"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.     

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.     

Spatial Autocorrelation Statistics of Areal Prevalence Rates under High Uncertainty in Denominator Data

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.     

The Size-adjusted Critical Region of Moran's I Test Statistics for Spatial Autocorrelation and Its Application to Geographical Areas

In this article, we explore the expression of the asymptotic approximation of the distribution function of Moran's I test statistic for the check of spatial autocorrelation, and we derive a more accurate critical value, which gives the rejection region similar to significant level α to the order of N^?1 (N = sample size). We show that in some cases our procedure effectively finds the significance of positive spatial autocorrelation in the problem testing for the lack of the spatial autocorrelation. Compared with our method, the testing procedure of Cliff and Ord (1971) is clearly ad hoc and should not be applied blindly, as they pointed out. Our procedure is derived from the theory of asymptotic expansion. We numerically analyze four types of county systems with rectangular lattices and three regional areas with irregular lattices.     

Map Resolution and Spatial Autocorrelation

Moran's I, a measure of spatial autocorrelation, is affected by map resolution and map scale. This study uses a geographic information system (GIS) to examine the resolution effects. Empirical distribution of wildland fires in Idyllwild, California, and hypothetical distributions of ordered patterns are analyzed. The results indicate that Moran's I increases systematically with the resolution level. The resolution effects can be summarized by a log-linear function relating the I coefficients to resolution levels. Empirical tests that compare the distribution of fire activity in a vegetation map and in a topographic map confirm the resolution effects observed.     

Spatial Autocorrelation and Impacts on Criminology

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.     

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 History of the Concept of Spatial Autocorrelation: A Geographer's Perspective

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.     

Spatial Autocorrelation in Ecological Studies: A Legacy of Solutions and Myths

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.     

Applying Measures of Spatial Autocorrelation: Computation and Simulation

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.     

A Structural Equation Approach to Models with Spatial Dependence

We introduce the class of structural equation models (SEMs) and corresponding estimation procedures into a spatial dependence framework. SEM allows both latent and observed variables within one and the same (causal) model. Compared with models with observed variables only, this feature makes it possible to obtain a closer correspondence between theory and empirics, to explicitly account for measurement errors, and to reduce multicollinearity. We extend the standard SEM maximum likelihood estimator to allow for spatial dependence and propose easily accessible SEM software like LISREL 8 and Mx. We present an illustration based on Anselin's Columbus, OH, crime data set. Furthermore, we combine the spatial lag model with the latent multiple-indicators–multiple-causes model and discuss estimation of this latent spatial lag model. We present an illustration based on the Anselin crime data set again.     

The Bearing Correlogram: A New Method of Analyzing Directional Spatial Autocorrelation

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.     

A Generalized Randomization Approach to Local Measures of Spatial Association

This article establishes a unified randomization significance testing framework upon which various local measures of spatial association are commonly predicated. The generalized randomization approach presented is composed of two testing procedures, the extended Mantel test and the generalized vector randomization test. These two procedures employ different randomization assumptions, namely total and conditional randomization, according to the way in which they incorporate local measures. By properly specifying necessary matrices and vectors for a particular local measure of spatial association under a particular randomization assumption, the generalized randomization approach as a whole yields a reliable set of equations for expected values and variances, which then is confirmed by a Monte Carlo simulation utilizing random permutations.     

Impacts of Spatial Autocorrelation in Georeferenced Beta and Multinomial Random Variables

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.     

Exploring Relationships Between the Global and Regional Measures of Spatial Autocorrelation

Abstract The objective of this research is to investigate dimensions of geographic variation in spatial dependency contained within large multilevel data sets. We calculate 1990 population density by census block group, county, and state for the 48 coterminous states and the District of Columbia of the United States, calculations of interest to a wide variety of spatial scientists. We explore relations between these levels and their variation across the nation. The empirical findings generated by this work furnish implications concerning the Modifiable Areal Unit Problem (MAUP), spatial autocorrelation statistics, scale effects, and resolution.