Local Spatial Autocorrelation in a Biological Model

We review the recently developed local spatial autocorrelation statistics I_i, c_i, G_i, and G_i*. We discuss two alternative randomization assumptions, total and conditional, and then newly derive expectations and variances under conditional randomization for I_i and c_i, as well as under total randomization for c_i. The four statistics are tested by a biological simulation model from population genetics in which a population lives on a 21 × 21 lattice of stepping stones (sixty-four individuals per stone) and reproduces and disperses over a number of generations. Some designs model global spatial autocorrelation, others spatially random surfaces. We find that spatially random designs give reliable test results by permutational methods of testing significance. Globally autocorrelated designs do not fit expectations by any of the three tests we employed. Asymptotic methods of testing significance failed consistently, regardless of design. Because most biological data sets are autocorrelated, significance testing for local spatial autocorrelation is problematic. However, the statistics are informative when employed in an exploratory manner. We found that hotspots (positive local autocorrelation) and coldspots (negative local autocorrelation) are successfully distinguished in spatially autocorrelated, biologically plausible data sets.     

Population and productivity in the teotihuacan valley: Changing patterns of spatial association in prehispanic central Mexico

Since its emergence earlier this century, cultural ecology has played a key role in attempts to understand the complex interrelations between cultural and environmental systems. Although rarely examined, a crucial aspect of cultural adaptation is the explicit spatial relationship between the distribution of human populations and the various resources available to them. In the following essay, we examine this particular question in terms of regional subsistence potential with settlement system remains from the Middle Horizon (Classic) and Late Horizon (Aztec) periods in the Teotihuacan Valley, Mexico. As a first step, traditional measures of association (Pearson's, Spearman's, and Kendall's) are employed to assess the point-to-point relationships between population and productivity potential. Then, recently developed randomization procedures are introduced and applied to evaluate the spatial association of these two variables. The results of these complementary avenues of analysis help to increase our understanding of the role of space in the cultural ecology of the prehispanic Teotihuacan Valley. In addition, they reaffirm the potentially powerful impact of cultural mechanisms on strategies of regional adaptation.     

A Classification of Spatial Distributions Based upon Several Cell Sizes

Several procedures, based upon cell count analysis, have been proposed for classifying spatial distributions, or maps, associated with some region R. Such procedures are rather imprecise and are known to depend upon the sixes and shapes of the cells in the particular partition of R under consideration. In this paper, the problem is considered from the point of view of hypothesis testing. A test of randomness based upon an arbitrary number of partitions of R is giuen. If the hypothesis of randomness is rejected, additional tests may be performed to classify the map into one of two categories, clustered or regular. These tests provide a number of advantages over existing procedures. Based upon multiple partitions of R, they decrease the dependence upon any particular partition, and the colresponding classification is precise since the null hypothesis distribution of the test statistic is (asymptotically) known. Finally, they allow a great deal of flexibility in testing for certain alternatives to randomness, and are applicable to one-, two-, and three- dimensional maps.     

Measuring Spatial Autocorrelation of Vectors

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.     

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.     

Maximum Getis–Ord Statistic Adjusted for Spatially Autocorrelated Data

Local statistics test the null hypothesis of no spatial association or clustering around the vicinity of a location. To carry out statistical tests, it is assumed that the observations are independent and that they exhibit no global spatial autocorrelation. In this article, approaches to account for global spatial autocorrelation are described and illustrated for the case of the Getis–Ord statistic with binary weights. Although the majority of current applications of local statistics assume that the spatial scale of the local spatial association (as specified via weights) is known, it is more often the case that it is unknown. The approaches described here cover the cases of testing local statistics for the cases of both known and unknown weights, and they are based upon methods that have been used with aspatial data, where the objective is to find changepoints in temporal data. After a review of the Getis–Ord statistic, the article provides a review of its extension to the case where the objective is to choose the best set of binary weights to estimate the spatial scale of the local association and assess statistical significance. Modified approaches that account for spatially autocorrelated data are then introduced and discussed. Finally, the method is illustrated using data on leukemia in central New York, and some concluding comments are made.     

A Surface-Based Approach to Measuring Spatial Segregation

Quantitative indices of residential segregation have been with us for half a century, but suffer significant limitations. While useful for comparison among regions, summary indices fail to reveal spatial aspects of segregation. Such measures generally consider only the population mix within zones, not between them. Zone boundaries are treated as impenetrable barriers to interaction between population subgroups, so that measurement of segregation is constrained by the zoning system, which bears no necessary relation to interaction among population subgroups. A segregation measurement approach less constrained by the chosen zoning system, which enables visualization of segregation levels at the local scale and accounts for the spatial dimension of segregation, is required. We propose a kernel density estimation approach to model spatial aspects of segregation. This provides an explicitly geographical framework for modeling and visualizing local spatial segregation. The density estimation approach lends itself to development of an index of spatial segregation with the advantage of functional compatibility with the most widely used index of segregation (the dissimilarity index D ). We provide a short review of the literature on measuring segregation, briefly describe the kernel density estimation method, and illustrate how the method can be used for measuring segregation. Examples using a simulated landscape and two empirical cases in Washington, DC and Philadelphia, PA are presented.     

A Scale‐Sensitive Test of Attraction and Repulsion Between Spatial Point Patterns

There exist a variety of tests for attraction and repulsion effects between spatial point populations, most notably those involving either nearest‐neighbor or cell‐count statistics. Diggle and Cox (1981) showed that for the nearest‐neighbor approach, a powerful test could be constructed using Kendall's rank correlation coefficient. In the present paper, this approach is extended to cell‐count statistics in a manner paralleling the K‐function approach of Lotwick and Silverman (1982). The advantage of the present test is that, unlike nearest‐neighbors, one can identify the spatial scales at which repulsion or attraction are most significant. In addition, it avoids the torus‐wrapping restrictions implicit in the Monte Carlo testing procedure of Lotwick and Silverman. Examples are developed to show that this testing procedure can in fact identify both attraction and repulsion between the same pair of point populations at different scales of analysis.     

A Shape‐Based Local Spatial Association Measure (LISShA): A Case Study in Maritime Anomaly Detection

The explicit consideration of the shape of geographic features has been largely ignored in existing spatial association measures. The primary contribution of this work is the development of a new local spatial association measure—a Local Indicator of Spatial and Shape Association (LISShA). The LISShA measure is modeled after local Geary's Spatial Autocorrelation measure with distance between shapes, calculated using the Small–Le metric, replacing difference between attribute values and the spatial neighborhood defined by Fréchet distance. We provide some explanation of these metrics and show, in detail, how the LISShA and proposed moments are calculated in a one‐dimensional context in a case study of maritime anomaly detection.     

Spatial autocorrelation of ABO serotypes in mediaeval cemeteries as an indicator of ethnic and familial structure

Hungarian cemeteries in an effort to detect familial structure. Different ethnic groups buried in separate areas dominated the spatial pattern in two cemeteries, leaving insufficient power to test for familial patterns. In a third, ethnically homogeneous cemetery, no evidence of familial structure was found. A simulation showed that familial structure could readily be detected by the methods applied when it exists in an ethnically homogeneous population. The spatial autocorrelation methods employed would have detected the ethnic diversity in the two cemeteries containing graves from different populations, even in the absence of archaeological information to that effect. A restricted randomization procedure was developed to test two alternative hypotheses concerning the ethnic designations of the occupants of the cemetery at Szentendre. As a result of this test, the hypothesis that graves located in a rough circle (putative Lombards) differ serologically from those located at the periphery (putative nonLombards) is strongly preferred over a second hypothesis based on grave goods which would imply spatially random placement of the graves of the two ethnic units.     

A New Perspective about Moran's Coefficient: Spatial Autocorrelation as a Linear Regression Problem. Moran系数的新视角：空间自相关视为线性回归问题

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.     

Exploring Spatial Patterns Using an Expanded Spatial Autocorrelation Framework

Spatial autocorrelation (SA) is regarded as an important dimension of spatial pattern. SA measures usually consist of two components: measuring the similarity of attribute values and defining the spatial relationships among observations. The latter component is often represented by a spatial weights matrix that predefines spatial relationship between observations in most measures. Therefore, SA measures, in essence, are measures of attribute similarity, conditioned by spatial relationship. Another dimension of spatial pattern can be explored by controlling observations to be compared based upon the degree of attribute similarity. The resulting measures are spatial proximity measures of observations, meeting predefined attribute similarity criteria. Proposed measures reflect degrees of clustering or dispersion for observations meeting certain levels of attribute similarity. An existing spatial autocorrelation framework is expanded to a general framework to evaluate spatial patterns and can accommodate the proposed approach measuring proximity. Analogous to the concept of variogram, clustergram is proposed to show the levels of spatial clustering over a range of attribute similarity, or attribute lags. Specific measures based on the proposed approach are formulated and applied to a hypothetical landscape and an empirical example, showing that these new measures capture spatial pattern information not reflected by traditional spatial autocorrelation measures.     

Elementary Spatial Systems of Industrial Production

An elementary spatial system of industrial production is conceptualized as the primary geographical formation in industry. Such an elementary production system (EPS) is viewed as consisting of raw-material and consumption nodes connected by lines of movement with a single production node. Besides the measures proposed by Kansky (University of Chicago, Research Paper 84, 1963), such a system, represented in the form of a connected, directed graph, can be described by measures of material and spatial transformation and by indices of dispersion of aggregate nodes. Characteristic properties of EPS are the spatial attraction between nodes and the spatial coincidence and intersection of common nodes of two or more systems. The usual Weberian movement-minimization approach to the building of a model of such an elementary production system is judged to be inadequate because it ignores local differences in production costs, and a “field of potential costs” is proposed instead to take account of the total cost picture.     

Local Indicators of Spatial Association—LISA

The capabilities for visualization, rapid data retrieval, and manipulation in geographic information systems (GIS) have created the need for new techniques of exploratory data analysis that focus on the “spatial” aspects of the data. The identification of local patterns of spatial association is an important concern in this respect. In this paper, I outline a new general class of local indicators of spatial association (LISA) and show how they allow for the decomposition of global indicators, such as Moran's I, into the contribution of each observation. The LISA statistics serve two purposes. On one hand, they may be interpreted as indicators of local pockets of nonstationarity, or hot spots, similar to the G_i and G*_i statistics of Getis and Ord (1992). On the other hand, they may be used to assess the influence of individual locations on the magnitude of the global statistic and to identify “outliers,” as in Anselin's Moran scatterplot (1993a). An initial evaluation of the properties of a LISA statistic is carried out for the local Moran, which is applied in a study of the spatial pattern of conflict for African countries and in a number of Monte Carlo simulations.     

Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation

A fundamental concern of spatial analysts is to find patterns in spatial data that lead to the identification of spatial autocorrelation or association. Further, they seek to identify peculiarities in the data set that signify that something out of the ordinary has occurred in one or more regions. In this paper we provide a statistic that tests for local spatial autocorrelation in the presence of the global autocorrelation that is characteristic of heterogeneous spatial data. After identifying the structure of global autocorrelation, we introduce a new measure that may be used to test for local structure. This new statistic Oi is asymptotically normally distributed and allows for straightforward tests of hypotheses. We provide several numerical examples that illustrate the performance of this statistic and compare it with another measure that does not account for global structure.     

SPATIAL DEPENDENCE AND SPATIAL STRUCTURAL INSTABILITY IN APPLIED REGRESSION ANALYSIS*

The stability of regression coefficients over the observation set (“regional homogeneity”) is typically assessed by means of a Chow test or within a seemingly unrelated regression (SUR) framework. When spatial error autocorrelation is present in cross-sectional equations the traditional tests are no longer applicable. I evaluate this both in formal terms as well as empirically. I introduce a taxonomy of spatial effects in models for structural instability, and discuss its implication for testing. I compare the performance of traditional tests, robust approaches, maximum-likelihood procedures and pretest techniques by means of a series of simple Monte Carlo experiments.     

TESTING THE SPATIAL SCALE AND THE DYNAMIC STRUCTURE IN REGIONAL MODELS (A CONTRIBUTION TO SPATIAL ECONOMETRIC SPECIFICATION ANALYSIS)*

This article addresses the problem of specification uncertainty in modeling spatial economic theories in stochastic form. It is ascertained that the traditional approach to spatial econometric modeling does not adequately deal with the type and extent of specification uncertainty commonly encountered in spatial economic analyses. Two alternative spatial econometric modeling procedures proposed in the literature are reviewed and shown to be suitable for analyzing systematically two sources of specification uncertainty, viz., the level of aggregation and the spatio-temporal dynamic structure in multiregional econometric models. The usefulness of one of these specification procedures is illustrated by the construction of a simple multiregional model for The Netherlands.     

Comparison of Moran's I and Geary's c in Multivariate Spatial Pattern Analysis

This article compares multivariate spatial analysis methods that include not only multivariate covariance, but also spatial dependence of the data explicitly and simultaneously in model design by extending two univariate autocorrelation measures, namely Moran's I and Geary's c. The results derived from the simulation datasets indicate that the standard Moran component analysis is preferable to Geary component analysis as a tool for summarizing multivariate spatial structures. However, the generalized Geary principal component analysis developed in this study by adding variance into the optimization criterion and solved as a trace ratio optimization problem performs as well as, if not better than its counterpart the Moran principal component analysis does. With respect to the sensitivity in detecting subtle spatial structures, the choice of the appropriate tool is dependent on the correlation and variance of the spatial multivariate data. Finally, the four techniques are applied to the Social Determinants of Health dataset to analyze its multivariate spatial pattern. The two generalized methods detect more urban areas and higher autocorrelation structures than the other two standard methods, and provide more obvious contrast between urban and rural areas due to the large variance of the spatial component.     

Optimal Weights for Focused Tests of Clustering Using the Local Moran Statistic

Local spatial statistics measure and test for spatial association for a variable or variables of interest in a geographic neighborhood surrounding a predefined location. Most applications adopt a single scale of analysis but give little attention to the scale of the process generating the data. Alternatively, when the researcher is uncertain about the process scale, local statistics may examine a number of scales. In these cases, it is important to include a correction for multiple testing when evaluating the statistical significance of each local statistic, something that is rarely done. Consequently, local statistics are more likely to identify significant relationships, even when no meaningful spatial association exists. In this article, we develop a methodology for the local Moran statistic that provides both an empirical estimate of the spatial scale of association and an assessment of the significance of the statistic for that scale. The key idea is to test a number of possible choices for the statistic's weight matrix and then account for the multiple testing associated with the number of weight matrices examined. Unlike previous research, our statistic avoids the use of simulation to determine statistical significance in the presence of multiple testing. To test the validity of our approach, we constructed a numerical example to assess the statistic's performance and conducted an empirical study using leukemia data from central New York state. The developed statistic addresses the need for the empirical determination of weights and spatial scale. The test therefore addresses the common weakness of many applications, where weights are defined exogenously, with little or no thought given to either the definition or its implications. Los indicadores locales (local spatial statistics) evalúan la asociación espacial de una o varias variables de interés dada un área predefinida y sus áreas vecinas. La mayoría de dichas medidas utilizan una escala única de análisis y prestan poca atención a la escala del proceso de generación de los datos. En los casos en los que el investigador no está seguro de la escala del proceso, las los indicadores locales pueden ser evaluados a varias escalas. En dichos casos, cuando se hace la evaluación de la significancia estadística de cada indicador local, es importante incorporar una corrección para pruebas múltiples (multiple tests), un ajuste que raramente se realiza en la gran mayoría de estudios. Debido al problema de pruebas múltiples, los indicadores locales son más propensos a identificar relaciones significativas, incluso cuando no existe asociación espacial significativa alguna. En este artículo los autores desarrollan una metodología que produce un índice local de Moran que proporciona tanto una estimación empírica de la escala espacial de la asociación así como una evaluación de la importancia del indicador para dicha escala. La idea clave es poner a prueba una serie de opciones posibles para la definición de la matriz de pesos espaciales (spatial weight matrix) del índice y luego tomar en cuenta las pruebas múltiples asociadas con el número de matrices de peso examinadas. A diferencia de métodos anteriores, el indicador local propuesto evita el uso de simulaciones para determinar la significancia estadística con pruebas múltiples. Para probar la validez del enfoque propuesto, se construyó un ejemplo numérico con el fin de evaluar el desempeño del nuevo índice y se llevó a cabo un estudio comparativo a partir de datos del centro de leucemia del estado de Nueva York. El índice desarrollado responde a la necesidad de definir las ponderaciones (pesos) empíricamente y la escala espacial. De esta forma el método propuesto supera limitaciones comúnmente halladas de muchas aplicaciones en las cuales los pesos son definidos exógenamente, con poca o ninguna atención a su definición o su implicancias. 局部空间统计量可用于度量和检验预定地理区域周围邻域的空间关联。大多数情况下仅采用单一尺度的分析而较少关注数据生成过程的尺度。而当其过程尺度无法确定时,局部统计量却可能检测出多个尺度。在这些案例中,对单个局部统计量统计显著性评估建立多重检验的修正是重要的,而这却鲜有实施。因此,即使存在无意义的空间关联时,局部统计也更可能识别出显著的相关性。 本文发展了一种基于局部Moran统计的方法,提供了空间尺度关联性的经验估计以及对该尺度下统计显著性的评估。其核心思想是测试统计权重矩阵的可能选择,然后考虑与权重矩阵检验数量数目相关的多重检验。与以往研究不同,该方法在多重检验情况中避免了采用模拟来确定统计显著性。为检验其有效性,采用了数值案例来评估其统计性能,并基于纽约州中部的血癌数据进行比较研究。该方法解决了权重和空间尺度确定经验估计的需求,通过验证也相应地解决了很多应用中的普遍弱点,即权重被定义成外生变量,而很少或根本没有考虑其定义或含义。