Y = b0 + sum(bi*Xi) + E
X = {X1, X2, ..., Xm} m = number of explaining variables Y = {y1, y2, ..., yn} Xi = {xi1, xi2, ..., xin} E = {e1, e2, ..., en} n = number of observations (cases)
Y ~ sum(bi*Xi) b0 is the intercept, X0 is set to 1
The β coefficients are localized, i.e. determined for each cell individually. These β coefficients are the most important output of r.gwr. Spatial patterns and localized outliers in these coefficients can reveal details of the relation of Y to X. Outliers in the β coefficients can also be caused by a small bandwidth and can be removed by increasing the bandwidth.
Geographically weighted regressions should be used as a diagnostic tool and not as an interpolation method. If a geographically weighted regression provides a higher R squared than the corresponding global regression, then a crucial predictor is missing in the model. If that missing predictor can not be estimated or is supposed to behave randomly, a geographically weighted regression might be used for interpolation, but the result, in particular the variation of the β coefficients needs to be judged according to prior assumptions. See also the manual and the examples of the R package spgwr.
r.gwr is designed for large datasets that can not be processed in R. A p value is therefore not provided, because even very small, meaningless effects will become significant with a large number of cells. Instead it is recommended to judge by the amount of variance explained (R squared for a given variable) and the gain in AIC (AIC without a given variable minus AIC global must be positive) whether the inclusion of a given explaining variable in the model is justified.
The F score for each explaining variable allows to test if the inclusion of this variable significantly increases the explaining power of the model, relative to the global model excluding this explaining variable. That means that the F value for a given explaining variable is only identical to the F value of the R-function summary.aov if the given explaining variable is the last variable in the R-formula. While R successively includes one variable after another in the order specified by the formula and at each step calculates the F value expressing the gain by including the current variable in addition to the previous variables, r.gwr calculates the F-value expressing the gain by including the current variable in addition to all other variables, not only the previous variables.
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