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Geographically Weighted Regression: The Analysis of Spatially Varying Relationships

TL;DR: In this paper, the basic GWR model is extended to include local statistics and local models for spatial data, and a software for Geographically Weighting Regression is described. But this software is not suitable for the analysis of large scale data.
Abstract: Acknowledgements.Local Statistics and Local Models for Spatial Data. Geographically Weighted Regression: The Basics. Extensions to the Basic GWR Model. Statistical Inference and Geographically Weighted Regression. GWR and Spatial Autocorrelation. Scale Issues and Geographically Weighted Regression. Geographically Weighted Local Statistics. Extensions of Geographically Weighting. Software for Geographically Weighted Regression. Epilogue. Bibliography.Index.

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TL;DR: In this paper, the authors describe six different statistical approaches to infer correlates of species distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations.
Abstract: Species distributional or trait data based on range map (extent-of-occurrence) or atlas survey data often display spatial autocorrelation, i.e. locations close to each other exhibit more similar values than those further apart. If this pattern remains present in the residuals of a statistical model based on such data, one of the key assumptions of standard statistical analyses, that residuals are independent and identically distributed (i.i.d), is violated. The violation of the assumption of i.i.d. residuals may bias parameter estimates and can increase type I error rates (falsely rejecting the null hypothesis of no effect). While this is increasingly recognised by researchers analysing species distribution data, there is, to our knowledge, no comprehensive overview of the many available spatial statistical methods to take spatial autocorrelation into account in tests of statistical significance. Here, we describe six different statistical approaches to infer correlates of species’ distributions, for both presence/absence (binary response) and species abundance data (poisson or normally distributed response), while accounting for spatial autocorrelation in model residuals: autocovariate regression; spatial eigenvector mapping; generalised least squares; (conditional and simultaneous) autoregressive models and generalised estimating equations. A comprehensive comparison of the relative merits of these methods is beyond the scope of this paper. To demonstrate each method’s implementation, however, we undertook preliminary tests based on simulated data. These preliminary tests verified that most of the spatial modeling techniques we examined showed good type I error control and precise parameter estimates, at least when confronted with simplistic simulated data containing

2,820 citations

Journal ArticleDOI
TL;DR: For example, a hump-shaped altitudinal species-richness pattern is the most typical (c. 50%), with a monotonic decreasing pattern also frequently reported, but the relative distribution of patterns changes readily with spatial grain and extent.
Abstract: Despite two centuries of exploration, our understanding of factors determining the distribution of life on Earth is in many ways still in its infancy. Much of the disagreement about governing processes of variation in species richness may be the result of differences in our perception of species-richness patterns. Until recently, most studies of large-scale species-richness patterns assumed implicitly that patterns and mechanisms were scale invariant. Illustrated with examples and a quantitative analysis of published data on altitudinal gradients of species richness (n = 204), this review discusses how scale effects (extent and grain size) can influence our perception of patterns and processes. For example, a hump-shaped altitudinal species-richness pattern is the most typical (c. 50%), with a monotonic decreasing pattern (c. 25%) also frequently reported, but the relative distribution of patterns changes readily with spatial grain and extent. If we are to attribute relative impact to various factors influencing species richness and distribution and to decide at which point along a spatial and temporal continuum they act, we should not ask only how results vary as a function of scale but also search for consistent patterns in these scale effects. The review concludes with suggestions of potential routes for future analytical exploration of species-richness patterns.

1,211 citations

Journal ArticleDOI
TL;DR: SAM (Spatial Analysis in Macroecology) as discussed by the authors ) is a freeware application that offers a comprehensive array of spatial statistical methods, focused primarily on surface pattern spatial analysis.
Abstract: SAM (Spatial Analysis in Macroecology) is a freeware application that offers a comprehensive array of spatial statistical methods, focused primarily on surface pattern spatial analysis. SAM is a compact, but powerful stand-alone software, with a user-friendly, menu-driven graphical interface. The methods available in SAM are the most commonly used in macroecology and geographical ecology, and range from simple tools for exploratory graphical analysis (e.g. mapping and graphing) and descriptive statistics of spatial patterns (e.g. autocorrelation metrics), to advanced spatial regression models (e.g. autoregression and eigenvector filtering). Download of the software, along with the user manual, can be downloaded online at the SAM website: (permanent URL at ).

1,123 citations

Journal ArticleDOI
TL;DR: It is recommended that block cross-validation be used wherever dependence structures exist in a dataset, even if no correlation structure is visible in the fitted model residuals, or if the fitted models account for such correlations.
Abstract: Ecological data often show temporal, spatial, hierarchical (random effects), or phylogenetic structure. Modern statistical approaches are increasingly accounting for such dependencies. However, when performing cross-validation, these structures are regularly ignored, resulting in serious underestimation of predictive error. One cause for the poor performance of uncorrected (random) cross-validation, noted often by modellers, are dependence structures in the data that persist as dependence structures in model residuals, violating the assumption of independence. Even more concerning, because often overlooked, is that structured data also provides ample opportunity for overfitting with non-causal predictors. This problem can persist even if remedies such as autoregressive models, generalized least squares, or mixed models are used. Block cross-validation, where data are split strategically rather than randomly, can address these issues. However, the blocking strategy must be carefully considered. Blocking in space, time, random effects or phylogenetic distance, while accounting for dependencies in the data, may also unwittingly induce extrapolations by restricting the ranges or combinations of predictor variables available for model training, thus overestimating interpolation errors. On the other hand, deliberate blocking in predictor space may also improve error estimates when extrapolation is the modelling goal. Here, we review the ecological literature on non-random and blocked cross-validation approaches. We also provide a series of simulations and case studies, in which we show that, for all instances tested, block cross-validation is nearly universally more appropriate than random cross-validation if the goal is predicting to new data or predictor space, or for selecting causal predictors. We recommend that block cross-validation be used wherever dependence structures exist in a dataset, even if no correlation structure is visible in the fitted model residuals, or if the fitted models account for such correlations.

998 citations

Journal ArticleDOI
TL;DR: In this paper, a q-statistic method is proposed to measure the degree of spatial stratified heterogeneity and to test its significance, and the q value is within [0, 1] (0 if a spatial stratification of heterogeneity is not significant, and 1 if there is a perfect spatial stratifying of heterogeneity).

879 citations