Speeding up estimation of spatially varying coefficients models

Spatially varying coefficients models, like GWR (Brunsdon et al., 1996; McMillen, 1996), are widely used in various application domains for which it may be of interest to consider the spatial heterogeneity of the coefficient?s model (house market, land use, population ecology, seis- mology, mining research, ...). In most application areas and disciplines, the continuous increase in spatial data sample sizes, both in terms of volume and richness of explanatory variables, has raised new method- ological issues. The two main issues concern here the time required to calculate each local coefficients and the memory requirements imposed for storing the hat matrix of size n × n for estimating the variance of parameters. To answer to these two issues, various avenues have been explored (Harris et al., 2010; Pozdnoukhov and Kaiser, 2011; Tran et al., 2016; Geniaux and Martinetti, 2018; Li et al., 2019; Murakami et al., 2020). The use of a subset of target points for local regressions which dates from the first explorations of weighted local regressions (Cleveland and Devlin, 1988; Loader, 1999) has been widely studied in the field of nonparametric econometrics. It has been little explored for varying coefficient models, and in particular for GWR where a single 2D ker- nel is used for all the coefficients. McMillen (2012) and McMillen and Soppelsa (2015) used a direct transposition of the Loader 1999' pro- posal in the context of the GWR for selecting target points. In this paper, we propose an original two-stage method. We select a set of tar- get points based on spatial smooth of residuals of a first stage regression to perform a GWR only on this subsample. In a second stage we use spatial smoothing method for extrapolating the remaining GWR coef- ficients. In addition to using an effective sample of target points, we explore the computational gain provided by using rough gaussian ker- nel. Monte Carlo experiments show that this way of selecting target points outperforms selection based on density of points or with ran- dom selection. Simulation results also show that using target points can even reduce the Bias and the RMSE of β coefficients compared to classic GWR by allowing to select more accurate bandwidth size. Our best estimator appears to be scalable under two conditions: the use of a ratio of target points that provides satisfactory approxima- tion of coefficients (10 to 20 % of locations) and an optimal bandwidth that remains within a reasonable neighborhood (