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.     

The Assessment of Spatial Autocorrelation through Constrained Multiple Regression

Most published measures of spatial autocorrelation (SA) can be recast as a (normalized) cross-product statistic that indexes the degree of relationship between corresponding entries from two matrices—one specifying the spatial connections among a set of n locations, and the other reflecting a very explicit definition of similarity between the set of values on some variable x realized over the n locations. We first give a very brief sketch of the basic cross-product approach to the evaluation of SA, and then generalize this strategy to include less restrictive specifications for the notion of similarity between the values on x. Using constrained multiple regression, the characterization of variate similarity basic to any assessment of SA can itself be framed according to the information present in the measure of spatial separation. These extensions obviate the inherent arbitrariness in how SA is usually evaluated, which now results from the requirement of a very restrictive definition of variate similarity before a cross-product index can be obtained.     

Testing for Regional Differences in Means: Distinguishing Inherent from Spurious Spatial Autocorrelation by Restricted Randomization

Tests for differences among regional means are typically carried out by analysis of variance (ANOVA). When such data are spatially autocorrelated (SA), the assumptions of ANOVA are not met, giving rise to excessive type I error rates. Two spatially adjusted ANOVA methods, Griffith's and COCOPAN, have been proposed to overcome this problem. In this study we show, by means of extensive simulations, the magnitude of the error rates introduced by SA induced in isolation-by-distance models typical of those used in population genetics. For data suspected of exhibiting such SA, we propose a strategy for distinguishing between inherent SA, generated within the data by a contagious process, and spurious SA, introduced by regional differences in means. The approach adopted is that of restricted randomization of distance matrices. We also furnish error rates and power estimates for both Griffith's method and COCOPAN. In addition to the simulated data, the methods are applied to an actual example from plant population biology.     

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.     

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.     

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.     

"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.     

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.     

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.     

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.     

Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity

Spatial nonstationarity is a condition in which a simple “global” model cannot explain the relationships between some sets of variables. The nature of the model must alter over space to reflect the structure within the data. In this paper, a technique is developed, termed geographically weighted regression, which attempts to capture this variation by calibrating a multiple regression model which allows different relationships to exist at different points in space. This technique is loosely based on kernel regression. The method itself is introduced and related issues such as the choice of a spatial weighting function are discussed. Following this, a series of related statistical tests are considered which can be described generally as tests for spatial nonstationarity. Using Monte Carlo methods, techniques are proposed for investigating the null hypothesis that the data may be described by a global model rather than a non-stationary one and also for testing whether individual regression coefficients are stable over geographic space. These techniques are demonstrated on a data set from the 1991 U.K. census relating car ownership rates to social class and male unemployment. The paper concludes by discussing ways in which the technique can be extended.     

Properties of Tests for Spatial Dependence in Linear Regression Models

Based on a large number of Monte Carlo simulation experiments on a regular lattice, we compare the properties of Moran's I and Lagrange multiplier tests for spatial dependence, that is, for both spatial error autocorrelation and for a spatially lagged dependent variable. We consider both bias and power of the tests for six sample sizes, ranging from twenty-five to 225 observations, for different structures of the spatial weights matrix, for several underlying error distributions, for misspecified weights matrices, and for the situation where boundary effects are present. The results provide an indication of the sample sizes for which the asymptotic properties of the tests can be considered to hold. They also illustrate the power of the Lagrange multiplier tests to distinguish between substantive spatial dependence (spatial lag) and spatial dependence as a nuisance (error autocorrelation).     

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.     

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.