Bayesian areal wombling via adjacency modeling

Recently, public health professionals and other geostatistical researchers have shown increasing interest in boundary analysis, the detection or testing of zones or boundaries that reveal sharp changes in the values of spatially oriented variables. For areal data (i.e., data which consist only of sums or averages over geopolitical regions), Lu and Carlin (Geogr Anal 37: 265–285, 2005) suggested a fully model-based framework for areal wombling using Bayesian hierarchical models with posterior summaries computed using Markov chain Monte Carlo (MCMC) methods, and showed the approach to have advantages over existing non-stochastic alternatives. In this paper, we develop Bayesian areal boundary analysis methods that estimate the spatial neighborhood structure using the value of the process in each region and other variables that indicate how similar two regions are. Boundaries may then be determined by the posterior distribution of either this estimated neighborhood structure or the regional mean response differences themselves. Our methods do require several assumptions (including an appropriate prior distribution, a normal spatial random effect distribution, and a Bernoulli distribution for a set of spatial weights), but also deliver more in terms of full posterior inference for the boundary segments (e.g., direct probability statements regarding the probability that a particular border segment is part of the boundary). We illustrate three different remedies for the computing difficulties encountered in implementing our method. We use simulation to compare among existing purely algorithmic approaches, the Lu and Carlin (2005) method, and our new adjacency modeling methods. We also illustrate more practical modeling issues (e.g., covariate selection) in the context of a breast cancer late detection data set collected at the county level in the state of Minnesota.     

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R². We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).     

Fitting complex population models by combining particle filters with Markov chain Monte Carlo

We show how a recent framework combining Markov chain Monte Carlo (MCMC) with particle filters (PFMCMC) may be used to estimate population state-space models. With the purpose of utilizing the strengths of each method, PFMCMC explores hidden states by particle filters, while process and observation parameters are estimated using an MCMC algorithm. PFMCMC is exemplified by analyzing time series data on a red kangaroo (Macropus rufus) population in New South Wales, Australia, using MCMC over model parameters based on an adaptive Metropolis-Hastings algorithm. We fit three population models to these data; a density-dependent logistic diffusion model with environmental variance, an unregulated stochastic exponential growth model, and a random-walk model. Bayes factors and posterior model probabilities show that there is little support for density dependence and that the random-walk model is the most parsimonious model. The particle filter Metropolis-Hastings algorithm is a brute-force method that may be used to fit a range of complex population models. Implementation is straightforward and less involved than standard MCMC for many models, and marginal densities for model selection can be obtained with little additional effort. The cost is mainly computational, resulting in long running times that may be improved by parallelizing the algorithm.     

Efficient Markov chain Monte Carlo sampling for hierarchical hidden Markov models

Traditional Markov chain Monte Carlo (MCMC) sampling of hidden Markov models (HMMs) involves latent states underlying an imperfect observation process, and generates posterior samples for top-level parameters concurrently with nuisance latent variables. When potentially many HMMs are embedded within a hierarchical model, this can result in prohibitively long MCMC runtimes. We study combinations of existing methods, which are shown to vastly improve computational efficiency for these hierarchical models while maintaining the modeling flexibility provided by embedded HMMs. The methods include discrete filtering of the HMM likelihood to remove latent states, reduced data representations, and a novel procedure for dynamic block sampling of posterior dimensions. The first two methods have been used in isolation in existing application-specific software, but are not generally available for incorporation in arbitrary model structures. Using the NIMBLE package for R, we develop and test combined computational approaches using three examples from ecological capture–recapture, although our methods are generally applicable to any embedded discrete HMMs. These combinations provide several orders of magnitude improvement in MCMC sampling efficiency, defined as the rate of generating effectively independent posterior samples. In addition to being computationally significant for this class of hierarchical models, this result underscores the potential for vast improvements to MCMC sampling efficiency which can result from combinations of known algorithms.     

Multinomial mixture model with heterogeneous classification probabilities

Royle and Link (Ecology 86(9):2505?C2512, 2005) proposed an analytical method that allowed estimation of multinomial distribution parameters and classification probabilities from categorical data measured with error. While useful, we demonstrate algebraically and by simulations that this method yields biased multinomial parameter estimates when the probabilities of correct category classifications vary among sampling units. We address this shortcoming by treating these probabilities as logit-normal random variables within a Bayesian framework. We use Markov chain Monte Carlo to compute Bayes estimates from a simulated sample from the posterior distribution. Based on simulations, this elaborated Royle-Link model yields nearly unbiased estimates of multinomial and correct classification probability estimates when classification probabilities are allowed to vary according to the normal distribution on the logit scale or according to the Beta distribution. The method is illustrated using categorical submersed aquatic vegetation data.     

Analysis of behavioral changes of zebrafish (Danio rerio) in response to formaldehyde using Self-organizing map and a hidden Markov model

Two computational methods were applied to classification of movement patterns of zebrafish (Danio rerio) to elucidate Markov processes in behavioral changes before and after treatment of formaldehyde (0.1 mg/L) in semi-natural conditions. The complex data of the movement tracks were initially classified by the Self-organizing map (SOM) to present different behavioral states of test individuals. Transition probabilities between behavioral states were further evaluated to fit Markov processes by using the hidden Markov model (HMM). Emission transition probability was also obtained from the observed variables (i.e., speed) for training with the HMM. Experimental transition and emission probability matrices were successfully estimated with the HMM for recognizing sequences of behavioral states with accuracy rates in acceptable ranges at central and boundary zones before (77.3-81.2%) and after (70.1-76.5%) treatment. A heuristic algorithm and a Markov model were efficiently combined to analyze movement patterns and could be a means of in situ behavioral monitoring tool.     

A single-chain-based multidimensional Markov chain model for subsurface characterization

Multidimensional Markov chain models in geosciences were often built on multiple chains, one in each direction, and assumed these 1-D chains to be independent of each other. Thus, unwanted transitions (i.e., transitions of multiple chains to the same location with unequal states) inevitably occur and have to be excluded in estimating the states at unobserved locations. This consequently may result in unreliable estimates, such as underestimation of small classes (i.e., classes with smaller than average areas) in simulated realizations. This paper presents a single-chain-based multidimensional Markov chain model for estimation (i.e., prediction and conditional stochastic simulation) of spatial distribution of subsurface formations with borehole data. The model assumes that a single Markov chain moves in a lattice space, interacting with its nearest known neighbors through different transition probability rules in different cardinal directions. The conditional probability distribution of the Markov chain at the location to be estimated is formulated in an explicit form by following the Bayes’ Theorem and the conditional independence of sparse data in cardinal directions. Since no unwanted transitions are involved, the model can estimate all classes fairly. Transiogram models (i.e., 1-D continuous Markov transition probability diagrams) are used to provide transition probability input with needed lags to generalize the model. Therefore, conditional simulation can be conducted directly and efficiently. The model provides an alternative for heterogeneity characterization of subsurface formations.

Prediction of potential areas of species distributions based on presence-only data

We introduce a methodology to infer zones of high potential for the habitat of a species, useful for management of biodiversity, conservation, biogeography, ecology, or sustainable use. Inference is based on a set of sites where the presence of the species has been reported. Each site is associated with covariate values, measured on discrete scales. We compute the predictive probability that the species is present at each node of a regular grid. Possible spatial bias for sites of presence is accounted for. Since the resulting posterior distribution does not have a closed form, a Markov chain Monte Carlo (MCMC) algorithm is implemented. However, we also describe an approximation to the posterior distribution, which avoids MCMC. Relevant features of the approach are that specific notions of data acquisition such as sampling intensity and detectability are accounted for, and that available a priori information regarding areas of distribution of the species is incorporated in a clear-cut way. These concepts, arising in the presence-only context, are not addressed in alternative methods. We also consider an uncertainty map, which measures the variability for the predictive probability at each node on the grid. A simulation study is carried out to test and compare our approach with other standard methods. Two case studies are also presented.     

Identifying fishing trip behaviour and estimating fishing effort from VMS data using Bayesian Hidden Markov Models

Recent advances in technologies have lead to a vast influx of data on movements, based on discrete recorded position of animals or fishing boats, opening new horizons for future analyses. However, most of the potential interest of tracking data depends on the ability to develop suitable modelling strategies to analyze trajectories from discrete recorded positions. A serious modelling challenge is to infer the evolution of the true position and the associated spatio-temporal distribution of behavioural states using discrete, error-prone and incomplete observations. In this paper, a Bayesian Hierarchical Model (HBM) using Hidden Markov Process (HMP) is proposed as a template for analyzing fishing boats trajectories based on data available from satellite-based vessel monitoring systems (VMS). The analysis seeks to enhance the definition of the fishing pressure exerted on fish stocks, by discriminating between the different behavioural states of a fishing trip, and also by quantifying the relative importance of each of these states during a fishing trip. The HBM approach is tested to analyse the behaviour of pelagic trawlers in the Bay of Biscay. A hidden Markov chain with a regular discrete time step is used to model transitions between successive behavioural states (e.g., fishing, steaming, stopping (at Port or at sea)) of each vessel. The parameters of the movement process (speed and turning angles) are defined conditionally upon the behavioural states. Bayesian methods are used to integrate the available data (typically VMS position recorded at discrete time) and to draw inferences on any unknown parameters of the model. The model is first tested on simulated data with different parameters structures. Results provide insights on the potential of HBM with HMP to analyze VMS data. They show that if VMS positions are recorded synchronously with the instants at which the process switch from one behavioural state to another, the estimation method provides unbiased and precise inferences on behavioural states and on associated movement parameters. However, if the observations are not gathered with a sufficiently high frequency, the performance of the estimation method could be drastically impacted when the discrete observations are not synchronous with the switching instants. The model is then applied to real pathways to estimate variables of interest such as the number of operations per trip, time and distance spent fishing or travelling.     

An evaluation of three statistical methods used to model resource selection

The performance of statistical methods for modeling resource selection by animals is difficult to evaluate with field data because true selection patterns are unknown. Simulated data based on a known probability distribution, though, can be used to evaluate statistical methods. Models should estimate true selection patterns if they are to be useful in analyzing and interpreting field data. We used simulation techniques to evaluate the effectiveness of three statistical methods used in modeling resource selection. We generated 25 use locations per animal and included 10, 20, 40, or 80 animals in samples of use locations. To simulate species of different mobility, we generated use locations at four levels according to a known probability distribution across DeSoto National Wildlife Refuge (DNWR) in eastern Nebraska and western Iowa, USA. We either generated 5 random locations per use location or 10,000 random locations (total) within 4 predetermined areas around use locations to determine how the definition of availability and the number of random locations affected results. We analyzed simulated data using discrete choice, logistic-regression, and a maximum entropy method (Maxent). We used a simple linear regression of estimated and known probability distributions and area under receiver operating characteristic curves (AUC) to evaluate the performance of each method. Each statistical method was affected differently by number of animals and random locations used in analyses, level at which selection of resources occurred, and area considered available. Discrete-choice modeling resulted in precise and accurate estimates of the true probability distribution when the area in which use locations were generated was ≥ the area defined to be available. Logistic-regression models were unbiased and precise when the area in which use locations were generated and the area defined to be available were the same size; the fit of these models improved with increased numbers of random locations. Maxent resulted in unbiased and precise estimates of the known probability distribution when the area in which use locations were generated was small (home-range level) and the area defined to be available was large (study area). Based on AUC analyses, all models estimated the selection distribution better than random chance. Results from AUC analyses, however, often contradicted results of the linear regression method used to evaluate model performance. Discrete-choice modeling was best able to estimate the known selection distribution in our study area regardless of sample size or number of random locations used in the analyses, but we recommend further studies using simulated data over different landscapes and different resource metrics to confirm our results. Our study offers an approach and guidance for others interested in assessing the utility of techniques for modeling resource selection in their study area.     

Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

Hidden process models for animal population dynamics.

Hidden process models are a conceptually useful and practical way to simultaneously account for process variation in animal population dynamics and measurement errors in observations and estimates made on the population. Process variation, which can be both demographic and environmental, is modeled by linking a series of stochastic and deterministic subprocesses that characterize processes such as birth, survival, maturation, and movement. Observations of the population can be modeled as functions of true abundance with realistic probability distributions to describe observation or estimation error. Computer-intensive procedures, such as sequential Monte Carlo methods or Markov chain Monte Carlo, condition on the observed data to yield estimates of both the underlying true population abundances and the unknown population dynamics parameters. Formulation and fitting of a hidden process model are demonstrated for Sacramento River winter-run chinook salmon (Oncorhynchus tshawytsha).     

A hierarchical Bayesian model for forecasting state-level corn yield

Historically, the National Agricultural Statistics Service crop forecasts and estimates have been determined by a group of commodity experts called the Agricultural Statistics Board (ASB). The corn yield forecasts for the “speculative region,” ten states that account for approximately 85 % of corn production, are based on two sets of monthly surveys, a farmer interview survey and a field measurement survey. The members of the ASB subjectively determine a forecast on the basis of a discussion of the survey data and auxiliary information about weather, average planting dates, and crop maturity. The ASB uses an iterative procedure, where initial state estimates are adjusted so that the weighted sum of the final state estimates is equal to a previously-determined estimate for the speculative region. Deficiencies of the highly subjective ASB process are lack of reproducibility and a measure of uncertainty. This paper describes the use of Bayesian methods to model the ASB process in a way that leads to objective forecasts and estimates of the corn yield. First, we use small area estimation techniques to obtain state-level forecasts. Second, we describe a way to adjust the state forecasts so that the weighted sum of the state forecasts is equal to a previously-determined regional forecast. We use several diagnostic techniques to assess the goodness of fit of various models and their competitors. We use Markov chain Monte Carlo methods to fit the models to both historic and current data from the two monthly surveys. Our results show that our methodology can provide reasonable and objective forecasts of corn yields for states in the speculative region.     

Cluster detection using Bayes factors from overparameterized cluster models

In this paper, we consider the use of a partition model to estimate regional disease rates and to detect spatial clusters. Formal inference regarding the number of partitions (or clusters) can be obtained with a reversible jump Markov chain Monte Carlo algorithm. As an alternative, we consider the ability of models with a fixed, but overly large, number of partitions to estimate regional disease rates and to provide informal inferences about the number and locations of clusters using local Bayes factors. We illustrate and compare these two approaches using data on leukemia incidence in upstate New York and data on breast cancer incidence in Wisconsin.     

Combining multistate capture-recapture data with tag recoveries to estimate demographic parameters

Matrix population models that allow an animal to occupy more than one state over time are important tools for population and evolutionary ecologists. Definition of state can vary, including location for metapopulation models and breeding state for life history models. For populations whose members can be marked and subsequently reencountered, multistate mark-recapture models are available to estimate the survival and transition probabilities needed to construct population models. Multistate models have proved extremely useful in this context, but they often require a substantial amount of data and restrict estimation of transition probabilities to those areas or states subjected to formal sampling effort. At the same time, for many species, there are considerable tag recovery data provided by the public that could be modeled in order to increase precision and to extend inference to a greater number of areas or states. Here we present a statistical model for combining multistate capture-recapture data (e.g., from a breeding ground study) with multistate tag recovery data (e.g., from wintering grounds). We use this method to analyze data from a study of Canada Geese (Branta canadensis) in the Atlantic Flyway of North America. Our analysis produced marginal improvement in precision, due to relatively few recoveries, but we demonstrate how precision could be further improved with increases in the probability that a retrieved tag is reported.     

Modeling and spatial prediction of pre-settlement patterns of forest distribution using witness tree data

At the time of European settlement, land surveys were conducted progressively westward throughout the United States. Outside of the original 13 colonies, surveys generally followed the Public Land Survey system in which trees, called witness trees, were regularly recorded at 1 mi by 1 mi grid intersections. This unintentional sampling provides insight into the composition and structure of pre-European settlement forests, which is used as baseline data to assess forest change following settlement. In this paper, a model for the Public Land Surveys of east central Alabama is developed. Assuming that the locations of trees of each species are realized from independent Poisson processes whose respective log intensities are linear functions of environmental covariates (i.e., elevation, landform, and physiographic province), the species observed at the survey grid intersections are independently sampled from a generalized logistic regression model. If all 68 species found in the survey were included, the model would be highly over-parameterized, so only the distribution of the most common taxon, pines, will be considered at this time. To assess the impact of environmental factors not included in the model, a hidden Gaussian random field shall be added as a random effect. A Markov Chain Monte Carlo algorithm is developed for Bayesian inference on model parameters, and for Bayes posterior prediction of the spatial distribution of pines in east central Alabama. Received: June 2004 / Revised: November 2004     

Selecting a binary Markov model for a precipitation process

This paper uses rth-order categorical Markov chains to model the probability of precipitation. Several stationary and non-stationary high-order Markov models are proposed and compared using BIC. The number of parameters increases exponentially by adding the Markov order. Several classes of high-order Markov models are proposed which their increase of number of parameters are modest. For example models that use the number of precipitation days in a period prior to date, temperature of the previous day and sines/cosines periodic functions (to model the seasonality) are considered. The theory of partial likelihood is used to estimate the parameters. Parsimonious non-stationary first order Markov models with few seasonal terms are found optimal using BIC and temperature does not turn out to be a useful covariate. However BIC seems to underestimate the number of seasonal terms. We have also compared the results with AIC in some cases which tends to pick parsimonious models with more seasonal terms and higher order. We also show that ignoring seasonal terms result in picking higher order Markov chains. Finally we apply the methods to build confidence intervals for the probability of periods with no precipitation or low number of precipitation days in Calgary using historical data from 1980 to 2000.     

The value of information for integrated assessment models of climate change

We estimate the value of information (VOI) for three key parameters of climate integrated assessment models (IAMs): marginal damages at low temperature anomalies, marginal damages at high temperature anomalies, and equilibrium climate sensitivity. Most empirical studies of climate damages have examined temperature anomalies up to 3 °C, while some recent theoretical studies emphasize the risks of “climate catastrophes,” which depend on climate sensitivity and on marginal damages at higher temperature anomalies. We use a new IAM to estimate the VOI for each parameter over a range of assumed levels of study precision based on prior probability distributions calibrated using results from previous studies. We measure the VOI as the maximum fixed fraction of consumption that a social planner would be willing to pay to conduct a new study before setting a carbon tax. Our central results suggest that the VOI is greatest for marginal damages at high temperature anomalies.     

Space-time Bayesian small area disease risk models: development and evaluation with a focus on cluster detection

This paper extends the spatial local-likelihood model and the spatial mixture model to the space-time (ST) domain. For comparison, a standard random effect space-time (SREST) model is examined to allow evaluation of each model’s ability in relation to cluster detection. To pursue this evaluation, we use the ST counterparts of spatial cluster detection diagnostics. The proposed criteria are based on posterior estimates (e.g., misclassification rate) and some are based on post-hoc analysis of posterior samples (e.g., exceedance probability). In addition, we examine more conventional model fit criteria including mean square error (MSE). We illustrate the methodology with a real ST dataset, Georgia throat cancer mortality data for the years 1994–2005, and a simulated dataset where different levels and shapes of clusters are embedded. Overall, it is found that conventional SREST models fair well in ST cluster detection and in goodness-of-fit, while for extreme risk detection the local likelihood ST model does best.     

A process-convolution approach to modelling temperatures in the North Atlantic Ocean

This paper develops a process-convolution approach for space-time modelling. With this approach, a dependent process is constructed by convolving a simple, perhaps independent, process. Since the convolution kernel may evolve over space and time, this approach lends itself to specifying models with non-stationary dependence structure. The model is motivated by an application from oceanography: estimation of the mean temperature field in the North Atlantic Ocean as a function of spatial location and time. The large amount of this data poses some difficulties; hence computational considerations weigh heavily in some modelling aspects. A Bayesian approach is taken here which relies on Markov chain Monte Carlo for exploring the posterior distribution.