Geographically Weighted Regression (GWR) is increasingly used in spatial analyses of social and environmental data. It allows spatial heterogeneities in processes and relationships to be investigated through a series of local regression models rather than a single global one. Standard GWR assumes that relationships between the response and predictor variables operate at the same spatial scale, which is frequently not the case. To address this, several GWR variants have been proposed. This paper describes a route map to decide whether to use a GWR model or not, and if so which of three core variants to apply: a standard GWR, a mixed GWR or a multiscale GWR (MS‐GWR). The route map comprises 3 primary steps that should always be undertaken: (1) a basic linear regression, (2) a MS‐GWR, and (3) investigations of the results of these in order to decide whether to use a GWR approach, and if so for determining the appropriate GWR variant. The paper also highlights the importance of investigating a number of secondary issues at global and local scales including collinearity, the influence of outliers, and dependent error terms. Code and data for the case study used to illustrate the route map are provided.

This study downscales the population and gross domestic product (GDP) scenarios given under Shared Socioeconomic Pathways (SSPs) into 0.5-degree grids. Our downscale approach has the following features. (i) It explicitly considers spatial and socioeconomic interactions among cities, (ii) it utilizes auxiliary variables, including road network and land cover, (iii) it endogenously estimates the influence from each factor by a model ensemble approach, and (iv) it allows us to control urban shrinkage/dispersion depending on SSPs. It is confirmed that our downscaling results are consistent with scenario assumptions (e.g., concentration in SSP1 and dispersion in SSP3). Besides, while existing grid-level scenarios tend to have overly-smoothed population distributions in nonurban areas, ours does not suffer from the problem, and captures the difference in urban and nonurban areas in a more reasonable manner. Our gridded dataset, including population counts and gross productivities by 0.5 degree grids by 10 years, are available from http://www.cger.nies.go.jp/gcp/population-and-gdp.html.

Historical and future spatially explicit population and gross domestic product (GDP) data are essential for the analysis of future climate risks. Unlike population projections that are generally available, GDP projections—particularly for scenarios compatible with shared socioeconomic pathways (SSPs)—are limited. Our objective is to perform a high-resolution and long-term GDP estimation under SSPs utilizing a wide variety of geographic auxiliary information. We estimated the GDP in a 1/12-degree grid scale. The estimation is done through downscaling of historical GDP data for 1850–2010 and SSP future scenario data for 2010–2100. In the downscaling, we first modeled the spatial and economic interactions among cities and projected different future urban growth patterns according to the SSPs. Subsequently, the projected patterns and other auxiliary geographic data were used to estimate the gridded GDP distributions. Finally, the GDP projections were visualized via three-dimensional mapping to enhance the clarity for multiple stakeholders. Our results suggest that the spatial pattern of urban and peri-urban GDP depends considerably on the SSPs; the GDP of the existing major cities grew rapidly under SSP1, moderately grew under SSP 2 and SSP4, slowly grew under SSP3, and dispersed growth under SSP5.

Although spatially varying coefficient (SVC) models have attracted considerable attention in applied science, they have been criticized as being unstable. The objective of this study is to show that capturing the “spatial scale” of each data relationship is crucially important to make SVC modeling more stable and, in doing so, adds flexibility. Here, the analytical properties of six SVC models are summarized in terms of their characterization of scale. Models are examined through a series of Monte Carlo simulation experiments to assess the extent to which spatial scale influences model stability and the accuracy of their SVC estimates. The following models are studied: (1) geographically weighted regression (GWR) with a fixed distance or (2) an adaptive distance bandwidth (GWRa); (3) flexible bandwidth GWR (FB-GWR) with fixed distance or (4) adaptive distance bandwidths (FB-GWRa); (5) eigenvector spatial filtering (ESF); and (6) random effects ESF (RE-ESF). Results reveal that the SVC models designed to capture scale dependencies in local relationships (FB-GWR, FB-GWRa, and RE-ESF) most accurately estimate the simulated SVCs, where RE-ESF is the most computationally efficient. Conversely, GWR and ESF, where SVC estimates are naïvely assumed to operate at the same spatial scale for each relationship, perform poorly. Results also confirm that the adaptive bandwidth GWR models (GWRa and FB-GWRa) are superior to their fixed bandwidth counterparts (GWR and FB-GWR). Key Words: flexible bandwidth geographically weighted regression, Monte Carlo simulation, nonstationarity, random effects eigenvector spatial filtering, spatial scale.

As an established spatial analytical tool, Geographically Weighted Regression (GWR) has been applied across a variety of disciplines. However, its usage can be challenging for large datasets, which are increasingly prevalent in today’s digital world. In this study, we propose two high-performance R solutions for GWR via Multi-core Parallel (MP) and Compute Unified Device Architecture (CUDA) techniques, respectively GWR-MP and GWR-CUDA. We compared GWR-MP and GWR-CUDA with three existing solutions available in Geographically Weighted Models (GWmodel), Multi-scale GWR (MGWR) and Fast GWR (FastGWR). Results showed that all five solutions perform differently across varying sample sizes, with no single solution a clear winner in terms of computational efficiency. Specifically, solutions given in GWmodel and MGWR provided acceptable computational costs for GWR studies with a relatively small sample size. For a large sample size, GWR-MP and FastGWR provided coherent solutions on a Personal Computer (PC) with a common multi-core configuration, GWR-MP provided more efficient computing capacity for each core or thread than FastGWR. For cases when the sample size was very large, and for these cases only, GWR-CUDA provided the most efficient solution, but should note its I/O cost with small samples. In summary, GWR-MP and GWR-CUDA provided complementary high-performance R solutions to existing ones, where for certain data-rich GWR studies, they should be preferred.