Spatial Smoothing Using Graph Laplacian Penalized Filter

Abstract

This paper considers a filter for smoothing spatial data. It can be used to smooth data on the vertices of arbitrary undirected graphs with arbitrary non-negative spatial weights. It consists of a quantity analogous to Geary’s $c$, which is one of the most prominent measures of spatial autocorrelation. In addition, the quantity can be represented by a matrix called the graph Laplacian in spectral graph theory. We show mathematically how spatial data becomes smoother as a parameter, called the smoothing parameter, increases from 0 and is fully smoothed as the parameter goes to infinity, except for the case where the spatial data is originally fully smoothed. We also illustrate the results numerically and apply the spatial filter to climatological/meteorological data. In addition, as supplementary investigations, we examine how the sum of squared residuals and the effective degrees of freedom vary with the smoothing parameter. Finally, we review two closely related literatures to the spatial filter. One is the intrinsic conditional autoregressive model and the other is the eigenvector spatial filter. We clarify how the spatial filter considered in this paper relates to them. We then mention future research.