The Spatial Weights Matrix and ESF

Abstract

A prominent feature differentiating between conventional and spatial statistics/econometrics is that the configuration of observations impacts results (i.e., their ordering is important). Accordingly, a spatial analysis needs to include some quantified version of this configuration, which has become known as the spatial weights matrix. In its simplest form, this matrix is topologically based and constructed as an n-by-n table of binary 0–1 indicator variables, one for each location on a map: a cell entry of 1 denotes that the table row and column locations are neighbors (or adjacent), and 0 denotes otherwise. One very common meaning of neighbor—the rook’s definition, based on an analogy with chess—is that two areal units in a surface partitioning share a common non-zero-length border. Sometimes the meaning is that neighbors share both zero- and non-zero-length borders—the queen’s definition. Other definitions involve a certain number of nearest neighbors, relative inter-centroid distance, and dividing ones by their corresponding row totals (i.e., row standardization). This chapter presents analytical comparisons of these different definitions of a spatial weights matrix, with special reference to those definitions preferred for spatial autoregression and eigenvector spatial filtering.