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The statistics Gi(d) and Gi*(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.

Introduced in this paper is a family of statistics, G, that can be used as a measure of spatial association in a number of circumstances. The basic statistic is derived, its properties are identified, and its advantages explained. Several of the G statistics make it possible to evaluate the spatial association of a variable within a specified distance of a single point. A comparison is made between a general G statistic and Moran’s I for similar hypothetical and empirical conditions. The empirical work includes studies of sudden infant death syndrome by county in North Carolina and dwelling unit prices in metropolitan San Diego by zip-code districts. Results indicate that G statistics should be used in conjunction with I in order to identify characteristics of patterns not revealed by the I statistic alone and, specifically, the G i and G i ∗ statistics enable us to detect local “pockets” of dependence that may not show up when using global statistics.

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.

The concept, spatial autocorrelation, is central to many concerns expressed in Regional Science and Urban Economics. The term, first used in 1967, had precursors embodied in such ideas as distance-decay and spatial interaction. Today, spatial autocorrelation is the mainstay of the burgeoning field of spatial econometrics. Its study has many advantages such as providing tests on model misspecification. The Moran's I statistic is one of many ways spatial autocorrelation can be represented. Interestingly, the concept has yet to make a significant appearance in the mainstream econometrics literature.

Spatial association means connectedness or relationship between and among variables over space (the Earth's surface). A single variable may be spatially autocorrelated ; that is, values of the variable are somehow connected or related spatially. Many variables may be associated with one another at one or more sites. If there is spatial interaction, there is also spatial association. Maps can depict spatial association. A mathematical shorthand technique can be used to represent, in general, measures of spatial association. Scientists test or theorize about variables to determine whether spatial association, either observed or expected, actually can be confirmed. Statistical procedures that have been developed for identifying and measuring the existence of spatial association are outlined.

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.

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 hypothe- ses. 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.

A cross-product statistic is used to demonstrate that spatial interaction models are a special case of a general model of spatial autocorrelation. A series of traditional measures of spatial autocorrelation is shown to have a cross-product form. Several interaction models are shown to have a similar form. A general spatial statistic is developed which indicates that the relationship between the two types of models is particularly strong when the focus is on measurements from a single point.

In this paper, we discuss a number of general issues that pertain to the interface between GIS and spatial analysis. In particular, we focus on the various paradigms for spatial data analysis that follow from the existence of this interface. We outline a series of questions that need to be confronted in the analysis of spatial data, and the extent to which a GIS can facilitate their resolution. We also review a number of exploratory and confirmatory techniques that we feel should form the core of a spatial analysis module for a GIS.

Applied geography has undergone remarkable changes in the last 20 years. Powerful new technologies have emerged that greatly improve the ability to collect, store, manage, view, analyze, and utilize information regarding the critical issues of our time. These technologies include geographic information systems (GIS), global positioning systems (GPS), satellite-base remote sensing, and a great variety of remarkable software that allows for the analysis of the compelling problems. The issues include globalization, global warming, pollution, security, crime, public health, transportation, energy supplies, and population growth. Geographic Information Science (GIScience) has given rise to an essentially multidisciplinary approach to applied problems. No single person is expert in all of these areas. It is necessary to emphasize coordination and collaboration and to find the bridges that reduce the barriers between disciplines. In this chapter we briefly discuss the new technologies and the way in which they are being used to solve the critical issues. We then make suggestions for an applied geography future vis-a-vis the geographic information sciences.

The origins of modern spatial analysis lie in the development of quantitative geography and regional science in the late 1950s. The use of quantitative procedures and techniques to analyse patterns of points, lines, areas and surfaces depicted on analogue maps or defined by co-ordinates in two-or three-dimensional space characterise the initial stage. Later on, more emphasis was placed on the indigenous features of geographical space, on spatial choices and processes, and their implications for the spatio-temporal evolution of complex spatial systems. Spatial analysis, as it has become over the past four decades, is more than spatial statistics and data analysis, and goes far beyond data sampling, data manipulation, exploratory and confirmatory spatial data analysis, into areas of spatial modelling encompassing a large and diverse set of models in both the environmental and the social sciences. In the environmental sciences, they range from physical dispersion models (e.g. for suspended particulates), chemical reaction models (e.g. photochemical smog), and biological systems models (e.g. for ecosystems in water), through deterministic process and stochastic process models of regional environmental quality management, and integrated models for environment-energy-economic assessment. For many models, especially the physical dispersion models, chemical reaction models, and biological systems models, the geographical location of the site is not considered to be of overriding importance. The variation in model results over space can be achieved by obtaining different inputs to the area in question. Process models include deterministic versions which attempt to describe a particular sequence in terms of known physical laws, and stochastic model versions which aim to describe a particular process such as erosion, groundwater movement and absorption of pollutants in terms of probability theory. For such processes, there is an interaction between the spatial process and a substrate. This substrate provides a one-, two-or three-dimensional framework within which the process model can operate. The space in which the process is modelled is disaggregated into a set of finite elements which are usually assumed to be internally homogeneous. Two-dimensional models generally use as their data small pixels (see Fischer et al., 1996).

In a recent paper Getis (1990), I develop a rationale for filtering spatially dependent variables into spatially independent variables and demonstrate a technique for changing one to the other. In that paper, the transformation is a multi-step procedure based on Ripley’s second order statistic (1981). In this chapter, I will briefly review the argument for the filtering procedure and propose a simplified method based on a spatial statistic developed by Getis and Ord (1992). The chapter is divided into four parts: (1) a short discussion of the rationale for filtering spatially dependent variables into spatially independent variables, (2) a review of a Getis–Ord statistic, (3) an outline of the filtering procedure, and (4) three examples taken from the literature on urban crime, regional inequality, and government expenditures.

In this paper structural elements are identified for the preparation of GIS-based data for use in econometric or statistical modeling. These elements include the need to know about the special characteristics of spatial data, such as map scale, spatial dependence, spatial variance heterogeneity and spatial trend heterogeneity, and the usual problems faced by modelers, such as nonspherical disturbances, stationarity of data, heteroscedasticity, and temporally and spatially autocorrelated disturbances. Detective work proceeds on the basis of the varying structures implied by the cross product statistic. These include measures of spatial differences, covariance, and interaction, and the exploratory data analysis functions included in the S-Plus statistical package.

Spacing analysis is introduced as a supplement or alternative to the traditional density gradient analysis for urban areas. Weaknesses in the use of the natural loglinear model of density are outlined. The spacing analysis is based on a second-order approach to spatial statistics. As an example, the population of Des Moines for the period 1960-1980 is analyzed. Results reveal that spacing analysis avoids many of the problems inherent in traditional density gradient analysis.

A causal model is developed that links the effects changes in employment and unemployment have on response variables that represent the economic health of 184 municipalities in the Chicago SMSA for 1980. Two causal series of paths are particularly helpful in identifying economic health. One represents the current level of health; the other, more relevant, path represents the trend during the period 1970-1980. By use of an index based on the path coefficients the municipalities are ranked with regard to level and trend. A second-order type of analysis is employed to identify spatial autocorrelation among the communities with regard to the two series. Results indicate that declining communities with regard to the two series. Results indicate that declining communities are more strongly associated spatially than are communities that are in relatively good economic health.