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Because eigenvector spatial filtering (ESF) provides a relatively simple and successful method to account for spatial autocorrelation in regression, increasingly it has been adopted in various fields. Although ESF can be easily implemented with a stepwise procedure, such as traditional stepwise regression, its computational efficiency can be further improved. Two major computational components in ESF are extracting eigenvectors and identifying a subset of these eigenvectors. This paper focuses on how a subset of eigenvectors can be efficiently and effectively identified. A simulation experiment summarized in this paper shows that, with a well-prepared candidate eigenvector set, ESF can effectively account for spatial autocorrelation and achieve computational efficiency. This paper further proposes a nonlinear equation for constructing an ideal candidate eigenvector set based on the results of the simulation experiment. [ABSTRACT FROM AUTHOR]

Typical ecological gradient analyses consider variation in the response of plants along a gradient of covariate values, but generally constrain themselves to predetermined response curves and ignore spatial autocorrelation. In this paper, we develop a formal spatial gradient analysis. We adopt the mathematical definition of gradients as directional rates of change with regard to a spatial surface. We view both the response and the covariate as spatial surfaces over a region of interest with respective gradient behavior. The gradient analysis we propose enables local comparison of these gradients. At any spatial location, we compare the behavior of the response surface with the behavior of the covariate surface to provide a novel form of sensitivity analysis. More precisely, we first fit a joint hierarchical Bayesian spatial model for a response variable and an environmental covariate. Then, after model fitting, at a given location, for each variable, we can obtain the posterior distribution of the derivative in any direction. We use these distributions to compute spatial sensitivities and angular discrepancies enabling a more detailed picture of the spatial nature of the response-covariate relationship. This methodology is illustrated using species presence probability as a response to elevation for two species of South African protea. We also offer a comparison with sensitivity analysis using geographically weighted regression. We show that the spatial gradient analysis allows for more extensive inference and provides a much richer description of the spatially varying relationships. [ABSTRACT FROM AUTHOR]

There is a common belief that the presence of residual spatial autocorrelation in ordinary least squares (OLS) regression leads to inflated significance levels in beta coefficients and, in particular, inflated levels relative to the more efficient spatial error model (SEM). However, our simulations show that this is not always the case. Hence, the purpose of this paper is to examine this question from a geometric viewpoint. The key idea is to characterize the OLS test statistic in terms of angle cosines and examine the geometric implications of this characterization. Our first result is to show that if the explanatory variables in the regression exhibit no spatial autocorrelation, then the distribution of test statistics for individual beta coefficients in OLS is independent of any spatial autocorrelation in the error term. Hence, inferences about betas exhibit all the optimality properties of the classic uncorrelated error case. However, a second more important series of results show that if spatial autocorrelation is present in both the dependent and explanatory variables, then the conventional wisdom is correct. In particular, even when an explanatory variable is statistically independent of the dependent variable, such joint spatial dependencies tend to produce 'spurious correlation' that results in over-rejection of the null hypothesis. The underlying geometric nature of this problem is clarified by illustrative examples. The paper concludes with a brief discussion of some possible remedies for this problem. [ABSTRACT FROM AUTHOR]

Although the assumption of independence among interaction flows frequently is engaged in spatial interaction modeling, in many circumstances it leads to misspecified models and incorrect inferences. An informed approach is to explicitly incorporate an assumed relationship structure among the interaction flows, and to explicitly model the network autocorrelation. This paper illustrates such an approach in the context of U.S. interstate migration flows. Behavioral assumptions, similar to those of the intervening opportunities or the competing destinations concepts, exemplify how to specify network flows that are related to particular origin–destination combinations. The stepwise incorporation of eigenvectors, which are extracted from a network link matrix, captures the network autocorrelation in a Poisson regression model specification context. Spatial autocorrelation in Poisson regression is measured by the test statistic of Jacqmin-Gadda et al. (Stat Med 16(11):1283–1297, 1997). Results show that estimated regression parameters in the spatial filtering interaction model become more intuitively interpretable. [ABSTRACT FROM AUTHOR]

Studies on small-world networks have received intensive interdisciplinary attention during the past several years. It is well-known among researchers that a small-world network is often characterized by high connectivity and clustering, but so far there exist few effective approaches to evaluate small-world properties, especially for spatial networks. This paper proposes a method to examine the small-world properties of spatial networks from the perspective of network autocorrelation. Two network autocorrelation statistics, Moran’s I and Getis–Ord’s G, are used to monitor the structural properties of networks in a process of “rewiring” networks from a regular to a random network. We discovered that Moran’s I and Getis–Ord’s G tend to converge and have relatively low values when properties of small-world networks emerge. Three transportation networks at the national, metropolitan, and intra-city levels are analyzed using this approach. It is found that spatial networks at these three scales possess small-world properties when the correlation lag distances reach certain thresholds, implying that the manifestation of small-world phenomena result from the interplay between the network structure and the dynamics taking place on the network. [ABSTRACT FROM AUTHOR]

The eigenfunction spatial filter derives from the Moran Coefficient that indexes spatial autocorrelation. Mean, variance and statistical distribution characterizations and descriptions of georeferenced random variables and their interrelationships are derived in terms of the eigenfunction spatial filter. Selected comparisons are made with spatial autoregressive model results. Implications of the eigenfunction spatial filter are outlined for simulation experiments, variance components of correlation coefficients, missing values estimation, and non-normal georeferenced random variables. Particularly unanticipated findings include: spatial filtering reveals the potential need to employ a spatial autocorrelation reduction stepwise regression variable selection criterion, the standard error of a univariate mean can be expressed in terms of a conventional regression variance inflation factor that captures spatial autocorrelation effects, spatial autocorrelation can simultaneously inflate and deflate a bivariate correlation coefficient, simulation experiments can be easily designed to involve specific map patterns associated with non-zero spatial autocorrelation, spatial filtering furnishes a convenient way to estimate missing georeferenced values, and the asymptotic variance of the Moran Coefficient is unaltered by spatial filtering. [ABSTRACT FROM AUTHOR]

Abstract. The Moran Coefficient spatial autocorrelation index can be decomposed into orthogonal map pattern components. This decomposition relates it directly to standard linear regression, in which corresponding eigenvectors can be used as predictors. This paper reports comparative results between these linear regressions and their auto-Gaussian counterparts for the following georeferenced data sets: Columbus (Ohio) crime, Ottawa-Hull median family income, Toronto population density, southwest Ohio unemployment, Syracuse pediatric lead poisoning, and Glasgow standard mortality rates, and a small remotely sensed image of the High Peak district. This methodology is extended to auto-logistic and auto-Poisson situations, with selected data analyses including percentage of urban population across Puerto Rico, and the frequency of SIDs cases across North Carolina. These data analytic results suggest that this approach to georeferenced data analysis offers considerable promise. [ABSTRACT FROM AUTHOR]