Multivariate Bayesian analysis of atmosphere–ocean general circulation models

Numerical experiments based on atmosphere–ocean general circulation models (AOGCMs) are one of the primary tools in deriving projections for future climate change. Although each AOGCM has the same underlying partial differential equations modeling large scale effects, they have different small scale parameterizations and different discretizations to solve the equations, resulting in different climate projections. This motivates climate projections synthesized from results of several AOGCMs’ output. We combine present day observations, present day and future climate projections in a single highdimensional hierarchical Bayes model. The challenging aspect is the modeling of the spatial processes on the sphere, the number of parameters and the amount of data involved. We pursue a Bayesian hierarchical model that separates the spatial response into a large scale climate change signal and an isotropic process representing small scale variability among AOGCMs. Samples from the posterior distributions are obtained with computer-intensive MCMC simulations. The novelty of our approach is that we use gridded, high resolution data covering the entire sphere within a spatial hierarchical framework. The primary data source is provided by the Coupled Model Intercomparison Project (CMIP) and consists of 9 AOGCMs on a 2.8 by 2.8 degree grid under several different emission scenarios. In this article we consider mean seasonal surface temperature and precipitation as climate variables. Extensions to our model are also discussed.     

Interpolation performance of a spatio-temporal model with spatially varying coefficients: application to PM<Subscript>10</Subscript> concentrations in Rio de Janeiro

In this work we present a Bayesian analysis in linear regression models with spatially varying coefficients for modeling and inference in spatio-temporal processes. This kind of model is particularly appealing in situations where the effect of one or more explanatory processes on the response present substantial spatial heterogeneity. We describe for this model how to make inference about the regression coefficients and response processes under two scenarios: when the explanatory processes are known throughout the study region, and when they are known only at the sampling locations. Using a simulation experiment we investigate how parameter inference and interpolation performance are affected by some features of the data and prior distribution that is used. The proposed methodology is used to model the dataset on PM₁₀ levels in the metropolitan region of Rio de Janeiro presented in Paez and Gamerman (2003).     

Modeling species-habitat relationships with spatially autocorrelated observation data.

Spatial autocorrelation in wildlife observation data arises when extrinsic environmental processes and patterns that influence the spatial distribution of wildlife are themselves spatially structured, or when species are subject to intrinsic population processes, causing contagion or dispersion effects. Territoriality, Allee effects, dispersal limitations, and social clustering are examples of intrinsic processes. Both forms of autocorrelation can violate the assumptions of generalized linear regression models, resulting in biased estimation of model coefficients and diminished predictive performance. Such consequences may be avoided for extrinsic autocorrelation when autocorrelated environmental variables are available for use as model covariates, whereas intrinsic spatial autocorrelation requires an alternative modeling approach. The autologistic model provides an approach suited to the binary observations often obtained in wildlife surveys, but its performance has not been tested across widely varying sampling intensities or strengths of intrinsic spatial structure. Here we use simulated data to test the autologistic model under a range of sampling conditions. The autologistic model obtains better fits and substantially better predictive performance than the standard logistic regression model over the full range of sampling designs and intensities tested. We provide a simple Bayesian implementation of the autologistic model, which until now has not been achieved with standard statistical software alone. A step-by-step procedure is given for characterizing and modeling spatial autocorrelation in binary observation data, along with computer code for fitting autologistic models in WinBUGS, a freeware Bayesian analysis package. This approach avoids normal approximations to the pseudo-likelihood, in contrast to previous Bayesian applications of the autologistic model. We provide a sample application of the autologistic model, fitted to survey data for a gliding marsupial in southeastern Australia.     

Using hidden Markov chains and empirical Bayes change-point estimation for transect data

Consider a lattice of locations in one dimension at which data are observed. We model the data as a random hierarchical process. The hidden process is assumed to have a (prior) distribution that is derived from a two-state Markov chain. The states correspond to the mean values (high and low) of the observed data. Conditional on the states, the observations are modelled, for example, as independent Gaussian random variables with identical variances. In this model, there are four free parameters: the Gaussian variance, the high and low mean values, and the transition probability in the Markov chain. A parametric empirical Bayes approach requires estimation of these four parameters from the marginal (unconditional) distribution of the data and we use the EM-algorithm to do this. From the posterior of the hidden process, we use simulated annealing to find the maximum a posteriori (MAP) estimate. Using a Gibbs sampler, we also obtain the maximum marginal posterior probability (MMPP) estimate of the hidden process. We use these methods to determine where change-points occur in spatial transects through grassland vegetation, a problem of considerable interest to plant ecologists.     

Trends in biodiversity and habitat quantification tools used for market-based conservation in the United States

Market-based conservation mechanisms are designed to facilitate the mitigation of harm to and conservation of habitats and biodiversity. Their potential is partly hindered, however, by the quantification tools used to assess habitat quality and functionality. Of specific concern are the lack of transparency and standardization in tool development and gaps in tool availability. To address these issues, we collected information via internet and literature searchers and through conversations with tool developers and users on tools used in U.S. conservation mechanisms, such as payments for ecosystem services (PES) and ecolabel programs, conservation banking, and habitat exchanges. We summarized information about tools and explored trends among and within mechanisms based on criteria detailing geographic, ecological, and technical features of tools. We identified 69 tools that assessed at least 34 species and 39 habitat types. Where tools reported pricing, 98% were freely available. More tools were applied to states along the U.S. West Coast than elsewhere, and the level of tool transferability varied markedly among mechanisms. Tools most often incorporated conditions at numerous spatial scales, frequently addressed multiple risks to site viability, and required 1–83 data inputs. Most tools required a moderate or greater level of user skill. Average tool-complexity estimates were similar among all mechanisms except PES programs. Our results illustrate the diversity among tools in their ecological features, data needs, and geographic application, supporting concerns about a lack of standardization. However, consistency among tools in user skill requirements, incorporation of multiple spatial scales, and complexity highlight important commonalities that could serve as a starting point for establishing more standardized tool development and feature-incorporation processes. Greater standardization in tool design may expand market participation and facilitate a needed assessment of the effectiveness of market-based conservation.     

Toward accurate and precise estimates of lion density

Reliable estimates of animal density are fundamental to understanding ecological processes and population dynamics. Furthermore, their accuracy is vital to conservation because wildlife authorities rely on estimates to make decisions. However, it is notoriously difficult to accurately estimate density for wide‐ranging carnivores that occur at low densities. In recent years, significant progress has been made in density estimation of Asian carnivores, but the methods have not been widely adapted to African carnivores, such as lions (Panthera leo). Although abundance indices for lions may produce poor inferences, they continue to be used to estimate density and inform management and policy. We used sighting data from a 3‐month survey and adapted a Bayesian spatially explicit capture‐recapture (SECR) model to estimate spatial lion density in the Maasai Mara National Reserve and surrounding conservancies in Kenya. Our unstructured spatial capture‐recapture sampling design incorporated search effort to explicitly estimate detection probability and density on a fine spatial scale, making our approach robust in the context of varying detection probabilities. Overall posterior mean lion density was estimated to be 17.08 (posterior SD 1.310) lions >1 year old/100 km², and the sex ratio was estimated at 2.2 females to 1 male. Our modeling framework and narrow posterior SD demonstrate that SECR methods can produce statistically rigorous and precise estimates of population parameters, and we argue that they should be favored over less reliable abundance indices. Furthermore, our approach is flexible enough to incorporate different data types, which enables robust population estimates over relatively short survey periods in a variety of systems. Trend analyses are essential to guide conservation decisions but are frequently based on surveys of differing reliability. We therefore call for a unified framework to assess lion numbers in key populations to improve management and policy decisions.     

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.     

Estimating the spatial covariance structure using the geoadditive model

In geostatistics, both kriging and smoothing splines are commonly used to generate an interpolated map of a quantity of interest. The geoadditive model proposed by Kammann and Wand (J R Stat Soc: Ser C (Appl Stat) 52(1):1–18, 2003) represents a fusion of kriging and penalized spline additive models. Complex data issues, including non-linear covariate trends, multiple measurements at a location and clustered observations are easily handled using the geoadditive model. We propose a likelihood based estimation procedure that enables the estimation of the range (spatial decay) parameter associated with the penalized splines of the spatial component in the geoadditive model. We present how the spatial covariance structure (covariogram) can be derived from the geoadditive model. In a simulation study, we show that the underlying spatial process and prediction of the spatial map are estimated well using the proposed likelihood based estimation procedure. We present several applications of the proposed methods on real-life data examples.     

On spline-based approaches to spatial linear regression for geostatistical data

For spatial linear regression, the traditional approach to capture spatial dependence is to use a parametric linear mixed-effects model. Spline surfaces can be used as an alternative to capture spatial variability, giving rise to a semiparametric method that does not require the specification of a parametric covariance structure. The spline component in such a semiparametric method, however, impacts the estimation of the regression coefficients. In this paper, we investigate such an impact in spatial linear regression with spline-based spatial effects. Statistical properties of the regression coefficient estimators are established under the model assumptions of the traditional spatial linear regression. Further, we examine the empirical properties of the regression coefficient estimators under spatial confounding via a simulation study. A data example in precision agriculture research regarding soybean yield in relation to field conditions is presented for illustration.     

Modelling dependence structures of extreme wind speed using bivariate distribution: a Bayesian approach

When investigating extremes of weather variables, it is seldom that a single weather station determines the damage, and extremes may be caused from the combined behaviour of several weather stations. To investigate joint dependence of extreme wind speed, a bivariate generalised extreme value distribution (BGEVD) was considered from frequentist and Bayesian approaches to analyse the extremes of component-wise monthly maximum wind speed at selected weather stations in South Africa. In the frequentist approach, the parameters of extreme value distributions (EVDs) were estimated with maximum likelihood, whereas in the Bayesian approach the Markov Chain Monte Carlo (MCMC) technique was used with the Metropolis–Hastings algorithm. The results showed that when fitted to component-wise maxima of extreme weather variables, the BGEVD provided apparent benefits over the univariate method, which allowed information to be pooled across stations and resulted in improved precision of the estimates for the parameters and return levels of the distributions. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from weather characteristics of four pairs of surrounding weather stations at various distances. The results from the Bayesian analysis showed that posterior inference might be affected by the choice of priors that were used to formulate the informative priors. From the results, it could be inferred that the Bayesian approach provides a satisfactory estimation strategy in terms of precision, compared with the frequentist approach, because it accounts for uncertainty in parameters and return level estimations.     

State space models with spatial deformation

Space deformation has been proposed to model space-time varying observation processes with non-stationary spatial covariance structure under the hypothesis of temporal stationarity. In real applications, however, the temporal stationarity assumption is inappropriate and unrealistic. In this work we propose a spatial-temporal model whose temporal trend is modeled through state space models and a spatially varying anisotropy is modeled through spatial deformation, under the Bayesian approach. A distinctive feature of our approach is the consideration of model uncertainty in an unified framework. Our model has a clear advantage over the ones proposed so far in the literature when the main objective of the study is to perform spatial interpolation for fixed points in time. Approximations of the posterior distributions of the model parameters are obtained via Markov chain Monte Carlo methods. This allows for prediction of the process values in space and time as well as handling of missing values. Two applications are presented: the first one to model concentrations of sulfur dioxide in the eastern United States and the second one to model monthly minimum temperatures in the State of Rio de Janeiro.     

Zero-inflated models with application to spatial count data

Count data arises in many contexts. Here our concern is with spatial count data which exhibit an excessive number of zeros. Using the class of zero-inflated count models provides a flexible way to address this problem. Available covariate information suggests formulation of such modeling within a regression framework. We employ zero-inflated Poisson regression models. Spatial association is introduced through suitable random effects yielding a hierarchical model. We propose fitting this model within a Bayesian framework considering issues of posterior propriety, informative prior specification and well-behaved simulation based model fitting. Finally, we illustrate the model fitting with a data set involving counts of isopod nest burrows for 1649 pixels over a portion of the Negev desert in Israel.     

Analysis of non-separable spatiotemporal data with application to enterovirus study in Taiwan

We propose an estimation procedure to incorporate non-separable spatiotemporal correlation into a generalized linear mixed model. The motivation of this paper is from a study of enterovirus infection with spatial-temporal correlation. The proposed method underlying a working estimating equation comes from a generalization of weighted least squares approaches. With an iterative two-stage estimation procedure, we may address the non-identifiability problem caused by latent random effects. Under certain regularity conditions, we show that the proposed estimate has consistency and asymptotic normality for spatiotemporal data. We also conduct a model-based simulation and apply the method to the enterovirus data in Taiwan.     

Bayesian modelling of extreme wind speed at Cape Town,South Africa

In the framework of generalized extreme value (GEV) distribution, the frequentist and Bayesian methods have been used to analyse the extremes of annual maxima wind speed recorded by automatic weather stations in Cape Town, Western Cape, South Africa. In the frequentist approach, the GEV distribution parameters were estimated using maximum likelihood, whereas in the Bayesian method the Markov Chain Monte Carlo technique with the Metropolis–Hastings algorithm was used. The results show that the GEV model with trend in the location parameter appears to be a better model for annual maxima data. The paper also discusses a method to construct informative priors empirically using historical data of the underlying process from other weather stations. The results from the Bayesian analysis show that posterior inference might be affected by the choice of priors and hence by the distance between a weather station used to formulate the priors and the point of interest.     

Local likelihood disease clustering: development and evaluation

This paper illustrates a method based on local likelihood (LL) for detecting disease clusters. The approach is based on estimating a lasso distance for each region: within which regions are considered to be clustered. An important advantage in implementing this approach is that it does not require any special Monte Carlo Markov Chain (MCMC) algorithm, e.g., reversible jump MCMC, which is essential in hidden Markov model approach. Another advantage is that extending the model to incorporate covariates is straightforward. We illustrate three ways of doing this by using Eastern Germany lip cancer data. By using simulated data, we have made a comparison with the BYM model [Besag et al. (1991) Annals of the Institute of Statistical Mathematics, 43, 1–59] and the mixture model [Lawson and Clark (2002) Disease Mapping and Risk Assessment for Public Health, Chapman and Hall]. We also did a limited examination of the ability of the LL model to recover true relative risk under different priors for lasso parameter. In order to check the edge effects, which has been overlooked in many spatial clustering models for disease mapping but deserves special attention as it lacks observable neighbors, we have adapted here a simple approach to observe the changes in relative risks when the edge regions are omitted. An erratum to this article is available at .     

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.     

Major PM10 source location by a spatial multivariate receptor model

We present a multivariate receptor model for identifying the spatial location of major PM₁₀ pollution sources through the concentrations at multiple monitoring stations. We build on a mixed multiplicative log-normal factor model adjusting the source contributions for meteorological covariates and for temporal correlation and considering source profiles as compositional Gaussian random fields, to account for the variability induced by the spatial distribution of the monitoring sites. Taking a Bayesian approach to estimation, the proposed hierarchical model is implemented and used to analyze average daily PM₁₀ concentration measurements from 13 monitoring sites in Taranto, Italy, for the period April–December 2005. Three major sources of pollution are identified and characterized in terms of their spatial and temporal behavior and in relation to meteorological data.     

Evaluating approaches for scaling-up community-based marine-protected areas into socially equitable and ecologically representative networks

Marine-protected areas (MPAs) are vital to marine conservation, but their coverage and distribution is insufficient to address declines in global biodiversity and fisheries. In response, many countries have committed through the Aichi Target 11 of the Convention on Biological Diversity to conserve 10% of the marine environment through ecologically representative and equitably managed MPAs by 2020. The rush to fulfill this commitment has raised concerns on how increasing MPA coverage will affect other elements of Target 11, including representation and equity. We examined a Philippines case study to assess and compare 3 MPA planning approaches for biodiversity representation and equitable distribution of costs to small-scale fishers. In the opportunistic approach, MPAs were identified and supported by coastal communities. The donor-assisted approach used local knowledge to select MPAs through a national-scale and donor-assisted conservation project. The systematic conservation planning approach identified MPA locations with the spatial prioritization software Marxan with Zones to achieve biodiversity objectives with minimal costs to fishers. We collected spatial data on biodiversity and fisheries features and performed a gap analysis to evaluate MPAs derived from different approaches. We assessed representation based on the proportion of biodiversity features conserved in MPAs and distribution equity by the distribution of opportunity costs (fishing areas lost in MPAs) among fisher stakeholder groups. The opportunistic approach did not ineffectively represent biodiversity and resulted in inequitable costs to fishers. The donor-assisted approach affected fishers disproportionately but provided near-optimal regional representation. Only the systematic approach achieved all representation targets with minimal and equitable costs to fishers. Our results demonstrate the utility of systematic conservation planning to address key elements of Target 11 and highlight opportunities (e.g., integration of local and scientific knowledge can address representation and equity concerns) and pitfalls (e.g., insufficient stakeholder considerations can exacerbate social inequalities) for planning MPAs in similar contexts.