15.2 Non-Parametric Approaches

An alternative to the spatial weights and cross-product formulation of a spatial autocorrelation statistic is to establish the connection between attribute (dis)similarity and locational similarity in a different way (Section 13.4.2). This is implemented by expressing a measure of attribute (dis)similarity directly as a function of the distance that separates pairs of observations: \[f(x_i,x_j) = g(d_{ij}),\] where \(g\) is a function that expresses the distance decay in the strength of association between the values observed at a pair of locations \(i, j\).

In this chapter, two approaches are outlined that let the function \(g\) be specified in a non-parametric way, directly from the data. This contrasts with the standard practice in geo-statistics, where a specific function (semi-variogram) is fit. The spatial correlogram approach uses a cross-product measure of attribute similarity, whereas the smoothed distance scatter plot is based on a squared difference dissimilarity.