and 15%, these low levels are attributable to the expected level of variation between epidemics, and are comparable to the agreement between two independent model simulations. By contrast, while the accuracy of predicting culls is higher (20–30%), this is lower than expected from the comparison between model epidemics. These results generally support the contention that the type of the model used in 2001 was a reliable representation of the epidemic process, but highlight the difficulties of predicting the complex human response, in terms of control strategies to the perceived epidemic risk.

WinBUGS for population ecologists: Bayesian modeling using Markov Chain Monte Carlo methods.

The computer package WinBUGS is introduced. We first give a brief introduction to Bayesian theory and its implementation using Markov chain Monte Carlo (MCMC) algorithms. We then present three case studies showing how WinBUGS can be used when classical theory is difficult to implement. The first example uses data on white storks from Baden Wurttemberg, Germany, to demonstrate the use of mark-recapture models to estimate survival, and also how to cope with unexplained variance through random effects. Recent advances in methodology and also the WinBUGS software allow us to introduce (i) a flexible way of incorporating covariates using spline smoothing and (ii) a method to deal with missing values in covariates. The second example shows how to estimate population density while accounting for detectability, using distance sampling methods applied to a test dataset collected on a known population of wooden stakes. Finally, the third case study involves the use of state-space models of wildlife population dynamics to make inferences about density dependence in a North American duck species. Reversible Jump MCMC is used to calculate the probability of various candidate models. For all examples, data and WinBUGS code are provided.

Bayesian computation: a statistical revolution

The 1990s saw a statistical revolution sparked predominantly by the phenomenal advances in computing technology from the early 1980s onwards. These advances enabled the development of powerful new computational tools, which reignited interest in a philosophy of statistics that had lain almost dormant since the turn of the century. In this paper we briefly review the historic and philosophical foundations of the two schools of statistical thought, before examining the implications of the reascendance of the Bayesian paradigm for both current and future statistical practice.

The Analysis of Ring-Recovery Data Using Random Effects

We show how random terms, describing both yearly variation and overdispersion, can easily be incorporated into models for mark-recovery data, through the use of Bayesian methods. For recovery data on lapwings, we show that the incorporation of the random terms greatly improves the goodness of fit. Omitting the random terms can lead to overestimation of the significance of weather on survival, and overoptimistic prediction intervals in simulations of future population behavior. Random effects models provide a natural way of modeling overdispersion-which is more satisfactory than the standard classical approach of scaling up all standard errors by a uniform inflation factor. We compare models by means of Bayesian p-values and the deviance information criterion (DIC).

Finite mixture models for proportions

Six data sets recording fetal control mortality in mouse litters are presented. The data are clearly overdispersed, and a standard approach would be to describe the data by means of a beta-binomial model or to use quasi-likelihood methods. For five of the examples, we show that beta-binomial model provides a reasonable description but that the fit can be significantly improved by using a mixture of a beta-binomial model with a binomial distribution. This mixture provides two alternative solutions, in one of which the binomial component indicates a high probability of death but is selected infrequently; this accounts for outlying litters with high mortality. The influence of the outliers on the beta-binomial fits is also demonstrated. The location and nature of the two main maxima to the likelihood are investigated through profile log-likelihoods. Comparisons are made with the performance of finite mixtures of binomial distributions.

Weak Identifiability in Models for Mark-Recapture-Recovery Data

The percentage overlap between prior and posterior distributions is obtained easily from the output of MCMC samplers. A 35% guideline for overlap between univariate marginal prior and posterior distributions has been suggested as an indicator of weak identifiability of a parameter. As long as uniform prior distributions are adopted for all of the model parameters, then the suggested guideline has been found to work well for a range of models of mark-recapture-recovery data, where all the parameters are probabilities. Its use is illustrated on models for ring-recovery data on male mallards, and the Cormack-Jolly-Seber model for capture-recapture data on dippers.

Bayesian methods for analysing ringing data

A major recent development in statistics has been the use of fast computational methods of Markov chain Monte Carlo. These procedures allow Bayesian methods to be used in quite complex modelling situations. In this paper, we shall use a range of real data examples involving lapwings, shags, teal, dippers, and herring gulls, to illustrate the power and range of Bayesian techniques. The topics include: prior sensitivity; the use of reversible-jump MCMC for constructing model probabilities and comparing models, with particular reference to models with random effects; model-averaging; and the construction of Bayesian measures of goodness-of-fit. Throughout, there will be discussion of the practical aspects of the work - for instance explaining when and when not to use the BUGS package.

Effect of data quality on estimates of farm infectiousness trends in the UK 2001 foot-and-mouth disease epidemic

Most of the mathematical models that were developed to study the UK 2001 foot-and-mouth disease epidemic assumed that the infectiousness of infected premises was constant over their infectious periods. However, there is some controversy over whether this assumption is appropriate. Uncertainty about which farm infected which in 2001 means that the only method to determine if there were trends in farm infectiousness is the fitting of mechanistic mathematical models to the epidemic data. The parameter values that are estimated using this technique, however, may be influenced by missing and inaccurate data. In particular to the UK 2001 epidemic, this includes unreported infectives, inaccurate farm infection dates and unknown farm latent periods. Here, we show that such data degradation prevents successful determination of trends in farm infectiousness.