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Geographically weighted regression (GWR) is a spatial statistical technique that recognizes that traditional 'global' regression models may be limited when spatial processes vary with spatial context. GWR captures process spatial heterogeneity by allowing effects to vary over space. To do this, GWR calibrates an ensemble of local linear models at any number of locations using 'borrowed' nearby data. This provides a surface of location-specific parameter estimates for each relationship in the model that is allowed to vary spatially, as well as a single bandwidth parameter that provides intuition about the geographic scale of the processes. A recent extension to this framework allows each relationship to vary according to a distinct spatial scale parameter, and is therefore known as multiscale (M)GWR. This paper introduces mgwr, a Python-based implementation of MGWR that explicitly focuses on the multiscale analysis of spatial heterogeneity. It provides novel functionality for inference and exploratory analysis of local spatial processes, new diagnostics unique to multi-scale local models, and drastic improvements to efficiency in estimation routines. We provide two case studies using mgwr, in addition to reviewing core concepts of local models. We present this in a literate programming style, providing an overview of the primary software functionality and demonstrations of suggested usage alongside the discussion of primary concepts and demonstration of the improvements made in mgwr.A recent extension to the GWR framework allows each relationship in the model to vary at a unique spatial scale and is therefore known as multiscale (M)GWR [3]. MGWR is much less restrictive in its assumptions than GWR, since the relationship between the response and a covariate is allowed to vary locally, vary regionally, and or not vary at all. Eliminating the restriction that all relationships vary at the same spatial scale can minimize over-fitting, reduce bias in the parameter estimates, and mitigate concurvity (i.e., collinearity due to similar functional transformations). Therefore, MGWR has been suggested as the default local model specification when using GWR to investigate process spatial heterogeneity and scale. This paper introduces mgwr (throughout this manuscript mgwr refers to the software implementation, while MGWR refers to the technique more generally), a Python-based software package for deploying GWR and MGWR models. Though there are existing software options, they are limited in terms of available functionality, computational efficiency, or both. For example, there is a GWR tool in the spatial analyst toolbox within ArcGIS [4] and there are several options within the R ecosystem, such as spgwr [5] and gwrr [6]. However, none of these implementations offers capabilities to calibrate an MGWR model nor the ability to compute the hat matrix (i.e., projection matrix) and the associated novel model diagnostics described in [7], which includes covariate-specific indicators of scale and inference framework. The R-based GWm...

A recent paper expands the well‐known geographically weighted regression (GWR) framework significantly by allowing the bandwidth or smoothing factor in GWR to be derived separately for each covariate in the model—a framework referred to as multiscale GWR (MGWR). However, one limitation of the MGWR framework is that, until now, no inference about the local parameter estimates was possible. Formally, the so‐called “hat matrix,” which projects the observed response vector into the predicted response vector, was available in GWR but not in MGWR. This paper addresses this limitation by reframing GWR as a Generalized Additive Model, extending this framework to MGWR and then deriving standard errors for the local parameters in MGWR. In addition, we also demonstrate how the effective number of parameters can be obtained for the overall fit of an MGWR model and for each of the covariates within the model. This statistic is essential for comparing model fit between MGWR, GWR, and traditional global models, as well as for adjusting multiple hypothesis tests. We demonstrate these advances to the MGWR framework with both a simulated data set and a real‐world data set and provide a link to new software for MGWR (MGWR1.0) which includes the novel inferential framework for MGWR described here.

Geographically weighted regression (GWR) is a spatial statistical technique that recognizes traditional 'global' regression models may be limited when spatial processes vary with spatial context. GWR captures process spatial heterogeneity via an operationalization of Tobler's first law of geography: "everything is related to everything else, but near things are more related than distant things" (1970). An ensemble of local linear models are calibrated at any number of locations by 'borrowing' nearby data. The result is a surface of location-specific parameter estimates for each relationship in the model that may vary spatially, as well as a single bandwidth parameter that provides intuition about the geographic scale of the processes. A recent extension to this framework allows each relationship to vary according to a distinct spatial scale parameter, and is therefore known as multiscale (M)GWR. This paper introduces mgwr, a Python-based implementation for efficiently calibrating a variety of (M)GWR models and a selection of associated diagnostics. It reviews some core concepts, introduces the primary software functionality, and demonstrates suggested usage on several example datasets.

A recent paper (Fotheringham et al. 2017) expands the well-known Geographically Weighted Regression (GWR) framework significantly by allowing the bandwidth or smoothing factor in GWR to be derived separately for each covariate in the model – a framework referred to as Multiscale GWR (MGWR). However, one limitation of the MGWR framework is that, until now, no inference about the local parameter estimates was possible. Formally, the so-called “hat matrix,” which projects the observed response vector into the predicted response vector, was available in GWR but not in MGWR. This paper addresses this limitation by reframing GWR as a Generalized Additive Model (GAM), extending this framework to MGWR and then deriving standard errors for the local parameters in MGWR. In addition, we also demonstrate how the effective number of parameters (ENP) can be obtained for the overall fit of an MGWR model and for each of the covariates within the model. This statistic is essential for comparing model fit between MGWR, GWR, and traditional global models, as well as adjusting for multiple hypothesis tests. We demonstrate these advances to the MGWR framework with both a simulated data set and a real-world data set.