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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]

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]

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]

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]

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]

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 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]

The author discusses how the 1969 article "The Problem of Spatial Autocorrelation," by A. Cliff and J. Ord has affected general science and theoretical geography. He claims that the 1969 article was not that successful in promoting the use of nonbinary geographic weights, but was successful in advocating the spatial autocorrelation indices. The author expresses his disappointment that no analytical spatial autocorrelation results have been developed for nonnormal random variables.

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