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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.

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.

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.

Scale is a central concept in the geographical sciences and is an intrinsic property of many spatial systems. It also serves as an essential thread in the fabric of many other physical and social sciences, which has contributed to the use of different terminology for similar manifestations of what we refer to as ‘scale’, leading to a surprising amount of diversity around this fundamental concept and its various ‘multiscale’ extensions. To address this, we review common abstractions about spatial scale and how they are employed in quantitative research. We also explore areas where the conceptualizations of multiple spatial scales can be differentiated. This is achieved by first bridging terminology and concepts, and then conducting a scoping review of the topic. A typology for spatial scale is discussed that can be used to categorize its multifarious meanings and measures. This typology is then used to distinguish what we term ‘process scale,’ from other types of spatial scale and to highlight current trends in uncovering aspects of process scale. We end with suggestions on how to further build knowledge regarding spatial processes through the lens of spatial scale.

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 longstanding inquiry.

This chapter introduces the concept of local versus global models and describes one type of local model, Geographically Weighted Regression, and its recent successor, Multiscale Geographically Weighted Regression. The conceptual basis for this type of model is explained in terms of data-borrowing. An empirical example is given to demonstrate both the value of local regression models and freely available software for their calibration.

The development of ‘route maps’ for spatial analytical methods is a pursuit with important ramifications. Comber et al. (2022) propose a route map to guide applications of geographically weighted regression (GWR) consisting of a 3-step primary pathway and a series of secondary arterials. This comment first highlights some concerns about the underlying ‘map’ (i.e., experimental setup and assumptions) and then with the proposed ‘route’ (i.e., core decisions and evaluation criteria). It closes by suggesting a more general focus on identifying modeling issues with the highest impact and facilitating consensus-building, which could improve the future production of route maps for navigating the methodological landscape in spatial analysis.

The concept of spatial interaction (SI) encapsulates the domain of human activities that occur between a set of locations embedded within geographical space. Data about such processes are essential for studying a wide spectrum of geographic phenomena that are important to society, such as the accessibility of services, product demand, transportation trends, and demographic dynamics. In particular, SI models seek to explore, explain, and predict aggregate movements or flows that occur across an abstract or physical network, which can be useful on its own, as well as a factor within other regional models. As the number and nature of SI modeling applications have grown, the associated theory and tools have simultaneously evolved to consider more complex spatial relationships, resulting in numerous expansions of the modeling paradigm. In this chapter, some foundations of SI modeling are first laid out before presenting a simple demonstration and then describing several extensions to the core modeling methodology.

Massive amounts of data that characterize how people meet their economic needs, interact within social communities, and utilize shared resources are being produced by cities. Harnessing these ever-increasing data streams is crucial for understanding urban dynamics. Within the context of transportation modeling it still remains largely unknown whether or not these new data sources provide the opportunity to better understand spatial processes. Therefore, in this paper, the usefulness of a recently available big transport dataset - the New York City (NYC) taxi trip data - is evaluated within a spatial interaction modeling framework. This is done by first comparing parameter estimates from a model using the taxi data to parameter estimates from a model using a traditional commuting dataset. In addition, the high temporal resolution of the taxi data provide an exciting means to explore potential dynamics in movement behavior. It is demonstrated how parameter estimates can be obtained for temporal subsets of data and compared over time to investigate mobility dynamics. The results of this work indicate that a pitfall of big transport data is that it is less useful for modeling distinct phenomena; however, there is a strong potential for modeling high frequency temporal dynamics of diverse urban activities.

Scale is a central concept in the geographical sciences and is an intrinsic property of many spatial systems. It also serves as an essential thread in the fabric of many other physical and social sciences, which has contributed to the use of different terminology for similar manifestations of what we refer to as ‘scale’, leading to a surprising amount of diversity around this fundamental concept and its various ‘multiscale’ extensions. To address this, we review common abstractions about spatial scale and how they are employed in quantitative research. We also explore areas where the conceptualizations of multiple spatial scales can be differentiated. This is achieved by first bridging terminology and concepts, and then conducting a scoping review of the topic. A typology for spatial scale is discussed that can be used to categorize its multifarious meanings and measures. This typology is then used to distinguish what we term ‘process scale,’ from other types of spatial scale and to highlight current trends in uncovering aspects of process scale. We end with suggestions on how to further build knowledge regarding spatial processes through the lens of spatial scale.