spdep (version 0.4-9)

invIrM: Compute SAR generating operator

Description

Computes the matrix used for generating simultaneous autoregressive random variables, for a given value of rho, a neighbours list object, a chosen coding scheme style, and optionally a list of general weights corresponding to neighbours.

Usage

invIrM(neighbours, rho, glist=NULL, style="W", method="solve",
 feasible=NULL)
invIrW(listw, rho, method="solve", feasible=NULL)

Arguments

neighbours

an object of class nb

rho

autoregressive parameter

glist

list of general weights corresponding to neighbours

style

style can take values W, B, C, and S

method

default solve, can also take value chol

feasible

if NULL, the given value of rho is checked to see if it lies within its feasible range, if TRUE, the test is not conducted

listw

a listw object from for example nb2listw

Value

• An nxn matrix with a "call" attribute.

Details

The function generates the full weights matrix V, checks that rho lies in its feasible range between 1/min(eigen(V)) and 1/max(eigen(V)), and returns the nxn inverted matrix $$(I - \rho V)^{-1}$$. With method=chol, Cholesky decomposition is used, thanks to contributed code by Markus Reder and Werner Mueller.

References

Tiefelsdorf, M., Griffith, D. A., Boots, B. 1999 A variance-stabilizing coding scheme for spatial link matrices, Environment and Planning A, 31, pp. 165-180; Tiefelsdorf, M. 2000 Modelling spatial processes, Lecture notes in earth sciences, Springer, p. 76; Haining, R. 1990 Spatial data analysis in the social and environmental sciences, Cambridge University Press, p. 117; Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 152; Reder, M. and Mueller, W. (2007) An Improvement of the invIrM Routine of the Geostatistical R-package spdep by Cholesky Inversion, Statistical Projects, LV No: 238.205, SS 2006, Department of Applied Statistics, Johannes Kepler University, Linz

See Also

nb2listw

Examples

nb7rt <- cell2nb(7, 7, torus=TRUE)
set.seed(1)
x <- matrix(rnorm(500*length(nb7rt)), nrow=length(nb7rt))
res0 <- apply(invIrM(nb7rt, rho=0.0, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res2 <- apply(invIrM(nb7rt, rho=0.2, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res4 <- apply(invIrM(nb7rt, rho=0.4, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res6 <- apply(invIrM(nb7rt, rho=0.6, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res8 <- apply(invIrM(nb7rt, rho=0.8, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res9 <- apply(invIrM(nb7rt, rho=0.9, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
plot(density(res9), col="red", xlim=c(-0.01, max(density(res9)$x)),
  ylim=range(density(res0)$y),
  xlab="estimated variance of the mean",
  main=expression(paste("Effects of spatial autocorrelation for different ",
    rho, "values")))
lines(density(res0), col="black")
lines(density(res2), col="brown")
lines(density(res4), col="green")
lines(density(res6), col="orange")
lines(density(res8), col="pink")
legend(c(-0.02, 0.01), c(7, 25),
 legend=c("0.0", "0.2", "0.4", "0.6", "0.8", "0.9"),
 col=c("black", "brown", "green", "orange", "pink", "red"), lty=1, bty="n")
system.time(invIrM(nb7rt, rho=0.9, method="chol", feasible=TRUE))
system.time(invIrM(nb7rt, rho=0.9, method="chol", feasible=NULL))
system.time(invIrM(nb7rt, rho=0.9, method="solve", feasible=TRUE))
system.time(invIrM(nb7rt, rho=0.9, method="solve", feasible=NULL))