"The Problem of Spatial Autocorrelation" and Local Spatial Statistics

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.     

Evaluation of a Spatial Relationship by the Concept of Intrinsic Spatial Distance

We propose the concept of intrinsic spatial distance (ISD) for the study of a spatial relationship between any two points in space. ISD is a distance measure that takes into account the separation of two points with respect to their physical and attribute closeness. We construct an algorithm to implement this concept as an ISD measurement. Based on the ISD concept, two points in space are related through a transitional path linking one to the other. As an ISD measurement decreases, the spatial relationship between two points becomes increasingly stronger. We argue theoretically and demonstrate empirically that the ISD concept is not predisposed in favor of the first law of geography, but directly considers variance of nearness in physical distance and attribute distance to derive the extent to which two points are spatially associated. Specifically, in single attribute cases, the information uncovered by an ISD measurement is more elaborate than that revealed by Moran's I, local Moran's I, and a semivariogram, giving a meticulous account of relatedness in both local and global contexts. The ISD concept is also sufficiently general to be used to study multiple attributes of relationships. Proponemos el concepto de distancia espacial intrínseca (intrinsic spatial distance ‐ISD) para el estudio de la relación espacial entre dos puntos en el espacio. La ISD es una medición de distancia que tiene en cuenta la separación de dos puntos con respecto a su cercanía física y de atributo. Construimos un algoritmo para aplicar este concepto y crear una medición ISD. De acuerdo con en el concepto ISD, dos puntos en el espacio están relacionadas a través de un camino de transición que los vincula uno al otro. A medida que la medición ISD disminuye, la relación espacial entre dos puntos se refuerza. Argumentamos teóricamente y demostramos empíricamente que el concepto ISD no está predispuesto a favor de la primera ley de la geografía (de Tobler), pero considera directamente la varianza de la cercanía de la distancia física y la distancia del valor del atributo para derivar una medición que cuantifica el grado en el que dos puntos están asociados espacialmente. En concreto, en los casos de atributos individuales, la información revelada por una medición ISD es más sofisticada que la que proporcionada por el índice de Moran, el índice de Moran local, y el semivariograma, pues da cuanta minuciosa de la relación espacial y de atributo en contextos locales y globales. El concepto de ISD también es lo suficientemente general para ser usado en el estudio de relaciones entre múltiples atributos. 本文提出了内蕴空间距离(ISD)来研究空间中任意两点之间的空间关系。ISD是考虑空间上分离的两点各自物理和属性接近度的一种距离度量。通过构造一个算法将该概念转变为ISD可度量方式。基于ISD概念,空间上的两点通过连接两者迁移路径相关联。当ISD度量减小时,两点之间空间关系变得越来越强。理论论证和实例演示表明ISD概念并不倾向于符合地理学第一定律,但直接考虑两点间物理距离和属性距离临近性的变化可以推导两点在空间上的联系程度。特别是在单一属性案例中,相比于Moran’s I,局部Moran’s I和半方差图,ISD度量揭示了全局与局部环境中详尽的空间相关信息。同时,ISD概念在研究多属性的空间关系中也是足够通用的。     

基于ESDA的新疆旅游经济发展空间分异规律研究

在ArcGIS空间分析技术支持下,借助GeoDA空间自相关分析方法,以新疆85个县(市)旅游总收入及其空间差异为研究对象,分析新疆旅游经济发展空间分异格局,并用地学统计方法研究其空间分异规律,得出:(1)新疆旅游经济具有较强的空间依赖性,全局空间自相关特征明显,表现为集聚度高、异质性强、差异性大的空间极化分异态势;(2)局部上则表现为高高相聚和低低相随的两极分化格局,热点区以乌鲁木齐为中心,以昌吉州和吐鲁番、阿勒泰地区为延伸,沿交通网络向周边地区递减的空间分异格局,喀什、和田地区为冷点区;(3)交通可达性及旅游基础设施数量对新疆旅游经济发展起决定性作用,高速公路尤其是铁路对旅游经济影响最大;旅游资源优势度对旅游实现更加重要;交通区位对新疆旅游经济空间分异起着基础性作用.     

Distance in Spatial Analysis: Measurement,Bias, and Alternatives

Distance is an important and basic concept in geography. Many theories, methods, and applications involve distance explicitly or implicitly. While measuring the distance between two locations is a straightforward task, many geographical processes involve areal units, where the distance measurement can be complicated. This research investigates distance measurement between a location (point) and an area (polygon). We find that traditional polygon-to-point distance measurements, which involve abstracting a polygon into a central or representative point, could be problematic and may lead to biased estimates in regression analysis. To solve this issue, we propose a new polygon-to-point distance metric along with two algorithms to compute the new distance metric. Simulation analysis shows the effectiveness of the new distance metric in providing unbiased estimates in linear regression.     

The SpatialARMED Framework: Handling Complex Spatial Components in Spatial Association Rule Mining

Recent research has identified spatial association rule (SAR) mining as a promising technique for geographic pattern mining and knowledge discovery. Nevertheless, important spatial components embedded in the studied phenomenon, in particular complex spatial functional relations such as neighborhood effects and spatial spillover effects have largely been neglected. This article unravels this specific problem to enhance the effective application of SAR mining practices in spatial data analytics. The main discussion focuses on the specification of complex spatial components by means of spatial dependence properties of the data and on how to integrate them in the process of SAR mining. A comprehensive framework dubbed SpatialARMED is proposed for the effective extraction of spatial patterns. The framework is then showcased through its application to crime analysis.     

Borders Moderating Distance: A Social Network Analysis of Spatial Effects on Policy Interaction

The present paper examines the importance of integrating geographical effects into the analysis of social networks. Specifically, we study the impacts of spatial distance and territorial borders on information exchange within two European cross-border regions where there is evidence of extensive cross-border political interaction in the domain of public transportation. We use exponential random graph models to review how these spatial factors influence policy interaction. We show how the effect of distance on the exchange of information between organizations is less obvious than it seems at first sight, when we control for endogenous networks effects with which it tends to be confounded. We also explore moderating effects among the spatial factors studied and in particular how the effect of distance affects information exchange by territoriality. While the probability of interaction decreases with distance in the context of domestic ties, distance does not exert the same friction when it comes to cross-border ties. These findings suggest that borders are not necessarily a barrier to interaction and that peripheral actors located in border regions may bridge distance in order to access strategic information they do not have access to within their territorial jurisdiction and is likely concentrated in foreign decision centers.     

A Spatial Economic Analysis of Urban Land Use and Obesity*

Abstract. We analyze an urban spatial model to examine the possible link between urban land use and obesity. Households maximize utility defined over housing, weight, and food subject to a fixed time budget allocated to commuting, calorie expenditure, and work. Our model explains the observed correspondence between high obesity rates and low development densities, but implies that these are determined endogenously in a spatial market equilibrium. We study the sorting of residents by attributes such as income, initial weight, and weight preferences, and examine the impacts on weight and density of urban design modifications that lower the costs of calorie expenditure.     

Toward a Theory of Spatial Statistics: Another Step Forward

An update is presented to an earlier piece outlining some steps that needed to be taken toward the establishment of a theory of spatial statistics. Findings that have appeared since this first paper are summarized and interpreted, and extensions and suggestions are offered for the further establishment of a basis for a theory of spatial statistics. Topics include boundary considerations, the role of latent spatial dependencies, and small-sample-size issues. These topics embrace the problems of data transformations, edge effect bias, reference sampling distributions, multivariate autocorrelation models, conditional expectations, and higher-order autoregressive structures. In part, a course is charted for the next step to be taken.     

The Role of Bose-Einstein Statistics in Point Pattern Analysis

Bose-Einstein statistics are introduced as the least-biased definition of equal likelihood or randomness available to geographers studying point patterns. The Bose-Einstein distribution is shown to be an entropy-maximizing probability distribution which enables the information theoretic concept of redundancy to be used as a measure of the extent to which an unknown spatial process influences the evolution of the frequency distribution of points. Finally the method is applied to the frequency distribution of karst depressions in limestone regions and the distributions of factories in Greater Manchester.     

Controlling the False Discovery Rate: A New Application to Account for Multiple and Dependent Tests in Local Statistics of Spatial Association

Assessing the significance of multiple and dependent comparisons is an important, and often ignored, issue that becomes more critical as the size of data sets increases. If not accounted for, false-positive differences are very likely to be identified. The need to address this issue has led to the development of a myriad of procedures to account for multiple testing. The simplest and most widely used technique is the Bonferroni method, which controls the probability that a true null hypothesis is incorrectly rejected. However, it is a very conservative procedure. As a result, the larger the data set the greater the chances that truly significant differences will be missed. In 1995, a new criterion, the false discovery rate (FDR), was proposed to control the proportion of false declarations of significance among those individual deviations from null hypotheses considered to be significant. It is more powerful than all previously proposed methods. Multiple and dependent comparisons are also fundamental in spatial analysis. As the number of locations increases, assessing the significance of local statistics of spatial association becomes a complex matter. In this article we use empirical and simulated data to evaluate the use of the FDR approach in appraising the occurrence of clusters detected by local indicators of spatial association. Results show a significant gain in identification of meaningful clusters when controlling the FDR, in comparison to more conservative approaches. When no control is adopted, false clusters are likely to be identified. If a conservative approach is used, clusters are only partially identified and true clusters are largely missed. In contrast, when the FDR approach is adopted, clusters are fully identified. Incorporating a correction for spatial dependence to conservative methods improves the results, but not enough to match those obtained by the FDR approach.     

Large Sample-Size Distribution of Statistics Used In Testing for Spatial Correlation

Test statistics for testing for spatial correlation in continuous variables have been given by both Moran and Geary and have subsequently been generalized. It has been conjectured for a long time that under the hypothesis of no spatial correlations all these statistics are normally distributed when the sample size is large. This paper proves a very general theorem on the large sample normality of quadratic forms. As corollaries to the theorem the asymptotic normality, under the hypothesis, of all the above-mentioned statistics is established. The necessary conditions are quite unrestrictive. It is also shown, by means of a counter example, that the conditions given in a similar theorem (Cliff and Ord) are inadequate to ensure normality.     

A Programming Approach to Minimizing and Maximizing Spatial Autocorrelation Statistics

A programming approach is presented for identifying the form of the weights matrix W which either minimizes or maximizes the value of Moran's spatial autocorrelation statistic for a specified vector of data values. Both nonlinear and linear programming solutions are presented. The former are necessary when the sum of the links in W is unspecified while the latter can be used if this sum is fixed. The approach is illustrated using data examined in previous studies for two variables measured for the counties of Eire. While programming solutions involving different sets of constraints are derived, all yield solutions in which the number of nonzero elements in W is considerably smaller than that in W defined using the contiguity relationships between the counties. In graph theory terms, all of the Ws derived define multicomponent graphs. Other characteristics of the derived Ws are also presented.     

Local Spatial Autocorrelation Statistics: Distributional Issues and an Application

The statistics G_i(d) and G_i*(d), introduced in Getis and Ord (1992) for the study of local pattern in spatial data, are extended and their properties further explored. In particular, nonbinary weights are allowed and the statistics are related to Moran's autocorrelation statistic, I. The correlations between nearby values of the statistics are derived and verified by simulation. A Bonferroni criterion is used to approximate significance levels when testing extreme values from the set of statistics. An example of the use of the statistics is given using spatial-temporal data on the AIDS epidemic centering on San Francisco. Results indicate that in recent years the disease is intensifying in the counties surrounding the city.