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The concept of scale is inherent to, and consequential for, the modeling of geographical processes. However, scale also causes huge problems because the results of many types of spatial analysis appear to be dependent on the scale of the units for which data are reported (measurement scale). Consequently, when the same spatial models are calibrated at different scales of aggregations, the results are often vastly different (the well-known Modifiable Areal Unit Problem or MAUP). With the advent of local models and the fundamental difference in their scale of application compared to global models, this issue is further exacerbated in unexpected ways. For example, a global model and local model calibrated using data measured at the same aggregation scale can also result in different and sometimes contradictory inferences (the classic Simpson's Paradox). Here we provide a geographical perspective on why and how contrasting inferences might result from the calibration of a local and global model using the same data. Further, we examine the viability of such an occurrence using a synthetic experiment and two empirical examples. Finally, we discuss how such a perspective might inform the analyst’s conundrum: when the respective inferences run counter to one another, do we believe the local or global model results?

The concept of scale is inherent to, and consequential for, the modeling of geographical processes. However, scale also causes huge problems because the results of many types of spatial analysis appear to be dependent on the scale of the units for which data are reported (measurement scale). With the advent of local models and the fundamental difference in their scale of application compared to global models, this issue is exacerbated in unexpected ways. For example, a global model and local model calibrated using data measured at the same aggregation scale can also result in different and sometimes contradictory inferences (the classic Simpson’s Paradox). Here we provide a geographical perspective on why and how contrasting inferences might result from the calibration of a local and global model using the same data. Further, we examine the viability of such an occurrence using a synthetic experiment and two empirical examples. Finally, we discuss how such a perspective might inform the analyst’s conundrum: when the respective inferences run counter to one another, do we believe the local or global model results?

Studies of spatially varying parameter estimates obtained in the calibration of various types of local statistical models are commonplace. The variation in such estimates is typically explained in terms of spatially varying processes. This paper highlights that an alternative explanation for spatially varying parameter estimates, in terms of non-linearity, should be examined prior to relating such variation to spatially varying processes. This can be achieved by a simple screening procedure which is described and demonstrated and which can easily be applied to the results of any local model. The problem is highlighted, and the solution demonstrated, with a set of simulated data and then with a real-world data set. The paper also highlights the obverse situation whereby the inappropriate application of a GAM produces spurious nonlinear results when the real relationships are linear but spatially varying.

We discuss the nature of processes relating to human behaviour and how to model such processes when they vary over space. In so doing, we describe the role of local modelling and how the bandwidth ...

This article attempts to identify and separate the role of spatial “context” in shaping voter preferences from the role of other socioeconomic determinants. It does this by calibrating a multiscale...

This introduction provides a brief review of the motivation, background, and context of the Forum. It explains the roles of the six papers in the Forum and the importance of reproducibility and rep...

A cornerstone of the scientific method, the ability to reproduce and replicate the results of research has gained widespread attention across the sciences in recent years. A corresponding burst of ...

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.

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 paper, we find that bandwidth selection is subject to uncertainty in both single-scale and multi-scale 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.