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.

Background: Obesity rates are recognized to be at epidemic levels throughout much of the world, posing significant threats to both the health and financial security of many nations. The causes of obesity can vary but are often complex and multifactorial, and while many contributing factors can be targeted for intervention, an understanding of where these interventions are needed is necessary in order to implement effective policy. This has prompted an interest in incorporating spatial context into the analysis and modeling of obesity determinants, especially through the use of geographically weighted regression (GWR). Method: This paper provides a critical review of previous GWR models of obesogenic processes and then presents a novel application of multiscale (M)GWR using the Phoenix metropolitan area as a case study. Results: Though the MGWR model consumes more degrees of freedom than OLS, it consumes far fewer degrees of freedom than GWR, ultimately resulting in a more nuanced analysis that can incorporate spatial context but does not force every relationship to become local a priori. In addition, MGWR yields a lower AIC and AICc value than GWR and is also less prone to issues of multicollinearity. Consequently, MGWR is able to improve our understanding of the factors that influence obesity rates by providing determinant-specific spatial contexts. Conclusion: The results show that a mix of global and local processes are able to best model obesity rates and that MGWR provides a richer yet more parsimonious quantitative representation of obesity rate determinants compared to both GWR and ordinary least squares.

Models designed to capture spatially varying processes are now employed extensively in the social and environmental sciences. The main strength of such models is their ability to represent relationships that vary across locations through locally varying parameter estimates. However, local models of spatial processes also provide information on the nature of these spatially varying relationships through the estimation of a ‘bandwidth’ parameter. This paper examines bandwidth at a conceptual, operational and empirical level within the framework of geographically weighted regression, one of the more frequently employed local spatial models. We outline how bandwidth relates to three characteristics of spatial processes: variation, dependence and strength.

Bandwidth, a key parameter in geographically weighted regression models, is closely related to the spatial scale at which the underlying spatially heterogeneous processes being examined take place. Generally, a single optimal bandwidth (geographically weighted regression) or a set of covariate-specific optimal bandwidths (multiscale geographically weighted regression) is chosen based on some criterion, such as the Akaike information criterion (AIC), and then parameter estimation and inference are conditional on the choice of this bandwidth. In this article, we find that bandwidth selection is subject to uncertainty in both single-scale and multiscale geographically weighted regression models and demonstrate that this uncertainty can be measured and accounted for. Based on simulation studies and an empirical example of obesity rates in Phoenix, we show that bandwidth uncertainties can be quantitatively measured by Akaike weights and confidence intervals for bandwidths can be obtained. Understanding bandwidth uncertainty offers important insights about the scales over which different processes operate, especially when comparing covariate-specific bandwidths. Additionally, unconditional parameter estimates can be computed based on Akaike weights accounts for bandwidth selection uncertainty.

Spatial interaction and spatial structure are foundational geographical abstractions, though there is often variation in how they are conceptualized and deployed in quantitative models. In particular, the last five decades have produced an exceptional diversity regarding the role of spatial structure within spatial interaction models. This is explored by outlining the initiation and development of the notion of spatial structure within spatial interaction modeling and critically reviewing four methodological approaches that emerged from ongoing debate about the topic. The outcome is a comprehensive coverage of the past and a sketch of one potential path forward for advancing this long-standing inquiry.

Multiscale geographically weighted regression (MGWR) is an important method that is used across many disciplines for exploring spatial heterogeneity and modeling local spatial processes. This book introduces the concepts behind local spatial modeling and explains how to model heterogeneous spatial processes within a regression framework. It starts with the basic ideas and fundamentals of local spatial modeling followed by a detailed discussion of scale issues and statistical inference related to MGWR. A comprehensive guide to free, user-friendly, software for MGWR is provided, as well as an example of the application of MGWR to understand voting behavior in the 2020 US Presidential election. Multiscale Geographically Weighted Regression: Theory and Practice is the definitive guide to local regression modeling and the analysis of spatially varying processes, a very cutting-edge, hands-on, and innovative resource. Features Provides a balance between conceptual and technical introduction to local models Explains state-of-the-art spatial analysis technique for multiscale regression modeling Describes best practices and provides a detailed walkthrough of freely available software, through examples and comparisons with other common spatial data modeling techniques Includes a detailed case study to demonstrate methods and software Takes a new and exciting angle on local spatial modeling using MGWR, an innovation to the previous local modeling ‘bible’ GWR The book is ideal for senior undergraduate and graduate students in advanced spatial analysis and GIS courses taught in any spatial science discipline as well as for researchers, academics, and professionals who want to understand how location can affect human behavior through local regression modeling. <!----> <!----> <!----> <!---->