gwr: Geographically weighted regression

In spgwr: Geographically Weighted Regression

Geographically weighted regression

Description

The function implements the basic geographically weighted regression approach to exploring spatial non-stationarity for given global bandwidth and chosen weighting scheme.

Usage

gwr(formula, data=list(), coords, bandwidth, gweight=gwr.Gauss, 
	adapt=NULL, hatmatrix = FALSE, fit.points, longlat=NULL, 
	se.fit=FALSE, weights, cl=NULL, predictions = FALSE, 
        fittedGWRobject = NULL, se.fit.CCT = TRUE)
## S3 method for class 'gwr'
print(x, ...)

Arguments

formula	regression model formula as in lm
data	model data frame, or SpatialPointsDataFrame or SpatialPolygonsDataFrame as defined in package sp
coords	matrix of coordinates of points representing the spatial positions of the observations; may be omitted if the object passed through the data argument is from package sp
bandwidth	bandwidth used in the weighting function, possibly calculated by gwr.sel
gweight	geographical weighting function, at present gwr.Gauss() default, or gwr.gauss(), the previous default or gwr.bisquare()
adapt	either NULL (default) or a proportion between 0 and 1 of observations to include in weighting scheme (k-nearest neighbours)
hatmatrix	if TRUE, return the hatmatrix as a component of the result, ignored if fit.points given
fit.points	an object containing the coordinates of fit points; often an object from package sp; if missing, the coordinates given through the data argument object, or the coords argument are used
longlat	TRUE if point coordinates are longitude-latitude decimal degrees, in which case distances are measured in kilometers; if x is a SpatialPoints object, the value is taken from the object itself
se.fit	if TRUE, return local coefficient standard errors - if hatmatrix is TRUE and no fit.points are given, two effective degrees of freedom sigmas will be used to generate alternative coefficient standard errors
weights	case weights used as in weighted least squares, beware of scaling issues, probably unsafe
cl	if NULL, ignored, otherwise cl must be an object describing a “cluster” created using makeCluster in the parallel package. The cluster will then be used to hand off the calculation of local coefficients to cluster nodes, if fit points have been given as an argument, and hatmatrix=FALSE
predictions	default FALSE; if TRUE and no fit points given, return GW fitted values at data points, if fit points given and are a Spatial*DataFrame object containing the RHS variables in the formula, return GW predictions at the fit points
fittedGWRobject	a fitted gwr object with a hatmatrix (optional), if given, and if fit.points are given and if se.fit is TRUE, two effective degrees of freedom sigmas will be used to generate alternative coefficient standard errors
se.fit.CCT	default TRUE, compute local coefficient standard errors using formula (2.14), p. 55, in the GWR book
x	an object of class "gwr" returned by the gwr function
...	arguments to be passed to other functions

Details

The function applies the weighting function in turn to each of the observations, or fit points if given, calculating a weighted regression for each point. The results may be explored to see if coefficient values vary over space. The local coefficient estimates may be made on a multi-node cluster using the cl argument to pass through a parallel cluster. The function will then divide the fit points (which must be given separately) between the clusters for fitting. Note that each node will need to have the “spgwr” package present, so initiating by clusterEvalQ(cl, library(spgwr)) may save a little time per node. The function clears the global environment on the node of objects sent. Using two nodes reduces timings to a little over half the time for a single node.

The section of the examples code now includes two simulation scenarios, showing how important it is to check that mapped pattern in local coefficients is actually there, rather than being an artefact.

Value

A list of class “gwr”:

SDF	a SpatialPointsDataFrame (may be gridded) or SpatialPolygonsDataFrame object (see package "sp") with fit.points, weights, GWR coefficient estimates, R-squared, and coefficient standard errors in its "data" slot.
lhat	Leung et al. L matrix
lm	Ordinary least squares global regression on the same model formula, as returned by lm.wfit().
bandwidth	the bandwidth used.
this.call	the function call used.

Author(s)

Roger Bivand [e-mail]

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2002, Geographically Weighted Regression, Chichester: Wiley; Paez A, Farber S, Wheeler D, 2011, "A simulation-based study of geographically weighted regression as a method for investigating spatially varying relationships", Environment and Planning A 43(12) 2992-3010; http://gwr.nuim.ie/

See Also

gwr.sel, gwr.gauss, gwr.bisquare

Examples

data(columbus, package="spData")
col.lm <- lm(CRIME ~ INC + HOVAL, data=columbus)
summary(col.lm)
col.bw <- gwr.sel(CRIME ~ INC + HOVAL, data=columbus,
  coords=cbind(columbus$X, columbus$Y))
col.gauss <- gwr(CRIME ~ INC + HOVAL, data=columbus,
  coords=cbind(columbus$X, columbus$Y), bandwidth=col.bw, hatmatrix=TRUE)
col.gauss
col.d <- gwr.sel(CRIME ~ INC + HOVAL, data=columbus,
  coords=cbind(columbus$X, columbus$Y), gweight=gwr.bisquare)
col.bisq <- gwr(CRIME ~ INC + HOVAL, data=columbus,
  coords=cbind(columbus$X, columbus$Y), bandwidth=col.d, 
  gweight=gwr.bisquare, hatmatrix=TRUE)
col.bisq
data(georgia)
g.adapt.gauss <- gwr.sel(PctBach ~ TotPop90 + PctRural + PctEld + PctFB + 
 PctPov + PctBlack, data=gSRDF, adapt=TRUE)
res.adpt <- gwr(PctBach ~ TotPop90 + PctRural + PctEld + PctFB + PctPov + 
 PctBlack, data=gSRDF, adapt=g.adapt.gauss)
res.adpt
pairs(as(res.adpt$SDF, "data.frame")[,2:8], pch=".")
brks <- c(-0.25, 0, 0.01, 0.025, 0.075)
cols <- grey(5:2/6)
plot(res.adpt$SDF, col=cols[findInterval(res.adpt$SDF$PctBlack, brks,
 all.inside=TRUE)])

# simulation scenario with patterned dependent variable
set.seed(1)
X0 <- runif(nrow(gSRDF)*3)
X1 <- matrix(sample(X0), ncol=3)
X1 <- prcomp(X1, center=FALSE, scale.=FALSE)$x
gSRDF$X1 <- X1[,1]
gSRDF$X2 <- X1[,2]
gSRDF$X3 <- X1[,3]
bw <- gwr.sel(PctBach ~ X1 + X2 + X3, data=gSRDF, verbose=FALSE)
out <- gwr(PctBach ~ X1 + X2 + X3, data=gSRDF, bandwidth=bw, hatmatrix=TRUE)
out
spplot(gSRDF, "PctBach", col.regions=grey.colors(20))
spplot(gSRDF, c("X1", "X2", "X3"), col.regions=grey.colors(20))
# pattern in the local coefficients
spplot(out$SDF, c("X1", "X2", "X3"), col.regions=grey.colors(20))
# but no "significant" pattern
spplot(out$SDF, c("X1_se", "X2_se", "X3_se"), col.regions=grey.colors(20))
out$SDF$X1_t <- out$SDF$X1/out$SDF$X1_se
out$SDF$X2_t <- out$SDF$X2/out$SDF$X2_se
out$SDF$X3_t <- out$SDF$X3/out$SDF$X3_se
spplot(out$SDF, c("X1_t", "X2_t", "X3_t"), col.regions=grey.colors(20))
# simulation scenario with random dependent variable
yrn <- rnorm(nrow(gSRDF))
gSRDF$yrn <- sample(yrn)
bw <- gwr.sel(yrn ~ X1 + X2 + X3, data=gSRDF, verbose=FALSE)
# bandwidth selection maxes out at 620 km, equal to upper bound
# of line search
out <- gwr(yrn ~ X1 + X2 + X3, data=gSRDF, bandwidth=bw, hatmatrix=TRUE)
out
spplot(gSRDF, "yrn", col.regions=grey.colors(20))
spplot(gSRDF, c("X1", "X2", "X3"), col.regions=grey.colors(20))
# pattern in the local coefficients
spplot(out$SDF, c("X1", "X2", "X3"), col.regions=grey.colors(20))
# but no "significant" pattern
spplot(out$SDF, c("X1_se", "X2_se", "X3_se"), col.regions=grey.colors(20))
out$SDF$X1_t <- out$SDF$X1/out$SDF$X1_se
out$SDF$X2_t <- out$SDF$X2/out$SDF$X2_se
out$SDF$X3_t <- out$SDF$X3/out$SDF$X3_se
spplot(out$SDF, c("X1_t", "X2_t", "X3_t"), col.regions=grey.colors(20))
# end of simulations


data(meuse)
coordinates(meuse) <- c("x", "y")
meuse$ffreq <- factor(meuse$ffreq)
data(meuse.grid)
coordinates(meuse.grid) <- c("x", "y")
meuse.grid$ffreq <- factor(meuse.grid$ffreq)
gridded(meuse.grid) <- TRUE
xx <- gwr(cadmium ~ dist, meuse, bandwidth = 228, hatmatrix=TRUE)
xx
x <- gwr(cadmium ~ dist, meuse, bandwidth = 228, fit.points = meuse.grid,
 predict=TRUE, se.fit=TRUE, fittedGWRobject=xx)
x
spplot(x$SDF, "pred")
spplot(x$SDF, "pred.se")

## Not run: 
  g.bw.gauss <- gwr.sel(PctBach ~ TotPop90 + PctRural + PctEld + PctFB +
    PctPov + PctBlack, data=gSRDF)
  res.bw <- gwr(PctBach ~ TotPop90 + PctRural + PctEld + PctFB + PctPov +
    PctBlack, data=gSRDF, bandwidth=g.bw.gauss)
  res.bw
  pairs(as(res.bw$SDF, "data.frame")[,2:8], pch=".")
  plot(res.bw$SDF, col=cols[findInterval(res.bw$SDF$PctBlack, brks,
    all.inside=TRUE)])
  g.bw.gauss <- gwr.sel(PctBach ~ TotPop90 + PctRural + PctEld + PctFB +
    PctPov + PctBlack, data=gSRDF, longlat=TRUE)
  data(gSRouter)
#  require(maptools)
#  SG <- GE_SpatialGrid(gSRouter, maxPixels = 100)
  if (require(sf, quietly=TRUE) && require(stars, quietly=TRUE)) {
    SG_0 <- st_as_stars(st_bbox(st_as_sf(gSRouter)), nx=87, ny=100)
    SG <- as(SG_0, "Spatial")
    SPxMASK0 <- over(SG, gSRouter)
    SGDF <- SpatialGridDataFrame(slot(SG, "grid"),
      data=data.frame(SPxMASK0=SPxMASK0),
      proj4string=CRS(proj4string(gSRouter)))
    SPxDF <- as(SGDF, "SpatialPixelsDataFrame")
    res.bw <- gwr(PctBach ~ TotPop90 + PctRural + PctEld + PctFB + PctPov +
      PctBlack, data=gSRDF, bandwidth=g.bw.gauss, fit.points=SPxDF,
      longlat=TRUE)
    res.bw
    res.bw$timings
    spplot(res.bw$SDF, "PctBlack")
    require(parallel)
    cl <- makeCluster(detectCores())
    res.bwc <- gwr(PctBach ~ TotPop90 + PctRural + PctEld + PctFB + PctPov +
      PctBlack, data=gSRDF, bandwidth=g.bw.gauss, fit.points=SPxDF,
      longlat=TRUE, cl=cl)
    res.bwc
    res.bwc$timings
    stopCluster(cl)
  }

## End(Not run)

spgwr documentation built on July 9, 2023, 6:04 p.m.