How GWR Regression works

Geographically Weighted Regression (GWR) is one of several spatial regression techniques, increasingly used in geography and other disciplines. GWR provides a local model of the variable or process you are trying to understand/predict by fitting a regression equation to every feature in the dataset. GWR constructs these separate equations by incorporating the dependent and explanatory variables of features falling within the bandwidth of each target feature. The shape and size of the bandwidth is dependent on user input for the Kernal Type, Bandwidth Method, Distance, and Number of Features.

Implementation Notes

In global regression models, such as OLS, results are unreliable when two or more variables exhibit multicollinearity (when two or more variables are redundant or together tell the same "story"). GWR builds a local regression equation for each feature in the dataset. When the values for a particular explanatory variable cluster spatially, you will very likely have problems with local multicolliearity.

Results are unstable in the presence of local collinearity as indicated by a condition number greater than 30. Condition numbers indicate how sensitive a linear equation solution is to small changes in matrix coefficients. Individual feature results when the condition number is greater than 30 are not included in the variance of the parameter estimates; this impacts standard error diagnostics, global sigma, and standardized residuals.

The user may change this condition number threshold by resetting the registry:

[HKEY_CURRENT_USER\Software\ESRI\GeoStatisticalExtension\DefaultParams\GWR]

"ConditionNumberThreshold"="40"

Parameter estimates and predicted values for GWR are computed using the following spatial weighting function: exp(-d^2/b^2). There may be differences in this weighting function among various GWR software implementations. Consequently, results from the ESRI GWR tool may not match results of other GWR software packages exactly.

See Regression Analysis Basics and Interpreting GWR Regression Results.

Additional Resources:

Fotheringham, Stewart A., Brunsdon, Chris, and Charlton, Martin.. Geographically Weighted Regression: the analysis of spatially varying relationships. John Wiley & Sons, 2002.

Mitchell, Andy. The ESRI Guide to GIS Analysis, Volume 2. ESRI Press, 2005.