Maximum entropy modeling of species geographic distributions

The availability of detailed environmental data, together with inexpensive and powerful computers, has fueled a rapid increase in predictive modeling of species environmental requirements and geographic distributions. For some species, detailed presence/absence occurrence data are available, allowing the use of a variety of standard statistical techniques. However, absence data are not available for most species. In this paper, we introduce the use of the maximum entropy method (Maxent) for modeling species geographic distributions with presence-only data. Maxent is a general-purpose machine learning method with a simple and precise mathematical formulation, and it has a number of aspects that make it well-suited for species distribution modeling. In order to investigate the efficacy of the method, here we perform a continental-scale case study using two Neotropical mammals: a lowland species of sloth, Bradypus variegatus, and a small montane murid rodent, Microryzomys minutus. We compared Maxent predictions with those of a commonly used presence-only modeling method, the Genetic Algorithm for Rule-Set Prediction (GARP). We made predictions on 10 random subsets of the occurrence records for both species, and then used the remaining localities for testing. Both algorithms provided reasonable estimates of the species’ range, far superior to the shaded outline maps available in field guides. All models were significantly better than random in both binomial tests of omission and receiver operating characteristic (ROC) analyses. The area under the ROC curve (AUC) was almost always higher for Maxent, indicating better discrimination of suitable versus unsuitable areas for the species. The Maxent modeling approach can be used in its present form for many applications with presence-only datasets, and merits further research and development.     

Modeling resource selection using polytomous logistic regression and kernel density estimates

Wildlife resource selection studies typically compare used to available resources; selection or avoidance occurs when use is disproportionately greater or less than availability. Comparing used to available resources is problematic because results are often greatly influenced by what is considered available to the animal. Moreover, placing relocation points within resource units is often difficult due to radiotelemetry and mapping errors. Given these problems, we suggest that an animal’s resource use be summarized at the scale of the home range (i.e., the spatial distribution of all point locations of an animal) rather than by individual points that are considered used or available. To account for differences in use-intensity throughout an animal’s home range, we model resource selection using kernel density estimates and polytomous logistic regression. We present a case study of elk (Cervus elaphus) resource selection in South Dakota to illustrate the procedure. There are several advantages of our proposed approach. First, resource availability goes undefined by the investigator, which is a difficult and often arbitrary decision. Instead, the technique compares the intensity of animal use throughout the home range. This technique also avoids problems with classifying locations rigidly as used or unused. Second, location coordinates do not need to be placed within mapped resource units, which is problematic given mapping and telemetry error. Finally, resource use is considered at an appropriate scale for management because most wildlife resource decisions are made at the level of the patch. Despite the advantages of this use-intensity procedure, future research should address spatial autocorrelation and develop spatial models for ordered categorical variables.     

Analyzing animal movements using Brownian bridges

By studying animal movements, researchers can gain insight into many of the ecological characteristics and processes important for understanding population-level dynamics. We developed a Brownian bridge movement model (BBMM) for estimating the expected movement path of an animal, using discrete location data obtained at relatively short time intervals. The BBMM is based on the properties of a conditional random walk between successive pairs of locations, dependent on the time between locations, the distance between locations, and the Brownian motion variance that is related to the animal's mobility. We describe two critical developments that enable widespread use of the BBMM, including a derivation of the model when location data are measured with error and a maximum likelihood approach for estimating the Brownian motion variance. After the BBMM is fitted to location data, an estimate of the animal's probability of occurrence can be generated for an area during the time of observation. To illustrate potential applications, we provide three examples: estimating animal home ranges, estimating animal migration routes, and evaluating the influence of fine-scale resource selection on animal movement patterns.     

Identifying taxonomic and functional surrogates for spring biodiversity conservation

Surrogate approaches are widely used to estimate overall taxonomic diversity for conservation planning. Surrogate taxa are frequently selected based on rarity or charisma, whereas selection through statistical modeling has been applied rarely. We used boosted‐regression‐tree models (BRT) fitted to biological data from 165 springs to identify bryophyte and invertebrate surrogates for taxonomic and functional diversity of boreal springs. We focused on these 2 groups because they are well known and abundant in most boreal springs. The best indicators of taxonomic versus functional diversity differed. The bryophyte Bryum weigelii and the chironomid larva Paratrichocladius skirwithensis best indicated taxonomic diversity, whereas the isopod Asellus aquaticus and the chironomid Macropelopia spp. were the best surrogates of functional diversity. In a scoring algorithm for priority‐site selection, taxonomic surrogates performed only slightly better than random selection for all spring‐dwelling taxa, but they were very effective in representing spring specialists, providing a distinct improvement over random solutions. However, the surrogates for taxonomic diversity represented functional diversity poorly and vice versa. When combined with cross‐taxon complementarity analyses, surrogate selection based on statistical modeling provides a promising approach for identifying groundwater‐dependent ecosystems of special conservation value, a key requirement of the EU Water Framework Directive.     

Weighted distributions and estimation of resource selection probability functions

Understanding how organisms selectively use resources is essential for designing wildlife management strategies. The probability that an individual uses a given resource, as characterized by environmental factors, can be quantified in terms of the resource selection probability function (RSPF). The present literature on the topic has claimed that, except when both used and unused sites are known, the RSPF is non-estimable and that only a function proportional to RSPF, namely, the resource selection function (RSF) can be estimated. This paper describes a close connection between the estimation of the RSPF and the estimation of the weight function in the theory of weighted distributions. This connection can be used to obtain fully efficient, maximum likelihood estimators of the resource selection probability function under commonly used survey designs in wildlife management. The method is illustrated using GPS collar data for mountain goats (Oreamnos americanus de Blainville 1816) in northwest British Columbia, Canada.     

Nonparametric spatial regression of survival probability: visualization of population sinks in Eurasian woodcock

Both evolutionary ecologists and wildlife managers make inference based on how fitness and demography vary in space. Spatial variation in survival can be difficult to assess in the wild because (1) multisite study designs are not well suited to populations that are continuously distributed across a large area and (2) available statistical models accounting for detectability less than 1.0 do not easily cope with geographical coordinates. Here we use penalized splines within a Bayesian state-space modeling framework to estimate and visualize survival probability in two dimensions. The approach is flexible in that no parametric form for the relationship between survival and coordinates need be specified a priori. To illustrate our method, we study a game species, the Eurasian Woodcock Scolopax rusticola, based on band recovery data (5000 individuals) collected over a > 50 000-km2 area in west-central France with contrasted habitats and hunting pressures. We find that spatial variation in survival probability matches an index of hunting pressure and creates a mosaic of population sources and sinks. Such analyses could provide guidance concerning the spatial management of hunting intensity or could be used to identify pathways of spatial variation in fitness, for example, to study adaptation to changing landscape and climate.     

Modelling species distributions with penalised logistic regressions: A comparison with maximum entropy models

An important aspect of species distribution modelling is the choice of the modelling method because a suboptimal method may have poor predictive performance. Previous comparisons have found that novel methods, such as Maxent models, outperform well-established modelling methods, such as the standard logistic regression. These comparisons used training samples with small numbers of occurrences per estimated model parameter, and this limited sample size may have caused poorer predictive performance due to overfitting. Our hypothesis is that Maxent models would outperform a standard logistic regression because Maxent models avoid overfitting by using regularisation techniques and a standard logistic regression does not. Regularisation can be applied to logistic regression models using penalised maximum likelihood estimation. This estimation procedure shrinks the regression coefficients towards zero, causing biased predictions if applied to the training sample but improving the accuracy of new predictions. We used Maxent and logistic regression (standard and penalised) to analyse presence/pseudo-absence data for 13 tree species and evaluated the predictive performance (discrimination) using presence-absence data. The penalised logistic regression outperformed standard logistic regression and equalled the performance of Maxent. The penalised logistic regression may be considered one of the best methods to develop species distribution models trained with presence/pseudo-absence data, as it is comparable to Maxent. Our results encourage further use of the penalised logistic regression for species distribution modelling, especially in those cases in which a complex model must be fitted to a sample with a limited size.     

Generalized functional responses for species distributions

Researchers employing resource selection functions (RSFs) and other related methods aim to detect correlates of space-use and mitigate against detrimental environmental change. However, an empirical model fit to data from one place or time is unlikely to capture species responses under different conditions because organisms respond nonlinearly to changes in habitat availability. This phenomenon, known as a functional response in resource selection, has been debated extensively in the RSF literature but continues to be ignored by practitioners for lack of a practical treatment. We therefore extend the RSF approach to enable it to estimate generalized functional responses (GFRs) from spatial data. GFRs employ data from several sampling instances characterized by diverse profiles of habitat availability. By modeling the regression coefficients of the underlying RSF as functions of availability, GFRs can account for environmental change and thus predict population distributions in new environments. We formulate the approach as a mixed-effects model so that it is estimable by readily available statistical software. We illustrate its application using (1) simulation and (2) wolf home-range telemetry. Our results indicate that GFRs can offer considerable improvements in estimation speed and predictive ability over existing mixed-effects approaches.     

Spatial memory, habitat auto-facilitation and the emergence of fractal home range patterns

Animals interact with their habitat in a manner which involves both negative and positive feedback mechanisms. We apply a specific modeling approach, “multi-scaled random walk”, for the scenario where a spatially explicit positive feedback process emerges from a combination of a spatial memory-dependent tendency to return to familiar patches and a consequently objective or subjective improvement of the quality of these patches (habitat auto-facilitation). In addition to the potential for local resource improvement from physically altering a patch, primarily known from the ecology of grazing ungulates, auto-facilitation from site fidelity may also embed more subtle subjective, individual-specific advantages from patch familiarity. Under the condition of resource superabundance, fitness gain from intra-home range patch fidelity creates a self-reinforcing use of the preferred patches on expense of a broader foraging in a priori equally favorable patches. Through this process, our simulations show that a spatially fractal dispersion of accumulated locations of the individual will emerge under the given model assumptions. Based on a conjecture that intra-home range patch fidelity depends on spatial memory we apply the multi-scaled random walk model to construct a spatially explicit habitat suitability parameter H_ij, which quantifies the dispersion of the generally most constraining resource from the individual's perspective. An intra-home range set of observed H-scores, H_obs, can then be estimated from a simple 2-scale calculation that is derived from the local dispersion of fixes. We show how the spatially explicit habitat utilization index H_obs not necessarily correlates positively with the local density fluctuations of fixes. The H-index solves some well-known problems from using the pattern of local densities of telemetry fixes - the classic utilization distribution - as a proxy variable for relative intra-home range habitat quality and resource selection. A pilot study on a set of telemetry fixes collected from a herd of free-ranging domestic sheep with overlapping summer home ranges illustrates how the H-index may be estimated and interpreted as a first-level approach towards a more extensive analysis of intra-home range habitat resource availability and patch preferences. Spatial memory in combination with site fidelity requires a modeling framework that explicitly describes the property of positive feedback mechanism under auto-facilitation in a spatio-temporally explicit manner.     

Finite population properties of individual predictors based on spatial patterns

In this study a conceptual framework for assessing the statistical properties of a non-stochastic spatial interpolator is developed through the use of design-based finite population inference tools. By considering the observed locations as the result of a probabilistic sampling design, we propose a standardized weighted predictor for spatial data starting from a deterministic interpolator that usually does not provide uncertainty measures. The information regarding the coordinates of the spatial locations is known at the population level and is directly used in constructing the weighting system. Our procedure captures the spatial pattern by means of the Euclidean distances between locations, which are fixed and do not require any further assessment after the sample has been drawn. The predictor for any individual value turns in a ratio of design-based random quantities. We illustrate the predictor design-based statistical properties, i.e. asymptotically p-unbiasedness and p-consistency, for simple random sampling without replacement. An application to a couple of environmental datasets is presented, for assessing predictor performances in correspondence of different population characteristics. A comparison with the equivalent non-spatial predictor is presented.     

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.     

Impact of the quality of climate models for modelling species occurrences in countries with poor climatic documentation: a case study from Bolivia

The quality of climate models has largely been overlooked as a possible source of uncertainty that may affect the outcomes of species distribution models, especially in the tropics, where comparatively few climatic stations are available. We compared the geographical discrepancies and potential conservation implications of using two different climate models (Saga and Worldclim) in combination with the species modelling approach Maxent in Bolivia. We estimated ranges of selected bird and fern species biogeographically restricted to either humid montane forest of the northern Bolivian Andes or seasonal dry tropical forests (in the Andes and southern lowlands). Saga and Worldclim predicted roughly similar climate patterns of temperature that were significantly correlated. Precipitation layers of both climate models were also roughly similar, but showed important differences. Species ranges estimated with Worldclim and Saga likewise produced different results. Ranges of species endemic to humid montane forests estimated with Saga had higher AUC (Area under the curve) values than those estimated with Worldclim, which for example predicted the occurrence of humid montane forest bird species near Lake Titicaca, an area that is clearly unsuitable for these species. Likewise, Worldclim overpredicted the occurrence of fern and bird species in the lowlands of the Chapare region and well south of the Andean Elbow, where more seasonal biomes occur. By contrast, Saga predictions were coherent with the known distribution of humid montane forests in the northern Bolivian Andes. Estimated ranges of species endemic to seasonal dry tropical forests predicted with Saga and Worldclim were not statistically different in most cases. However, detailed comparisons revealed that Saga was able to distinguish fragments of seasonal dry tropical forests in rain-shadow valleys of the northern Bolivian Andes, whereas Worldclim was not. These differences highlight the neglected influence of climate layers on modelling results and the importance of using the most accurate climate data available when modelling species distributions.     

Modeling a spatially restricted distribution in the Neotropics: How the size of calibration area affects the performance of five presence-only methods

We here examine species distribution models for a Neotropical anuran restricted to ombrophilous areas in the Brazilian Atlantic Forest hotspot. We extend the known occurrence for the treefrog Hypsiboas bischoffi (Anura: Hylidae) through GPS field surveys and use five modeling methods (BIOCLIM, DOMAIN, OM-GARP, SVM, and MAXENT) and selected bioclimatic and topographic variables to model the species distribution. Models were first trained using two calibration areas: the Brazilian Atlantic Forest (BAF) and the whole of South America (SA). All modeling methods showed good levels of predictive power and accuracy with mean AUC ranging from 0.77 (BIOCLIM/BAF) to 0.99 (MAXENT/SA). MAXENT and SVM were the most accurate presence-only methods among those tested here. All but the SVM models calibrated with SA predicted larger distribution areas when compared to models calibrated in BAF. OM-GARP dramatically overpredicted the species distribution for the model calibrated in SA, with a predicted area around 10⁶ km² larger than predicted by other SDMs. With increased calibration area (and environmental space), OM-GARP predictions followed changes in the environmental space associated with the increased calibration area, while MAXENT models were more consistent across calibration areas. MAXENT was the only method that retrieved consistent predictions across calibration areas, while allowing for some overprediction, a result that may be relevant for modeling the distribution of other spatially restricted organisms.     

Assessing the effects of pseudo-absences on predictive distribution model performance

Modelling species distributions with presence data from atlases, museum collections and databases is challenging. In this paper, we compare seven procedures to generate pseudo-absence data, which in turn are used to generate GLM-logistic regressed models when reliable absence data are not available. We use pseudo-absences selected randomly or by means of presence-only methods (ENFA and MDE) to model the distribution of a threatened endemic Iberian moth species (Graellsia isabelae). The results show that the pseudo-absence selection method greatly influences the percentage of explained variability, the scores of the accuracy measures and, most importantly, the degree of constraint in the distribution estimated. As we extract pseudo-absences from environmental regions further from the optimum established by presence data, the models generated obtain better accuracy scores, and over-prediction increases. When variables other than environmental ones influence the distribution of the species (i.e., non-equilibrium state) and precise information on absences is non-existent, the random selection of pseudo-absences or their selection from environmental localities similar to those of species presence data generates the most constrained predictive distribution maps, because pseudo-absences can be located within environmentally suitable areas. This study shows that if we do not have reliable absence data, the method of pseudo-absence selection strongly conditions the obtained model, generating different model predictions in the gradient between potential and realized distributions.     

Spectrum of selection: new approaches to detecting the scale-dependent response to habitat

Detecting habitat selection depends on the spatial scale of analysis, but multi-scale studies have been limited by the use of a few, spatially variable, hierarchical levels. We developed spatially explicit approaches to quantify selection along a continuum of scales using spatial (coarse-graining) and geostatistical (variogram) pattern analyses at multiple levels of habitat use (seasonal range, travel routes, feeding areas, and microsites). We illustrate these continuum-based approaches by applying them to winter habitat selection by woodland caribou (Rangifer tarandus caribou) using two key habitat components, Cladina lichens and snow depth. We quantified selection as the reduction in variance in used relative to available sites, thus avoiding reliance on correlations between organism and habitat, for which interpretation can be impeded by cross-scale correlations. By consistently selecting favorable habitat features, caribou experienced reduced variance in these features. The degree to which selection was accounted for by the travel route, feeding area, or microsite levels varied across the scale continuum. Caribou selected for Cladina within a 13-km scale domain and selected shallower snow at all scales. Caribou responded most strongly at the dominant scales of patchiness, implicating habitat heterogeneity as an underlying cause of multi-scale habitat selection. These novel approaches enable a spatial understanding of resource selection behavior.     

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.     

Bandwidth determination for kernel density analysis of wildfire events at forest sub-district scale

Forest fire is regarded as one of the most significant factors leading to land degradation. While evaluating fire hazard or producing fire risk zone maps, quantitative analyses using historic fire data is often required, and during all these modeling and multi-criteria analysis processes, the fire event itself is taken as the dependent variable. However, there are two main problematic issues in analyzing historic fire data. The first difficulty arises from the fact that it is in point format, whereas a continuous surface is frequently needed for statistically analyzing the relationship of fire events with other factors, such as anthropogenic, topographic and climatic conditions. Another, and probably the most bothersome challenge is to overcome inaccuracy inherent in historic fire data in point format, since the exact coordinates of ignition points are mostly unknown. In this study, kernel density mapping, a widely used method for converting discrete point data into a continuous raster surface, was used to map the historic fire data in Mumcular Forest Sub-district in Mu?la, Turkey. The historic fire data was transferred onto the digital forest stand map of the study area, where the exact locations of ignition points are unknown; however, the exact number of ignition points in each compartment of the forest stand map is known. Different random distributions of ignition points were produced, and for each random distribution, kernel density maps were produced by applying two distinct kernel functions with several smoothing parameter options. The obtained maps were compared through correlation analysis in order to illustrate the effect of randomness, choice of kernel function and smoothing parameter. The proposed method gives a range of values rather than a single bandwidth value; however, it provides a more reliable way than comparing the maps with different bandwidths subjectively by eye.     

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.