Version:	0.6-37
Date:	2024-06-10
Title:	Geographically Weighted Regression
Encoding:	UTF-8
Depends:	R (≥ 2.14), sp (≥ 0.8-3), spData (≥ 2.3.1)
Imports:	stats, methods
Suggests:	spdep, parallel, sf, knitr, rmarkdown, tinytest
Description:	Functions for computing geographically weighted regressions are provided, based on work by Chris Brunsdon, Martin Charlton and Stewart Fotheringham.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
BugReports:	https://github.com/rsbivand/spgwr/issues
VignetteBuilder:	knitr
URL:	https://github.com/rsbivand/spgwr/, https://rsbivand.github.io/spgwr/
RoxygenNote:	6.1.1
NeedsCompilation:	yes
Packaged:	2024-06-10 08:53:59 UTC; rsb
Author:	Roger Bivand [cre, aut], Danlin Yu [aut], Tomoki Nakaya [ctb], Miquel-Angel Garcia-Lopez [ctb]
Maintainer:	Roger Bivand <Roger.Bivand@nhh.no>
Repository:	CRAN
Date/Publication:	2024-06-10 13:20:02 UTC

Global tests of geographical weighted regressions

Description

Four related test statistics for comparing OLS and GWR models based on bapers by Brunsdon, Fotheringham and Charlton (1999) and Leung et al (2000), and a development from the GWR book (2002).

Usage

LMZ.F3GWR.test(go)
LMZ.F2GWR.test(x)
LMZ.F1GWR.test(x)
BFC99.gwr.test(x)
BFC02.gwr.test(x, approx=FALSE)
## S3 method for class 'gwr'
anova(object, ..., approx=FALSE)

Arguments

Details

The papers in the references give the background for the analyses of variance presented.

Value

BFC99.GWR.test, BFC02.gwr.test, LMZ.F1GWR.test and LMZ.F2GWR.test return "htest" objects, LMZ.F3GWR.test a matrix of test results.

Author(s)

Roger Bivand Roger.Bivand@nhh.no and Danlin Yu

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2002, Geographically Weighted Regression, Chichester: Wiley; http://gwr.nuim.ie/

See Also

gwr

Examples

data(columbus, package="spData")
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)
BFC99.gwr.test(col.gauss)
BFC02.gwr.test(col.gauss)
BFC02.gwr.test(col.gauss, approx=TRUE)
anova(col.gauss)
anova(col.gauss, approx=TRUE)
## Not run: 
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)
BFC99.gwr.test(col.bisq)

## End(Not run)

Georgia census data set (SpatialDataFramePolygons)

Description

The Georgia census data set from Fotheringham et al. (2002) in GPKG format.

Usage

data(georgia)

Format

A SpatialPolygonsDataFrame object.

The "data" slot is a data frame with 159 observations on the following 13 variables.

AreaKey

a numeric vector

Latitude

a numeric vector

Longitud

a numeric vector

TotPop90

a numeric vector

PctRural

a numeric vector

PctBach

a numeric vector

PctEld

a numeric vector

PctFB

a numeric vector

PctPov

a numeric vector

PctBlack

a numeric vector

ID

a numeric vector

X

a numeric vector

Y

a numeric vector

Details

Variables are from GWR3 file GeorgiaData.csv.

Source

Originally: http://www.census.gov/geo/cob/bdy/co/co90shp/co13_d90_shp.zip, currently behind: https://www.census.gov/geographies/mapping-files/time-series/geo/carto-boundary-file.1990.html choosing 1990 Census and Georgia; http://gwr.nuim.ie/

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2002, Geographically Weighted Regression: The Analysis of Spatially Varying Relationships, Chichester: Wiley.

Examples

data(georgia)
plot(gSRDF)
data(gSRouter)

Generalised geographically weighted regression

Description

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

Usage

ggwr(formula, data = list(), coords, bandwidth, gweight = gwr.Gauss,
 adapt = NULL, fit.points, family = gaussian, longlat = NULL, type = 
c("working", "deviance", "pearson", "response"))

Arguments

Value

A list of class “gwr”:

Note

The use of GWR on GLM is only at the initial proof of concept stage, nothing should be treated as an accepted method at this stage.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2002, Geographically Weighted Regression, Chichester: Wiley; http://gwr.nuim.ie/

See Also

ggwr.sel, gwr

Examples

if (require(sf)) {
xx <- as(st_read(system.file("shapes/sids.gpkg", package="spData")[1]), "Spatial")
bw <- 144.4813
## Not run: 
bw <- ggwr.sel(SID74 ~ I(NWBIR74/BIR74) + offset(log(BIR74)), data=xx,
  family=poisson(), longlat=TRUE)

## End(Not run)
nc <- ggwr(SID74 ~ I(NWBIR74/BIR74) + offset(log(BIR74)), data=xx,
  family=poisson(), longlat=TRUE, bandwidth=bw)
nc
## Not run: 
nc <- ggwr(SID74 ~ I(NWBIR74/10000) + offset(log(BIR74)), data=xx,
  family=poisson(), longlat=TRUE, bandwidth=bw)
nc
nc <- ggwr(SID74 ~ I(NWBIR74/10000) + offset(log(BIR74)), data=xx,
  family=quasipoisson(), longlat=TRUE, bandwidth=bw)
nc

## End(Not run)
}

Crossvalidation of bandwidth for generalised GWR

Description

The function finds a bandwidth for a given generalised geographically weighted regression by optimzing a selected function. For cross-validation, this scores the root mean square prediction error for the generalised geographically weighted regressions, choosing the bandwidth minimizing this quantity.

Usage

ggwr.sel(formula, data = list(), coords, adapt = FALSE, gweight = gwr.Gauss,
 family = gaussian, verbose = TRUE, longlat = NULL, RMSE=FALSE,
 tol=.Machine$double.eps^0.25)

Arguments

Value

returns the cross-validation bandwidth.

Note

The use of GWR on GLM is only at the initial proof of concept stage, nothing should be treated as an accepted method at this stage.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2002, Geographically Weighted Regression, Chichester: Wiley; http://gwr.nuim.ie/

See Also

gwr.sel, ggwr

Examples

if (require(sf)) {
xx <- as(st_read(system.file("shapes/sids.gpkg", package="spData")[1]), "Spatial")
bw <- ggwr.sel(SID74 ~ I(NWBIR74/BIR74) + offset(log(BIR74)), data=xx,
  family=poisson(), longlat=TRUE)
bw
}

Adaptive kernel for GWR

Description

The function constructs weights using an adaptive kernal for geographically weighted regression

Usage

gw.adapt(dp, fp, quant, longlat=FALSE)

Arguments

Value

a vector of weights

Author(s)

Roger Bivand Roger.Bivand@nhh.no

Geographically weighted local statistics

Description

The function provides an implementation of geographically weighted local statistics based on Chapter 7 of the GWR book - see references. Local means, local standard deviations, local standard errors of the mean, standardised differences of the global and local means, and local covariances and if requested correlations, are reported for the chosed fixed or adaptive bandwidth and weighting function.

Usage

gw.cov(x, vars, fp, adapt = NULL, bw, gweight = gwr.bisquare,
 cor = TRUE, var.term = FALSE, longlat = NULL)

Arguments

Value

If argument fp is given, and it is a SpatialPixels object, a SpatialPixelsDataFrame is returned, if it is any other coordinate object, a SpatialPointsDataFrame is returned. If argument fp is not given, the object returned will be the class of object x. The data slot will contain a data frame with local means, local standard deviations, local standard errors of the mean, standardised differences of the global and local means, and local covariances and if requested correlations.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2002, Geographically Weighted Regression, Chichester: Wiley (chapter 7); http://gwr.nuim.ie/

See Also

gwr

Examples

data(georgia)
SRgwls <- gw.cov(gSRDF, vars=6:11, bw=2, longlat=FALSE)
names(SRgwls$SDF)
spplot(SRgwls$SDF, "mean.PctPov")
spplot(SRgwls$SDF, "sd.PctPov")
spplot(SRgwls$SDF, "sem.PctPov")
spplot(SRgwls$SDF, "diff.PctPov")
spplot(SRgwls$SDF, "cor.PctPov.PctBlack.")
SRgwls <- gw.cov(gSRDF, vars=6:11, bw=150, longlat=TRUE)
names(SRgwls$SDF)
spplot(SRgwls$SDF, "mean.PctPov")
spplot(SRgwls$SDF, "sd.PctPov")
spplot(SRgwls$SDF, "sem.PctPov")
spplot(SRgwls$SDF, "diff.PctPov")
spplot(SRgwls$SDF, "cor.PctPov.PctBlack.")
data(gSRouter)
#gGrid <- sample.Polygons(slot(gSRouter, "polygons")[[1]], 5000,
gGrid <- spsample(slot(gSRouter, "polygons")[[1]], 5000,
  type="regular")
gridded(gGrid) <- TRUE
SGgwls <- gw.cov(gSRDF, vars=6:11, fp=gGrid, bw=150, longlat=TRUE)
names(SGgwls$SDF)
spplot(SGgwls$SDF, "mean.PctPov")
spplot(SGgwls$SDF, "sd.PctPov")
spplot(SGgwls$SDF, "sem.PctPov")
spplot(SGgwls$SDF, "diff.PctPov")
spplot(SGgwls$SDF, "cor.PctPov.PctBlack.")

set.seed(1)
pts <- data.frame(x=runif(100, 0, 5), y=runif(100, 0, 5), z=rnorm(100))
coordinates(pts) <- c("x", "y")
proj4string(pts) <- CRS("+proj=longlat +ellps=WGS84")
fps <- SpatialPoints(cbind(x=runif(100, 0, 5), y=runif(100, 0, 5)),
 proj4string=CRS("+proj=longlat +ellps=WGS84"))
t0 <- gw.cov(pts, "z", fp=fps, adapt=0.2, longlat=TRUE)

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

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”:

Author(s)

Roger Bivand Roger.Bivand@nhh.no

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)

GWR bisquare weights function

Description

The function returns a vector of weights using the bisquare scheme:

w_{ij}(g) = (1 - (d_{ij}^2/d^2))^2

if d_{ij} <= d else w_{ij}(g) = 0, where d_{ij} are the distances between the observations and d is the distance at which weights are set to zero.

Usage

gwr.bisquare(dist2, d)

Arguments

Value

matrix of weights.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2000, Quantitative Geography, London: Sage; C. Brunsdon, A.Stewart Fotheringham and M.E. Charlton, 1996, "Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity", Geographical Analysis, 28(4), 281-298; http://gwr.nuim.ie/

See Also

gwr.sel, gwr

Examples

plot(seq(-10,10,0.1), gwr.bisquare(seq(-10,10,0.1)^2, 6.0), type="l")

GWR Gaussian weights function

Description

The gwr.gauss function returns a vector of weights using the Gaussian scheme:

w(g) = e^{{-(d/h)}^2}

where d are the distances between the observations and h is the bandwidth.

The default (from release 0.5) gwr.Gauss function returns a vector of weights using the Gaussian scheme:

w(g) = e^{-(1/2) {{(d/h)}^2}}

Usage

gwr.gauss(dist2, bandwidth)
gwr.Gauss(dist2, bandwidth)

Arguments

Value

vector of weights.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2000, Quantitative Geography, London: Sage; C. Brunsdon, A.Stewart Fotheringham and M.E. Charlton, 1996, "Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity", Geographical Analysis, 28(4), 281-298; http://gwr.nuim.ie/

See Also

gwr.sel, gwr

Examples

plot(seq(-10,10,0.1), gwr.Gauss(seq(-10,10,0.1)^2, 3.5), type="l")

Moran's I for gwr objects

Description

The function returns Leung et al. (2000) three moment approximation for Moran's I, for a gwr object calculated with argument hatmatrix=TRUE. This implementation should not be regarded as authoritative, as it involves assumptions about implied methods and about estimated degrees of freedom.

Usage

gwr.morantest(x, lw, zero.policy = FALSE)

Arguments

Value

a “htest” object with the results of testing the GWR residuals

Author(s)

Roger Bivand

References

Leung Y, Mei C-L, Zhang W-X 2000 Testing for spatial autocorrelation among the residuals of the geographically weighted regression, Environment and Planning A, 32, 871-890.

Examples

if (suppressWarnings(require(spData)) && suppressWarnings(require(spdep))) {
  data(columbus, package="spData")
  bw <- gwr.sel(CRIME ~ INC + HOVAL, data=columbus, coords=coords)
  col0 <- gwr(CRIME ~ INC + HOVAL, data=columbus, coords=coords,
    bandwidth=bw, hatmatrix=TRUE)
  gwr.morantest(col0, nb2listw(col.gal.nb))
}

Crossvalidation of bandwidth for geographically weighted regression

Description

The function finds a bandwidth for a given geographically weighted regression by optimzing a selected function. For cross-validation, this scores the root mean square prediction error for the geographically weighted regressions, choosing the bandwidth minimizing this quantity.

Usage

gwr.sel(formula, data=list(), coords, adapt=FALSE, gweight=gwr.Gauss,
 method = "cv", verbose = TRUE, longlat=NULL, RMSE=FALSE, weights,
 tol=.Machine$double.eps^0.25, show.error.messages = FALSE)

Arguments

Details

If the regression contains little pattern, the bandwidth will converge to the upper bound of the line search, which is the diagonal of the bounding box of the data point coordinates for “adapt=FALSE”, and 1 for “adapt=TRUE”; see the simulation block in the examples below.

Value

returns the cross-validation bandwidth.

Note

Use of method="aic" results in the creation of an n by n matrix, and should not be chosen when n is large.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

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.bisquare, gwr.gauss

Examples

data(columbus, package="spData")
gwr.sel(CRIME ~ INC + HOVAL, data=columbus,
  coords=cbind(columbus$X, columbus$Y))
gwr.sel(CRIME ~ INC + HOVAL, data=columbus,
  coords=cbind(columbus$X, columbus$Y), gweight=gwr.bisquare)
## Not run: 
data(georgia)
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]
yrn <- rnorm(nrow(gSRDF))
gSRDF$yrn <- sample(yrn)
bw <- gwr.sel(yrn ~ X1 + X2 + X3, data=gSRDF, method="cv", adapt=FALSE, verbose=FALSE)
bw
bw <- gwr.sel(yrn ~ X1 + X2 + X3, data=gSRDF, method="aic", adapt=FALSE, verbose=FALSE)
bw
bw <- gwr.sel(yrn ~ X1 + X2 + X3, data=gSRDF, method="cv", adapt=TRUE, verbose=FALSE)
bw
bw <- gwr.sel(yrn ~ X1 + X2 + X3, data=gSRDF, method="aic", adapt=TRUE, verbose=FALSE)
bw

## End(Not run)

GWR tricube weights function

Description

The function returns a vector of weights using the tricube scheme:

w_{ij}(g) = (1 - (d_{ij}/d)^3)^3

if d_{ij} <= d else w_{ij}(g) = 0, where d_{ij} are the distances between the observations and d is the distance at which weights are set to zero.

Usage

gwr.tricube(dist2, d)

Arguments

Value

matrix of weights.

Author(s)

Roger Bivand Roger.Bivand@nhh.no

References

Fotheringham, A.S., Brunsdon, C., and Charlton, M.E., 2000, Quantitative Geography, London: Sage; C. Brunsdon, A.Stewart Fotheringham and M.E. Charlton, 1996, "Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity", Geographical Analysis, 28(4), 281-298; http://gwr.nuim.ie/

See Also

gwr.sel, gwr

Examples

plot(seq(-10,10,0.1), gwr.tricube(seq(-10,10,0.1)^2, 6.0), type="l")