A First Course in Linear Model Theory

Chapter 3 considers three extensions: mixed GWR models, in which some regression coefŽ cients vary spatially but others are global; robust GWR, in which spatial outliers are downweighted; and spatially heteroscedastic GWR models, for which the residual variance also is spatially varying. Chapter 4 describes procedures for constructing conŽ dence intervals on regression coefŽ cients and for comparing models. Chapter 5 explores some relationships between GWR and classical measures of spatial autocorrelation. Chapter 6 examines the sensitivity of GWR results to variations in the scale of analysis (a version of the modiŽ able areal unit problem) and the extent to which GWR can suggest an appropriate scale of analysis. Chapter 7 proposes localized versions of various simple statistics (e.g., means, correlation coefŽ cients, odds ratios) via geographical weighting, whereas Chapter 8 further extends localization to produce geographically weighted versions of generalized linear models, principal components, and density estimation. Chapter 9 describes and illustrates computer software for GWR written by the authors (and made available from them at essentially no cost). The Ž nal chapter is a short epilogue. The authors are enthusiastic proponents of GWR, and their enthusiasm is infectious. Nevertheless, I feel that they sell random-coefŽ cients models and spatial regression a little too short. For example, they correctly point out that in the presence of nonstationarity, a global statistic produced by spatial regression may be very far off target locally; however, they do not mention the opposite problem, that of unnecessarily allowing for nonstationarity when the process is stationary. In the latter situation, local statistics, although unbiased, are less reliable than global statistics, and some overinterpretation of spurious patterns may occur. Moreover, the authors imply that spatial regression is limited to stationary models, when in fact there have been some important recent developments in nonstationary geostatistical modeling. For example, the local kriging and co-kriging methodology of Haas (1995, 1996) is a geostatistical alternative for handling spatially varying multivariate relationships, but the authors either are unaware of it or choose not to mention it. Another relevant but very recent development is the Bayesian approach of Gelfand, Kim, Sirmans, and Banerjee (2003). I detected almost no typographical errors, but there are a few substantive slip-ups. For example, on page 55, an expression is given for the variancecovariance matrix of the vector of location-speciŽ c regression coefŽ cients that involves a population parameter 3⁄4 2 , and then 3⁄4 2 is incorrectly expressed as a function of the data. Also incorrect, and potentially quite confusing, is representation (5.26) for a simultaneous autoregressive model; moreover, the conditions on the spatial weights given there for the existence of conditional or simultaneous autoregressive models are necessary but not sufŽ cient. In Section 2.7.4, it would have been worth noting that the formulas for standard errors of estimated regression coefŽ cients given previously are not strictly correct when the data are used to choose an optimal bandwidth. Notwithstanding these minor criticisms, I found Geographically Weighted Regression an enjoyable, easy-to-read exposition of GWR, and I highly recommend it for geographers, environmental scientists, and others interested in modeling relationships among spatially distributed variables.