spdep (version 0.1-10)

localG: G and Gstar local spatial statistics

Description

The local spatial statistic G is calculated for each zone based on the spatial weights object used. The value returned is a Z-value, and may be used as a diagnostic tool. High positive values indicate the posibility of a local cluster of high values of the variable being analysed, very low relative values a similar cluster of low values. For inference, a Bonferroni-type test is suggested in the references, where tables of critical values may be found (see also details below).

Usage

localG(x, listw, zero.policy=FALSE, spChk=NULL)

Arguments

x

a numeric vector the same length as the neighbours list in listw

listw

a listw object created for example by nb2listw

zero.policy

if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA

spChk

should the data vector names be checked against the spatial objects for identity integrity, TRUE, or FALSE, default NULL to use get.spChkOption()

Value

• A vector of G or Gstar values, with attributes "gstari" set to TRUE or FALSE, "call" set to the function call, and class "localG".

Details

If the neighbours member of listw has a "self.included" attribute set to TRUE, the Gstar variant, including the self-weight $w_{ii} > 0$, is calculated and returned. The returned vector will have a "gstari" attribute set to TRUE. Self-weights can be included by using the include.self function in the spweights package before converting the neighbour list to a spatial weights list with nb2listw as shown below in the example.

The critical values of the statistic under assumptions given in the references for the 95th percentile are for n=1: 1.645, n=50: 3.083, n=100: 3.289, n=1000: 3.886.

References

Ord, J. K. and Getis, A. 1995 Local spatial autocorrelation statistics: distributional issues and an application. Geographical Analysis, 27, 286--306; Getis, A. and Ord, J. K. 1996 Local spatial statistics: an overview. In P. Longley and M. Batty (eds) Spatial analysis: modelling in a GIS environment (Cambridge: Geoinformation International), 261--277.

Examples

data(getisord)
xycoords <- cbind(xyz$x, xyz$y)
nb30 <- dnearneigh(xycoords, 0, 30)
G30 <- localG(spNamedVec("val", xyz), nb2listw(nb30, style="B"))
G30[length(xyz$val)-136]
nb60 <- dnearneigh(xycoords, 0, 60)
G60 <- localG(spNamedVec("val", xyz), nb2listw(nb60, style="B"))
G60[length(xyz$val)-136]
nb90 <- dnearneigh(xycoords, 0, 90)
G90 <- localG(spNamedVec("val", xyz), nb2listw(nb90, style="B"))
G90[length(xyz$val)-136]
nb120 <- dnearneigh(xycoords, 0, 120)
G120 <- localG(spNamedVec("val", xyz), nb2listw(nb120, style="B"))
G120[length(xyz$val)-136]
nb150 <- dnearneigh(xycoords, 0, 150)
G150 <- localG(spNamedVec("val", xyz), nb2listw(nb150, style="B"))
G150[length(xyz$val)-136]
brks <- seq(-5,5,1)
cm.col <- cm.colors(length(brks)-1)
image(x, y, t(matrix(G30, nrow=16, ncol=16, byrow=TRUE)),
  breaks=brks, col=cm.col, asp=1)
text(xyz$x, xyz$y, round(G30, digits=1), cex=0.7)
polygon(c(195,225,225,195), c(195,195,225,225), lwd=2)
title(main=expression(paste("Values of the ", G[i], "statistic")))
G30s <- localG(spNamedVec("val", xyz), nb2listw(include.self(nb30),
 style="B"))
cat("value according to Getis and Ord's eq. 14.2, p. 263 (1996)
")
G30s[length(xyz$val)-136]
cat(paste("value given by Getis and Ord (1996), p. 267",
  "(division by n-1 rather than n 
 in variance)
"))
G30s[length(xyz$val)-136] *
  (sqrt(sum(scale(xyz$val, scale=FALSE)^2)/length(xyz$val)) /
  sqrt(var(xyz$val)))
image(x, y, t(matrix(G30s, nrow=16, ncol=16, byrow=TRUE)),
  breaks=brks, col=cm.col, asp=1)
text(xyz$x, xyz$y, round(G30s, digits=1), cex=0.7)
polygon(c(195,225,225,195), c(195,195,225,225), lwd=2)
title(main=expression(paste("Values of the ", G[i]^"*", "statistic")))