In This Article Spatial Autocorrelation

  • Introduction
  • General Overviews
  • Reference Resources
  • Journals
  • Spatial Statistics versus Spatial Econometrics
  • Mixed Models: Spatially Structured and Spatially Unstructured Random Effects

Geography Spatial Autocorrelation
by
Dan Griffith
  • LAST MODIFIED: 31 March 2016
  • DOI: 10.1093/obo/9780199874002-0128

Introduction

Spatial autocorrelation (SA)—the correlation among georeferenced observations arising from their relative locations in geographic space—has a history dating to the mid-1900s, although conceptual awareness of it dates back to the early 1900s. But SA is everywhere. It manifests itself in one- and two-dimensional synchronizations, exemplified by the experiment involving multiple metronomes sitting on a board that rests on two soda cans (illustrating an indirect, common factor SA source), or the aggregate flashing of fireflies created by their emission into the air of chemicals that stimulate nearby fireflies to light (illustrating a direct spatial interaction SA source). This latter outcome also can arise from mimicking behavior, as occurs with bandit bumble bees in a single meadow (i.e., when they rob a yellow rattler’s flower of nectar, their entry holes side in a given field tend to be unambiguously on only its left or right hand). The degree of organization in the geographic patterns that emerge signifies the level of positive SA. Such SA resulted in Heckscher discovering a new species of firefly in 2013. This type of SA is the basis of Nobel winner Schelling’s models of segregation, and The Economist (20 April 2013, p. 16) stating that regardless of class in Britain, geographical clusters of voters act like “political opinions derive from the air people breathe.” Moderate positive SA characterizes slider puzzles, magnetic sculpture toys, and television pictures. Meanwhile, negative SA relates to spatial patterns of competition. Although this nature of SA is rarely encountered in practice, it is illustrated by the Grand Prairie Independent School District’s (GPISD) attempts to increase the amount of money it receives from the state of Texas by holding annual events to attract students from surrounding school districts to attend its schools (Dallas Morning News, 9 January 2014); GPISD attempts to increase its enrollments by decreasing enrollments in its neighboring school districts. A timeline for the evolution of the SA concept helps establish its historically relevant literature. In the early 1800s, Laplace recognized autocorrelation—albeit serial for time series—by acknowledging that between day variations in barometric pressure readings tend to be much greater than within day readings. From 1914 to 1935, spatial series observational correlations were recognized by Student, then Yule, then both Stephan and Neprash, and then Fisher. This recognition set the stage for establishing the concept of SA. Moran and Geary did so in the early 1950s. In parallel, writing in French, Matheron and Krige also did so within the context of geostatistics. Next, more formal models of SA were formulated, first by Whittle, then by Mead, and finally by Cliff and Ord, whose numerous publications popularized the concept in the 1970s. One outcome of Cliff and Ord’s work was the coining of the phrase spatial econometrics in 1979 by Paelinck and Klaassen. Finally, as the century drew to a close, Griffith established the foundation of eigenvector spatial filtering, which extends SA analysis to the entire family of non-normal random variables.

General Overviews

Given the ubiquity of SA, surprisingly few comprehensive or reader-friendly introductions to this concept exist. The title of Cliff and Ord 1973 seemed like such a misnomer that they renamed its 1981 revised edition Spatial Processes (see Cliff and Ord 1981, cited under Statistical Distribution Theory for Global Spatial Autocorrelation Measures). Nevertheless, eight introductory surveys eventually appeared. Sensing a gap in the literature, three short monographs were published by disparate sources in the late 1980s: Goodchild 1986, Griffith 1987, and Odland 1988. The next publications to appear were more comprehensive conceptual articles, Griffith 1992 and Legendre 1993. Getis 2008 furnished a historical perspective about the concept. Griffith 2005 and Griffith 2009 are encyclopedia entries. As awareness of SA becomes more universal, papers describing it, such as that by Valcu and Kempenaers 2010, should appear across the scientific literature.

  • Cliff, A., and J. Ord. Spatial Autocorrelation. London: Pion, 1973.

    E-mail Citation »

    The original edition of Spatial Processes 1981.

  • Getis, A. “A History of the Concept of Spatial Autocorrelation: A Geographer’s Perspective.” Geographical Analysis 40 (2008): 297–309.

    DOI: 10.1111/j.1538-4632.2008.00727.xE-mail Citation »

    A historical overview of the concept of SA within the context of spatial analysis. This history is traced from the 1960s, with emphasis on geography and spatial econometrics. SA is cast as the central theme of spatial statistics and spatial econometrics. Some thoughts are offered about possible future SA-focused research.

  • Goodchild, M. Spatial Autocorrelation. Concepts and Techniques in Modern Geography 47. Norwich, UK: Geo Books, 1986.

    E-mail Citation »

    Introduces the concept of SA, focusing on the Moran Coefficient, Geary Ratio, and join counts. Differentiates between SA summary and test statistics, arguing that their confidence intervals usually are more helpful than hypothesis testing about them. Includes empirical examples, links between SA, spectral analysis, kriging, and computer code.

  • Griffith, D. Spatial Autocorrelation. Washington, DC: Association of American Geographers, 1987.

    E-mail Citation »

    Introduces the concept of SA, focusing on the Moran Coefficient, Geary Ratio, and join counts. Pays attention to the implications of autocorrelation for regression and analysis of variance, particularly consequences of autocorrelation in residuals. Presents examples of crime in Buffalo, New York, and computer code for simulating and calculating indices of SA.

  • Griffith, D. “What is Spatial Autocorrelation? Reflections on the Past 25 Years of Spatial Statistics.” L’Espace Géographique 21 (1992): 265–280.

    DOI: 10.3406/spgeo.1992.3091E-mail Citation »

    Reviews the nine fundamental interpretations of SA: a spatial process mechanism, a diagnostic tool, a nuisance parameter, a spatial spillover effect, an outcome of areal unit demarcation, redundant information, map pattern, a missing variables indicator/surrogate, and self-correlation. Each definition is spelled out and illustrated with a numerical or conceptual example.

  • Griffith, D. “Spatial Autocorrelation.” In Encyclopedia of Social Measurement. Vol. 3. Edited by K. Kempf-Leonard, 581–590. Amsterdam: Elsevier, 2005.

    E-mail Citation »

    Presents correlation as an interest of social scientists, with its special SA form an interest of social scientists dealing with georeferenced data. Addresses the importance of measuring and accounting for SA. Surveys impacts of SA on statistical estimator properties. Summarizes quantification of SA effects. And outlines common auto-models.

  • Griffith, D. “Methods: Spatial Autocorrelation.” In International Encyclopedia of Human Geography. Edited by R. Kitchin and N. Thrift, 396–402. New York: Elsevier, 2009.

    E-mail Citation »

    A more contemporary treatment of SA. It includes a basic definition and summary of the nine fundamental interpretations and empirical examples of this concept. Both positive and negative SA are treated, as are SA indices and visualization techniques as well as prominent statistical properties.

  • Legendre, P. “Spatial Autocorrelation: Trouble or New Paradigm?” Ecology 74 (1993): 1659–1673.

    DOI: 10.2307/1939924E-mail Citation »

    Includes description and measurement of SA in ecological variables, proper statistical testing in the presence of SA, and ways of explicitly introducing spatial structures into ecological models. Casts SA as a general statistical property of ecological variables observed across geographic space, most often tending to materialize as patches and gradients.

  • Odland, J. Spatial Autocorrelation. Thousand Oaks, CA: SAGE, 1988.

    E-mail Citation »

    Introduces the concept of SA, focusing on the Moran Coefficient, Geary Ratio, and join counts. Presents these statistics as special cases of general cross-product statistics for both categorical and continuous data. Treats construction of linear regression models for spatial data, and testing for, and elimination of, SA in regression residuals.

  • Valcu, M., and B. Kempenaers. “Spatial Autocorrelation: An Overlooked Concept in Behavioral Ecology.” Behavioral Ecology 21 (2010): 902–905.

    DOI: 10.1093/beheco/arq107E-mail Citation »

    Argues that behavioral ecologists could benefit from accounting for SA in their data.

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