Spatial Statistics

by

Daniel A. Griffith, Yongwan Chun

• LAST REVIEWED: 26 November 2019
• LAST MODIFIED: 26 November 2019
• DOI: 10.1093/obo/9780199363445-0125

Introduction

Spatial statistics (SS)—statistics that address and account for the correlations among georeferenced observations arising from their relative locations in geographic space (i.e., spatial autocorrelation [SA])—has a formal history dating to the mid-1900s, although conceptual awareness of it dates back to the very early 1900s. It is a special case of correlated data. It arises from a relaxation of the independent observations assumption of classical statistics. Its development emphasized the following three themes: point pattern analysis, spatial autoregression, and geostatistics. Point pattern analysis was a precursor to spatial autoregression, whereas these latter two themes evolved in parallel, with little cross-fertilization during most of their first 40 years of development, primarily because spatial autoregression was the preferred interest of English-speaking scholars, whereas geostatistics was the preferred interest of French-speaking scholars. Much of the early work treated point patterns; spatial autoregression and geostatistics began eclipsing this emphasis around 1990. SS is best understood after completion of a course in multivariate statistical analysis. Spatial Autocorrelation, the book by Cliff and Ord in 1973 initiated a popularizing of SS in the early 1970s; later, Spatial Econometrics, by Paelinck and Klaassen in 1979, extended it to spatial econometrics. The important message is that accounting for SA in georeferenced data really matters in the environmental sciences.

General Overviews

A map is a critical component of spatial statistics (SS): attribute data need to be tagged to, usually, a two-dimensional surface (i.e., the surface of the Earth). Accordingly, an important aspect of SS is the set of points or polygons constituting observations that house attribute values. This tagging is a necessary, but not a sufficient, data feature for SS. A spatial statistical measure has to vary as the relative locations of points/polygons change. Consequently, SS must incorporate arrangement properties of points/polygons, such as the proximity of (or distance between, connectivity among) these observations, the area and/or shape of polygons, the length of lines, the orientation of point/attribute value alignments, and the centrality/peripheralness of observations, among other things. The Esri’s GIS Dictionary (Redlands, CA: Esri, 2005) comments that a spatial statistic uses some of these properties in its arithmetic calculations most often directed at pattern and/or shape analysis, surface modeling and/or prediction, and spatial regression. Given the contemporary popularity of SS, surprisingly few comprehensive reader-friendly introductions to this field exist. Ripley 1981 covers the three fundamental SS themes; Ripley 1988, which addresses the question asking what is special about SS, covers aspects of spatial autoregression. Bailey and Gatrell 1995 discusses fundamental components and methodologies for point pattern analysis, geostatistical analysis, and area data analysis. Getis 1999 furnishes one of the first readable dedicated descriptions of SS. Gaetan and Guyon 2010 provides a more contemporary treatment of SS, one that includes discussion of simulation. Meanwhile, Gelfand, et al. 2010 provides a comprehensive examination of both classical and state-of-the-art SS topics. Although these sources supply a range of SS coverages, the single treatise that tends to be favored is Cressie 1993, titled Statistics for Spatial Data; this book is not an easy read without having completed selected master’s level coursework in statistics, but it is excellent (it now is contained in the Wiley Classics Library). Chun and Griffith 2013 is a useful primer for Cressie 1993. Meanwhile, natural extensions to SS include space-time data methodology (Cressie and Wikle 2011).

• Bailey, T. C., and A. C. Gatrell. 1995. Interactive spatial data analysis. Essex, UK: Pearson Education.

  This is one of the books that have been widely used for classroom teaching of spatial data analysis. It discusses fundamental concepts and technical details for point pattern analysis, geostatistical analysis, and area data analysis. Also, this book discusses some methods to model geographical flow data.

• Chiles, J.-P., and P. Delfiner. 2009. Geostatistics: Modeling spatial uncertainty. New York: John Wiley & Sons.

  This book provides a rich discussion about geostatistical models and methods, including variograms, kriging, and its advanced variations. Furthermore, it covers multivariate models, nonlinear models, and sampling issues in geostatistical analysis.

• Chun, Y., and D. Griffith. 2013. Spatial statistics and geostatistics. Thousand Oaks, CA: SAGE.

  Its goal is to allow a reader to obtain a practical understanding of SS.

• Cressie, N. 1993. Statistics for spatial data. New York: Wiley.

  DOI: 10.1002/9781119115151

  A Wiley Classics Library book that establishes a general spatial model that unifies geostatistics, spatial autoregression, point pattern analysis, and random sets. Chapter 6 presents the derivations of spatial models, including the conditional and simultaneous autoregressive Gaussian models, as well as non-Gaussian spatial models. It also presents inferential statistics for these models in chapter 7.

• Cressie, N., and C. Wikle. 2011. Statistics for spatio-temporal data. New York: Wiley.

  A sequel to Cressie 1993. This book indicates the next step in spatial data analysis, namely, space-time data analysis.

• Gaetan, C., and X. Guyon. 2010. Spatial statistics and models. New York: Springer.

  DOI: 10.1007/978-0-387-92257-7

  A treatment of geostatistics in terms of second-order spatial models, spatial dependency on networks, point processes, geography-based simulation, and the primary statistical methods used to estimate spatial model parameters and their uncertainties.

• Gelfand, A., P. Diggle, P. Guttorp, and M. Fuentes, eds. 2010. Handbook of spatial statistics. Boca Raton, FL: Chapman & Hall.

  A description of the evolution of SS, followed by an overview of the three fundamental themes of SS supplemented with a section about space-time data.

• Getis, A. 1999. Spatial statistics. In Geographical information systems. Vol. 1. 2d ed. Edited by Longley M. Goodchild, D. Maguire, and D. Rhind, 239–251. New York: Wiley.

  A narrative focusing on Tobler’s first law of geography: everything is related to everything else, but nearby phenomena are more strongly related than distant phenomena. Getis highlights that SS involves spatial association, pattern analysis, scale and zoning, geostatistics, classification, spatial sampling, and the spatial weights matrix.

• Haining, R. P. 2003. Spatial data analysis: Theory and practice. Cambridge, UK: Cambridge University Press.

  DOI: 10.1017/CBO9780511754944

  This book provides ample discussion about spatial data analysis. It begins with the context for spatial data analysis, including spatial processes and the nature of spatial data. It discusses spatial data collection and quality and covers exploratory spatial data analysis and statistical modeling methods.

• Ripley, B. 1981. Spatial statistics. New York: Wiley.

  DOI: 10.1002/0471725218

  The time-dependent purpose of this book was to furnish a comprehensive guide to SS. Chapters cover the three fundamental themes of SS, with about half of the book devoted to point pattern analysis topics.

• Ripley, B. 1988. Statistical Inference for Spatial Processes. Cambridge, UK: Cambridge University Press.

  The purpose of this book was to outline spatial statistical inference theory while furnishing some new insights into classical statistics.