Package 'spgwr' reference manual

Title:	Geographically Weighted Regression
Description:	Functions for computing geographically weighted regressions are provided, based on work by Chris Brunsdon, Martin Charlton and Stewart Fotheringham.
Authors:	Roger Bivand [cre, aut] , Danlin Yu [aut] , Tomoki Nakaya [ctb], Miquel-Angel Garcia-Lopez [ctb]
Maintainer:	Roger Bivand <[email protected]>
License:	GPL (>= 2)
Version:	0.6-37
Built:	2024-10-08 05:33:04 UTC
Source:	https://github.com/rsbivand/spgwr

Georgia census data set (SpatialDataFramePolygons)

Description

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

Usage

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

Arguments

`formula`	regression model formula as in `glm`
`data`	model data frame as in `glm`, or may be a SpatialPointsDataFrame or SpatialPolygonsDataFrame object as defined in package sp
`coords`	matrix of coordinates of points representing the spatial positions of the observations
`bandwidth`	bandwidth used in the weighting function, possibly calculated by `ggwr.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)
`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
`family`	a description of the error distribution and link function to be used in the model, see `glm`
`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
`type`	the type of residuals which should be returned. The alternatives are: "working" (default), "pearson", "deviance" and "response"

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, dispersion if a "quasi"-family is used, and the residuals of type "type" in its "data" slot.
`lhat`	Leung et al. L matrix, here set to NA
`lm`	GLM global regression on the same model formula.
`bandwidth`	the bandwidth used.
`this.call`	the function call used.

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 [email protected]

References

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

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

Arguments

`formula`	regression model formula as in `glm`
`data`	model data frame as in `glm`, or may be a SpatialPointsDataFrame or SpatialPolygonsDataFrame object as defined in package sp
`coords`	matrix of coordinates of points representing the spatial positions of the observations
`adapt`	either TRUE: find the proportion between 0 and 1 of observations to include in weighting scheme (k-nearest neighbours), or FALSE — find global bandwidth
`gweight`	geographical weighting function, at present `gwr.Gauss()` default, or `gwr.gauss()`, the previous default or `gwr.bisquare()`
`family`	a description of the error distribution and link function to be used in the model, see `glm`
`verbose`	if TRUE (default), reports the progress of search for bandwidth
`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
`RMSE`	default FALSE to correspond with CV scores in newer references (sum of squared CV errors), if TRUE the previous behaviour of scoring by LOO CV RMSE
`tol`	the desired accuracy to be passed to `optimize`

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 [email protected]

References

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

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
}

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

Arguments

`dp`	data points coordinates
`fp`	fit points coordinates
`quant`	proportion of data points to include in the weights
`longlat`	if TRUE, use distances on an ellipse with WGS84 parameters

Value

a vector of weights

Author(s)

Roger Bivand [email protected]

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

Arguments

`x`	`x` should be a SpatialPolygonsDataFrame object or a SpatialPointsDataFrame object
`vars`	`vars` is a vector of column names of the data frame in the data slot of x
`fp`	`fp` if given an object inheriting from “Spatial” that contains the fit points to be used, for example a SpatialPixels object describing the grid of points to be used
`adapt`	`adapt` if given should lie between 0 and 1, and indicates the proportion of observations to be included in the weighted window - it cannot be selected automatically
`bw`	`bw` when adapt is not given, the bandwidth chosen to suit the data set - it cannot be selected automatically
`gweight`	`gweight` default gwr.bisquare - the weighting function to use
`cor`	`cor` default TRUE, report correlations in addition to covariances
`var.term`	`var.term` default FALSE, if TRUE apply a correction to the variance term
`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

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 [email protected]

References

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

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)
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, ...)
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 [email protected]

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/

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)
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)
gwr.bisquare(dist2, d)

Arguments

`dist2`	vector of squared distances between observations
`d`	distance at which weights are set to zero

Value

matrix of weights.

Author(s)

Roger Bivand [email protected]

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/

Examples

plot(seq(-10,10,0.1), gwr.bisquare(seq(-10,10,0.1)^2, 6.0), type="l")
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)
gwr.gauss(dist2, bandwidth)
gwr.Gauss(dist2, bandwidth)

Arguments

`dist2`	vector of squared distances between observations and fit point
`bandwidth`	bandwidth

Value

vector of weights.

Author(s)

Roger Bivand [email protected]

References

Examples

plot(seq(-10,10,0.1), gwr.Gauss(seq(-10,10,0.1)^2, 3.5), type="l")
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)
gwr.morantest(x, lw, zero.policy = FALSE)

Arguments

`x`	a `gwr` object returned by `gwr()` with argument hatmatrix=TRUE
`lw`	a `listw` object created for example by `nb2listw` in the spdep package
`zero.policy`	if TRUE assign zero to the lagged value of zones without neighbours, if FALSE (default) assign NA

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

Arguments

`formula`	regression model formula as in `lm`
`data`	model data frame as in `lm`, or may be a SpatialPointsDataFrame or SpatialPolygonsDataFrame object as defined in package sp
`coords`	matrix of coordinates of points representing the spatial positions of the observations
`adapt`	either TRUE: find the proportion between 0 and 1 of observations to include in weighting scheme (k-nearest neighbours), or FALSE — find global bandwidth
`gweight`	geographical weighting function, at present `gwr.Gauss()` default, or `gwr.gauss()`, the previous default or `gwr.bisquare()`
`method`	default "cv" for drop-1 cross-validation, or "aic" for AIC optimisation (depends on assumptions about AIC degrees of freedom)
`verbose`	if TRUE (default), reports the progress of search for bandwidth
`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
`RMSE`	default FALSE to correspond with CV scores in newer references (sum of squared CV errors), if TRUE the previous behaviour of scoring by LOO CV RMSE
`weights`	case weights used as in weighted least squares, beware of scaling issues — only used with the cross-validation method, probably unsafe
`tol`	the desired accuracy to be passed to `optimize`
`show.error.messages`	default FALSE; may be set to TRUE to see error messages if `gwr.sel` returns without a value

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 [email protected]

References

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)
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)
gwr.tricube(dist2, d)

Arguments

`dist2`	vector of squared distances between observations
`d`	distance at which weights are set to zero

Value

matrix of weights.

Author(s)

Roger Bivand [email protected]

References

Examples

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

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)
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

`go`, `x`, `object`	a `gwr` object returned by `gwr()`
`...`	arguments passed through (unused)
`approx`	default FALSE, if TRUE, use only (n - tr(S)) instead of (n - 2*tr(S) - tr(S'S)) as the GWR degrees of freedom

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 [email protected] and Danlin Yu

References

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

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)
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)

Package 'spgwr'

Help Index

Georgia census data set (SpatialDataFramePolygons)

Description

Usage

Format

Details

Source

References

Examples

Generalised geographically weighted regression

Description

Usage

Arguments

Value

Note

Author(s)

References

See Also

Examples

Crossvalidation of bandwidth for generalised GWR

Description

Usage

Arguments

Value

Note

Author(s)

References

See Also

Examples

Adaptive kernel for GWR

Description

Usage

Arguments

Value

Author(s)

Geographically weighted local statistics

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Geographically weighted regression

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

GWR bisquare weights function

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

GWR Gaussian weights function

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Moran's I for gwr objects

Description

Usage

Arguments

Value

Author(s)

References