Geographically weighted regression: a natural evolution of the expansion method for spatial data analysis

Geographically weighted regression: a natural evolution of the expansion method for spatial data analysis

1998, volume 30, pages 1905 – 1927 | A S Fotheringham, M E Charlton, C Brunsdon
The paper discusses the application of two statistical techniques, Geographically Weighted Regression (GWR) and the Expansion Method (EM), to examine spatial variability in regression results. Both methods allow for the production of local regression results, providing a set of mappable statistics that denote local relationships, rather than relying on global regression results. The authors compare the application of these techniques to health data from northeast England, demonstrating that GWR produces more informative results regarding parameter variation over space. The Expansion Method measures parameter 'drift' by expanding the global model to include spatial coordinates, allowing trends in parameter estimates over space to be measured. However, this method is limited to displaying trends and is restricted by the complexity of the expansion equations. GWR extends the traditional regression framework by allowing local parameters to be estimated, recognizing that relationships may vary spatially. It uses a weighted least squares approach, where observations closer to the point of interest are given more weight. The choice of the spatial weighting function, such as a Gaussian kernel, is crucial and can be optimized using cross-validation to balance bias and variance. The paper also discusses the choice of the spatial weighting function, the bias-variance trade-off, and the testing for spatial nonstationarity. An empirical comparison using data on the distribution of limiting long-term illness (LLTI) in northeast England shows that GWR provides more detailed insights into the spatial variations in relationships compared to the Expansion Method. Overall, the paper highlights the advantages of GWR in capturing local spatial variations and its potential as an analytical tool for spatial data analysis.The paper discusses the application of two statistical techniques, Geographically Weighted Regression (GWR) and the Expansion Method (EM), to examine spatial variability in regression results. Both methods allow for the production of local regression results, providing a set of mappable statistics that denote local relationships, rather than relying on global regression results. The authors compare the application of these techniques to health data from northeast England, demonstrating that GWR produces more informative results regarding parameter variation over space. The Expansion Method measures parameter 'drift' by expanding the global model to include spatial coordinates, allowing trends in parameter estimates over space to be measured. However, this method is limited to displaying trends and is restricted by the complexity of the expansion equations. GWR extends the traditional regression framework by allowing local parameters to be estimated, recognizing that relationships may vary spatially. It uses a weighted least squares approach, where observations closer to the point of interest are given more weight. The choice of the spatial weighting function, such as a Gaussian kernel, is crucial and can be optimized using cross-validation to balance bias and variance. The paper also discusses the choice of the spatial weighting function, the bias-variance trade-off, and the testing for spatial nonstationarity. An empirical comparison using data on the distribution of limiting long-term illness (LLTI) in northeast England shows that GWR provides more detailed insights into the spatial variations in relationships compared to the Expansion Method. Overall, the paper highlights the advantages of GWR in capturing local spatial variations and its potential as an analytical tool for spatial data analysis.
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