Advanced mapping of environmental data: Geostatistics, Machine Learning and Bayesian Maximum Entropy

This book combines geostatistics and global mapping systems to present an up-to-the-minute study of environmental data. Featuring numerous case studies, the reference covers model dependent (geostatistics) and data driven (machine learning algorithms) analysis techniques such as risk mapping, conditional stochastic simulations, descriptions of spatial uncertainty and variability, artificial neural networks (ANN) for spatial data, Bayesian maximum entropy (BME), and more.

Evaluating the influence of spatial uncertainty in locality points for species distributional modeling

1. Species distribution modelling is used increasingly in both applied and theoretical research to predict how species are distributed and to understand attributes of species' environmental requirements. In species distribution modelling, various statistical methods are used that combine species occurrence data with environmental spatial data layers to predict the suitability of any site for that species. While the number of data sharing initiatives involving species' occurrences in the scientific community has increased dramatically over the past few years, various data quality and methodological concerns related to using these data for species distribution modelling have not been addressed adequately. 2. We evaluated how uncertainty in georeferences and associated locational error in occurrences influence species distribution modelling using two treatments: (1) a control treatment where models were calibrated with original, accurate data and (2) an error treatment where data were first degraded spatially to simulate locational error. To incorporate error into the coordinates, we moved each coordinate with a random number drawn from the normal distribution with a mean of zero and a standard deviation of 5 km. We evaluated the influence of error on the performance of 10 commonly used distributional modelling techniques applied to 40 species in four distinct geographical regions. 3. Locational error in occurrences reduced model performance in three of these regions; relatively accurate predictions of species distributions were possible for most species, even with degraded occurrences. Two species distribution modelling techniques, boosted regression trees and maximum entropy, were the best performing models in the face of locational errors. The results obtained with boosted regression trees were only slightly degraded by errors in location, and the results obtained with the maximum entropy approach were not affected by such errors. 4. Synthesis and applications. To use the vast array of occurrence data that exists currently for research and management relating to the geographical ranges of species, modellers need to know the influence of locational error on model quality and whether some modelling techniques are particularly robust to error. We show that certain modelling techniques are particularly robust to a moderate level of locational error and that useful predictions of species distributions can be made even when occurrence data include some error.

What do we gain from simplicity versus complexity in species distribution models?

Species distribution models (SDMs) are widely used to explain and predict species ranges and environmental niches. They are most commonly constructed by inferring species' occurrence-environment relationships using statistical and machine-learning methods. The variety of methods that can be used to construct SDMs (e.g. generalized linear/additive models, tree-based models, maximum entropy, etc.), and the variety of ways that such models can be implemented, permits substantial flexibility in SDM complexity. Building models with an appropriate amount of complexity for the study objectives is critical for robust inference. We characterize complexity as the shape of the inferred occurrence-environment relationships and the number of parameters used to describe them, and search for insights into whether additional complexity is informative or superfluous. By building 'under fit' models, having insufficient flexibility to describe observed occurrence-environment relationships, we risk misunderstanding the factors shaping species distributions. By building 'over fit' models, with excessive flexibility, we risk inadvertently ascribing pattern to noise or building opaque models. However, model selection can be challenging, especially when comparing models constructed under different modeling approaches. Here we argue for a more pragmatic approach: researchers should constrain the complexity of their models based on study objective, attributes of the data, and an understanding of how these interact with the underlying biological processes. We discuss guidelines for balancing under fitting with over fitting and consequently how complexity affects decisions made during model building. Although some generalities are possible, our discussion reflects differences in opinions that favor simpler versus more complex models. We conclude that combining insights from both simple and complex SDM building approaches best advances our knowledge of current and future species ranges.

Active Learning of Very-High Resolution Optical Imagery with SVM: Entropy vs Margin Sampling

An active learning method is proposed for the semi-automatic selection of training sets in remote sensing image classification. The method adds iteratively to the current training set the unlabeled pixels for which the prediction of an ensemble of classifiers based on bagged training sets show maximum entropy. This way, the algorithm selects the pixels that are the most uncertain and that will improve the model if added in the training set. The user is asked to label such pixels at each iteration. Experiments using support vector machines (SVM) on an 8 classes QuickBird image show the excellent performances of the methods, that equals accuracies of both a model trained with ten times more pixels and a model whose training set has been built using a state-of-the-art SVM specific active learning method

Influence of environmental conditions on the distribution of Central Asian green toads with three ploidy levels.

We studied the distribution of Palearctic green toads (Bufo viridis subgroup), an anuran species group with three ploidy levels, inhabiting the Central Asian Amudarya River drainage. Various approaches (one-way, multivariate, components variance analyses and maximum entropy modelling) were used to estimate the effect of altitude, precipitation, temperature and land vegetation covers on the distribution of toads. It is usually assumed that polyploid species occur in regions with harsher climatic conditions (higher latitudes, elevations, etc.), but for the green toads complex, we revealed a more intricate situation. The diploid species (Bufo shaartusiensis and Bufo turanensis) inhabit the arid lowlands (from 44 to 789 m a.s.l.), while tetraploid Bufo pewzowi were recorded in mountainous regions (340-3492 m a.s.l.) with usually lower temperatures and higher precipitation rates than in the region inhabited by diploid species. The triploid species Bufo baturae was found in the Pamirs (Tajikistan) at the highest altitudes (2503-3859 m a.s.l.) under the harshest climatic conditions.

BME-based uncertainty assessment of the Chernobyl fallout

The vast territories that have been radioactively contaminated during the 1986 Chernobyl accident provide a substantial data set of radioactive monitoring data, which can be used for the verification and testing of the different spatial estimation (prediction) methods involved in risk assessment studies. Using the Chernobyl data set for such a purpose is motivated by its heterogeneous spatial structure (the data are characterized by large-scale correlations, short-scale variability, spotty features, etc.). The present work is concerned with the application of the Bayesian Maximum Entropy (BME) method to estimate the extent and the magnitude of the radioactive soil contamination by 137Cs due to the Chernobyl fallout. The powerful BME method allows rigorous incorporation of a wide variety of knowledge bases into the spatial estimation procedure leading to informative contamination maps. Exact measurements (?hard? data) are combined with secondary information on local uncertainties (treated as ?soft? data) to generate science-based uncertainty assessment of soil contamination estimates at unsampled locations. BME describes uncertainty in terms of the posterior probability distributions generated across space, whereas no assumption about the underlying distribution is made and non-linear estimators are automatically incorporated. Traditional estimation variances based on the assumption of an underlying Gaussian distribution (analogous, e.g., to the kriging variance) can be derived as a special case of the BME uncertainty analysis. The BME estimates obtained using hard and soft data are compared with the BME estimates obtained using only hard data. The comparison involves both the accuracy of the estimation maps using the exact data and the assessment of the associated uncertainty using repeated measurements. Furthermore, a comparison of the spatial estimation accuracy obtained by the two methods was carried out using a validation data set of hard data. Finally, a separate uncertainty analysis was conducted that evaluated the ability of the posterior probabilities to reproduce the distribution of the raw repeated measurements available in certain populated sites. The analysis provides an illustration of the improvement in mapping accuracy obtained by adding soft data to the existing hard data and, in general, demonstrates that the BME method performs well both in terms of estimation accuracy as well as in terms estimation error assessment, which are both useful features for the Chernobyl fallout study.

Effects of sample size on the performance of species distribution models

A wide range of modelling algorithms is used by ecologists, conservation practitioners, and others to predict species ranges from point locality data. Unfortunately, the amount of data available is limited for many taxa and regions, making it essential to quantify the sensitivity of these algorithms to sample size. This is the first study to address this need by rigorously evaluating a broad suite of algorithms with independent presence-absence data from multiple species and regions. We evaluated predictions from 12 algorithms for 46 species (from six different regions of the world) at three sample sizes (100, 30, and 10 records). We used data from natural history collections to run the models, and evaluated the quality of model predictions with area under the receiver operating characteristic curve (AUC). With decreasing sample size, model accuracy decreased and variability increased across species and between models. Novel modelling methods that incorporate both interactions between predictor variables and complex response shapes (i.e. GBM, MARS-INT, BRUTO) performed better than most methods at large sample sizes but not at the smallest sample sizes. Other algorithms were much less sensitive to sample size, including an algorithm based on maximum entropy (MAXENT) that had among the best predictive power across all sample sizes. Relative to other algorithms, a distance metric algorithm (DOMAIN) and a genetic algorithm (OM-GARP) had intermediate performance at the largest sample size and among the best performance at the lowest sample size. No algorithm predicted consistently well with small sample size (n < 30) and this should encourage highly conservative use of predictions based on small sample size and restrict their use to exploratory modelling.

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The quality of environmental data analysis and propagation of errors are heavily affected by the representativity of the initial sampling design [CRE 93, DEU 97, KAN 04a, LEN 06, MUL07]. Geostatistical methods such as kriging are related to field samples, whose spatial distribution is crucial for the correct detection of the phenomena. Literature about the design of environmental monitoring networks (MN) is widespread and several interesting books have recently been published [GRU 06, LEN 06, MUL 07] in order to clarify the basic principles of spatial sampling design (monitoring networks optimization) based on Support Vector Machines was proposed. Nonetheless, modelers often receive real data coming from environmental monitoring networks that suffer from problems of non-homogenity (clustering). Clustering can be related to the preferential sampling or to the impossibility of reaching certain regions.

Impact of model complexity on cross-temporal transferability in Maxent species distribution models: An assessment using paleobotanical data

Maximum entropy modeling (Maxent) is a widely used algorithm for predicting species distributions across space and time. Properly assessing the uncertainty in such predictions is non-trivial and requires validation with independent datasets. Notably, model complexity (number of model parameters) remains a major concern in relation to overfitting and, hence, transferability of Maxent models. An emerging approach is to validate the cross-temporal transferability of model predictions using paleoecological data. In this study, we assess the effect of model complexity on the performance of Maxent projections across time using two European plant species (Alnus giutinosa (L.) Gaertn. and Corylus avellana L) with an extensive late Quaternary fossil record in Spain as a study case. We fit 110 models with different levels of complexity under present time and tested model performance using AUC (area under the receiver operating characteristic curve) and AlCc (corrected Akaike Information Criterion) through the standard procedure of randomly partitioning current occurrence data. We then compared these results to an independent validation by projecting the models to mid-Holocene (6000 years before present) climatic conditions in Spain to assess their ability to predict fossil pollen presence-absence and abundance. We find that calibrating Maxent models with default settings result in the generation of overly complex models. While model performance increased with model complexity when predicting current distributions, it was higher with intermediate complexity when predicting mid-Holocene distributions. Hence, models of intermediate complexity resulted in the best trade-off to predict species distributions across time. Reliable temporal model transferability is especially relevant for forecasting species distributions under future climate change. Consequently, species-specific model tuning should be used to find the best modeling settings to control for complexity, notably with paleoecological data to independently validate model projections. For cross-temporal projections of species distributions for which paleoecological data is not available, models of intermediate complexity should be selected.

Fourth order pseudo maximum likelihood methods

We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in Gourieroux et al. (1984) to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods [Authors]

Official Positions for FRAX(®) clinical regarding glucocorticoids: the impact of the use of glucocorticoids on the estimate by FRAX(®) of the 10 year risk of fracture from Joint Official Positions Development Conference of the International Society for Clinical Densitometry and International Osteoporosis Foundation on FRAX(®).

Given the significant impact the use of glucocorticoids can have on fracture risk independent of bone density, their use has been incorporated as one of the clinical risk factors for calculating the 10-year fracture risk in the World Health Organization's Fracture Risk Assessment Tool (FRAX(®)). Like the other clinical risk factors, the use of glucocorticoids is included as a dichotomous variable with use of steroids defined as past or present exposure of 3 months or more of use of a daily dose of 5 mg or more of prednisolone or equivalent. The purpose of this report is to give clinicians guidance on adjustments which should be made to the 10-year risk based on the dose, duration of use and mode of delivery of glucocorticoids preparations. A subcommittee of the International Society for Clinical Densitometry and International Osteoporosis Foundation joint Position Development Conference presented its findings to an expert panel and the following recommendations were selected. 1) There is a dose relationship between glucocorticoid use of greater than 3 months and fracture risk. The average dose exposure captured within FRAX(®) is likely to be a prednisone dose of 2.5-7.5 mg/day or its equivalent. Fracture probability is under-estimated when prednisone dose is greater than 7.5 mg/day and is over-estimated when the prednisone dose is less than 2.5 mg/day. 2) Frequent intermittent use of higher doses of glucocorticoids increases fracture risk. Because of the variability in dose and dosing schedule, quantification of this risk is not possible. 3) High dose inhaled glucocorticoids may be a risk factor for fracture. FRAX(®) may underestimate fracture probability in users of high dose inhaled glucocorticoids. 4) Appropriate glucocorticoid replacement in individuals with adrenal insufficiency has not been found to increase fracture risk. In such patients, use of glucocorticoids should not be included in FRAX(®) calculations.

Adaptively weighted maximum likelihood estimation of discrete distributions

SummaryDiscrete data arise in various research fields, typically when the observations are count data.I propose a robust and efficient parametric procedure for estimation of discrete distributions. The estimation is done in two phases. First, a very robust, but possibly inefficient, estimate of the model parameters is computed and used to indentify outliers. Then the outliers are either removed from the sample or given low weights, and a weighted maximum likelihood estimate (WML) is computed.The weights are determined via an adaptive process such that if the data follow the model, then asymptotically no observation is downweighted.I prove that the final estimator inherits the breakdown point of the initial one, and that its influence function at the model is the same as the influence function of the maximum likelihood estimator, which strongly suggests that it is asymptotically fully efficient.The initial estimator is a minimum disparity estimator (MDE). MDEs can be shown to have full asymptotic efficiency, and some MDEs have very high breakdown points and very low bias under contamination. Several initial estimators are considered, and the performances of the WMLs based on each of them are studied.It results that in a great variety of situations the WML substantially improves the initial estimator, both in terms of finite sample mean square error and in terms of bias under contamination. Besides, the performances of the WML are rather stable under a change of the MDE even if the MDEs have very different behaviors.Two examples of application of the WML to real data are considered. In both of them, the necessity for a robust estimator is clear: the maximum likelihood estimator is badly corrupted by the presence of a few outliers.This procedure is particularly natural in the discrete distribution setting, but could be extended to the continuous case, for which a possible procedure is sketched.RésuméLes données discrètes sont présentes dans différents domaines de recherche, en particulier lorsque les observations sont des comptages.Je propose une méthode paramétrique robuste et efficace pour l'estimation de distributions discrètes. L'estimation est faite en deux phases. Tout d'abord, un estimateur très robuste des paramètres du modèle est calculé, et utilisé pour la détection des données aberrantes (outliers). Cet estimateur n'est pas nécessairement efficace. Ensuite, soit les outliers sont retirés de l'échantillon, soit des faibles poids leur sont attribués, et un estimateur du maximum de vraisemblance pondéré (WML) est calculé.Les poids sont déterminés via un processus adaptif, tel qu'asymptotiquement, si les données suivent le modèle, aucune observation n'est dépondérée.Je prouve que le point de rupture de l'estimateur final est au moins aussi élevé que celui de l'estimateur initial, et que sa fonction d'influence au modèle est la même que celle du maximum de vraisemblance, ce qui suggère que cet estimateur est pleinement efficace asymptotiquement.L'estimateur initial est un estimateur de disparité minimale (MDE). Les MDE sont asymptotiquement pleinement efficaces, et certains d'entre eux ont un point de rupture très élevé et un très faible biais sous contamination. J'étudie les performances du WML basé sur différents MDEs.Le résultat est que dans une grande variété de situations le WML améliore largement les performances de l'estimateur initial, autant en terme du carré moyen de l'erreur que du biais sous contamination. De plus, les performances du WML restent assez stables lorsqu'on change l'estimateur initial, même si les différents MDEs ont des comportements très différents.Je considère deux exemples d'application du WML à des données réelles, où la nécessité d'un estimateur robuste est manifeste : l'estimateur du maximum de vraisemblance est fortement corrompu par la présence de quelques outliers.La méthode proposée est particulièrement naturelle dans le cadre des distributions discrètes, mais pourrait être étendue au cas continu.

Robust response transformations based on optimal prediction

Nonlinear regression problems can often be reduced to linearity by transforming the response variable (e.g., using the Box-Cox family of transformations). The classic estimates of the parameter defining the transformation as well as of the regression coefficients are based on the maximum likelihood criterion, assuming homoscedastic normal errors for the transformed response. These estimates are nonrobust in the presence of outliers and can be inconsistent when the errors are nonnormal or heteroscedastic. This article proposes new robust estimates that are consistent and asymptotically normal for any unimodal and homoscedastic error distribution. For this purpose, a robust version of conditional expectation is introduced for which the prediction mean squared error is replaced with an M scale. This concept is then used to develop a nonparametric criterion to estimate the transformation parameter as well as the regression coefficients. A finite sample estimate of this criterion based on a robust version of smearing is also proposed. Monte Carlo experiments show that the new estimates compare favorably with respect to the available competitors.