Multiscale Geographically Weighted Regression (MGWR)

Scale is a fundamental geographic concept, and a substantial literature exists discussing the various roles that scale plays in different geographical contexts. Relatively little work exists, though, that provides a means of measuring the geographic scale over which different processes operate. Here we demonstrate how geographically weighted regression (GWR) can be adapted to provide such measures. GWR explores the potential spatial nonstationarity of relationships and provides a measure of the spatial scale at which processes operate through the determination of an optimal bandwidth. Classical GWR assumes that all of the processes being modeled operate at the same spatial scale, however. The work here relaxes this assumption by allowing different processes to operate at different spatial scales. This is achieved by deriving an optimal bandwidth vector in which each element indicates the spatial scale at which a particular process takes place. This new version of GWR is termed multiscale geographically weighted regression (MGWR), which is similar in intent to Bayesian nonseparable spatially varying coefficients (SVC) models, although potentially providing a more flexible and scalable framework in which to examine multiscale processes. Model calibration and bandwidth vector selection in MGWR are conducted using a back-fitting algorithm. We compare the performance of GWR and MGWR by applying both frameworks to two simulated data sets with known properties and to an empirical data set on Irish famine. Results indicate that MGWR not only is superior in replicating parameter surfaces with different levels of spatial heterogeneity but provides valuable information on the scale at which different processes operate.