Statistical Methods: Spatial Statistics

Spatial statistics, also termed as spatial analysis or spatial data analysis, refers to a branch of statistics that aims to study entities based on their topological, geometric, or geographic properties. Among the many topics of spatial statistics, we focus on metrics of representing spatial patterns as well as spatial regression methods. The metrics we often use include (local) Moran's I, Geary's C, and Getis' G statistics that help us quantify the extent of spatial autocorrelation at local or global levels (e.g., of regression residuals—see our LTM-ESF). We also heavily rely on the so many landscape pattern metrics that are available in Fragstats software.

Among spatial regression methods, we use more (and teach) geographically weighted regression (GWR) (Brunsdon et al. 1998, Fotheringham et al. 2002) to study the landscape or human-environment processes that vary over space. GWR assumes no spatial stationarity, put another way, the process under investigation are NOT constant over space. Rather, the process is local context dependent: At each location (usually characterized by its x and y coordinates), the coefficient(s) for one or more than one independent variable is unique and dependent on the location. Such spatial variation could arise from many reasons such as lack of some important predictor variable(s). One key concept is to select a spatial kernel (i.e., weights assigned to other locations centered on the location of interest): if too broad, then the GWR degrades to a global model where all points have the same regression coefficients across the space; if too narrow, then the GWR bears too much variability.

We use GWR for explanatory or exploratory purpose more than for predictive purpose although both usages are meaningful from a pure methodological perspective. The software we use is the Geographically Weighted Regression package developed by the authors or the one available at ArcGIS under Spatial Statistics Tools. Other than GWR, we use eigenvector spatial filtering (ESF) and multilevel modeling (MLM) to handle issues arising from spatial autocorrelation.

Readings and References:

Brunsdon, C., A.S. Fotheringham, and M. Charlton (1998). Geographically Weighted Regression. Journal of the Royal Statistical Society: Series D (The Statistician) 47(3): 431–443.

Fotheringham, A.S., C. Brunsdon, and M. Charlton (2002). Geographically Weighted Regression. John Wiley & Sons: West Sussex, UK.