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.     

Species distribution modelling—Effect of design and sample size of pseudo-absence observations

We explored the effect of varying pseudo-absence data in species distribution modelling using empirical data for four real species and simulated data for two imaginary species. In all analyses we used a fixed study area, a fixed set of environmental predictors and a fixed set of presence observations. Next, we added pseudo-absence data generated by different sampling designs and in different numbers to assess their relative importance for the output from the species distribution model. The sampling design strongly influenced the predictive performance of the models while the number of pseudo-absences had minimal effect on the predictive performance. We attribute much of these results to the relationship between the environmental range of the pseudo-absences (i.e. the extent of the environmental space being considered) and the environmental range of the presence observations (i.e. under which environmental conditions the species occurs). The number of generated pseudo-absences had a direct effect on the predicted probability, which translated to different distribution areas. Pseudo-absence observations that fell within grid cells with presence observations were purposely included in our analyses. We discourage the practice of excluding certain pseudo-absence data because it involves arbitrary assumptions about what are (un)suitable environments for the species being modelled.     

Specimen‐Based Modeling,Stopping Rules,and the Extinction of the Ivory‐Billed Woodpecker

Abstract: Assessing species survival status is an essential component of conservation programs. We devised a new statistical method for estimating the probability of species persistence from the temporal sequence of collection dates of museum specimens. To complement this approach, we developed quantitative stopping rules for terminating the search for missing or allegedly extinct species. These stopping rules are based on survey data for counts of co‐occurring species that are encountered in the search for a target species. We illustrate both these methods with a case study of the Ivory‐billed Woodpecker (Campephilus principalis), long assumed to have become extinct in the United States in the 1950s, but reportedly rediscovered in 2004. We analyzed the temporal pattern of the collection dates of 239 geo‐referenced museum specimens collected throughout the southeastern United States from 1853 to 1932 and estimated the probability of persistence in 2011 as <6.4 × 10^?5, with a probable extinction date no later than 1980. From an analysis of avian census data (counts of individuals) at 4 sites where searches for the woodpecker were conducted since 2004, we estimated that at most 1–3 undetected species may remain in 3 sites (one each in Louisiana, Mississippi, Florida). At a fourth site on the Congaree River (South Carolina), no singletons (species represented by one observation) remained after 15,500 counts of individual birds, indicating that the number of species already recorded (56) is unlikely to increase with additional survey effort. Collectively, these results suggest there is virtually no chance the Ivory‐billed Woodpecker is currently extant within its historical range in the southeastern United States. The results also suggest conservation resources devoted to its rediscovery and recovery could be better allocated to other species. The methods we describe for estimating species extinction dates and the probability of persistence are generally applicable to other species for which sufficient museum collections and field census results are available.     

Obtaining Environmental Favourability Functions from Logistic Regression

Logistic regression is a statistical tool widely used for predicting species’ potential distributions starting from presence/absence data and a set of independent variables. However, logistic regression equations compute probability values based not only on the values of the predictor variables but also on the relative proportion of presences and absences in the dataset, which does not adequately describe the environmental favourability for or against species presence. A few strategies have been used to circumvent this, but they usually imply an alteration of the original data or the discarding of potentially valuable information. We propose a way to obtain from logistic regression an environmental favourability function whose results are not affected by an uneven proportion of presences and absences. We tested the method on the distribution of virtual species in an imaginary territory. The favourability models yielded similar values regardless of the variation in the presence/absence ratio. We also illustrate with the example of the Pyrenean desman’s (Galemys pyrenaicus) distribution in Spain. The favourability model yielded more realistic potential distribution maps than the logistic regression model. Favourability values can be regarded as the degree of membership of the fuzzy set of sites whose environmental conditions are favourable to the species, which enables applying the rules of fuzzy logic to distribution modelling. They also allow for direct comparisons between models for species with different presence/absence ratios in the study area. This makes them more useful to estimate the conservation value of areas, to design ecological corridors, or to select appropriate areas for species reintroductions. Received: June 2005 / Revised: July 2005     

Use of Coarse‐Resolution Models of Species’ Distributions to Guide Local Conservation Inferences

Abstract: Distribution models are used increasingly for species conservation assessments over extensive areas, but the spatial resolution of the modeled data and, consequently, of the predictions generated directly from these models are usually too coarse for local conservation applications. Comprehensive distribution data at finer spatial resolution, however, require a level of sampling that is impractical for most species and regions. Models can be downscaled to predict distribution at finer resolutions, but this increases uncertainty because the predictive ability of models is not necessarily consistent beyond their original scale. We analyzed the performance of downscaled, previously published models of environmental favorability (a generalized linear modeling technique) for a restricted endemic insectivore, the Iberian desman (Galemys pyrenaicus), and a more widespread carnivore, the Eurasian otter (Lutra lutra), in the Iberian Peninsula. The models, built from presence–absence data at 10 × 10 km resolution, were extrapolated to a resolution 100 times finer (1 × 1 km). We compared downscaled predictions of environmental quality for the two species with published data on local observations and on important conservation sites proposed by experts. Predictions were significantly related to observed presence or absence of species and to expert selection of sampling sites and important conservation sites. Our results suggest the potential usefulness of downscaled projections of environmental quality as a proxy for expensive and time‐consuming field studies when the field studies are not feasible. This method may be valid for other similar species if coarse‐resolution distribution data are available to define high‐quality areas at a scale that is practical for the application of concrete conservation measures.     

Relative density of finds for assessing similarity-based maps of orchid occurrence

Similarity-based mapping of the expected distribution of 10 orchid species was conducted in a study area covering 300 km² in south-eastern Estonia. The observation track and species finds were recorded during fieldwork. Absence locations were generated on the line of observation track. Both presence and absence sites having an in-between distance of at least 100 m were used as training data. Expected presence/absence of a species was calculated according to similarity between the predictable location and selected observations (examples) of presence and absence sites. For each species, the machine learning system identified the best predictive sets by selecting the most useful variables out of 136 map and remote sensing features. Similarity-based estimations were evaluated both by training fit and by independent verification data. Reliability of the predictive maps was expressed also by usefulness ratios—the densities of validation find sites (1) in the predicted presence area relative to the density of those in the predicted absence area, and (2) relative to the share of the observation track in the predicted presence area and in the predicted absence area. The predictive mapping was most efficient for Dactylorhiza incarnata, D. russowii, Epipactis palustris, and Goodyera repens. We conclude that the profound coverage of observations on any larger area is unrealistic and the reliability of similarity-based predictive maps depends on the representativity of existing records relative to the diversity of the study area. The investigation showed that the studied species are much more common in nature than the records in the national database indicate.     

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.     

Model based grouping of species across environmental gradients

We present a novel approach to the statistical analysis and prediction of multispecies data. The approach allows the simultaneous grouping and quantification of multiple species’ responses to environmental gradients. The underlying statistical model is a finite mixture model, where mixing is performed over the individual species’ responses to environmental gradients. Species with similar responses are grouped with minimal information loss. We term these groups species archetypes. Each species archetype has an associated GLM that can be used to predict distributions with appropriate measures of uncertainty. Initially, we illustrate the concept and method using artificial data and then with application to real data comprising 200 species from the Great Barrier Reef (GBR) lagoon on 13 oceanographic and geological gradients from 12°S to 24°S. The 200 species from the GBR are well represented by 15 species archetypes. The model is interpreted through maps of the probability of presence for a fine scale set of locations throughout the study area. Maps of uncertainty are also produced to provide statistical context. The presence of each species archetype was strongly influenced by oceanographic gradients, principally temperature, oxygen and salinity. The number of species in each group ranged from 4 to 34. The method has potential application to the analysis of multispecies distribution patterns and for multispecies management.     

Monitoring,imperfect detection,and risk optimization of a Tasmanian devil insurance population

Most species are imperfectly detected during biological surveys, which creates uncertainty around their abundance or presence at a given location. Decision makers managing threatened or pest species are regularly faced with this uncertainty. Wildlife diseases can drive species to extinction; thus, managing species with disease is an important part of conservation. Devil facial tumor disease (DFTD) is one such disease that led to the listing of the Tasmanian devil (Sarcophilus harrisii) as endangered. Managers aim to maintain devils in the wild by establishing disease‐free insurance populations at isolated sites. Often a resident DFTD‐affected population must first be removed. In a successful collaboration between decision scientists and wildlife managers, we used an accessible population model to inform monitoring decisions and facilitate the establishment of an insurance population of devils on Forestier Peninsula. We used a Bayesian catch‐effort model to estimate population size of a diseased population from removal and camera trap data. We also analyzed the costs and benefits of declaring the area disease‐free prior to reintroduction and establishment of a healthy insurance population. After the monitoring session in May–June 2015, the probability that all devils had been successfully removed was close to 1, even when we accounted for a possible introduction of a devil to the site. Given this high probability and the baseline cost of declaring population absence prematurely, we found it was not cost‐effective to carry out any additional monitoring before introducing the insurance population. Considering these results within the broader context of Tasmanian devil management, managers ultimately decided to implement an additional monitoring session before the introduction. This was a conservative decision that accounted for uncertainty in model estimates and for the broader nonmonetary costs of mistakenly declaring the area disease‐free.     

Incorporating detectability of threatened species into environmental impact assessment

Environmental impact assessment (EIA) is a key mechanism for protecting threatened plant and animal species. Many species are not perfectly detectable and, even when present, may remain undetected during EIA surveys, increasing the risk of site‐level loss or extinction of species. Numerous methods now exist for estimating detectability of plants and animals. Despite this, regulations concerning survey protocol and effort during EIAs fail to adequately address issues of detectability. Probability of detection is intrinsically linked to survey effort; thus, minimum survey effort requirements are a useful way to address the risks of false absences. We utilized 2 methods for determining appropriate survey effort requirements during EIA surveys. One method determined the survey effort required to achieve a probability of detection of 0.95 when the species is present. The second method estimated the survey effort required to either detect the species or reduce the probability of presence to 0.05. We applied these methods to Pimelea spinscens subsp. spinescens, a critically endangered grassland plant species in Melbourne, Australia. We detected P. spinescens in only half of the surveys undertaken at sites where it was known to exist. Estimates of the survey effort required to detect the species or demonstrate its absence with any confidence were much higher than the effort traditionally invested in EIA surveys for this species. We argue that minimum survey requirements be established for all species listed under threatened species legislation and hope that our findings will provide an impetus for collecting, compiling, and synthesizing quantitative detectability estimates for a broad range of plant and animal species. Incorporación de la Capacidad de Detectar una Especie Amenazada a la Evaluación de Impacto Ambiental     

Simultaneous-count models to estimate abundance from counts of unmarked individuals with imperfect detection

We developed a method to estimate population abundance from simultaneous counts of unmarked individuals over multiple sites. We considered that at each sampling occasion, individuals in a population could be detected at 1 of the survey sites or remain undetected and used either multinomial or binomial simultaneous-count models to estimate abundance, the latter being equivalent to an N-mixture model with one site. We tested model performance with simulations over a range of detection probabilities, population sizes, growth rates, number of years, sampling occasions, and sites. We then applied our method to 3 critically endangered vulture species in Cambodia to demonstrate the real-world applicability of the model and to provide the first abundance estimates for these species in Cambodia. Our new approach works best when existing methods are expected to perform poorly (i.e., few sites and large variation in abundance among sites) and if individuals may move among sites between sampling occasions. The approach performed better when there were >8 sampling occasions and net probability of detection was high (>0.5). We believe our approach will be useful in particular for simultaneous surveys at aggregation sites, such as roosts. The method complements existing approaches for estimating abundance of unmarked individuals and is the first method designed specifically for simultaneous counts.     

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.     

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.     

Using the past to contextualize anthropogenic impacts on the present and future distribution of an endemic Caribbean mammal

Island species are difficult to conserve because they face the synergy of climate change, invasive species, deforestation, and increasing human population densities in areas where land mass is shrinking. The Caribbean island of Hispaniola presents particular challenges because of geopolitical complexities that span 2 countries and hinder coordinated management of species across the island. We employed species distribution modeling to evaluate the impacts of climatic change and anthropogenic activities on the distribution of an endemic mammal of conservation concern, the Hispaniolan solenodon (Solenodon paradoxus). We aggregated occurrence points for this poorly known species for the Last Glacial Maximum (LGM) and the present (1975–2016) based on museum collections, online biodiversity databases, and new field surveys. We quantified degree of overlap between periods and scenarios with Schoener's D. Through a conservation paleobiology lens, we found that over time humans played an increasing role in shaping the distribution of S. paradoxus, thus, providing a foundation for developing conservation strategies on appropriate spatiotemporal scales. Human population density was the single most important predictor of S. paradoxus occurrence. Densities >166 people/km² corresponded to a near-zero probability of occurrence. Models that accounted for climate but not anthropogenic variables falsely identified suitable habitat in Haiti, where on-the-ground surveys confirm habitat is unavailable. Climate-only models also significantly overestimated the potential for habitat connectivity between isolated populations. Our work highlights that alternative fates for S. paradoxus in the Anthropocene exist across the political border between the Dominican Republic and Haiti due to the fundamentally different economic and political realities of each country. Relationships in the fossil record confirm that Hispaniola's sociopolitical boundary is not biologically significant but instead represents one imposed on the island's fauna in the past 500 years by colonial activity. Our approach reveals how a paleontological perspective can contribute to concrete management insights.     

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.     

Movement distances enhance validity of predictive models

Including the distance species are able to move in predictive models improves conservation practice. Bird inventory projects carried out from 1993 to 2004 in Taiwan provide an opportunity to investigate the relationships among species distribution, movement distance, and the environment. We compared projected distributions of 17 Taiwanese endemic bird species using what we called the Standard Method (i.e. movement distance is zero) and what we called the Buffer Method (i.e. movement distance is longer than zero) in three presence-only models (GARP, MAXENT and LIVES). The Standard Method used species original occurrence records directly while the Buffer Method expanded the occurrence of species to areas 1 km² around each recorded location. We first tested the efficacy of the Buffer Method using ten common species of the 17, and then applied the method to two rare species of the 17. For both the common and rare species, the distributions predicted by the two methods showed slight but important differences. The Buffer Method for all species had a higher average predictive probability, while the Standard Method had a higher maximum predictive probability. Most of the values for the area under the curve (AUC) were over 0.8 with the exceptions of Taiwan Barbet (Megalaima nuchalis) and Taiwan Hwamei (Garrulax taewanus), which have recently separated from Indochinese Barbet (Megalaima annamensis) and Chinese Hwamei (Garrulax canorus), and since 2008 and 2006 have been regarded as species endemic to the study area. Kappa values showed good performance for all species using both methods. The Buffer Method, however, resulted in significantly higher sensitivity and accuracy values for all models of species (p < 0.05). We conclude that when modeling species distribution including the area where the species was censused along with areas within the minimum movement areas better defines the surrounding areas that might supplement core habitat requirements. Therefore, using the Buffer Method, species surrounding distribution can be obtained which provides a better understanding of the species distributions. Given that distribution size is a key to the conservation of species, we suggest the Buffer Method can be used in conservation planning.     

Estimation of abundance from presence–absence maps using cluster models

A presence–absence map consists of indicators of the occurrence or nonoccurrence of a given species in each cell over a grid, without counting the number of individuals in a cell once it is known it is occupied. They are commonly used to estimate the distribution of a species, but our interest is in using these data to estimate the abundance of the species. In practice, certain types of species (in particular flora types) may be spatially clustered. For example, some plant communities will naturally group together according to similar environmental characteristics within a given area. To estimate abundance, we develop an approach based on clustered negative binomial models with unknown cluster sizes. Our approach uses working clusters of cells to construct an estimator which we show is consistent. We also introduce a new concept called super-clustering used to estimate components of the standard errors and interval estimators. A simulation study is conducted to examine the performance of the estimators and they are applied to real data.     

Using multilevel spatial models to understand salamander site occupancy patterns after wildfire

Studies of the distribution of elusive forest wildlife have suffered from the confounding of true presence with the uncertainty of detection. Occupancy modeling, which incorporates probabilities of species detection conditional on presence, is an emerging approach for reducing observation bias. However, the current likelihood modeling framework is restrictive for handling unexplained sources of variation in the response that may occur when there are dependence structures such as smaller sampling units that are nested within larger sampling units. We used multilevel Bayesian occupancy modeling to handle dependence structures and to partition sources of variation in occupancy of sites by terrestrial salamanders (family Plethodontidae) within and surrounding an earlier wildfire in western Oregon, USA. Comparison of model fit favored a spatial N-mixture model that accounted for variation in salamander abundance over models that were based on binary detection/non-detection data. Though catch per unit effort was higher in burned areas than unburned, there was strong support that this pattern was due to a higher probability of capture for individuals in burned plots. Within the burn, the odds of capturing an individual given it was present were 2.06 times the odds outside the burn, reflecting reduced complexity of ground cover in the burn. Ther was weak support that true occupancy was lower within the burned area. While the odds of occupancy in the burn were 0.49 times the odds outside the burn among the five species, the magnitude of variation attributed to the burn was small in comparison to variation attributed to other landscape variables and to unexplained, spatially autocorrelated random variation. While ordinary occupancy models may separate the biological pattern of interest from variation in detection probability when all sources of variation are known, the addition of random effects structures for unexplained sources of variation in occupancy and detection probability may often more appropriately represent levels of uncertainty.     

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.