Improvements to the calibration of a Geographically Weighted Regression with Parameter-Specific Distance Metrics and Bandwidths

In standard geographically weighted regression (GWR), the spatially-varying relationships between the dependent and each independent variable are explored under a constant and fixed scale, but for many processes their variation intensity may differ with respect to location and direction. To address this short-coming, a GWR model with parameter-specific distance metrics (PSDM GWR) can be used, which by default, also specifies parameter specific bandwidths. In doing so, PSDM GWR provides a scale-dependent extension of GWR. Commonly however, an ideal distance metric for a given independent variable parameter is not immediately obvious. Thus, in this article, PSDM GWR is investigated with respect to distance metric choice. Here, it is demonstrated that the optimum (distance metric specific) bandwidth corresponding to a given independent variable remains essentially constant, independent of the choices made for the other independent variables. This result allows for a considerable saving in computational overheads, permitting a much simpler searching procedure for multiple bandwidth optimization. Results are first demonstrated empirically, and then a simulation experiment is conducted to objectively verify the same findings. Computational savings are vital to the uptake of PSDM GWR, where ultimately, it should be considered the default choice in any GWR-based study of spatially-varying relationships, as standard GWR, mixed (or semi-parametric) GWR, flexible bandwidth (or multi-scale) GWR and the global regression are specific cases thereof.