Bayesian hierarchical mixture models for otolith microchemistry analysis

The field of fisheries research commonly uses classical statistical classification methods to estimate the proportion of fish that return to natal spawning grounds to spawn. With the advent of otolith microchemical analysis, researchers are able to extract information from fish ear stones (otoliths) about the chemical composition of water in which fish have spent distinct periods of their lives. Here we present a method of analysis set in the Bayesian statistical paradigm which enables explicit incorporation of habitat information into the analysis. The ecological system is seen as arising from a mixture of disparate fish populations and information from the biological relationships inherent in otolith formation is exploited through the hierarchical model structure. We present the model and motivation, demonstrate the validity of the model through simulation studies, and conclude with an analysis of a data set from Lake Erie.     

A Bayesian hierarchical framework for calibrating aquatic biogeochemical models

Model practitioners increasingly place emphasis on rigorous quantitative error analysis in aquatic biogeochemical models and the existing initiatives range from the development of alternative metrics for goodness of fit, to data assimilation into operational models, to parameter estimation techniques. However, the treatment of error in many of these efforts is arguably selective and/or ad hoc. A Bayesian hierarchical framework enables the development of robust probabilistic analysis of error and uncertainty in model predictions by explicitly accommodating measurement error, parameter uncertainty, and model structure imperfection. This paper presents a Bayesian hierarchical formulation for simultaneously calibrating aquatic biogeochemical models at multiple systems (or sites of the same system) with differences in their trophic conditions, prior precisions of model parameters, available information, measurement error or inter-annual variability. Our statistical formulation also explicitly considers the uncertainty in model inputs (model parameters, initial conditions), the analytical/sampling error associated with the field data, and the discrepancy between model structure and the natural system dynamics (e.g., missing key ecological processes, erroneous formulations, misspecified forcing functions). The comparison between observations and posterior predictive monthly distributions indicates that the plankton models calibrated under the Bayesian hierarchical scheme provided accurate system representations for all the scenarios examined. Our results also suggest that the Bayesian hierarchical approach allows overcoming problems of insufficient local data by “borrowing strength” from well-studied sites and this feature will be highly relevant to conservation practices of regions with a high number of freshwater resources for which complete data could never be practically collected. Finally, we discuss the prospect of extending this framework to spatially explicit biogeochemical models (e.g., more effectively connect inshore with offshore areas) along with the benefits for environmental management, such as the optimization of the sampling design of monitoring programs and the alignment with the policy practice of adaptive management.     

Bayesian multi-species N-mixture models for unmarked animal communities

Environmental and Ecological Statistics - We propose an extension of the N-mixture model that enables the estimation of abundances of multiple species as well as the correlations between them. Our...     

A Bayesian framework for stable isotope mixing models

Stable isotope sourcing is used to estimate proportional contributions of sources to a mixture, such as in the analysis of animal diets and plant nutrient use. Statistical methods for inference on the diet proportions using stable isotopes have focused on the linear mixing model. Existing frequentist methods provide inferences when the diet proportion vector can be uniquely solved for in terms of the isotope ratios. Bayesian methods apply for arbitrary numbers of isotopes and diet sources but existing models are somewhat limited as they assume that trophic fractionation or discrimination is estimated without error or that isotope ratios are uncorrelated. We present a Bayesian model for the estimation of mean diet that accounts for uncertainty in source means and discrimination and allows correlated isotope ratios. This model is easily extended to allow the diet proportion vector to depend on covariates, such as time. Two data sets are used to illustrate the methodology. Code is available for selected analyses.     

Bayesian analysis of Jolly-Seber type models

We propose the use of finite mixtures of continuous distributions in modelling the process by which new individuals, that arrive in groups, become part of a wildlife population. We demonstrate this approach using a data set of migrating semipalmated sandpipers (Calidris pussila) for which we extend existing stopover models to allow for individuals to have different behaviour in terms of their stopover duration at the site. We demonstrate the use of reversible jump MCMC methods to derive posterior distributions for the model parameters and the models, simultaneously. The algorithm moves between models with different numbers of arrival groups as well as between models with different numbers of behavioural groups. The approach is shown to provide new ecological insights about the stopover behaviour of semipalmated sandpipers but is generally applicable to any population in which animals arrive in groups and potentially exhibit heterogeneity in terms of one or more other processes.     

A Hierarchical Bayesian model of wildfire in a Mediterranean biodiversity hotspot: Implications of weather variability and global circulation

In this study we combined an extensive database of observed wildfires with high-resolution meteorological data to build a novel spatially and temporally varying survival model to analyze fire regimes in the Mediterranean ecosystem in the Cape Floristic Region (CFR) of South Africa during the period 1980-2000. The model revealed an important influence of seasonally anomalous weather on fire probability, with increased probability of fire in seasons that are warmer and drier than average. In addition to these local-scale influences, the Antarctic Ocean Oscillation (AAO) was identified as an important large-scale influence or teleconnection to global circulation patterns. Fire probability increased in seasons during positive AAO phases, when the subtropical jet moves northward and low level moisture transport decreases. These results confirm that fire occurrence in the CFR is strongly affected by climatic variability at both local and global scales, and thus likely to respond sensitively to future climate change. Comparison of the modelled fire probability between two periods (1951-1975 and 1976-2000) revealed a 4-year decrease in an average fire return time. If, as currently forecasted, climate change in the region continues to produce higher temperatures, more frequent heat waves, and/or lower rainfall, our model thus indicates that fire frequency is likely to increase substantially. The regional implications of shorter fire return times include shifting community structure and composition, favoring species that tolerate more frequent fires.     

Hierarchical Bayesian Spatial Models for Multispecies Conservation Planning and Monitoring

Abstract: Biologists who develop and apply habitat models are often familiar with the statistical challenges posed by their data's spatial structure but are unsure of whether the use of complex spatial models will increase the utility of model results in planning. We compared the relative performance of nonspatial and hierarchical Bayesian spatial models for three vertebrate and invertebrate taxa of conservation concern (Church's sideband snails [Monadenia churchi], red tree voles [Arborimus longicaudus], and Pacific fishers [Martes pennanti pacifica]) that provide examples of a range of distributional extents and dispersal abilities. We used presence–absence data derived from regional monitoring programs to develop models with both landscape and site‐level environmental covariates. We used Markov chain Monte Carlo algorithms and a conditional autoregressive or intrinsic conditional autoregressive model framework to fit spatial models. The fit of Bayesian spatial models was between 35 and 55% better than the fit of nonspatial analogue models. Bayesian spatial models outperformed analogous models developed with maximum entropy (Maxent) methods. Although the best spatial and nonspatial models included similar environmental variables, spatial models provided estimates of residual spatial effects that suggested how ecological processes might structure distribution patterns. Spatial models built from presence–absence data improved fit most for localized endemic species with ranges constrained by poorly known biogeographic factors and for widely distributed species suspected to be strongly affected by unmeasured environmental variables or population processes. By treating spatial effects as a variable of interest rather than a nuisance, hierarchical Bayesian spatial models, especially when they are based on a common broad‐scale spatial lattice (here the national Forest Inventory and Analysis grid of 24 km² hexagons), can increase the relevance of habitat models to multispecies conservation planning.     

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.     

Evaluating the ecological benefits of wildfire by integrating fire and ecosystem simulation models

Fire managers are now realizing that wildfires can be beneficial because they can reduce hazardous fuels and restore fire-dominated ecosystems. A software tool that assesses potential beneficial and detrimental ecological effects from wildfire would be helpful to fire management. This paper presents a simulation platform called FLEAT (Fire and Landscape Ecology Assessment Tool) that integrates several existing landscape- and stand-level simulation models to compute an ecologically based measure that describes if a wildfire is moving the burning landscape towards or away from the historical range and variation of vegetation composition. FLEAT uses a fire effects model to simulate fire severity, which is then used to predict vegetation development for 1, 10, and 100 years into the future using a landscape simulation model. The landscape is then simulated for 5000 years using parameters derived from historical data to create an historical time series that is compared to the predicted landscape composition at year 1, 10, and 100 to compute a metric that describes their similarity to the simulated historical conditions. This tool is designed to be used in operational wildfire management using the LANDFIRE spatial database so that fire managers can decide how aggressively to suppress wildfires. Validation of fire severity predictions using field data from six wildfires revealed that while accuracy is moderate (30-60%), it is mostly dictated by the quality of GIS layers input to FLEAT. Predicted 1-year landscape compositions were only 8% accurate but this was because the LANDFIRE mapped pre-fire composition accuracy was low (21%). This platform can be integrated into current readily available software products to produce an operational tool for balancing benefits of wildfire with potential dangers.     

A Bayesian hierarchical model for monitoring harbor seal changes in Prince William Sound, Alaska

Bayesian hierarchical models were used to assess trends of harbor seals, Phoca vitulina richardsi, in Prince William Sound, Alaska, following the 1989 Exxon Valdez oil spill. Data consisted of 4–10 replicate observations per year at 25 sites over 10 years. We had multiple objectives, including estimating the effects of covariates on seal counts, and estimating trend and abundance, both per site and overall. We considered a Bayesian hierarchical model to meet our objectives. The model consists of a Poisson regression model for each site. For each observation the logarithm of the mean of the Poisson distribution was a linear model with the following factors: (1) intercept for each site and year, (2) time of year, (3) time of day, (4) time relative to low tide, and (5) tide height. The intercept for each site was then given a linear trend model for year. As part of the hierarchical model, parameters for each site were given a prior distribution to summarize overall effects. Results showed that at most sites, (1) trend is down; counts decreased yearly, (2) counts decrease throughout August, (3) counts decrease throughout the day, (4) counts are at a maximum very near to low tide, and (5) counts decrease as the height of the low tide increases; however, there was considerable variation among sites. To get overall trend we used a weighted average of the trend at each site, where the weights depended on the overall abundance of a site. Results indicate a 3.3% decrease per year over the time period.     

Linking chronic wasting disease to mule deer movement scales: a hierarchical Bayesian approach.

Observed spatial patterns in natural systems may result from processes acting across multiple spatial and temporal scales. Although spatially explicit data on processes that generate ecological patterns, such as the distribution of disease over a landscape, are frequently unavailable, information about the scales over which processes operate can be used to understand the link between pattern and process. Our goal was to identify scales of mule deer (Odocoileus hemionus) movement and mixing that exerted the greatest influence on the spatial pattern of chronic wasting disease (CWD) in northcentral Colorado, USA. We hypothesized that three scales of mixing (individual, winter subpopulation, or summer subpopulation) might control spatial variation in disease prevalence. We developed a fully Bayesian hierarchical model to compare the strength of evidence for each mixing scale. We found strong evidence that the finest mixing scale corresponded best to the spatial distribution of CWD infection. There was also evidence that land ownership and habitat use play a role in exacerbating the disease, along with the known effects of sex and age. Our analysis demonstrates how information on the scales of spatial processes that generate observed patterns can be used to gain insight when process data are sparse or unavailable.     

A Bayesian state-space formulation of dynamic occupancy models

Species occurrence and its dynamic components, extinction and colonization probabilities, are focal quantities in biogeography and metapopulation biology, and for species conservation assessments. It has been increasingly appreciated that these parameters must be estimated separately from detection probability to avoid the biases induced by non-detection error. Hence, there is now considerable theoretical and practical interest in dynamic occupancy models that contain explicit representations of metapopulation dynamics such as extinction, colonization, and turnover as well as growth rates. We describe a hierarchical parameterization of these models that is analogous to the state-space formulation of models in time series, where the model is represented by two components, one for the partially observable occupancy process and another for the observations conditional on that process. This parameterization naturally allows estimation of all parameters of the conventional approach to occupancy models, but in addition, yields great flexibility and extensibility, e.g., to modeling heterogeneity or latent structure in model parameters. We also highlight the important distinction between population and finite sample inference; the latter yields much more precise estimates for the particular sample at hand. Finite sample estimates can easily be obtained using the state-space representation of the model but are difficult to obtain under the conventional approach of likelihood-based estimation. We use R and WinBUGS to apply the model to two examples. In a standard analysis for the European Crossbill in a large Swiss monitoring program, we fit a model with year-specific parameters. Estimates of the dynamic parameters varied greatly among years, highlighting the irruptive population dynamics of that species. In the second example, we analyze route occupancy of Cerulean Warblers in the North American Breeding Bird Survey (BBS) using a model allowing for site-specific heterogeneity in model parameters. The results indicate relatively low turnover and a stable distribution of Cerulean Warblers which is in contrast to analyses of counts of individuals from the same survey that indicate important declines. This discrepancy illustrates the inertia in occupancy relative to actual abundance. Furthermore, the model reveals a declining patch survival probability, and increasing turnover, toward the edge of the range of the species, which is consistent with metapopulation perspectives on the genesis of range edges. Given detection/non-detection data, dynamic occupancy models as described here have considerable potential for the study of distributions and range dynamics.     

A Bayesian hierarchical model for assessing the impact of human activity on nitrogen dioxide concentrations in Europe

Ambient concentrations of many pollutants are associated with emissions due to human activity, such as road transport and other combustion sources. In this paper we consider air pollution as a multi-level phenomenon on a continental scale within a Bayesian hierarchical model. We examine different scales of variation in pollution concentrations ranging from large scale transboundary effects to more localised effects which are directly related to human activity. Specifically, in the first stage of the model, we isolate underlying patterns in pollution concentrations due to global factors such as underlying climate and topography, which are modelled together with spatial structure. At this stage measurements from monitoring sites located within rural areas are used which, as far as possible, are chosen to reflect background concentrations. Having isolated these global effects, in the second stage we assess the effects of human activity on pollution in urban areas. The proposed model was applied to concentrations of nitrogen dioxide measured throughout the EU for which significant increases are found to be associated with human activity in urban areas. The approach proposed here provides valuable information that could be used in performing health impact assessments and to inform policy.     

Age-dependent survival analyzed with Bayesian models of mark-recapture data

We describe a Bayesian random effects model of mark-recapture data that accounts for age-dependence in survival and individual heterogeneity in capture probabilities and survival. The model is applied to data on the Glanville fritillary butterfly (Melitaea cinxia) collected from a population enclosed in a large cage in the field. The cage population consisted of a mixture of butterflies originating from newly established and old populations in a large metapopulation in the Aland Islands in Finland. The explanatory variables in the model included the effects of temperature, sex, and population type (new vs. old) on capture probabilities, and the effects of age, sex, population type, and day vs. night on survival. We found that mortality rate increased with age, that mortality rate was much higher during the day than during the night, and that the life span of females originating from newly established populations was shorter than the life span of females from old populations. Capture probability decreased with increasing temperature and decreased with increasing mobility of individuals.     

A hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove

The spread of invasive species is a long studied subject that garners much interest in the ecological research community. Historically the phenomenon has been approached using a purely deterministic mathematical framework (usually involving differential equations of some form). These methods, while scientifically meaningful, are generally highly simplified and fail to account for uncertainty in the data and process, of which our knowledge could not possibly exist without error. We propose a hierarchical Bayesian model for population spread that accommodates data sources with errors, dependence structures between population dynamics parameters, and takes into account prior scientific understanding via non-linear relationships between model parameters and space-time response variables. We model the process (i.e., the bird population in this case) as a Poisson response with spatially varying diffusion coefficients as well as a logistic population growth term using a common reaction-diffusion equation that realistically mimics the ecological process. We focus the application on the ongoing invasion of the Eurasian Collared-Dove.     

Cryptic multiple hypotheses testing in linear models: overestimated effect sizes and the winner's curse

Fitting generalised linear models (GLMs) with more than one predictor has become the standard method of analysis in evolutionary and behavioural research. Often, GLMs are used for exploratory data analysis, where one starts with a complex full model including interaction terms and then simplifies by removing non-significant terms. While this approach can be useful, it is problematic if significant effects are interpreted as if they arose from a single a priori hypothesis test. This is because model selection involves cryptic multiple hypothesis testing, a fact that has only rarely been acknowledged or quantified. We show that the probability of finding at least one ‘significant’ effect is high, even if all null hypotheses are true (e.g. 40% when starting with four predictors and their two-way interactions). This probability is close to theoretical expectations when the sample size (N) is large relative to the number of predictors including interactions (k). In contrast, type I error rates strongly exceed even those expectations when model simplification is applied to models that are over-fitted before simplification (low N/k ratio). The increase in false-positive results arises primarily from an overestimation of effect sizes among significant predictors, leading to upward-biased effect sizes that often cannot be reproduced in follow-up studies (‘the winner's curse’). Despite having their own problems, full model tests and P value adjustments can be used as a guide to how frequently type I errors arise by sampling variation alone. We favour the presentation of full models, since they best reflect the range of predictors investigated and ensure a balanced representation also of non-significant results.     

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.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

An application of Bayesian variable selection to spatial concurrent linear models

Spatial concurrent linear models, in which the model coefficients are spatial processes varying at a local level, are flexible and useful tools for analyzing spatial data. One approach places stationary Gaussian process priors on the spatial processes, but in applications the data may display strong nonstationary patterns. In this article, we propose a Bayesian variable selection approach based on wavelet tools to address this problem. The proposed approach does not involve any stationarity assumptions on the priors, and instead we impose a mixture prior directly on each wavelet coefficient. We introduce an option to control the priors such that high resolution coefficients are more likely to be zero. Computationally efficient MCMC procedures are provided to address posterior sampling, and uncertainty in the estimation is assessed through posterior means and standard deviations. Examples based on simulated data demonstrate the estimation accuracy and advantages of the proposed method. We also illustrate the performance of the proposed method for real data obtained through remote sensing.