Tropical deforestation in Madagascar: analysis using hierarchical,spatially explicit,Bayesian regression models

Establishing cause–effect relationships for deforestation at various scales has proven difficult even when rates of deforestation appear well documented. There is a need for better explanatory models, which also provide insight into the process of deforestation. We propose a novel hierarchical modeling specification incorporating spatial association. The hierarchical aspect allows us to accommodate misalignment between the land-use (response) data layer and explanatory data layers. Spatial structure seems appropriate due to the inherently spatial nature of land use and data layers explaining land use. Typically, there will be missing values or holes in the response data. To accommodate this we propose an imputation strategy. We apply our modeling approach to develop a novel deforestation model for the eastern wet forested zone of Madagascar, a global rain forest “hot spot”. Using five data layers created for this region, we fit a suitable spatial hierarchical model. Though fitting such models is computationally much more demanding than fitting more standard models, we show that the resulting interpretation is much richer. Also, we employ a model choice criterion to argue that our fully Bayesian model performs better than simpler ones. To the best of our knowledge, this is the first work that applies hierarchical Bayesian modeling techniques to study deforestation processes. We conclude with a discussion of our findings and an indication of the broader ecological applicability of our modeling style.     

Addressing among-group variation in covariate effects using multilevel models

Multilevel models are used to model processes associated with hierarchical data structures. Despite infrequent use in the biological and environmental sciences, the use of these models with hierarchically-structured data conveys multiple advantages. These include the assessment of whether covariate effects differ among groups or clusters, and separate estimation of covariate effects by hierarchical level (thereby addressing atomistic and aggregation fallacy concerns). We illustrate these advantages using larval mayfly count data derived from annual surveys on the Mississippi River and a continuous covariate (water depth).     

Making more out of sparse data: hierarchical modeling of species communities

Community ecologists and conservation biologists often work with data that are too sparse for achieving reliable inference with species-specific approaches. Here we explore the idea of combining species-specific models into a single hierarchical model. The community component of the model seeks for shared patterns in how the species respond to environmental covariates. We illustrate the modeling framework in the context of logistic regression and presence-absence data, but a similar hierarchical structure could also be used in many other types of applications. We first use simulated data to illustrate that the community component can improve parameterization of species-specific models especially for rare species, for which the data would be too sparse to be informative alone. We then apply the community model to real data on 500 diatom species to show that it has much greater predictive power than a collection of independent species-specific models. We use the modeling approach to show that roughly one-third of distance decay in community similarity can be explained by two variables characterizing water quality, rare species typically preferring nutrient-poor waters with high pH, and common species showing a more general pattern of resource use.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

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.     

Inference about density and temporary emigration in unmarked populations

Few species are distributed uniformly in space, and populations of mobile organisms are rarely closed with respect to movement, yet many models of density rely upon these assumptions. We present a hierarchical model allowing inference about the density of unmarked populations subject to temporary emigration and imperfect detection. The model can be fit to data collected using a variety of standard survey methods such as repeated point counts in which removal sampling, double-observer sampling, or distance sampling is used during each count. Simulation studies demonstrated that parameter estimators are unbiased when temporary emigration is either "completely random" or is determined by the size and location of home ranges relative to survey points. We also applied the model to repeated removal sampling data collected on Chestnut-sided Warblers (Dendroica pensylvancia) in the White Mountain National Forest, U.S.A. The density estimate from our model, 1.09 birds/ha, was similar to an estimate of 1.11 birds/ha produced by an intensive spot-mapping effort. Our model is also applicable when processes other than temporary emigration affect the probability of being available for detection, such as in studies using cue counts. Functions to implement the model have been added to the R package unmarked.     

Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions

A stochastic model is applied to describe the spatial structure of a forest stand. We aim at quantifying the strength of the competition process between the trees in terms of interaction within and between different size classes of trees using multivariate Gibbs point processes with hierarchical interactions introduced in [Högmander, H., Särkkä, A., 1999. Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051–1058]. The new model overcomes the main limitation of the traditional use of the Gibbs models allowing to describe systems with non-symmetric interactions between different objects. When analyzing interactions between neighbouring trees it is natural to assume that the size of a tree determines its hierarchical level: the largest trees are not influenced by any other trees than the trees in the same size class, while trees in the other size classes are influenced by the other trees in the same class as well as by all larger trees. In this paper, we describe a wide range of Gibbs models with both hierarchical and non-hierarchical interactions as well as a simulation algorithm and a parameter estimation procedure for the hierarchical models. We apply the hierarchical interaction model to the analysis of forest data consisting of locations and diameters of tree stems.     

Predicting species distributions from samples collected along roadsides

Predictive models of species distributions are typically developed with data collected along roads. Roadside sampling may provide a biased (nonrandom) sample; however, it is currently unknown whether roadside sampling limits the accuracy of predictions generated by species distribution models. We tested whether roadside sampling affects the accuracy of predictions generated by species distribution models by using a prospective sampling strategy designed specifically to address this issue. We built models from roadside data and validated model predictions at paired locations on unpaved roads and 200 m away from roads (off road), spatially and temporally independent from the data used for model building. We predicted species distributions of 15 bird species on the basis of point-count data from a landbird monitoring program in Montana and Idaho (U.S.A.). We used hierarchical occupancy models to account for imperfect detection. We expected predictions of species distributions derived from roadside-sampling data would be less accurate when validated with data from off-road sampling than when it was validated with data from roadside sampling and that model accuracy would be differentially affected by whether species were generalists, associated with edges, or associated with interior forest. Model performance measures (kappa, area under the curve of a receiver operating characteristic plot, and true skill statistic) did not differ between model predictions of roadside and off-road distributions of species. Furthermore, performance measures did not differ among edge, generalist, and interior species, despite a difference in vegetation structure along roadsides and off road and that 2 of the 15 species were more likely to occur along roadsides. If the range of environmental gradients is surveyed in roadside-sampling efforts, our results suggest that surveys along unpaved roads can be a valuable, unbiased source of information for species distribution models.     

Clustering species using a model of population dynamics and aggregation theory

The high species diversity of some ecosystems like tropical rainforests goes in pair with the scarcity of data for most species. This hinders the development of models that require enough data for fitting. The solution commonly adopted by modellers consists in grouping species to form more sizeable data sets. Classical methods for grouping species such as hierarchical cluster analysis do not take account of the variability of the species characteristics used for clustering. In this study a clustering method based on aggregation theory is presented. It takes account of the variability of species characteristics by searching for the grouping that minimizes the quadratic error (square bias plus variance) of some model’s prediction. This method allows one to check whether the gain in variance brought by data pooling compensate for the bias that it introduces. This method was applied to a data set on 94 tree species in a tropical rainforest in French Guiana, using a Usher matrix model to predict species dynamics. An optimal trade-off between bias and variance was found when grouping species. Grouping species appeared to decrease the quadratic error, except when the number of groups was very small. This clustering method yielded species groups similar to those of the hierarchical cluster analysis using Ward’s method when variance was small, that is when the number of groups was small.     

Combining functional data with hierarchical Gaussian process models

Gaussian process models have been used in applications ranging from machine learning to fisheries management. In the Bayesian framework, the Gaussian process is used as a prior for unknown functions, allowing the data to drive the relationship between inputs and outputs. In our research, we consider a scenario in which response and input data are available from several similar, but not necessarily identical, sources. When little information is known about one or more of the populations it may be advantageous to model all populations together. We present a hierarchical Gaussian process model with a structure that allows distinct features for each source as well as shared underlying characteristics. Key features and properties of the model are discussed and demonstrated in a number of simulation examples. The model is then applied to a data set consisting of three populations of Rotifer Brachionus calyciflorus Pallas. Specifically, we model the log growth rate of the populations using a combination of lagged population sizes. The various lag combinations are formally compared to obtain the best model inputs. We then formally compare the leading hierarchical Gaussian process model with the inferential results obtained under the independent Gaussian process model.     

Accounting for matching uncertainty in two stage capture–recapture experiments using photographic measurements of natural marks

We propose a Bayesian hierarchical modeling approach for estimating the size of a closed population from data obtained by identifying individuals through photographs of natural markings. We assume that noisy measurements of a set of distinctive features are available for each individual present in a photographic catalogue. To estimate the population size from two catalogues obtained during two different sampling occasions, we embed the standard two-stage $M_t$ capture–recapture model for closed population into a multivariate normal data matching model that identifies the common individuals across the catalogues. In addition to estimating the population size while accounting for the matching process uncertainty, this hierarchical modelling approach allows to identify the common individuals by using the information provided by the capture–recapture model. This way, our model also represents a novel and reliable tool able to reduce the amount of effort researchers have to expend in matching individuals. We illustrate and motivate the proposed approach via a real data set of photo-identification of narwhals. Moreover, we compare our method with a set of possible alternative approaches by using both the empirical data set and a simulation study.     

A robust-design formulation of the incidence function model of metapopulation dynamics applied to two species of rails

The incidence function model (IFM) uses area and connectivity to predict metapopulation dynamics. However, false absences and missing data can lead to underestimates of the number of sites contributing to connectivity, resulting in overestimates of dispersal ability and turnovers (extinctions plus colonizations). We extend estimation methods for the IFM by using a hierarchical Bayesian model to account both for false absences due to imperfect detection and for missing data due to sites not surveyed in some years. We compare parameter estimates, measures of metapopulation dynamics, and forecasts using stochastic patch occupancy models (SPOMs) among three IFM models: (1) a Bayesian formulation assuming no false absences and omitting site-year combinations with missing data; (2) a hierarchical Bayesian formulation assuming no false absences but incorporating missing data; and (3) a hierarchical Bayesian formulation allowing for imperfect detection and incorporating missing data. We fit the models to multiyear data sets of occupancy for two bird species that differ in body size and presumed dispersal ability but inhabit the same network of sites: the small Black Rail (Laterallus jamaicensis) and the medium-sized Virginia Rail (Rallus limicola). Incorporating missing data affected colonization parameters and led to lower estimates of dispersal ability for the Black Rail. Detection rates were high for the Black Rail in most years but moderate for the Virginia Rail. Incorporating imperfect detection resulted in higher occupancy and lower turnover rates for both species, with largest effects for the Virginia Rail. Forecasts using SPOMs were sensitive to both missing data and false absences; persistence in models assuming no false absences was more optimistic than from robust models. Our results suggest that incorporating false absences and missing data into the IFM can improve (1) estimates of dispersal ability and the effect of connectivity on colonization, (2) the scaling of extinction risk with patch area, and (3) forecasts of occupancy and turnover rates.     

Predator–prey models: an application for the plankton dynamics of lake Geneva

In this paper we present a hierarchical Bayesian analysis for a predator–prey model applied to ecology considering the use of Markov Chain Monte Carlo methods. We consider the introduction of a random effect in the model and the presence of a covariate vector. An application to ecology is considered using a data set related to the plankton dynamics of lake Geneva for the year 1990. We also discuss some aspects of discrimination of the proposed models.     

A multiscale hierarchical Markov transition matrix model for generating and analyzing thematic raster maps

A model is described for generating hierarchically scaled spatial pattern as represented in a thematic raster map. The model involves a series of Markov transition matrices, one for each level in the scaling hierarchy. In full generality, the model allows the transition matrices to be different at each level, potentially making available a large number of parameters for landscape characterization. The model is self-similar when the transition matrices are all equal. A method is presented for fitting the model to data that take the form of a single-resolution thematic raster map. Explicit analytic solutions are obtained for the fitted parameters. The fitting method is based on a relationship between the hierarchical transitions in the model and spatial transitions at varying distance scales in the data map, a categorical analogy of the geostatistical variogram.     

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).     

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.     

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 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.     

Analyzing spatial count data,with an application to weed counts

Count data on a lattice may arise in observational studies of ecological phenomena. In this paper a hierarchical spatial model is used to analyze weed counts. Anisotropy is introduced, and a bivariate extension of the model is presented.     

Hierarchical modeling of abundance in closed population capture–recapture models under heterogeneity

Hierarchical modeling of abundance in space or time using closed-population mark-recapture under heterogeneity (model \(\hbox {M}_{\text {h}}\) ) presents two challenges: (i) finding a flexible likelihood in which abundance appears as an explicit parameter and (ii) fitting the hierarchical model for abundance. The first challenge arises because abundance not only indexes the population size, it also determines the dimension of the capture probabilities in heterogeneity models. A common approach is to use data augmentation to include these capture probabilities directly into the likelihood and fit the model using Bayesian inference via Markov chain Monte Carlo (MCMC). Two such examples of this approach are (i) explicit trans-dimensional MCMC, and (ii) superpopulation data augmentation. The superpopulation approach has the advantage of simple specification that is easily implemented in BUGS and related software. However, it reparameterizes the model so that abundance is no longer included, except as a derived quantity. This is a drawback when hierarchical models for abundance, or related parameters, are desired. Here, we analytically compare the two approaches and show that they are more closely related than might appear superficially. We exploit this relationship to specify the model in a way that allows us to include abundance as a parameter and that facilitates hierarchical modeling using readily available software such as BUGS. We use this approach to model trends in grizzly bear abundance in Yellowstone National Park from 1986 to 1998.