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.     

"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.     

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.     

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.     

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.     

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 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.     

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 and the Classic Maya Collapse: One Technique, One Conclusion

A pair of articles appearing recently in this journal (Whitley & Clark, Journal of Archaeological Science12, 377-395, 1985; Kvamme, Journal of Archaeological Science17, 197-207, 1990) apply spatial autocorrelation analysis to the distribution of terminal long-count dates from southern Lowland Classic Maya monuments. The authors employ similar techniques yet arrive at contradictory conclusions regarding the presence of geographical patterning in the collapse of the Classic Maya civilization in this region. Kvamme's contention, however, that Whitley & Clark conducted an inappropriate analysis and arrived at an erroneous conclusion is unsubstantiated. Both articles present appropriate analyses and report results which support the presence of spatial patterning in the Lowland Maya dates.     

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.     

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.     

华人网络的再认识：一个空间的视角

现有关于华人跨国网络的研究,为我们描绘了华人网络的三张面孔,即作为“文化载体”、“社会纽带”和“经济圈”的华人网络。本文从空间的角度,对以上研究进行了补充。它以香港及其邻近的东南亚地区的华人移民经验为依据,在检讨了滨下武志、王赓武和和斯金纳等人有关中国社会经济地理、文化中心和市场结构理论的基础上,提出了一个东南亚华人网络空间结构的理论模型。该模型以国际大都会为中心,形成了“去中心”、“多层次”和“多维度”的钻石型网络结构。在该模型中,华人网络的经济、社会和文化的面向获得了互动和统一。若干类似的子网络相互联结,共同构成了全球华人网络的宏观地域系统。     

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.     

The Hand-Pump Flotation System: A New Method for Archaeobotanical Recovery

Abstract

At the sites of Acquachiara in Italy, el-Hemmeh in Jordan, and Beisamoun in Israel, archaeobotanical recovery was accomplished using flotation systems powered by hand-pumps, as opposed to non-circulating systems or systems powered by electricity or gasoline. These flotation tanks were simple and inexpensive to construct, easily operated by a single individual, highly durable, and easily transported. The speed of water flow was both ample and could be controlled without difficulty. These systems had the capacity to process samples at a rate sufficient to keep pace with large-scale sampling programs at all three sites. In three case studies, we describe two tanks that recycle water and one that does not, list the parts and procedures necessary to construct a hand-pump system, and detail the results of a recovery test that demonstrates the efficacy of this technique. The type of flotation tank described would be appropriate for a range of research projects with various resource constraints, enabling substantial paleoethnobotanical data collection with only a minimal initial investment of time and money.