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This year is the 50th anniversary of Besag's classic auto‐models publication, a cornerstone in the development of modern‐day spatial statistics/econometrics. Besag struggled for nearly two decades to make his conceptualization collectively successful across a wide suite of random variables. But only his auto‐normal, and to a lesser degree his auto‐logistic/binomial, were workable. Others, like his auto‐Poisson, were effectively failures, whereas still others, such as potentials like an auto‐Weibull, defied even awkward mathematical incorporations of spatial lag terms. Besag circumvented this impediment by introducing an auto‐normal random effects components (within a Bayesian estimation context), building upon his single total success. This article describes an alternative approach, partly paralleling his reformulation while avoiding inserting spatial lag terms directly into probability density/mass functions, implanting spatial autocorrelation into cumulative distributions functions (CDFs), instead, via a spatially autocorrelated uniform distribution. The already existing probability integral transform and quantile function mathematical statistics theorems enable this mechanism to spatialize any random variable, with these new ones labeled sui‐models. [ABSTRACT FROM AUTHOR]

Inference procedures for spatial autocorrelation statistics assume that the underlying configurations of spatial units are fixed. However, sometimes this assumption can be disadvantageous, for example, when analyzing social media posts or moving objects. This article examines for the case of point geometries how a change from fixed to random spatial indexes affects inferences about global Moran's I, a popular spatial autocorrelation measure. Homogeneous and inhomogeneous Matérn and Thomas cluster processes are studied and for each of these processes, 10,000 random point patterns are simulated for investigating three aspects that are key in an inferential context: the null distributions of I when the underlying geometries are varied; the effect of the latter on critical values used to reject null hypotheses; and how the presence of point processes affects the statistical power of Moran's I. The results show that point processes affect all three characteristics. Inferences about spatial structure in relevant application contexts may therefore be different from conventional inferences when this additional source of randomness is taken into account. [ABSTRACT FROM AUTHOR]

Except for about a half dozen papers, virtually all (co)authored by Griffith, the existing literature lacks much content about the interface between spatial optimization, a popular form of geographic analysis, and spatial autocorrelation, a fundamental property of georeferenced data. The popular p‐median location‐allocation problem highlights this situation: the empirical geographic distribution of demand virtually always exhibits positive spatial autocorrelation. This property of geospatial data offers additional overlooked information for solving such spatial optimization problems when it actually relates to their solutions. With a proof‐of‐concept outlook, this paper articulates connections between the well‐known Majority Theorem of the 1‐median minisum problem and local indices of spatial autocorrelation; the LISA statistics appear to be the more useful of these later statistics because they better embrace negative spatial autocorrelation. The relationship articulation outlined here results in the positing of a new proposition labeled the egalitarian theorem. [ABSTRACT FROM AUTHOR]

Failure to account for global spatial autocorrelation when using scan statistics to find clusters generated by local processes will result in P‐values that are too low, and consequently, spurious findings of statistical significance are not uncommon. The presence of global spatial autocorrelation also decreases the ability to reject false null hypotheses and it is therefore more difficult to find local clusters when they exist. By estimating the degree of global autocorrelation and using that estimate to transform the data, it is then possible to apply scan statistics to the transformed data. This results in a reduction in the likelihood of spurious finding of statistical significance when local clusters do not exist. [ABSTRACT FROM AUTHOR]

This paper expands upon a GA 50thAnniversary panel presentation made at the 2018 New Orleans American Association of Geographers meeting. It notes that retrospectives for this topic already exist, both in terms of the journal and the sub‐discipline. This paper presents discussion about the following selected prominent transitional themes bridging the past and the future of geographical analysis: space–time data; geographic uncertainty and error; and, dimensional reduction spatial statistical modeling. It also presents discussion about the following three themes that are expected to receive considerable future attention: negative spatial autocorrelation; synergies between spatial statistics and spatial optimization; and, big spatial data. The first two of these topics essentially have been ignored in the literature to date. [ABSTRACT FROM AUTHOR]

Tobler's first law of geography is widely recognized as reflecting broad empirical realities in geography. Its key concepts of "near" and "related" are intuitive in a univariate setting. However, when moving to the joint consideration of spatial patterns among multiple variables, the combination of attribute similarity and geographical similarity that underlies the concept of spatial autocorrelation is much harder to deal with. This article uses the notion of distance in multiattribute space to explore and visualize the connection between "near" and "related" in a multivariate context. We approach this from a global, local, and regional perspective. We outline a number of ways to combine different visualization techniques and introduce a new local neighbor match test for multivariate local clusters. We illustrate the methods by means of Guerry's classic data set on moral statistics in 1833 France. [ABSTRACT FROM AUTHOR]

The literature is replete with acknowledgments that spatial autocorrelation (SA) inflates the variance of a random variable (RV), and that it also may alter other RV distributional properties. In most studies, impacts of SA have been examined only for the three most commonly used distributions: the normal, Poisson (and its negative binomial counterpart), and binomial distributions; much less is known about its effects on two other RVs that are utilized in GIScience research: the beta and the multinomial. The beta distribution—which is considered to be very flexible because it can mimic a uniform, exponential, sinusoidal, and normal RV—can be utilized to analyze the radiance of a remotely sensed image, for example. The multinomial distribution, a generalization of the binomial distribution, has been widely used for land use classification, and to describe land use change. The literature also suggests that RV impacts of negative SA, a neglected topic in spatial analysis, may differ from those of positive SA, at least for some RVs (e.g., the Poisson RV). The purpose of this article is to extend the investigation of effects of SA to beta and multinomial RVs, with both positive SA and negative SA assessed and contrasted with each other, using simulation experiments. The simulation experiments are designed to support this assessment. One of the major discoveries is that impacts of positive SA and negative SA behave similarly when a RV conforms to a normal distribution; however, maximum negative SA is unable to materialize for asymmetric RV, whereas positive SA always converges upon its maximum. [ABSTRACT FROM AUTHOR]

Residual spatial autocorrelation is a situation frequently encountered in regression analysis of spatial data. The statistical problems arising due to this phenomenon are well‐understood. Original developments in the field of statistical analysis of spatial data were meant to detect spatial pattern, in order to assess whether corrective measures were required. An early development was the use of residual autocorrelation as an exploratory tool to improve regression analysis of spatial data. In this note, we propose the use of spatial filtering and exploratory data analysis as a way to identify omitted but potentially relevant independent variables. We use an example of blood donation patterns in Toronto, Canada, to demonstrate the proposed approach. In particular, we show how an initial filter used to rectify autocorrelation problems can be progressively replaced by substantive variables. In the present case, the variables so retrieved reveal the impact of urban form, travel habits, and demographic and socio‐economic attributes on donation rates. The approach is particularly appealing for model formulations that do not easily accommodate positive spatial autocorrelation, but should be of interest as well for the case of continuous variables in linear regression. [ABSTRACT FROM AUTHOR]

We propose a new estimator of spatial autocorrelation of areal incidence or prevalence rates in small areas, such as crime and health indicators, for correcting spatially heterogeneous sampling errors in denominator data. The approach is dubbed the heteroscedasticity‐consistent empirical Bayes (HC‐EB) method. As American Community Survey (ACS) data have been released to the public for small census geographies, small‐area estimates now form the demographic landscape of neighborhoods. Meanwhile, there is growing awareness of the diminished statistical validity of global and local Moran's I when such small‐area estimates are used in denominator data. Using teen birth rates by census tracts in Mecklenburg County, North Carolina, we present comparisons of conventional and new HC‐EB estimates of Global and Local Moran's I statistics created on ACS data, along with estimates on ground truth values from the 2010 decennial census. Results show that the new adjustment method dramatically enhances the statistical validity of global and local spatial autocorrelation statistics. [ABSTRACT FROM AUTHOR]

In this simulation study, regressions specified with autocorrelation effects are compared against those with relationship heterogeneity effects, and in doing so, provides guidance on their use. Regressions investigated are: (1) multiple linear regression, (2) a simultaneous autoregressive error model, and (3) geographically weighted regression. The first is nonspatial and acts as a control, the second accounts for stationary spatial autocorrelation via the error term, while the third captures spatial heterogeneity through the modeling of nonstationary relationships between the response and predictor variables. The geostatistical‐based simulation experiment generates data and coefficients with known multivariate spatial properties, all within an area‐unit spatial setting. Spatial autocorrelation and spatial heterogeneity effects are varied and accounted for. On fitting the regressions, that each have different assumptions and objectives, to very different geographical processes, valuable insights to their likely performance are uncovered. Results objectively confirm an inherent interrelationship between autocorrelation and heterogeneity, that results in an identification problem when choosing one regression over another. Given this, recommendations on the use and implementation of these spatial regressions are suggested, where knowledge of the properties of real study data and the analytical questions being posed are paramount. [ABSTRACT FROM AUTHOR]

The goal is to predict the final extent of the Ebola epidemic in West Africa, 2014–2015, well before its end. Our models are based on the nature of the reported data, and the social, medical, and technological conditions that existed in real time during the course of the epidemic. The spatial and temporal nature of Ebola transmission is considered. A modified, classical compartmental model is used to develop variations of Gompertz and logistic‐type predictive models. A map analysis strongly hints at the existence of an initial rural component of the transmission of the disease followed by an urban component. Cumulative and weekly data for the three countries illustrate how the disease intensified and spread over the 90 weeks of the epidemic. A spatial autocorrelation study shows the clustering locational changes of the disease over time. A cross‐correlation study gives credence to both a rural and an urban transmission pattern. Our early estimates of the toll of the disease, crude as they are, stand in marked contrast to some of the official estimates at the time, which greatly inflated the number of new cases. [ABSTRACT FROM AUTHOR]

Geographically weighted regression (GWR) is a technique that explores spatial nonstationarity in data‐generating processes by allowing regression coefficients to vary spatially. It is a widely applied technique across domains because it is intuitive and conforms to the well‐understood framework of regression. An alternative method to GWR that has been suggested is spatial filtering, which it has been argued provides a superior alternative to GWR by producing spatially varying regression coefficients that are not correlated with each other and which display less spatial autocorrelation. It is, therefore, worthwhile to examine these claims by comparing the output from both methods. We do this by using simulated data that represent two sets of spatially varying processes and examining how well both techniques replicate the known local parameter values. The article finds no support that spatial filtering produces local parameter estimates with superior properties. The results indicate that the original spatial filtering specification is prone to overfitting and is generally inferior to GWR, while an alternative specification that minimizes the mean square error (MSE) of coefficient estimates produces results that are similar to GWR. However, since we generally do not know the true coefficients, the MSE minimizing specification is impractical for applied research. [ABSTRACT FROM AUTHOR]

The p‐regions is a mixed integer programming (MIP) model for the exhaustive clustering of a set of n geographic areas into p spatially contiguous regions while minimizing measures of intraregional heterogeneity. This is an NP‐hard problem that requires a constant research of strategies to increase the size of instances that can be solved using exact optimization techniques. In this article, we explore the benefits of an iterative process that begins by solving the relaxed version of the p‐regions that removes the constraints that guarantee the spatial contiguity of the regions. Then, additional constraints are incorporated iteratively to solve spatial discontinuities in the regions. In particular we explore the relationship between the level of spatial autocorrelation of the aggregation variable and the benefits obtained from this iterative process. The results show that high levels of spatial autocorrelation reduce computational times because the spatial patterns tend to create spatially contiguous regions. However, we found that the greatest benefits are obtained in two situations: (1) when n / p ≥ 3; and (2) when the parameter p is close to the number of clusters in the spatial pattern of the aggregation variable. [ABSTRACT FROM AUTHOR]

The most common indicator used to measure spatial dependence is Moran's I proposed by statistician Patrick A. P. Moran in 1950. The index is simple to use and applies the principle of the Pearson correlation coefficient, although it incorporates a proximity measure between elements. However, Moran's I tends to underestimate real spatial autocorrelation when the number of locations are few. This study aims to present a modified version of Moran's I that can measure real spatial autocorrelation even with small samples and check for spatial dependence. [ABSTRACT FROM AUTHOR]

Moran's Index for spatial autocorrelation and localized index for spatial association have been widely applied in many research fields as the first step to explore and assess the spatial dependency in a set of geographic events. This article presents extensions to the equations for calculating global and localized spatial autocorrelation so to include the temporal attribute values of the geographic events being analyzed. The extended equations were successfully implemented and tested with a real world data set. In addition, simulated data sets were used to reveal how the extended equations performed. Beyond the usefulness of the extended equations, we suggest that care be taken with regard to assessing spatiotemporal patterns under the normality and randomization assumptions as different outcomes from different assumptions would require different approaches for interpretation. [ABSTRACT FROM AUTHOR]

This study focuses on integration processes in European Research and Development ( R&D) by analyzing the spatiotemporal dimension of three different R&D collaboration networks across Europe. The studied networks cover different types of knowledge creation, namely project-based R&D networks within the European Union ( EU) Framework Programmes ( FPs), co-patent networks, and co-publication networks. Integration in European R&D-one of the main pillars of the EU Science Technology and Innovation policy-refers to the harmonization of fragmented national research systems across Europe and to the free movement of knowledge and researchers. The objective is to describe and compare spatiotemporal patterns at a regional level and to estimate the evolution of separation effects over the time period 1999-2006 that influence the probability of cross-region collaborations in the distinct networks under consideration. The study adopts a spatial interaction modeling perspective, econometrically specifying a panel generalized linear model relationship, taking into account spatial autocorrelation among flows using eigenfunction spatial filtering methods. The results show that geographical factors are a lower hurdle for R&D collaborations in the FP networks than in co-patent networks and co-publication networks. Furthermore, it is shown that the geographical integration is higher in the FP network. [ABSTRACT FROM AUTHOR]

Eigenvector-based spatial filtering is one of the often used approaches to model spatial autocorrelation among the observations or errors in a regression model. In this approach, a subset of eigenvectors extracted from a modified spatial weight matrix is added to the model as explanatory variables. The subset is typically specified via the selection procedure of the forward stepwise model, but it is disappointingly slow when the observations n take a large number. Hence, as a complement or alternative, the present article proposes the use of the least absolute shrinkage and selection operator ( LASSO) to select the eigenvectors. The LASSO model selection procedure was applied to the well-known Boston housing data set and simulation data set, and its performance was compared with the stepwise procedure. The obtained results suggest that the LASSO procedure is fairly fast compared with the stepwise procedure, and can select eigenvectors effectively even if the data set is relatively large ( n = 104), to which the forward stepwise procedure is not easy to apply. [ABSTRACT FROM AUTHOR]

This article introduces measures to quantify spatial autocorrelation for vectors. In contrast to scalar variables, spatial autocorrelation for vectors involves an assessment of both direction and magnitude in space. Extending conventional approaches, measures of global and local spatial associations for vectors are proposed, and the associated statistical properties and significance testing are discussed. The new measures are applied to study the spatial association of taxi movements in the city of Shanghai. Complications due to the edge effect are also examined. [ABSTRACT FROM AUTHOR]

Poisson models generally are utilized in analyzing spatial patterns of crime count data. When spatial autocorrelation is present, these models are extended to account for it. Among various methods, eigenvector spatial filtering ( ESF) furnishes an efficient means of analysis. However, because space-time crime data have temporal components as well as spatial components, Poisson models need to be further adjusted to reflect the two types of components simultaneously. This article discusses how the ESF method can be utilized to model space-time crime data, extending the generalized linear mixed model specification for it. This approach is illustrated with an application to space-time vehicle burglary incidents in the city of Plano, Texas, during 2004-2009. Los modelos de Poisson generalmente se utilizan en el análisis de los patrones espaciales de los datos de recuento de crimen. Cuando hay autocorrelación espacial, estos modelos son modificados para dar cuenta de ello. Entre los diversos métodos existentes, el método Eigenvector (autovector, vector propio) de filtrado espacial ( Eigenvector Spatial Filtering-ESF) proporciona un medio eficaz para dicho análisis. Sin embargo, dado que los datos de criminalidad espacio-temporales tienen tanto componentes temporales como espaciales, los modelos tipo Poisson requieren de un ajuste adicional para reflejar ambos tipos de componentes de manera simultánea. El artículo presente expone cómo el método ESF puede ser utilizado para modelar datos espacio-temporales sobre delitos mediante la modificación del modelo mixto lineal generalizado ( Generalized Linear Mixed Model-GLMM). El procedimiento propuesto se ilustra con el caso de incidentes espacio-temporales de robos de vehículos en la ciudad de Plano, Texas, durante 2004-2009. 泊松模型一般用于犯罪计数数据的空间模式分析中，当空间自相关关系呈现时，这类模型可扩展以解释潜在的分布特征。在各种模型中，特征向量空间滤波（ESF）提供了一种有效的分析方法。然而，由于时空犯罪数据包含时间和空间组分，因此泊松模型需要进一步调整以同时反映这两种不同类型的数据。本文讨论了如何利用特征向量空间滤波(ESF)模型对时空犯罪数据进行建模，并采用扩展广义线性混合模型(GLMM)进行规范。最后，以德克萨斯州普莱诺市2004-2009年的车辆盗窃案数据进行了实证验证。 [ABSTRACT FROM AUTHOR]

In the European Union ( EU), homogenous inflation forces are expected to prevail because of increased economic integration, especially after the creation of a single currency area. This expectation is directly related to the issue of inflation convergence, which has gained increasing attention by both academics and policy makers in Europe. Although the examination of core inflation is of great importance for macroeconomic policy, the role of disaggregate inflation indices, and especially food inflation, has also been emphasized in the literature. However, the issue of food inflation convergence has been largely ignored to date in empirical studies. This study explores the evolving distribution of food inflation rates in the EU-25 member states using distribution dynamics analysis and covering the period from January 1997 to March 2011. This analysis assumes that each country represents an independent observation, providing unique information that can be used to estimate the transition dynamics of inflation. We show that spatial autocorrelation prevails inside the EU-25, and, therefore, the independency assumption is violated. To ensure spatial independence, the Getis spatial filter is implemented prior to a distribution dynamics analysis. The results of this analysis confirm the existence of convergence trends, which are even clearer after the spatial filtering procedure, indicating, on the one hand, the influence of spatial effects on food inflation and, on the other hand, the effectiveness of the Getis spatial filtering technique. En la Unión Europea (UE), se espera que las fuerzas de inflación homogéneas prevalezcan debido a la mayor integración económica, sobre todo después de la creación de la zona de moneda única. Esta expectativa se relaciona directamente con el tema de la convergencia de la inflación, que ha ganado cada vez más atención por parte de los investigadores académicos y los decisores políticos europeos. Aunque el análisis de la inflación subyacente es de gran importancia para la política macroeconómica, el papel de los índices de inflación a niveles desagregados, sobre todo en el caso de la inflación de alimentos, ha sido un tema enfatizado por la literatura especializada. Sin embargo, la cuestión de la convergencia de la inflación de alimentos carece hasta la fecha de estudios empíricos. El artículo presente estudia la evolución de la distribución de las tasas de inflación de alimentos en los estados miembros de la UE-25, utilizando el método de análisis de la dinámica de distribución (distribution dynamic analysis) y abarca el período comprendido entre enero de 1997 a marzo de 2011. Este análisis supone que cada país representa una observación independiente que proporciona información única que se puede utilizada para estimar la dinámica de transición inflacionaria. El presente estudio demuestra que la autocorrelación espacial prevalece dentro de los estados UE- 25, y por lo tanto la hipótesis de independencia estadística de las observaciones es violada. Para garantizar la independencia espacial, el método de filtrado espacial Getis ( Getis Spatial Filter) es implementado antes de proceder con el análisis de la dinámica de distribución. Los resultados del análisis confirman la existencia de las tendencias de convergencia, que son aún más claras después de la aplicación del filtrado espacial. Estos resultados evidencian por un lado, la influencia de los efectos espaciales en la inflación de alimentos, y por otro lado, la eficacia de la técnica de filtrado espacial de Getis. 在欧盟中，由于区域经济一体化进程的推进，同质商品的通货膨胀盛行，特别是单一货币区域建立后，该趋势更为明显。这种演化态势和通货膨胀的收敛问题直接相关，已经引起了欧洲学术界和政策制定者的关注。虽然核心通胀的检测对宏观经济政策十分重要，但是分解通胀指数的作用，尤其是本文所强调的食品通胀也有文献中提及。然而，食品通胀的收敛问题却在实证研究中很大程度上被忽视。本文利用分布动力学方法对欧盟25国1997年1月至2011年3月的食品通胀率的演化分布进行分析。假设每个国家代表一个独立的观察个体并提供唯一的信息，可用于估计通胀的转变动态，研究发现空间自相关在欧盟25国中普遍存在，因此独立假设不成立。为了保证空间独立性，在进行分布动力学分析之前，先使用Getis空间滤波技术进行处理。分析结果证实了收敛趋势的存在，且该趋势在空间滤波程序处理后更为明显。它一方面显示出食品通胀空间效应的影响，另一方面表现出Getis空间滤波技术的有效性。 [ABSTRACT FROM AUTHOR]

In this article, a Poisson gravity model is introduced that incorporates spatial dependence of the explained variable without relying on restrictive distributional assumptions of the underlying data-generating process. The model comprises a spatially filtered component-including the origin-, destination-, and origin-destination-specific variables-and a spatial residual variable that captures origin- and destination-based spatial autocorrelation. We derive a two-stage nonlinear least-squares (NLS) estimator (2NLS) that is hetero-scedasticity-robust and, thus, controls for the problem of over- or underdispersion that often is present in the empirical analysis of discrete data or, in the case of overdispersion, if spatial autocorrelation is present. This estimator can be shown to have desirable properties for different distributional assumptions, like the observed flows or (spatially) filtered component being either Poisson or negative binomial. In our spatial autoregressive (SAR) model specification, the resulting parameter estimates can be interpreted as the implied total impact effects defined as the sum of direct and indirect spatial feedback effects. Monte Carlo results indicate marginal finite sample biases in the mean and standard deviation of the parameter estimates and convergence to the true parameter values as the sample size increases. In addition, this article illustrates the model by analyzing patent citation flows data across European regions. En el presente artículo, se introduce un modelo de gravedad Poisson, que incorpora la dependencia espacial de la variable explicada, sin apoyarse en presunciones de distribución restrictivas del proceso subyacente de generación de datos. El modelo comprende de un componente espacialmente filtrado, que incluye las variables de origen, destino y origen-destino específico; y una variable espacial residual que captura la auto-correlación espacial basada en el origen y destino. Se deriva del calculador (2NLS) de dos etapas no lineales de mínimos cuadrados (NLS), el cual es robusto en heterocedasticidad, y por ello controla el problema de sobre-dispersión o baja-dispersión (over and under dispersion), que a menudo se presenta en el análisis empírico de datos discretos; o, en el caso de de sobre-dispersión, cuando se presenta la auto correlación espacial. Este calculador puede demostrar tener propiedades deseables para diferentes supuestos distribucionales, como los flujos observados un componente (espacialmente) filtrado, ya sea Poisson o binomial negativo. En nuestra especificación de modelo espacial auto regresivo (SAR), las estimaciones de los parámetros resultantes se pueden interpretar como los efectos de impacto total implícitos, definidos como la suma de efectos espaciales, directos o indirectos, de retroalimentación ( feedback). Los resultados Monte Carlo indican sesgos marginales de muestras finitas en la media y la desviación estándar de los parámetros estimados, y la convergencia de los valores de los parámetros reales, a medida que aumenta el tamaño de muestra. Este artículo ilustra el modelo mediante el análisis de flujos de datos de citas de patentes, a través de las regiones europeas. 本文提出了一种蕴含空间依赖的泊松引力模型，该模型中解释变量无需依赖潜在数据生成过程的限制性分布假设。该模型由包含起点、终点、起点-终点特定变量的空间滤波组分和空间残差变量组成，能捕捉到基于起点和终点的空间自相关。我们推导出一个二阶非线性最小二乘（NLS）估计（2NLS），它对异方差具有鲁棒性，从而可控制对于离散或过离散数据经验性分析中经常出现的过离散和低离散问题。如果空间自相关存在，过离散数据分析就是一个例子。对于不同的分布假设，如或泊松分布或是负二项式分布的观测流或（空间）滤波组分，该估计量显示出令人满意的性能。在本文的空间自回归（SAR）模型设定中，参数估计结果可解释为隐含的全局影响效应，并可被定义为直接和间接的空间反馈效应之和。蒙特卡罗结果给出了参数估计中均值、标准差的临界有限样本偏差，且随样本量增大收敛于真正参数值。此外，本文基于欧洲地区专利引用的流数据进行了模型验证。 [ABSTRACT FROM AUTHOR]

The article discusses the use of an expanded spatial autocorrelation (SA) framework to explore spatial patterns. The components of SA measures are discussed, including defining the spatial relationships among observations and measurement of similarity of attribute values. The article proposes a clustergram to demonstrate the spatial clustering levels over an array of attribute similarity or lags.

The computation of Moran's index of spatial autocorrelation requires the definition of a spatial weighting matrix. The eigendecomposition of this doubly centered matrix (i.e., one that forces the sums of all rows and columns to equal zero) has interesting properties that have been exploited in various contexts: distribution properties of the Moran coefficient (MC), spatial filtering in linear models, generalized linear models, and multivariate analysis. In this article, this eigendecomposition is used to propose a new view of MC based on its interpretation in the simple context of linear regression. I use this interpretation to demonstrate the different properties of MC and also the inefficiency of this index in some situations involving simultaneous positive and negative spatial autocorrelation. I propose some new statistics and procedures for testing spatial autocorrelation, and conduct a simulation study to evaluate these new approaches. [ABSTRACT FROM AUTHOR]

We present a new linear regression model for use with aggregated, small area data that are spatially autocorrelated. Because these data are aggregates of individual-level data, we choose to model the spatial autocorrelation using a geostatistical model specified at the scale of the individual. The autocovariance of observed small area data is determined via the natural aggregation over the population. Unlike lattice-based autoregressive approaches, the geostatistical approach is invariant to the scale of data aggregation. We establish that this geostatistical approach also is a valid autoregressive model; thus, we call this approach the geostatistical autoregressive (GAR) model. An asymptotically consistent and efficient maximum likelihood estimator is derived for the GAR model. Finite sample evidence from simulation experiments demonstrates the relative efficiency properties of the GAR model. Furthermore, while aggregation results in less efficient estimates than disaggregated data, the GAR model provides the most efficient estimates from the data that are available. These results suggest that the GAR model should be considered as part of a spatial analyst's toolbox when aggregated, small area data are analyzed. More important, we believe that the GAR model's attention to the individual-level scale allows for a more flexible and theory-informed specification than the existing autoregressive approaches based on an area-level spatial weights matrix. Because many spatial process models, both in geography and in other disciplines, are specified at the individual level, we hope that the GAR covariance specification will provide a vehicle for a better informed and more interdisciplinary use of spatial regression models with area-aggregated data. [ABSTRACT FROM AUTHOR]

Biogeographical studies are often based on a statistical analysis of data sampled in a spatial context. However, in many cases standard analyses such as regression models violate the assumption of independently and identically distributed errors. In this article, we show that the theory of wavelets provides a method to remove autocorrelation in generalized linear models (GLMs). Autocorrelation can be described by smooth wavelet coefficients at small scales. Therefore, data can be decomposed into uncorrelated and correlated parts. Using an appropriate linear transformation, we are able to extend GLMs to autocorrelated data. We illustrate our new method, called the wavelet-revised model (WRM), by applying it to multiple regression with response variables conforming to various distributions. Results are presented for simulated data and real biogeographical data (species counts of the plant genus Utricularia [bladderworts] in grid cells throughout Germany). The results of our WRM are compared with those of GLMs and models based on generalized estimating equations. We recommend WRMs, especially as a method that allows for spatial nonstationarity. The technique developed for lattice data is applicable without any prior knowledge of the real autocorrelation structure. [ABSTRACT FROM AUTHOR]

This article reflects on the principal statistical methodological issues that quantitative geographers addressed in the 1950s and 1960s. The source of many of these methodological problems lies in the assumptions of classical statistics that data are independent and identically distributed. The correlation inherent in spatial data recorded at nearby locations and the irregular nature of spatial frameworks are contrary to these assumptions and impact upon the use of both descriptive and inferential statistics. Most of the wider developments in statistics at that time still fell short of meeting the needs of statistical geographers. What emerged were statistical tests for pattern, nonmodel based but capable of being adapted to an irregular areal framework and certain other factors considered important in any particular spatial analysis. This article notes that, while spatial autocorrelation was presented as a “problem” for geographers, it is also important to realize that its presence can represent an opportunity. [ABSTRACT FROM AUTHOR]

Knowing about the challenges and opportunities of spatial autocorrelation is one thing, but applying the measures to one's own data is another matter entirely. While manual computation of the measures for toy data sets is possible, applying them to small data sets required the use of computers and thus software. This article will shed some light on how the measures were and are implemented in software and on implementation issues that are still not fully resolved. [ABSTRACT FROM AUTHOR]

A major aim of including the spatial component in ecological studies is to characterize the nature and intensity of spatial relationships between organisms and their environment. The growing awareness by ecologists of the importance of including spatial structure in ecological studies (for hypothesis development, experimental design, statistical analyses, and spatial modeling) is beneficial because it promotes more effective research. Unfortunately, as more researchers perform spatial analysis, some misconceptions about the virtues of spatial statistics have been carried through the process and years. Some of these statistical concepts and challenges were already presented by Cliff and Ord in 1969. Here, we classify the most common misconceptions about spatial autocorrelation into three categories of challenges: (1) those that have no solutions, (2) those where solutions exist but are not well known, and (3) those where solutions have been proposed but are incorrect. We conclude in stressing where new research is needed to address these challenges. [ABSTRACT FROM AUTHOR]

This article outlines the context in geography and statistics in the mid 1960s, at the height of geography's so-called “quantitative revolution,” that led us into a long-term collaboration about spatial statistics, which has continued in surges and lulls for some 40 years. We focus upon problems in spatial autocorrelation, including the measurement of autocorrelation, distribution theory, and variable geographical lattices. This narrative may not describe how it was, but it does describe how we remember the events of the time. [ABSTRACT FROM AUTHOR]

The article discusses impacts and wider impacts on statistics of the seminal 1969 Cliff-Ord article on the problem of spatial autocorrelation published in London Papers in Regional Science. [ABSTRACT FROM AUTHOR]

This article examines the relationship between spatial dependency and spatial heterogeneity, two properties unique to spatial data. The property of spatial dependence has led to a large body of research into spatial autocorrelation and also, largely independently, into geostatistics. The property of spatial heterogeneity has led to a growing awareness of the limitation of global statistics and the value of local statistics and local statistical models. The article concludes with a discussion of how the two properties can be accommodated within the same modelling framework. [ABSTRACT FROM AUTHOR]

The impetus for Cliff and Ord's landmark article, “The Problem of Spatial Autocorrelation,” was to avoid the problem of topological invariance in the spatial weights matrix. The problem and three points of view on it – theoretical, topological, and empirical – are discussed. [ABSTRACT FROM AUTHOR]

In retrospect it is the word “problem” in the title that seems most remarkable about the Cliff and Ord article. Spatial autocorrelation is indeed a problem in standard inferential statistics, which was developed to handle controlled experiments, when these methods are used to generalize from natural experiments. From the perspective of geographic information science, however, spatial dependence is a defining characteristic of geographic data that makes many of the functions of geographic information systems possible. The almost universal presence of spatial heterogeneity in such data also argues against generalization and is made explicit in the recent development of place-based analytic techniques. The final section argues for a new approach to the teaching of quantitative methods in the environmental and social sciences that treats natural experiments, spatial dependence, and spatial heterogeneity as the norm. [ABSTRACT FROM AUTHOR]

Despite criminology's widespread application of geography, the full implications of Cliff and Ord's article have yet to be realized. In this essay the major types of spatial studies in criminology are outlined, followed by a depiction of the context of criminological research at the time the article was published. Next the major changes to the field occurring after the publication of Cliff and Ord's paper, focusing on technology advances and theory, are set out. Fortunately these changes mean the discipline is well placed to move beyond seeing the presence of spatial autocorrelation as a methodological issue that needs to be explained away. [ABSTRACT FROM AUTHOR]

Even though Cliff and Ord (1969) do not question “network autocorrelation,” their paper contains an interesting ambiguity that is here discussed. Hence, this article aims at discussing the problem of spatial autocorrelation in (perhaps preferably on) a network as well as at giving an overview of the existing relevant literature. A typology of network autocorrelations is suggested and it is hoped that this discussion initiates debates and points to new research challenges. [ABSTRACT FROM AUTHOR]

The influence in spatial epidemiology of the seminar work on autocorrelation by Cliff and Ord is discussed. Quantifying the evidence of spatial clustering was an important step in the development of modern statistical methods for analyzing spatial variations of diseases. Autocorrelation is nowadays mostly accounted for at a latent level within a hierarchical framework to small area disease mapping. The importance of accounting for autocorrelation in geographical correlation studies is also reviewed. [ABSTRACT FROM AUTHOR]

Spatial autocorrelation is a concept that helps to define the field of spatial analysis. It is central to studies using spatial statistics and spatial econometrics. In this paper, we trace the early development of the concept and explain the academic links that brought the concept to the fore in the late 1960s. In geography, the importance of the work of Michael F. Dacey, Andrew D. Cliff, and J. Keith Ord is emphasized. Later, with the publication of a volume on spatial econometrics by Luc Anselin, spatial research and the use of the concept of spatial autocorrelation received a considerable boost. These developments are outlined together with comments about recent and possible future trends in spatial autocorrelation-based research. [ABSTRACT FROM AUTHOR]

In recent years, there has been a growing interest in the problems caused by the existence of instability in cross-sectional regressions. The results about local autocorrelation measures are part of this debate, as are the proposals concerning the concept of geographically weighted regressions. This article also deals with the problem of stability (or the lack thereof), but focusing the discussion on the supposition of constancy in the parameter of spatial dependence. In most cases, this assumption is treated, with the risks that this involves, as a maintained hypothesis, which should be ascertained before continuing with the modeling exercise. In the article, we present a simple heterogeneity test for this type of parameters, based on the Lagrange Multiplier principle. To illustrate its use, we take the distribution of per capita income among the European regions as our discussion case. According to our results, there are clear signs of structural breaks in the spatial distribution of this variable and the scale factor and the autocorrelation coefficient appear to be principal actors. [ABSTRACT FROM AUTHOR]

The statistic known as Moran's I is widely used to test for the presence of spatial dependence in observations taken on a lattice. Under the null hypothesis that the data are independent and identically distributed normal random variates, the distribution of Moran's I is known, and hypothesis tests based on this statistic have been shown in the literature to have various optimality properties. Given its simplicity, Moran's I is also frequently used outside of the formal hypothesis-testing setting in exploratory analyses of spatially referenced data; however, its limitations are not very well understood. To illustrate these limitations, we show that, for data generated according to the spatial autoregressive (SAR) model, Moran's I is only a good estimator of the SAR model's spatial-dependence parameter when the parameter is close to 0. In this research, we develop an alternative closed-form measure of spatial autocorrelation, which we call APLE, because it is an approximate profile-likelihood estimator (APLE) of the SAR model's spatial-dependence parameter. We show that APLE can be used as a test statistic for, and an estimator of, the strength of spatial autocorrelation. We include both theoretical and simulation-based motivations (including comparison with the maximum-likelihood estimator), for using APLE as an estimator. In conjunction, we propose the APLE scatterplot, an exploratory graphical tool that is analogous to the Moran scatterplot, and we demonstrate that the APLE scatterplot is a better visual tool for assessing the strength of spatial autocorrelation in the data than the Moran scatterplot. In addition, Monte Carlo tests based on both APLE and Moran's I are introduced and compared. Finally, we include an analysis of the well-known Mercer and Hall wheat-yield data to illustrate the difference between APLE and Moran's I when they are used in exploratory spatial data analysis. [ABSTRACT FROM AUTHOR]

The creation of a spatial weights matrix by a procedure called AMOEBA, A Multidirectional Optimum Ecotope-Based Algorithm, is dependent on the use of a local spatial autocorrelation statistic. The result is (1) a vector that identifies those spatial units that are related and unrelated to contiguous spatial units and (2) a matrix of weights whose values are a function of the relationship of the ith spatial unit with all other nearby spatial units for which there is a spatial association. In addition, the AMOEBA procedure aids in the demarcation of clusters, called ecotopes, of related spatial units. Experimentation reveals that AMOEBA is an effective tool for the identification of clusters. A comparison with a scan statistic procedure (SaTScan) gives evidence of the value of AMOEBA. Total fertility rates in enumeration districts in Amman, Jordan, are used to show a real-world example of the use of AMOEBA for the construction of a spatial weights matrix and for the identification of clusters. Again, comparisons reveal the effectiveness of the AMOEBA procedure. [ABSTRACT FROM AUTHOR]

The Poisson model, discovered nearly two centuries ago, is the basis for analyses of rare events. Its first applications included descriptions of deaths from mule kicks. More than half a century ago the Poisson model began being used in geographical analysis. Its initial descriptions of geographic distributions of points, disease maps, and spatial flows were accompanied by an assumption of independence. Today this unrealistic assumption is replaced by one allowing for the presence of spatial autocorrelation in georeferenced counts. Contemporary statistical theory has led to the creation of powerful Poisson-based modeling tools for geographically distributed count data. [ABSTRACT FROM AUTHOR]

The auto-Poisson probability model furnishes an obvious tool for modeling counts of geographically distributed rare events. Unfortunately, its original specification can accommodate only negative spatial autocorrelation, which itself is a rare event. More recent alternative reformulations, namely, the Winsorized and spatial filter specifications, circumvent this drawback. A comparison of their performances presented in this article reveals some of their relative advantages and disadvantages. [ABSTRACT FROM AUTHOR]

This article presents an overview of GeoDa™, a free software program intended to serve as a user-friendly and graphical introduction to spatial analysis for non-geographic information systems (GIS) specialists. It includes functionality ranging from simple mapping to exploratory data analysis, the visualization of global and local spatial autocorrelation, and spatial regression. A key feature of GeoDa is an interactive environment that combines maps with statistical graphics, using the technology of dynamically linked windows. A brief review of the software design is given, as well as some illustrative examples that highlight distinctive features of the program in applications dealing with public health, economic development, real estate analysis, and criminology. [ABSTRACT FROM AUTHOR]

In many instances it is of interest to measure the degree of similarity between neighboring regions. Spatial autocorrelation measures are the most popular means of doing it. However, such measures only capture a global linear relationship between regions, whereas in many circumstances a more general instrument is required. For instance, in economic poverty analysis or environmental applications (and in other cases where we are interested in extreme events and threshold exceedances) we should be more interested in the spatial pattern in the tails of the joint distributions. In this article we introduce some exploratory tool that focuses on the bivariate joint tails behavior to detect a pattern of spatial regularities. The method will be illustrated with reference to simulated environmental data. [ABSTRACT FROM AUTHOR]

One approach to dealing with spatial autocorrelation in regression analysis involves the filtering of variables in order to separate spatial effects from the variables' total effects. In this paper we compare two filtering approaches, both of which allow spatial statistical analysts to use conventional linear regression models. Getis' filtering approach is based on the autocorrelation observed with the use of the G local statistic. Griffith's approach uses an eigenfunction decomposition based on the geographic connectivity matrix used to compute a Moran's I statistic. Economic data are used to compare the workings of the two approaches. A final comparison with an autoregressive model strengthens the conclusion that both techniques are effective filtering devices, and that they yield similar regression models. We do note, however, that each technique should be used in its appropriate context. [ABSTRACT FROM AUTHOR]

Extensions of nondirectional spatial autocorrelation techniques to two dimensions have existed for many years, but the results are difficult to compare to the traditional nondirectional techniques and often lack ease of interpretability. This paper reviews the traditional one- and two-dimensional spatial autocorrelation methods and proposes a new directional method which is both easier to compare to nondirectional methods and easier to interpret than previous directional methods. [ABSTRACT FROM AUTHOR]

The article responds to comments on the article "Extending Moran’s Index for Measuring Spatiotemporal Clustering of Geographic Events. It mentions an issue to consider for any further research in the realm of spatiotemporal autocorrelation is the very different ways events are related to each other spatially and temporally. It states that the study would stimulate others to further devise spatiotemporal analytics to benefit Geographic Information System (GIS) community.

The article comments on the article "Extending Moran’s Index for Measuring Spatiotemporal Clustering of Geographic Events" by J. Lee and S. Li. It formalizes the extended Moran coefficient (MC) space-time weights matrix for the two cases of contemporaneous and spatially lagged autocorrelation. It concludes that the Lee and Li paper should have had a much more comprehensive contextualization.

A correction to the article "Local Spatial Autocorrelation in a Biological Model" that was published in the October 1998 issue is presented.