From GWR by Roger Bivand:
Geographically weighted regression (GWR) is an exploratory technique
mainly intended to indicate where non-stationarity is taking place on
the map, that is where locally weighted regression coefficients move
away from their global values. Its basis is the concern that the
fitted coefficient values of a global model, fitted to all the data,
may not represent detailed local variations in the data adequately –
in this it follows other local regression implementations. It differs,
however, in not looking for local variation in ‘data’ space, but by
moving a weighted window over the data, estimating one set of
coefficient values at every chosen ‘fit’ point. The fit points are
very often the points at which observations were made, but do not have
to be. If the local coefficients vary in space, it can be taken as an
indication of non-stationarity.
The technique ... involves first selecting a bandwidth for an
isotropic spatial weights kernel, typically a Gaussian kernel with a
fixed bandwidth chosen by leave-one-out cross-validation. Choice of
the bandwidth can be very demanding, as n regressions must be fitted
at each step. Alternative techniques are available, for example for
adaptive bandwidths, but they may often be even more
compute-intensive.
> library(maptools)
> library(spdep)
> owd <- getwd()
> setwd(system.file("etc/shapes", package = "spdep"))
> NY8 <- readShapeSpatial("NY8_utm18")
> setwd(owd)
> library(spgwr)
> bwG <- gwr.sel(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = NY8, gweight = gwr.Gauss,
+ verbose = FALSE)
> gwrG <- gwr(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = NY8, bandwidth = bwG,
+ gweight = gwr.Gauss, hatmatrix = TRUE)
> gwrG
Once the bandwidth has been found, or chosen by hand, the gwr function
may be used to fit the model with the chosen local kernel and
bandwidth. If the data argument is passed a SpatialPolygonsDataFrame
or a SpatialPointsDataFrame object, the output object will contain a
component, which is an object of the same geometry populated with the
local coefficient estimates. If the input objects have polygon
support, the centroids of the spatial entities are taken as the basis
for analysis. The function also takes a fit.points argument, which
permits local coefficients to be created by geographically weighted
regression for other support than the data points.
