Being newly immersed in the upstream part of the oil business, I just recently had my first work session with data in ARC–GIS®. The project involves subsurface geographical modeling. Obviously I had considerable interest in discovering if the methodology in this book would enhance my modeling capabilities. The book is for social scientists, but I had no difficulty imagining my own important oil exploration application within the framework of geographically weighted regression (GWR). The first chapter nicely explains what is unique in this book. A standard regression model using geographically oriented data (the example is housing prices across all of England) is a global representation of a spatial relationship, an average that does not account for any local differences. In y = f (x), imagine a whole family of f ’s that are indexed by spatial location. That is the focus of this book. It is about one form of local spatial modeling, which is GWR. A more general resource for this topic is the earlier book by Fotheringham and Wegener (2000), which escaped the notice of Technometrics. Imagine a display of model parameters in a geographical information system (GIS) and you will understand the focus for this book. The authors note, “only where there is no significant spatial variation in a measured relationship can global models be accepted” (p. 10). The second chapter develops the basis of GWR. It analyzes the housing sales prices versus the 33 boroughs in London and begins by fitting a conventional multiple regression model versus housing characteristics. The GWR is motivated by differences in the regression models fitted separately by borough. The GWR is a spatial moving-window approach with all data distances weighted versus a specific data point using a weighting function and a bandwidth. A GIS can then be used to evaluate the spatial dependency of the parameters. As in kriging, local standard errors also are calculated. The chapter also provides all the math. Chapter 3 comprises several further considerations: parameters that are globally constant, outliers, and spatial heteroscedasticity. The first issue leads to hypothesis tests for model comparison using an Akaike information criterion (AIC). Local outliers are hard to detect. Studentized (deletion) residuals are recommended. The outliers can be plotted geographically. Robust regression is suggested as a less computationally intensive alternative. Hetereoscedasticity is harder to handle. Chapter 4 adds statistical inference to the capabilities of GWR: both a confidence interval approach using local likelihood and an AIC method. Four additional methodology chapters present various extensions of GWR. Chapter 5 considers the relationship between GWR and spatial autocorrelation, and includes a combined version of GWR and spatial regression using some complex hybrid models. Chapter 6 examines the relationship of scale and zoning problems in spatial analysis to GWR. Chapter 7 introduces the use of initial exploratory data analysis using geographically weighted statistics, which are based on the idea of using a kernel around each data point to create weights. Univariate statistics and correlation coefficients are defined for exploring local patterns in data. A final set of extensions in Chapter 8 discusses regression models with non-Gaussian errors, logistic regression, local principal components analysis, and local probability density estimation. The methods all use some kind of distributional model. The million-dollar question for me is always, “What about software?” The authors have a stand-alone program, GWR 3, available in CD–ROM by contacting the authors. Basically the drill with GWR 3 is to gather your data, use Excel to transform and reformat the data for GWR 3, use GWR 3 to produce a set of coefficients, and feed those coefficients to your favorite GIS to produce your maps. Forty pages of discussion about using the software are provided. A final epilogue chapter also discusses embedding GWR in R or Matlab and includes some references to people who have done that type of work. I probably would not have read this book if I had not happened to have had it in my briefcase on a visit with the exploration technologists. Though inclusive of appropriate mathematical development, this material is readily approachable because of the many illustrations and the pages and pages of GIS displays. The authors unabashedly present much of the material as their developmental work, so GWR offers a lot of opportunity for research and further development through novel applications and extensions.

Geographically Weighted Regression

GIS-based spatial modeling of COVID-19 incidence rate in the continental United States.

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.

Inference in Multiscale Geographically Weighted Regression

Machine learning algorithms such as Random Forest (RF) are being increasingly applied on traditionally geographical topics such as population estimation. Even though RF is a well performing and gen...

Geographical random forests: a spatial extension of the random forest algorithm to address spatial heterogeneity in remote sensing and population modelling

The Annals has been published since 1911 and is the journal of the American Association of Geographers (AAG) which changed its name from the Association of American Geographers effective 1 January, 2016. The title up to Issue 6, 2015 was the Annals of the Association of American Geographers. The title from Issue 1, 2016 is the Annals of the American Association of Geographers.

Annals of the American association of geographers

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 we...

Wenbai Yang

Papers

Multiscale Geographically Weighted Regression (MGWR)