Scalable GWR: A Linear-Time Algorithm for Large-Scale Geographically Weighted Regression with Polynomial Kernels

Although a number of studies have developed fast geographically weighted regression (GWR) algorithms for large samples, none of them has achieved linear-time estimation, which is considered a requisite for big data analysis in machine learning, geostatistics, and related domains. Against this backdrop, this study proposes a scalable GWR (ScaGWR) for large data sets. The key improvement is the calibration of the model through a precompression of the matrices and vectors whose size depends on the sample size, prior to the leave-one-out cross-validation, which is the heaviest computational step in conventional GWR. This precompression allows us to run the proposed GWR extension so that its computation time increases linearly with the sample size. With this improvement, the ScaGWR can be calibrated with 1 million observations without parallelization. Moreover, the ScaGWR estimator can be regarded as an empirical Bayesian estimator that is more stable than the conventional GWR estimator. We compare the ScaGWR with the conventional GWR in terms of estimation accuracy and computational efficiency using a Monte Carlo simulation. Then, we apply these methods to a U.S. income analysis. The code for ScaGWR is available in the R package scgwr. The code is embedded into C++ code and implemented in another R package, GWmodel.