Geographically Weighted Regression (GWR)

Geographically weighted regression (GWR) is a technique of spatial statistical modeling used to analyze spatially varying relationships between variables. Processes generating geographical patterns may vary under different geographical contexts. The identification of where and how such spatial heterogeneity in the processes appears on maps is a key in understanding complex geographical phenomena. To investigate this issue, several local spatial analysis techniques highlighting geographically “local” differences have been developed. Other techniques of local spatial analysis, such as Openshaw's GAM (geographical analysis machine) and Anselin's LISA (10cal indicators of spatial association), focus on the distribution of only one variable on a map. They are typically used for determining the geographical concentrations of high-risk diseases or crimes. GWR is also a common tool of local spatial analysis; however, it is unique with regard to the investigation and mapping of the distribution of local relationships between variables by using spatial weight.

When the geographical distribution of a variable to be explained is provided, regression modeling using geographically referenced explanatory variables can prove to be an effective method to investigate or confirm the plausible explanations of the distribution. GWR extends the conventional regression models to allow geographical drifts in coefficients by the introduction of local model fitting with geographical weighting. Based on a simple calibration procedure, this approach is computer intensive; however, it effectively enables the modeling of complex geographical variations in these relationships with the fewest restrictions on the functional form of the geographical variations. Thus, GWR is considered to be a useful geocomputation tool for exploratory spatial data analysis (ESDA) with regard to the association of the target variable with the explanatory variables under study. Since GWR models can be regarded to be a special form of nonparametric regression, statistical inferences of the GWR models are generally well established on the basis of theories on nonparametric regression.

GWR was originally proposed by Fotheringham, Brunsdon, and Charlton when they jointly worked at the University of Newcastle upon Tyne, in the United Kingdom. After the first publication of GWR in 1996 and to date, they have contributed most to the [Page 180]fundamental developments of GWR, including the release of Windows-based application software specialized for this method. Further, a great variety of empirical applications of GWR and theoretical tuning of the GWR approach to specific issues have been conducted in various fields, such as biology, climatology, epidemiology, marketing analysis, political science, and so on.

Geographically “Global” and “Local” Regression Models

Let us consider an example of health geography in which a researcher seeks the determinants of geographical inequality of health by associating the regional mortality rates with regional socioeconomic indicators, such as median income or residents' composition of social classes. The analyst may apply the following simple regression model:

where the dependent variable yi denotes the mortality rate in location i, the independent variable xi is the typical median income in the same location, and ∊i is the error term that follows a normal distribution with zero mean and σ2 variance.

In the first equation, β0 and β1 denote the coefficients to be estimated by using the least squares method.

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