Gradient-based optimization for multi-scale geographically weighted regression

Abstract

Multi-scale geographically weighted regression (MGWR) is among the most popular methods to analyze non-stationary spatial relationships. However, the current model calibration algorithm is computationally intensive: its runtime has a cubic growth with the sample size, while its memory use grows quadratically. We propose calibrating MGWR with gradient-based optimization. This is obtained by analytically deriving the gradient vector and the Hessian matrix of the corrected Akaike information criterion (AICc) and wrapping them with a trust-region optimization algorithm. We evaluate the model quality empirically. Our method converges to the same coefficients and produces the same inference as the current method but it has a substantial computational gain when the sample size is large. It reduces the runtime to quadratic convergence and makes the memory use linear with respect to sample size. Our new algorithm outperforms the existing alternatives and makes MGWR feasible for large spatial datasets.