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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.

Geographically weighted regression (GWR) extends the familiar regression framework by estimating a set of parameters for any number of locations within a study area, rather than producing a single parameter estimate for each relationship specified in the model. Recent literature has suggested that GWR is highly susceptible to the effects of multicollinearity between explanatory variables and has proposed a series of local measures of multicollinearity as an indicator of potential problems. In this paper, we employ a controlled simulation to demonstrate that GWR is in fact very robust to the effects of multicollinearity. Consequently, the contention that GWR is highly susceptible to multicollinearity issues needs rethinking.

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.

One of the key concerns in spatial analysis and modeling is to study and analyze the processes that generate our observations of the real world. The typical global models employed to do this, however, fail to identify spatial variations in these processes because they assume that the processes being investigated are spatially stationary. In many real-life situations, spatial variations in relationships seem plausible and at least worth examining so that the assumption of global stationarity is, at best, unhelpful and, at worst, unrealistic. In contrast, local spatial models allow for potential variations in relationships over space leading to greater insights into the data-generating processes. In this study, a framework for localizing spatial interaction models, based on geographically weighted techniques, is developed. Using the framework, we construct a family of spatially weighted interaction models (SWIM) that can help in detecting, visualizing, and analyzing spatial nonstationarity in spatial interaction processes. Using custom-built algorithms, we apply both traditional interaction models and SWIM to a journey-to-work data set in Switzerland. The results of the model calibrations are explored using matrix visualizations, which suggest that SWIM provide useful information on the nature of spatially nonstationary processes leading to spatial patterns of flows.

Background The COVID-19 pandemic caused substantial mortality in the United States with impacts unevenly distributed across the country. Official COVID-19-related death counts, however, almost certainly underrepresent the true impact of the pandemic due to underreporting, misclassification, and, particularly in the early stages of pandemic, limited testing and diagnosis [ 1 ]. Excess death rates, deaths above expected levels based on historical trends, arguably provide a more comprehensive measure of COVID-19 impacts by capturing both direct COVID-19 deaths and indirect fatalities related to pandemic disruptions. The goal of the study is to examine spatial and temporal disparities in COVID-19 excess mortality in 2020–2021 and 2021–2022 across the U.S., distinguishing between quantifiable sociodemographic influences and unmeasurable place-based factors through Multiscale Geographically Weighted Regression (MGWR). Methods Excess mortalities are examined in 2020–2021 and 2021–2022 to capture temporal and spatial shifts in COVID-19-related excess mortality patterns. MGWR is used to identify localized variations in the determinants of excess death rates using data on socioeconomic conditions, political affiliation, demographic factors, health status, and healthcare access. Results We present the results of calibrating both a global and a local model of excess death rates during two phases of the COVID-19 pandemic. In terms of the global results, in both time periods excess death rates were significantly higher in counties with high percentages of people below the poverty line, Republican-leaning residents, high proportions of elderly population, high levels of deprivation, high unemployment, and relatively high proportions of residents with diabetes. Rates were also significantly higher in counties with relatively high proportions of residents without health insurance, where there were more females than males, and where there were fewer younger adults, although these effects were not as strong as the previous associations. However, these macro-level conditioned associations can hide important local variations in the determinants of severe COVID-19-related health outcomes. Because COVID-19-related excess death rates exhibit strong spatial patterns, any covariate sharing a similar spatial distribution, even if coincidental, might spuriously be reported to have a significant impact on excess dates rates when examined globally. To examine this possibility, a local statistical model is calibrated which suggests some alternative views on the determinants of COVID-19-relates deaths. For instance, although excess death rates were strongly linked to Republican party support across the whole country in the first phase of the pandemic, this relationship was limited to the eastern seaboard and the Deep South in the second phase. There was a significant conditioned relationship between excess deaths and the elderly only across the southern half of the country in both phases of the pandemic. The impacts of being without health insurance were only severe in the western half of the country and only in the first phase of the pandemic. In contrast to the global finding, the positive association with diabetes was only found along the east coast and only in the first phase of the pandemic. In the first phase of the pandemic, excess mortality was only significantly positively associated with the proportion of Hispanics in the Southwest and was insignificant elsewhere, In the second phase of the pandemic, there were no significant positive relationships reported locally but there were significant negative relationships across the upper Midwest, the Northeast, and in Texas. In distinct contrast to the global results, the local conditioned relationship between excess death rates and percentage Black population was significantly positive across the country in both phases of the pandemic. In the first phase of the pandemic, conditioning on all the covariates in the model, excess deaths from COVID-19 were lower than expected in most parts of the country except for a cone-shaped set of states from Nebraska to Texas; in the second phase the unseen benefits of location were only experienced in the upper Midwest. The results support the use of local models to better understand the nature of pandemics and also that COVID-19 impacts arose from a complex interaction between both measurable factors and localized, often unobservable, influences. Conclusions Disparities in excess deaths during the COVID-19 pandemic reflect a combination of structural vulnerabilities and unmeasured local influences. To effectively reduce mortality gaps and strengthen preparedness for future health crises, public health interventions must be geographically tailored, targeting both region-specific risk factors and the contextual conditions that shape local outcomes.

Scale is a fundamental geographic concept, and a substantial literature exists discussing the various roles that scale plays in different geographical contexts. Relatively little work exists, though, that provides a means of measuring the geographic scale over which different processes operate. Here we demonstrate how geographically weighted regression (GWR) can be adapted to provide such measures. GWR explores the potential spatial nonstationarity of relationships and provides a measure of the spatial scale at which processes operate through the determination of an optimal bandwidth. Classical GWR assumes that all of the processes being modeled operate at the same spatial scale, however. The work here relaxes this assumption by allowing different processes to operate at different spatial scales. This is achieved by deriving an optimal bandwidth vector in which each element indicates the spatial scale at which a particular process takes place. This new version of GWR is termed multiscale geographically weighted regression (MGWR), which is similar in intent to Bayesian nonseparable spatially varying coefficients (SVC) models, although potentially providing a more flexible and scalable framework in which to examine multiscale processes. Model calibration and bandwidth vector selection in MGWR are conducted using a back-fitting algorithm. We compare the performance of GWR and MGWR by applying both frameworks to two simulated data sets with known properties and to an empirical data set on Irish famine. Results indicate that MGWR not only is superior in replicating parameter surfaces with different levels of spatial heterogeneity but provides valuable information on the scale at which different processes operate.

Aims To investigate spatial congruence between ecological niches and genotype in two allopatric species of desert tortoise that are species of conservation concern. Location Mojave and Sonoran Desert ecoregions; California, Nevada, Arizona, Utah, USA. Methods We compare ecological niches of Gopherus agassizii and Gopherus morafkai using species distribution modelling (SDM) and then calibrate a pooled‐taxa distribution model to explore local differences in species–environment relationships based on the spatial residuals of the pooled‐taxa model. We use multiscale geographically weighted regression (MGWR) applied to those residuals to estimate local species–environment relationships that can vary across the landscape. We identify multivariate clusters in these local species–environment relationships and compare them against models of (a) a geographically based taxonomic designation for two sister species and (b) an environmental ecoregion designation, with respect to their ability to predict a genotype association index for these two species. Results We find non‐identical niches for these species, with differences that span physiographic and vegetation niche dimensions. We find evidence for two distinct clusters of local species–environment relationships that when mapped, predict an index of genotype association for the two sister taxa better than did either the geographically based taxonomic designation or an environmental ecoregion designation. Main conclusions Exploring local species–environment relationships by coupling SDM and MGWR can benefit studies of biogeography and conservation. We find that niche separation in habitat selection conforms to genotypic differences between sister taxa of tortoise in a recent secondary contact zone. This result may inform decision making by agencies with regulatory or land management authority for the two sister taxa addressed here.

Geographically weighted regression (GWR) is used to investigate spatial relationships between freshwater acidification critical load data and contextual catchment data across Great Britain. Although this analysis is important in developing a greater understanding of the critical load process, the study also examines the application of the GWR technique itself. In particular, and unlike many previous presentations of GWR, the steps taken in choosing a particular GWR model form are presented in detail. A further important advance here is that the calibration results of the chosen GWR model are scrutinized for robustness to outlying observations. With respect to the critical load process itself, the results of this study largely agree with those of earlier research, where relationships between critical load and catchment data can vary across space. The more sophisticated spatial statistical models used here, however, are shown to be more flexible and informative, allowing a clearer picture of process heterogeneities to be revealed.

This article considers critically how one of the oldest and most widely applied statistical methods, principal components analysis (PCA), is employed with spatial data. We first provide a brief guide to how PCA works: This includes robust and compositional PCA variants, links to factor analysis, latent variable modeling, and multilevel PCA. We then present two different approaches to using PCA with spatial data. First we look at the nonspatial approach, which avoids challenges posed by spatial data by using a standard PCA on attribute space only. Within this approach we identify four main methodologies, which we define as (1) PCA applied to spatial objects, (2) PCA applied to raster data, (3) atmospheric science PCA, and (4) PCA on flows. In the second approach, we look at PCA adapted for effects in geographical space by looking at PCA methods adapted for first-order nonstationary effects (spatial heterogeneity) and second-order stationary effects (spatial autocorrelation). We also describe how PCA can be used to investigate multiple scales of spatial autocorrelation. Furthermore, we attempt to disambiguate a terminology confusion by clarifying which methods are specifically termed \"spatial PCA\" in the literature and how this term has different meanings in different areas. Finally, we look at a further three variations of PCA that have not been used in a spatial context but show considerable potential in this respect: simple PCA, sparse PCA, and multilinear PCA.

Scotland has an organised colorectal cancer screening programme; however, despite proactively offering screening opportunities free to the at-risk population, and also despite using a screening test which may be completed at home, screening participation levels are unequal. Understanding causal pathways linking participation with other population characteristics may be aided by identifying how relationships between the two patterns vary across different localities, and such knowledge may also inform decisions regarding geographical targeting of screening promotion efforts. In this analysis, models calibrated using multiscale geographically weighted regression enabled the assessment of spatial variations of determinants of screening participation levels. The models were calibrated for localities across west central Scotland (n=409), where participation levels were relatively low, using aggregated individual-level screening records within a two-year window (2009-2011). Area deprivation was found to have a strong negative impact on participation levels across the study area, and ethnic population concentration had a significant impact on male participation levels on localities within Glasgow city. Estimates of local intercepts pointed to a systemic difference in screening participation between the two health board regions in the study area. Overall the results suggest that work to increase screening participation was necessary. They also suggest that barriers to participation could be addressed locally, and that differences between health board regions required further investigation.