  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 datasets. The key
improvement is the calibration of the model through a pre-compression 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 pre-compression 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 one 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 US 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.
