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     

Fine-scale habitat modeling of a top marine predator: do prey data improve predictive capacity?

Predators and prey assort themselves relative to each other, the availability of resources and refuges, and the temporal and spatial scale of their interaction. Predictive models of predator distributions often rely on these relationships by incorporating data on environmental variability and prey availability to determine predator habitat selection patterns. This approach to predictive modeling holds true in marine systems where observations of predators are logistically difficult, emphasizing the need for accurate models. In this paper, we ask whether including prey distribution data in fine-scale predictive models of bottlenose dolphin (Tursiops truncatus) habitat selection in Florida Bay, Florida, U.S.A., improves predictive capacity. Environmental characteristics are often used as predictor variables in habitat models of top marine predators with the assumption that they act as proxies of prey distribution. We examine the validity of this assumption by comparing the response of dolphin distribution and fish catch rates to the same environmental variables. Next, the predictive capacities of four models, with and without prey distribution data, are tested to determine whether dolphin habitat selection can be predicted without recourse to describing the distribution of their prey. The final analysis determines the accuracy of predictive maps of dolphin distribution produced by modeling areas of high fish catch based on significant environmental characteristics. We use spatial analysis and independent data sets to train and test the models. Our results indicate that, due to high habitat heterogeneity and the spatial variability of prey patches, fine-scale models of dolphin habitat selection in coastal habitats will be more successful if environmental variables are used as predictor variables of predator distributions rather than relying on prey data as explanatory variables. However, predictive modeling of prey distribution as the response variable based on environmental variability did produce high predictive performance of dolphin habitat selection, particularly foraging habitat.     

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.     

Modeling kelp forest distribution and biomass along temperate rocky coastlines

We address the global deficit of data describing kelp forest distribution, relative covers and biomass by testing the ability of species distribution models to predict these attributes at locations where data are currently limited. We integrated biological ground truth data with high-resolution environmental datasets to develop generalized additive models that accurately predict the structure of Laminaria forests within the Bay of Morlaix (48°42′42″N, 3°55′40″W). Forest distribution and proportional covers were predicted using water depth, light availability, wave exposure and sediment dynamics. The biomass of individual kelp species was modeled by supplementing these same variables with measures of seafloor slope and benthic position. Biomass predictions for Laminaria digitata and Laminaria hyperborea contrast the physiological tolerances of these species to light and wave exposure gradients. As a direct management output, we produced high-resolution maps (25 m² grids) that closely match independent field data and provide vital information for marine spatial planning.     

Are existing vegetation maps adequate to predict bird distributions?

Bird species are selective on the vegetation types in which they are found but predictive models of bird distribution based on variables derived from land-use/land-cover maps tend to have limited success. It has been suggested that accuracy of existing maps used to derive predictors is in part responsible for the limited success of bird distribution models. In two areas of 4900 km² of Western Andalusia, Spain, we compared the predictive ability of bird distribution models derived from two existing general-purpose land-use/land-cover maps, which differ in their resolution and accuracy: a coarse scale vegetation map of Europe, the CORINE land-cover map, and a detailed regional map, the 1995 land-use/land-cover map of Andalusia from the SINAMBA (Consejerı́a de Medio Ambiente, Junta de Andalucı́a). We compared the bird distribution models derived from these general-purpose vegetation maps with models derived from two more accurate structural vegetation maps built considering directly variables that influence bird habitat selection, one built from satellite images for this study and another obtained by improving the resolution and accuracy of the SINAMBA map with satellite data. We sampled the presence/absence of bird species at 857 points using 15-min point surveys. Predictive models for 54 bird species were built with generalised additive models (GAMs), using as potential predictors the same set of landscape and vegetation structure variables measured on each map. We compared for each bird species the predictive accuracy of the best model derived from each map. Vegetation structure measured at bird sample points was used as ground-truth for comparing the accuracy of vegetation maps. Although maps differed in their resolution and accuracy, the results show that all of them produced similarly accurate bird distribution models, with a mixed map produced with both thematic and satellite information being the best. The models derived from the more accurate vegetation structure maps obtained from satellite data were not more accurate than those derived directly from the SINAMBA or CORINE maps. Our results suggest that some general-purpose land-use/land-cover maps are accurate enough to derive bird distribution models. There is a certain limit to improve vegetation maps above which there is no effect in their power to predict bird distribution.     

Predicting tree species presence and basal area in Utah: A comparison of stochastic gradient boosting,generalized additive models,and tree-based methods

Many efforts are underway to produce broad-scale forest attribute maps by modelling forest class and structure variables collected in forest inventories as functions of satellite-based and biophysical information. Typically, variants of classification and regression trees implemented in Rulequest's^© See5 and Cubist (for binary and continuous responses, respectively) are the tools of choice in many of these applications. These tools are widely used in large remote sensing applications, but are not easily interpretable, do not have ties with survey estimation methods, and use proprietary unpublished algorithms. Consequently, three alternative modelling techniques were compared for mapping presence and basal area of 13 species located in the mountain ranges of Utah, USA. The modelling techniques compared included the widely used See5/Cubist, generalized additive models (GAMs), and stochastic gradient boosting (SGB). Model performance was evaluated using independent test data sets. Evaluation criteria for mapping species presence included specificity, sensitivity, Kappa, and area under the curve (AUC). Evaluation criteria for the continuous basal area variables included correlation and relative mean squared error. For predicting species presence (setting thresholds to maximize Kappa), SGB had higher values for the majority of the species for specificity and Kappa, while GAMs had higher values for the majority of the species for sensitivity. In evaluating resultant AUC values, GAM and/or SGB models had significantly better results than the See5 models where significant differences could be detected between models. For nine out of 13 species, basal area prediction results for all modelling techniques were poor (correlations less than 0.5 and relative mean squared errors greater than 0.8), but SGB provided the most stable predictions in these instances. SGB and Cubist performed equally well for modelling basal area for three species with moderate prediction success, while all three modelling tools produced comparably good predictions (correlation of 0.68 and relative mean squared error of 0.56) for one species.     

Predicting the distributions of marine organisms at the global scale

We present and evaluate AquaMaps, a presence-only species distribution modelling system that allows the incorporation of expert knowledge about habitat usage and was designed for maximum output of standardized species range maps at the global scale. In the marine environment there is a significant challenge to the production of range maps due to large biases in the amount and location of occurrence data for most species. AquaMaps is compared with traditional presence-only species distribution modelling methods to determine the quality of outputs under equivalently automated conditions. The effect of the inclusion of expert knowledge to AquaMaps is also investigated. Model outputs were tested internally, through data partitioning, and externally against independent survey data to determine the ability of models to predict presence versus absence. Models were also tested externally by assessing correlation with independent survey estimates of relative species abundance. AquaMaps outputs compare well to the existing methods tested, and inclusion of expert knowledge results in a general improvement in model outputs. The transparency, speed and adaptability of the AquaMaps system, as well as the existing online framework which allows expert review to compensate for sampling biases and thus improve model predictions are proposed as additional benefits for public and research use alike.     

Uncertainty propagation in vegetation distribution models based on ensemble classifiers

Ensemble learning techniques are increasingly applied for species and vegetation distribution modelling, often resulting in more accurate predictions. At the same time, uncertainty assessment of distribution models is gaining attention. In this study, Random Forests, an ensemble learning technique, is selected for vegetation distribution modelling based on environmental variables. The impact of two important sources of uncertainty, that is the uncertainty on spatial interpolation of environmental variables and the uncertainty on species clustering into vegetation types, is quantified based on sequential Gaussian simulation and pseudo-randomization tests, respectively. An empirical assessment of the uncertainty propagation to the distribution modelling results indicated a gradual decrease in performance with increasing input uncertainty. The test set error ranged from 30.83% to 52.63% and from 30.83% to 83.62%, when the uncertainty ranges on spatial interpolation and on vegetation clustering, respectively, were fully covered. Shannon’s entropy, which is proposed as a measure for uncertainty of ensemble predictions, revealed a similar increasing trend in prediction uncertainty. The implications of these results in an empirical distribution modelling framework are further discussed with respect to monitoring setup, spatial interpolation and species clustering.     

A Cross-National Analysis of How Economic Inequality Predicts Biodiversity Loss

Abstract:  We used socioeconomic models that included economic inequality to predict biodiversity loss, measured as the proportion of threatened plant and vertebrate species, across 50 countries. Our main goal was to evaluate whether economic inequality, measured as the Gini index of income distribution, improved the explanatory power of our statistical models. We compared four models that included the following: only population density, economic footprint (i.e., the size of the economy relative to the country area), economic footprint and income inequality (Gini index), and an index of environmental governance. We also tested the environmental Kuznets curve hypothesis, but it was not supported by the data. Statistical comparisons of the models revealed that the model including both economic footprint and inequality was the best predictor of threatened species. It significantly outperformed population density alone and the environmental governance model according to the Akaike information criterion. Inequality was a significant predictor of biodiversity loss and significantly improved the fit of our models. These results confirm that socioeconomic inequality is an important factor to consider when predicting rates of anthropogenic biodiversity loss.     

Comparative performance of generalized additive models and multivariate adaptive regression splines for statistical modelling of species distributions

Two statistical modelling techniques, generalized additive models (GAM) and multivariate adaptive regression splines (MARS), were used to analyse relationships between the distributions of 15 freshwater fish species and their environment. GAM and MARS models were fitted individually for each species, and a MARS multiresponse model was fitted in which the distributions of all species were analysed simultaneously. Model performance was evaluated using changes in deviance in the fitted models and the area under the receiver operating characteristic curve (ROC), calculated using a bootstrap assessment procedure that simulates predictive performance for independent data. Results indicate little difference between the performance of GAM and MARS models, even when MARS models included interaction terms between predictor variables. Results from MARS models are much more easily incorporated into other analyses than those from GAM models. The strong performance of a MARS multiresponse model, particularly for species of low prevalence, suggests that it may have distinct advantages for the analysis of large datasets. Its identification of a parsimonious set of environmental correlates of community composition, coupled with its ability to robustly model species distributions in relation to those variables, can be seen as converging strongly with the purposes of traditional ordination techniques.     

Ecological Correlates and Conservation Implications of Overestimating Species Geographic Ranges

Abstract:  Species range maps based on extents of occurrence (EOO maps) have become the basis for many analyses in broad-scale ecology and conservation. Nevertheless, EOO maps are usually highly interpolated and overestimate small-scale occurrence, which may bias research outcomes. We evaluated geographical range overestimation and its potential ecological causes for 1158 bird species by quantifying EOO map occurrence across 4040 well-studied survey locations in Australia, North America, and southern Africa at the scale of 80–742 km². Most species occurred in only 40–70% of the range indicated by their EOO maps. The observed proportional range overestimation affected the range-size frequency distribution, indicating that species are more range-restricted than suggested by EOO maps. The EOO maps most strongly overestimated the distribution of narrow-ranging species and ecological specialists with narrow diet and habitat breadth. These relationships support basic ecological predictions about the relationship between niche breadth and the fine-scale occurrence of species. Consequently, at-risk species were subject to particularly high proportional range overestimation, on average 62% compared with 37% of nonthreatened species. These trends affect broad-scale ecological analyses and species conservation assessments, which will benefit from a careful consideration of potential biases introduced by range overestimation.     

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.     

Using habitat suitability index and particle dispersion models for early detection of marine invaders.

Eradication and control of invasive species are often possible only if populations are detected when they are small and localized. To be efficient, detection surveys should be targeted at locations where there is the greatest risk of incursions. We examine the utility of habitat suitability index (HSI) and particle dispersion models for targeting sampling for marine pests. Habitat suitability index models are a simple way to identify suitable habitat when species distribution data are lacking. We compared the performance of HSI models with statistical models derived from independent data from New Zealand on the distribution of two nonindigenous bivalves: Theora lubrica and Musculista senhousia. Logistic regression models developed using the HSI scores as predictors of the presence/absence of Theora and Musculista explained 26.7% and 6.2% of the deviance in the data, respectively. Odds ratios for the HSI scores were greater than unity, indicating that they were genuine predictors of the presence/ absence of each species. The fit and predictive accuracy of each logistic model were improved when simulated patterns of dispersion from the nearest port were added as a predictor variable. Nevertheless, the combined model explained, at best, 46.5% of the deviance in the distribution of Theora and correctly predicted 56% of true presences and 50% of all cases. Omission errors were between 6% and 16%. Although statistical distribution models built directly from environmental predictors always outperformed the equivalent HSI models, the gain in model fit and accuracy was modest. High residual deviance in both types of model suggests that the distributions realized by Theora and Musculista in the field data were influenced by factors not explicitly modeled as explanatory variables and by error in the environmental data used to project suitable habitat for the species. Our results highlight the difficulty of accurately predicting the distribution of invasive marine species that exhibit low habitat occupancy and patchy distributions in time and space. Although the HSI and statistical models had utility as predictors of the likely distribution of nonindigenous marine species, the level of spatial accuracy achieved with them may be well below expectations for sensitive surveillance programs.     

The role of local and landscape level measures of greenness in modelling boreal plant species richness

Measurements of primary productivity and its heterogeneity based on satellite images can provide useful estimates of species richness and distribution patterns. However, species richness at a given site may depend not only on local habitat quality and productivity but also on the characteristics of the surrounding landscape. In this study we investigated whether the predictions of species richness of plant families in northern boreal landscape in Finland can be improved by incorporating greenness information from the surrounding landscape, as derived from remotely sensed data (mean, maximum, standard deviation and range values of NDVI derived from Landsat ETM), into local greenness models. Using plant species richness data of 28 plant families from 440 grid cells of 25 ha in size, generalized additive models (GAMs) were fitted into three different sets of explanatory variables: (1) local greenness only, (2) landscape greenness only, and (3) combined local and landscape greenness. The derived richness–greenness relationships were mainly unimodal or positively increasing but varied between different plant families, and depended also on whether greenness was measured as mean or maximum greenness. Incorporation of landscape level greenness variables improved significantly both the explanatory power and cross-validation statistics of the models including only local greenness variables. Landscape greenness information derived from remote sensing data integrated with local information has thus the potentiality to improve predictive assessments of species richness over extensive and inaccessible areas, especially in high-latitude landscapes. Overall, the significant relationship between plants and surrounding landscape quality detected here suggests that landscape factors should be considered in preserving species richness of boreal environments, as well as in conservation planning for biodiversity in other environments.     

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.     

Updating coarse-scale species distribution models using small fine-scale samples

Large, fine-grained samples are ideal for predictive species distribution models used for management purposes, but such datasets are not available for most species and conducting such surveys is costly. We attempted to overcome this obstacle by updating previously available coarse-grained logistic regression models with small fine-grained samples using a recalibration approach. Recalibration involves re-estimation of the intercept or slope of the linear predictor and may improve calibration (level of agreement between predicted and actual probabilities). If reliable estimates of occurrence likelihood are required (e.g., for species selection in ecological restoration) calibration should be preferred to other model performance measures. This updating approach is not expected to improve discrimination (the ability of the model to rank sites according to species suitability), because the rank order of predictions is not altered. We tested different updating methods and sample sizes with tree distribution data from Spain. Updated models were compared to models fitted using only fine-grained data (refitted models). Updated models performed reasonably well at fine scales and outperformed refitted models with small samples (10-100 occurrences). If a coarse-grained model is available (or could be easily developed) and fine-grained predictions are to be generated from a limited sample size, updating previous models may be a more accurate option than fitting a new model. Our results encourage further studies on model updating in other situations where species distribution models are used under different conditions from their training (e.g., different time periods, different regions).     

Spatiotemporal hierarchical modelling of extreme precipitation in Western Australia using anisotropic Gaussian random fields

We discuss an approach for the statistical modelling of extreme precipitation events in South-West Australia over space and time, using a latent spatiotemporal process where precipitation maxima follow a generalised extreme value distribution. Temporal features are captured by modelling trends on the location and scale parameters. Spatial features are captured using anisotropic Gaussian random fields. Site specific explanatory variables are also incorporated. We fit several models using Bayesian inferential methods to precipitation extremes recorded at 36 weather stations around the Western Australian state capital city of Perth over the period 1907–2009. Model choice is performed using the DIC criterion. The best fitting model shows significant non-stationarity over time, with extreme precipitation events becoming less frequent. Extreme precipitation events are stronger at coastal locations, with the intensity decreasing as we head to the higher and drier areas to the North-East.     

Species distribution models and ecological theory: A critical assessment and some possible new approaches

Given the importance of knowledge of species distribution for conservation and climate change management, continuous and progressive evaluation of the statistical models predicting species distributions is necessary. Current models are evaluated in terms of ecological theory used, the data model accepted and the statistical methods applied. Focus is restricted to Generalised Linear Models (GLM) and Generalised Additive Models (GAM). Certain currently unused regression methods are reviewed for their possible application to species modelling.A review of recent papers suggests that ecological theory is rarely explicitly considered. Current theory and results support species responses to environmental variables to be unimodal and often skewed though process-based theory is often lacking. Many studies fail to test for unimodal or skewed responses and straight-line relationships are often fitted without justification.Data resolution (size of sampling unit) determines the nature of the environmental niche models that can be fitted. A synthesis of differing ecophysiological ideas and the use of biophysical processes models could improve the selection of predictor variables. A better conceptual framework is needed for selecting variables.Comparison of statistical methods is difficult. Predictive success is insufficient and a test of ecological realism is also needed. Evaluation of methods needs artificial data, as there is no knowledge about the true relationships between variables for field data. However, use of artificial data is limited by lack of comprehensive theory.Three potentially new methods are reviewed. Quantile regression (QR) has potential and a strong theoretical justification in Liebig's law of the minimum. Structural equation modelling (SEM) has an appealing conceptual framework for testing causality but has problems with curvilinear relationships. Geographically weighted regression (GWR) intended to examine spatial non-stationarity of ecological processes requires further evaluation before being used.Synthesis and applications: explicit theory needs to be incorporated into species response models used in conservation. For example, testing for unimodal skewed responses should be a routine procedure. Clear statements of the ecological theory used, the nature of the data model and sufficient details of the statistical method are needed for current models to be evaluated. New statistical methods need to be evaluated for compatibility with ecological theory before use in applied ecology. Some recent work with artificial data suggests the combination of ecological knowledge and statistical skill is more important than the precise statistical method used. The potential exists for a synthesis of current species modelling approaches based on their differing ecological insights not their methodology.