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The research of Art Getis has had many positive direct effects on the fields of regional science and spatial statistics. There already have been, and will continue to be, many indirect effects as well, as the repercussions of his work ripple through the field. With his contributions to spatial statistics, Art Getis laid out a rich research agenda for future work in the field. In this presentation, I describe several research problems inspired by his work. These include (a) the multiple testing problem, (b) the effects of global spatial autocorrelation on the power and use of local statistics, (c) the effects of weight misspecification on the power of local statistics, and (d) the statistical distribution of his local spatial heterogeneity (LOSH) statistic. In addition, there are a number of connections between the local statistics he developed and other statistics. These include the relationships between (a) the Getis-Ord statistic and the two-sample t statistic, (b) the Getis-Ord statistic and the maximum of Gaussian random fields, and (c) the LOSH statistic and the well-known Geary statistic. [ABSTRACT FROM AUTHOR]

Spatial cancer data analyses frequently utilize regression techniques to investigate associations between cancer incidences and potential covariates. Model specification, a process of formulating an appropriate model, is a well-recognized task in the literature. It involves a distributional assumption for a dependent variable, a proper set of predictor variables (i.e., covariates), and a functional form of the model, among other things. For example, one of the assumptions of a conventional statistical model is independence of model residuals, an assumption that can be easily violated when spatial autocorrelation is present in observations. A failure to account for spatial structure can result in unreliable estimation results. Furthermore, the difficulty of describing georeferenced data may increase with the presence of a positive and negative spatial autocorrelation mixture, because most current model specifications cannot successfully explain a mixture of spatial processes with a single spatial autocorrelation parameter. Particularly, properly accounting for a spatial autocorrelation mixture is challenging. This paper empirically investigates and uncovers a possible spatial autocorrelation mixture pattern in breast cancer incidences in Broward County, Florida, during 2000–2010, employing different model specifications. The analysis results show that Moran eigenvector spatial filtering provides a flexible method to examine such a mixture. [ABSTRACT FROM AUTHOR]

Being a hot topic in recent years, many studies have been conducted with spatial data containing massive numbers of observations. Because initial developments for classical spatial autocorrelation statistics are based on rather small sample sizes, in the context of massive spatial datasets, this paper presents extensions to efficiency and statistical power comparisons between the Moran coefficient and the Geary ratio for different variable distribution assumptions and selected geographic neighborhood definitions. The question addressed asks whether or not earlier results for small n extend to large and massively large n, especially for non-normal variables; implications established are relevant to big spatial data. To achieve these comparisons, this paper summarizes proofs of limiting variances, also called asymptotic variances, to do the efficiency analysis, and derives the relationship function between the two statistics to compare their statistical power at the same scale. Visualization of this statistical power analysis employs an alternative technique that already appears in the literature, furnishing additional understanding and clarity about these spatial autocorrelation statistics. Results include: the Moran coefficient is more efficient than the Geary ratio for most surface partitionings, because this index has a relatively smaller asymptotic as well as exact variance, and the superior power of the Moran coefficient vis-à-vis the Geary ratio for positive spatial autocorrelation depends upon the type of geographic configuration, with this power approaching one as sample sizes become increasingly large. Because spatial analysts usually calculate these two statistics for interval/ration data, this paper also includes comments about the join count statistics used for nominal data. [ABSTRACT FROM AUTHOR]

House price prediction is a hot topic in the economic literature. House price prediction has traditionally been approached using a-spatial linear (or intrinsically linear) hedonic models. It has been shown, however, that spatial effects are inherent in house pricing. This article considers parametric and semi-parametric spatial hedonic model variants that account for spatial autocorrelation, spatial heterogeneity and (smooth and nonparametrically specified) nonlinearities using penalized splines methodology. The models are represented as a mixed model that allow for the estimation of the smoothing parameters along with the other parameters of the model. To assess the out-of-sample performance of the models, the paper uses a database containing the price and characteristics of 10,512 homes in Madrid, Spain (Q1 2010). The results obtained suggest that the nonlinear models accounting for spatial heterogeneity and flexible nonlinear relationships between some of the individual or areal characteristics of the houses and their prices are the best strategies for house price prediction. [ABSTRACT FROM AUTHOR]

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]

Recent literature on border effect has fostered research on informal barriers to trade and the role played by network dependencies. In relation to social networks, it has been shown that intensity of trade in goods is positively correlated with migration flows between pairs of countries/regions. In this article, we investigate whether such a relation also holds for interregional trade of services. We also consider whether interregional trade flows in services linked with tourism exhibit spatial and/or social network dependence. Conventional empirical gravity models assume the magnitude of bilateral flows between regions is independent of flows to/from regions located nearby in space, or flows to/from regions related through social/cultural/ethic network connections. With this aim, we provide estimates from a set of gravity models showing evidence of statistically significant spatial and network (demographic) dependence in the bilateral flows of the trade of services considered. The analysis has been applied to the Spanish intra- and interregional monetary flows of services from the accommodation, restaurants and travel agencies for the period 2000-2009, using alternative datasets for the migration stocks and definitions of network effects. [ABSTRACT FROM AUTHOR]

In a weighted spatial network, as specified by an exchange matrix, the variances of the spatial values are inversely proportional to the size of the regions. Spatial values are no more exchangeable under independence, thus weakening the rationale for ordinary permutation and bootstrap tests of spatial autocorrelation. We propose an alternative permutation test for spatial autocorrelation, based upon exchangeable spatial modes, constructed as linear orthogonal combinations of spatial values. The coefficients obtain as eigenvectors of the standardized exchange matrix appearing in spectral clustering and generalize to the weighted case the concept of spatial filtering for connectivity matrices. Also, two proposals aimed at transforming an accessibility matrix into an exchange matrix with a priori fixed margins are presented. Two examples (inter-regional migratory flows and binary adjacency networks) illustrate the formalism, rooted in the theory of spectral decomposition for reversible Markov chains. [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]

This paper provides a survey of the existing literature on the specification and estimation of dynamic spatial panel data models, a collection of models for spatial panels extended to include one or more of the following variables and/or error terms: a dependent variable lagged in time, a dependent variable lagged in space, a dependent variable lagged in both space and time, independent variables lagged in time, independent variables lagged in space, serial error autocorrelation, spatial error autocorrelation, spatial-specific and time-period-specific effects. The survey also examines the reasoning behind different model specifications and the purposes for which they can be used, which should be useful for practitioners. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficiency improvement for the algorithm called AMOEBA, A Multidirectional Optimum Ecotope-Based Algorithm, devised by Aldstadt and Getis (Geogr Anal 38(4):327-343, ). AMOEBA embeds a local spatial autocorrelation statistic in an iterative procedure in order to identify spatial clusters (ecotopes) of related spatial units. We provide an analysis of the computational complexity of the original AMOEBA and develop an alternative formulation that reduces computational time without losing optimality. Empirical evidence is provided using georeferenced socio-demographic data in Accra, Ghana. [ABSTRACT FROM AUTHOR]

A spatio-temporal model of housing price trends is developed that focuses on individual housing sales over time. The model allows for both the spatio-temporal lag effects of previous sales in the vicinity of each housing sale, and for general autocorrelation effects over time. A key feature of this model is the recognition of the unequal spacing between individual housing sales over time. Hence the residuals are modeled as a first-order autoregressive process with unequally spaced events. The maximum-likelihood estimation of this model is developed in detail, and tested in terms of simulations based on selected data. In addition, the model is applied to a small data set in the Philadelphia area. [ABSTRACT FROM AUTHOR]

A vast majority of the recent literature on spatial hedonic analysis has been concerned with residential property values, with only very few examples of studies focused on commercial property prices. The dearth of studies can be attributed to some of the challenges faced in the analysis of commercial properties, in particular the scarcity of information compared to residential transactions. In order to address this issue, in this paper we propose the use of cokriging and housing prices as ancillary information to estimate commercial property prices. Cokriging takes into account the spatial autocorrelation structure of property prices, and the use of more abundant information on housing prices helps to improve the accuracy of property value estimates. A case study of Toledo in Spain, a city for which commercial activity stemming from tourism is one of the key elements of the economy in the city, demonstrates that substantial accuracy and precision gains can be obtained from the use of cokriging. [ABSTRACT FROM AUTHOR]

Since before the inception of work by Okabe, the intermingling of spatial autocorrelation (i.e., local distance and configuration) and distance decay (i.e., global distance) effects has been suspected in spatial interaction data. This convolution was first treated conceptually because technology and methodology did not exist at the time to easily or fully address spatial autocorrelation effects within spatial interaction model specifications. Today, however, sufficient computer power coupled with eigenfunction-based spatial filtering offers a means for accommodating spatial autocorrelation effects within a spatial interaction model for modest-sized problems. In keeping with Okabe’s more recent efforts to dissemination spatial analysis tools, this paper summarizes how to implement the methodology utilized to analyze a particular empirical flows dataset. [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]

We estimate spatiotemporal models of average neighborhood single family home prices to use in predicting individual property prices. Average home-price variations are explained in terms of changes in average neighborhood house attributes, spatial attributes, and temporal economic variables. Models adopting three different definitions of neighborhoods are estimated with quarterly cross-sectional data over the period 2000–2004 from four cities in Southern California. Heteroscedasticity and autocorrelation problems are detected and adjusted for via a sequential routine. Results of these models suggest that forecasts obtained using city neighborhood average price equations may have advantage over forecasts obtained using city aggregated price equations. [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]

Mostly lip service treatments of negative spatial autocorrelation (NSA) appear in the literature, although spatial scientists confront it in practice. NSA was detected serendipitously in recalcitrant empirical analyses containing a sizeable amount of global positive spatial autocorrelation (PSA) unaccounted for by standard spatial statistical models, and labeled hidden because conventional spatial statistical tools detected only PSA while giving absolutely not hint of NSA existing. The meaning of this phenomenon is explored empirically, with findings including: a better understanding of NSA, spatial filter model construction guidelines, effective illustrations of NSA, and how hidden NSA furnishes a diagnostic for model misspecification. [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]

It has long been believed that properties of spatial structure have a strong effect on trip distribution, which thus leads to a bias in the estimated distance decay parameters of spatial interaction models. This paper is an attempt to identify to what extent the spatial structure effect affects the trip distribution and determine whether the incorporation of a term to account for the relative location of destinations into the conventional gravity models, results in a model that can more correctly represent the actual trip distribution. The main focus is on the comparison of the origin–specific estimates of the distance decay parameter, calibrated from the traditional production-constrained model and the production-constrained competing destinations model. The results show that the competing destinations model is superior to the conventional model in both reproducing the interaction flows and giving behavioral explanation to the distance decay parameters, but the essential aim of the competing destinations model to remove the map pattern from the distance decay parameters of the conventional model has not been identified. [ABSTRACT FROM AUTHOR]

The detection of change in remotely sensed images is often carried out by designating a threshold to distinguish between areas of change and areas of no change. The choice of threshold is often arbitrary however. The purpose of this paper is to offer a statistical framework for the selection of thresholds. The framework accounts for the facts that one carries out multiple tests of the null hypothesis of no change, when searching for regions of change over an image with a large number of pixels. Special attention is given to global spatial autocorrelation, which can affect the selection of appropriate threshold values. [ABSTRACT FROM AUTHOR]

As either the spatial resolution or the spatial scale for a geographic landscape increases, both latent spatial dependence and spatial heterogeneity also will tend to increase. In addition, the amount of georeferenced data that results becomes massively large. These features of high spatial resolution hyperspectral data present several impediments to conducting a spatial statistical analysis of such data. Foremost is the requirement of popular spatial autoregressive models to compute eigenvalues for a row-standardized geographic weights matrix that depicts the geographic configuration of an image's pixels. A second drawback arises from a need to account for increased spatial heterogeneity. And a third concern stems from the usefulness of marrying geostatistical and spatial autoregressive models in order to employ their combined power in a spatial analysis. Research reported in this paper addresses all three of these topics, proposing successful ways to prevent them from hindering a spatial statistical analysis. For illustrative purposes, the proposed techniques are employed in a spatial analysis of a high spatial resolution hyperspectral image collected during research on riparian habitats in the Yellowstone ecosystem. [ABSTRACT FROM AUTHOR]

Abstract. This paper systematically investigates spatially autocorrelated patterns and the behaviour of their associated test statistic Moran's I in three bounded regular tessellations. These regular tessellations consist of triangles, squares, and hexagons, each of increasing size (n = 64;256; 1024). These tesselations can be downloaded at http://geo-www.sbs.ohio-state.edu/faculty/ tiefelsdorf/regspastruc/ in several GIS formats. The selection of squares is particularly motivated by their use in raster based GIS and remote sensing. In contrast, because of topological correspondences, the hexagons serve as excellent proxy tessellations for empirical maps in vector based GIS. For all three tessellations, the distributional characteristics and the feasibility of the normal approximation are examined for global Moran's I, Moran's I[sup (k)]associated with higher order spatial lags, and local Moran's I[sub I]. A set of eigenvectors can be generated for each tessellation and their spatial patterns can be mapped. These eigenvectors can be used as proxy variables to overcome spatial autocorrelation in regression models. The particularities and similarities in the spatial patterns of these eigenvectors are discussed. The results indicate that [ii the normal approximation for Moran's I is not always feasible; [ii] the three tessellations induce different distributional characteristics of Moran's I, and [iii] different spatial patterns of eigenvectors are associated with the three tessellations. [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]