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.     

Dynamic multi-resolution spatial models

Data from remote-sensing platforms play an important role in monitoring environmental processes, such as the distribution of stratospheric ozone. Remote-sense data are typically spatial, temporal, and massive. Existing prediction methods such as kriging are computationally infeasible. The multi-resolution spatial model (MRSM) captures nonstationary spatial dependence and produces fast optimal estimates using a change-of-resolution Kalman filter. However, past data can provide valuable information about the current status of the process being investigated. In this article, we incorporate the temporal dependence into the process by developing a dynamic MRSM. An application of the dynamic MRSM to a month of daily total column ozone data is presented, and on a given day the results of posterior inference are compared to those for the spatial-only MRSM. It is apparent that there are advantages to using the dynamic MRSM in regions where data are missing, such as when a whole swath of satellite data is missing.     

Graphical spatial models: a new view on interpreting spatial pattern

Graphical models provide an important tool for facilitating communication between scientists, decision-makers, and statisticians—many complicated ecological processes can be described in terms of “box-and-arrow” conceptual diagrams (e.g., Shipley in Cause and correlation in biology: a user’s guide to path analysis, structural equations and causal inferences, Cambridge Universtiy Press, Cambridge, 2000; Clark and Gelfand TRENDS in Ecology and Evolution 21:375–380, 2006). In particular, problems in landscape ecology often involve modeling relationships among multiple physical and/or biological variables that may operate on differing spatial scales (e.g., Rossi et al. in Ecol Monographs 62:277–314, 1992; Legendre et al. in Ecography 25:601–615, 2002; Overmars et al. in Ecol Model 164:257–270, 2003; Brown and Spector in J Appl Ecol 45:1639–1648, 2008; Koniak and Noy-Meir in Ecol Model 220:1148–1158, 2008). These problems are inherently multivariate, though researchers commonly rely on univariate methods, such as spatial regression models, to address them. In this paper, we introduce a multivariate method—graphical spatial models—that extends path analysis to incorporate spatial autocorrelation in one or more variables in a directed graph. We show how both exogenous and endogenous ecological processes as defined by Legendre et al. (Ecography 25:601–615, 2002) and Lichstein et al. (Ecol Monographs 72:445–463, 2002) can be represented in a graph. Most importantly, we show how to translate graphs representing these ecological processes into statistically estimable models. We motivate our theoretical results using an example of stream health data from the Willamette Valley, Oregon. For these data we are interested in the spatial pattern within both riparian land use and an index of stream health, and whether there is an association between land use and stream health, after accounting for these spatial patterns. We use a graphical spatial model to address these ecological questions simultaneously. We find that the health of a stream decreases as the percent of developed land within a 120-m riparian buffer increases; interestingly, there is only evidence of spatial pattern within land use.     

Predicting modes of spatial change from state-and-transition models

State-and-transition models (STMs) can represent many different types of landscape change, from simple gradient-driven transitions to complex, (pseudo-) random patterns. While previous applications of STMs have focused on individual states and transitions, this study addresses broader-scale modes of spatial change based on the entire network of states and transitions. STMs are treated as mathematical graphs, and several metrics from algebraic graph theory are applied—spectral radius, algebraic connectivity, and the S-metric. These indicate, respectively, the amplification of environmental change by state transitions, the relative rate of propagation of state changes through the landscape, and the degree of system structural constraints on the spatial propagation of state transitions. The analysis is illustrated by application to the Gualalupe/San Antonio River delta, Texas, with soil types as representations of system states. Concepts of change in deltaic environments are typically based on successional patterns in response to forcings such as sea level change or river inflows. However, results indicate more complex modes of change associated with amplification of changes in system states, relatively rapid spatial propagation of state transitions, and some structural constraints within the system. The implications are that complex, spatially variable state transitions are likely, constrained by local (within-delta) environmental gradients and initial conditions. As in most applications, the STM used in this study is a representation of observed state transitions. While the usual predictive application of STMs is identification of local state changes associated with, e.g., management strategies, the methods presented here show how STMs can be used at a broader scale to identify landscape scale modes of spatial change.     

Comparing CAR and P-spline models in spatial disease mapping

Smoothing risks is one of the main goals in disease mapping as classical measures, such as standardized mortality ratios, can be extremely variable. However, smoothing risks might hinder the detection of high risk areas, since these two objectives are somewhat contradictory. Most of the work on smoothing risks and detection of high risk areas has been derived using conditional autoregressive (CAR) models. In this work, penalized splines (P-splines) models are also investigated. Confidence intervals for the log-relative risk predictor will be derived as a tool to detect high-risk areas. The performance of P-spline and CAR models will be compared in terms of smoothing (relative bias), sensitivity (ability to detect high risk areas), and specificity (ability to discard false patterns created by noise) through a simulation study based on the well-known Scottish lip cancer data.     

Validation of ecological state space models using the Laplace approximation

Many statistical models in ecology follow the state space paradigm. For such models, the important step of model validation rarely receives as much attention as estimation or hypothesis testing, perhaps due to lack of available algorithms and software. Model validation is often based on a naive adaptation of Pearson residuals, i.e. the difference between observations and posterior means, even if this approach is flawed. Here, we consider validation of state space models through one-step prediction errors, and discuss principles and practicalities arising when the model has been fitted with a tool for estimation in general mixed effects models. Implementing one-step predictions in the R package Template Model Builder, we demonstrate that it is possible to perform model validation with little effort, even if the ecological model is multivariate, has non-linear dynamics, and whether observations are continuous or discrete. With both simulated data, and a real data set related to geolocation of seals, we demonstrate both the potential and the limitations of the techniques. Our results fill a need for convenient methods for validating a state space model, or alternatively, rejecting it while indicating useful directions in which the model could be improved.     

Linking 3D spatial models of fuels and fire: Effects of spatial heterogeneity on fire behavior

Crown fire endangers fire fighters and can have severe ecological consequences. Prediction of fire behavior in tree crowns is essential to informed decisions in fire management. Current methods used in fire management do not address variability in crown fuels. New mechanistic physics-based fire models address convective heat transfer with computational fluid dynamics (CFD) and can be used to model fire in heterogeneous crown fuels. However, the potential impacts of variability in crown fuels on fire behavior have not yet been explored. In this study we describe a new model, FUEL3D, which incorporates the pipe model theory (PMT) and a simple 3D recursive branching approach to model the distribution of fuel within individual tree crowns. FUEL3D uses forest inventory data as inputs, and stochastically retains geometric variability observed in field data. We investigate the effects of crown fuel heterogeneity on fire behavior with a CFD fire model by simulating fire under a homogeneous tree crown and a heterogeneous tree crown modeled with FUEL3D, using two different levels of surface fire intensity. Model output is used to estimate the probability of tree mortality, linking fire behavior and fire effects at the scale of an individual tree. We discovered that variability within a tree crown altered the timing, magnitude and dynamics of how fire burned through the crown; effects varied with surface fire intensity. In the lower surface fire intensity case, the heterogeneous tree crown barely ignited and would likely survive, while the homogeneous tree had nearly 80% fuel consumption and an order of magnitude difference in total net radiative heat transfer. In the higher surface fire intensity case, both cases burned readily. Differences for the homogeneous tree between the two surface fire intensity cases were minimal but were dramatic for the heterogeneous tree. These results suggest that heterogeneity within the crown causes more conditional, threshold-like interactions with fire. We conclude with discussion of implications for fire behavior modeling and fire ecology.     

Cross-validation of species distribution models: removing spatial sorting bias and calibration with a null model

Species distribution models are usually evaluated with cross-validation. In this procedure evaluation statistics are computed from model predictions for sites of presence and absence that were not used to train (fit) the model. Using data for 226 species, from six regions, and two species distribution modeling algorithms (Bioclim and MaxEnt), I show that this procedure is highly sensitive to "spatial sorting bias": the difference between the geographic distance from testing-presence to training-presence sites and the geographic distance from testing-absence (or testing-background) to training-presence sites. I propose the use of pairwise distance sampling to remove this bias, and the use of a null model that only considers the geographic distance to training sites to calibrate cross-validation results for remaining bias. Model evaluation results (AUC) were strongly inflated: the null model performed better than MaxEnt for 45% and better than Bioclim for 67% of the species. Spatial sorting bias and area under the receiver-operator curve (AUC) values increased when using partitioned presence data and random-absence data instead of independently obtained presence-absence testing data from systematic surveys. Pairwise distance sampling removed spatial sorting bias, yielding null models with an AUC close to 0.5, such that AUC was the same as null model calibrated AUC (cAUC). This adjustment strongly decreased AUC values and changed the ranking among species. Cross-validation results for different species are only comparable after removal of spatial sorting bias and/or calibration with an appropriate null model.     

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.     

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.     

Recursive residuals and model diagnostics for normal and non-normal state space models

Model diagnostics for normal and non-normal state space models are based on recursive residuals which are defined from the one-step ahead predictive distribution. Routine calculation of these residuals is discussed in detail. Various diagnostic tools are suggested to check, for example, for wrong observation distributions and for autocorrelation. The paper also discusses such topics as model diagnostics for discrete time series and model discrimination via Bayes factors. The case studies cover environmental applications such as analysing a time series of the number of daily rainfall occurrences and a time series of daily sulfur dioxide emissions.     

Exploring land use determinants in Italian municipalities: comparison of spatial econometric models

Environmental and Ecological Statistics - This study sets up a spatial econometric framework to explore the factors that best describe land consumption in Italy at the municipal level. By modelling...     

A new algorithm for estimating the parameters of the spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using a spatial generalized linear mixed model with location specific latent variables. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. So far, several numerical algorithms to solve the problem of calculating maximum likelihood estimates of this model have been presented. In this paper to estimate the parameters an approximate method is considered and a new algorithm is introduced that is much faster than existing algorithms but just as accurate. This is called the Approximate Expectation Maximization Gradient algorithm. The performance of the proposed algorithm and is illustrated with a simulation study and on a real data set.     

Null models reveal preferential sampling, spatial autocorrelation and overfitting in habitat suitability modelling

Nowadays, species are driven to extinction at a high rate. To reduce this rate it is important to delineate suitable habitats for these species in such a way that these areas can be suggested as conservation areas. The use of habitat suitability models (HSMs) can be of great importance for the delineation of such areas. In this study MaxEnt, a presence-only modelling technique, is used to develop HSMs for 223 nematode species of the Southern Bight of the North Sea. However, it is essential that these models are beyond discussion and they should be checked for potential errors. In this study we focused on two categories (1) errors which can be attributed to the database such as preferential sampling and spatial autocorrelation and (2) errors induced by the modelling technique such as overfitting, In order to quantify these adverse effects thousands of nulls models were created. The effect of preferential sampling (i.e. some areas where visited more frequenty than others) was investigated by comparing model outcomes based from null models sampling the actual sampling stations and null models sampling the entire mapping area (Raes and ter Steege, 2007). Overfitting is exposed by a fivefold cross-validation and the influence of spatial autocorrelation is assessed by separating test and training sets in space. Our results clearly show that all these effects are present: preferential sampling has a strong effect on the selection of non-random species models. Crossvalidation seems to have less influence on the model selection and spatial autocorrelation is also strongly present. It is clear from this study that predefined thresholds are not readily applicable to all datasets and additional tests are needed in model selection.     

Simulation of water quality in a flood detention area using models of different spatial discretization

Detention areas provide a means to lower peak discharges in rivers by temporarily storing excess water. In the case of extreme flood events, the storage effect reduces the risk of dike failures or extensive inundations for downstream reaches and near the site of abstraction. Due to the large amount of organic matter contained in the river water and the inundation of terrestrial vegetation in the detention area, a deterioration of water quality may occur. In particular, decay processes can cause a severe depletion of dissolved oxygen (DO) in the temporary water body. In this paper, we studied the potential of a water quality model to simulate the DO dynamics in a large but shallow detention area to be built at the Elbe River (Germany). Our focus was on examining the impact of spatial discretization on the model’s performance and usability. Therefore, we used a zero-dimensional (0D) and a two-dimensional (2D) modeling approach in parallel. The two approaches solely differ in their spatial discretization, while conversion processes, parameters, and boundary conditions were kept identical. The dynamics of DO simulated by the two models are similar in the initial flooding period but diverge when the system starts to drain. The deviation can be attributed to the different spatial discretization of the two models, leading to different estimates of flow velocities and water depths. Only the 2D model can account for the impact of spatial variability on the evolution of state variables. However, its application requires high efforts for pre- and post-processing and significantly longer computation times. The 2D model is, therefore, not suitable for investigating various flood scenarios or for analyzing the impact of parameter uncertainty. For practical applications, we recommend to firstly set up a fast-running model of reduced spatial discretization, e.g. a 0D model. Using this tool, the reliability of the simulation results should be checked by analyzing the parameter uncertainty of the water quality model. A particular focus may be on those parameters that are spatially variable and, therefore, believed to be better represented in a 2D approach. The benefit from the application of the more costly 2D model should be assessed, based on the analyses carried out with the 0D model. A 2D model appears to be preferable only if the simulated detention area has a complex topography, flow velocities are highly variable in space, and the parameters of the water quality model are well known.     

The application of predicted habitat models to investigate the spatial ecology of demersal fish assemblages

Benthic habitats are known to influence the abundance and richness of demersal fish assemblages; however, little is known about how habitat structure and composition influences these distributions at very fine scales. We examined how the benthic environment structures marine fish assemblages using high-resolution bathymetry and accurate predicted benthic habitat maps. Areas characterised by a mosaic of habitat patches supported the highest richness of demersal fishes. A total of 37.4% of the variation in the distribution of the fish assemblage was attributed to 6 significant variables. Depth explained 23.0% of the variation, with the boulders explaining 12.6% and relief 1.4%. The remaining measures (seawhips, light/exposure and solid reef) provided a small (<1.0%) but significant contribution. Identifying components of the benthic environment important in structuring fish assemblages and understanding how they influence the spatial distribution of marine fishes is imperative for better management of demersal fish populations.     

Correction to: Exploring land use determinants in Italian municipalities: comparison of spatial econometric models

Environmental and Ecological Statistics -     

Modelling the spatial distribution of montane and subalpine forests in the central Alps using digital elevation models

The analysis of complex interactions between spatial distribution patterns of site factors and vegetation types is crucial for understanding high mountain ecosystems, especially in the view of a changing climate. Therefore, in the present study, a GIS and remote sensing-based approach is followed to produce a vegetation map for a study area in the Western Alps (Switzerland). Two major forest alliances are chosen for analysis: subalpine coniferous forest Vaccinio-Piceion/Larici-Pinetum cembrae and montane oak forest Quercion pubescenti-petraeae. As spatial information on site factors is commonly lacking in mountain areas, the use of a digital elevation model (DEM) is a potential substitute for use in vegetation analyses: it highly correlates with temperature, moisture, geomorphological processes and disturbance factors. Thus, it is important to analyse the capabilities of a DEM for indicating habitat conditions in a landscape characterised by high topodiversity and a patchwork of microclimatic habitats.For the purpose of identifying the potential of landform parameters for the indication of forest habitat structures in the present study, 24 primary and secondary landform parameters have been derived, indicating temperature and moisture distribution, exposure towards wind, snow, etc. Quantitative analyses were performed using statistical means such as contingency correlation coefficients and principal components analysis. The results formed the basis for the development of parallel-epiped-vegetation models (PED) used to simulate the spatial distribution patterns of the subalpine coniferous and the montane oak forest. It can be shown that topographic variables derived from a DEM at a spatial resolution of 25 m are very useful for indicating habitats of large forest types. Additionally potential forest sites in the cultural landscape, removed by human logging, can be reconstructed.Inaccuracies within the simulation results can partly be attributed to the insufficient parameterisation of geomorphologic activity and to poor spatial resolution of the DEM as compared to the vegetation data. Although the lack of information on the human dimension leads to some uncertainties in the interpretation of spatial patterns of vegetation, the exclusive use of topographic variables in vegetation models for the indication of forest habitats is very promising.