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Although geographically weighted Poisson regression (GWPR) is a popular regression for spatially indexed count data, its development is relatively limited compared to that found for linear geographically weighted regression (GWR), where many extensions (e.g., multiscale GWR, scalable GWR) have been proposed. The weak development of GWPR can be attributed to the computational cost and identification problem in the underpinning Poisson regression model. This study proposes linearized GWPR (L-GWPR) by introducing a log-linear approximation into the GWPR model to overcome these bottlenecks. Because the L-GWPR model is identical to the Gaussian GWR model, it is free from the identification problem, easily implemented, computationally efficient, and offers similar potential for extension. Specifically, L-GWPR does not require a double-loop algorithm, which makes GWPR slow for large samples. Furthermore, we extended L-GWPR by introducing ridge regularization to enhance its stability (regularized L-GWPR). The results of the Monte Carlo experiments confirmed that regularized L-GWPR estimates local coefficients accurately and computationally efficiently. Finally, we compared GWPR and regularized L-GWPR through a crime analysis in Tokyo.

GWR is a popular approach for investigating the spatial variation in relationships between response and predictor variables, and critically for investigating and understanding process spatial heterogeneity. The geographically weighted (GW) framework is increasingly used to accommodate different types of models and analyses reflecting a wider desire to explore spatial variation in model parameters or components. However the growth in the use of GWR and different GW models has only been partially supported by package development in both R and Python, the major coding environments for spatial analysis. The result is that refinements have been inconsistently included (if at all) within GWR and GW functions in any given package. This paper outlines the structure of a new `gwverse` package, that will over time replace `GWmodel`, that takes advantage of recent developments in the composition of complex, integrated packages. It conceptualises `gwverse` as having a modular structure, that separates core GW functionality and applications such as GWR. It adopts a function factory approach, in which bespoke functions are created and returned to the user based on user-defined parameters. The paper introduces two demonstrator modules that can be used to undertake GWR and identifies a number of key considerations and next steps.

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 global one. Standard GWR assumes that the 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 inform the choice of 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 primary steps: a basic linear regression, a MS-GWR, and investigations of the results of these. The paper provides guidance for deciding whether to use a GWR approach, and if so for determining the appropriate GWR variant. It describes 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, and further considerations are described in an extensive Appendix., Comment: 34 pages, 8 figures

For a long time, there have been huge discrepancies between different models and experiments concerning the liquid-liquid phase transition (LLPT) in dense hydrogen. In this work, we present the results of extensive calculations of the LLPT in dense hydrogen using the most expensive first-principle path-integral molecular dynamics simulations available. The nonlocal density functional rVV10 and hybrid functional PBE0 are used to improve the description of the electronic structure of hydrogen. Of all the density functional theory calculations available, we report the most consistent results through quantum Monte Carlo simulations and coupled electron-ion Monte Carlo simulations of the LLPT in dense hydrogen. The critical point of the first-order LLPT is estimated above 2000 K according to the equation of state. Moreover, the metallization pressure obtained from the jump of dc electrical conductivity almost coincides with the plateau of equation of state.

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.

While 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: (i) geographically weighted regression (GWR) with a fixed distance or (ii) an adaptive distance bandwidth (GWRa), (iii) flexible bandwidth GWR (FB-GWR) with fixed distance or (iv) adaptive distance bandwidths (FB-GWRa), (v) eigenvector spatial filtering (ESF), and (vi) 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 naively 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).

In this study, we present a collection of local models, termed geographically weighted (GW) models, that can be found within the GWmodel R package. A GW model suits situations when spatial data are poorly described by the global form, and for some regions the localised fit provides a better description. The approach uses a moving window weighting technique, where a collection of local models are estimated at target locations. Commonly, model parameters or outputs are mapped so that the nature of spatial heterogeneity can be explored and assessed. In particular, we present case studies using: (i) GW summary statistics and a GW principal components analysis; (ii) advanced GW regression fits and diagnostics; (iii) associated Monte Carlo significance tests for non-stationarity; (iv) a GW discriminant analysis; and (v) enhanced kernel bandwidth selection procedures. General Election data sets from the Republic of Ireland and US are used for demonstration. This study is designed to complement a companion GWmodel study, which focuses on basic and robust GW models.