Find extreme eigenvalues of binary symmetric spatial weights

The functions find extreme eigenvalues of binary symmetric spatial weights, when these form planar graphs; general weights are not permiited. l_max finds the largest eigenvalue using Rayleigh quotient methods of any “listw” object. lextrB first calls l_max, and uses its output to find the smallest eigenvalue in addition for binary symmetric spatial weights. lextrW extends these to find the smallest eigenvalue for intrinsically symmetric row-standardized binary weights matrices (transformed to symmetric through similarity internally). lextrS does the same for variance-stabilized (“S” style) intrinsically symmetric binary weights matrices (transformed to symmetric through similarity internally).

Usage

lextrB(lw, zero.policy = TRUE, control = list())
lextrW(lw, zero.policy=TRUE, control=list())
lextrS(lw, zero.policy=TRUE, control=list())
l_max(lw, zero.policy=TRUE, control=list())

Arguments

lw: a binary symmetric listw object from, for example, nb2listw with style “B” for lextrB, style “W” for lextrW and style “S” for lextrS; for l_max, the object may be asymmetric and does not have to be binary
zero.policy: default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
control: a list of control arguments

Control arguments

trace: report values in while loops, default NULL assuming FALSE; logical
tol: tolerance for breaking while loops, default .Machine$double.eps^(1/2); numeric
maxiter: maximum number of iterations in while loops, default 6 * (length(lw$neighbours) - 2; integer
useC: use C code, default TRUE, logical (not in l_max)

Value

The functions return approximations to the extreme eigenvalues with the eigenvectors returned as attributes of this object.

References

Griffith, D. A. (2004). Extreme eigenfunctions of adjacency matrices for planar graphs employed in spatial analyses. Linear Algebra and its Applications, 388:201–219.

Author

Roger Bivand, Yongwan Chun, Daniel Griffith

Note

It may be necessary to modify control arguments if warnings about lack of convergence are seen.

Examples

data(boston, package="spData")
#require(spdep, quietly=TRUE)
ab.listb <- spdep::nb2listw(boston.soi, style="B")
er <- range(eigenw(ab.listb))
er
#> [1] -3.039465  5.306204
res_1 <- lextrB(ab.listb)
c(res_1)
#>  lambda_n  lambda_1 
#> -3.039374  5.306203 
run <- FALSE
if (require("RSpectra", quietly=TRUE)) run <- TRUE
if (run) {
B <- as(ab.listb, "CsparseMatrix")
eigs(B, k=1, which="SR")$values
}
#> [1] -3.039465
if (run) {
eigs(B, k=1, which="LR")$values
}
#> [1] 5.306204
k5 <- spdep::knn2nb(spdep::knearneigh(boston.utm, k=5))
c(l_max(spdep::nb2listw(k5, style="B")))
#> [1] 5
max(Re(eigenw(spdep::nb2listw(k5, style="B"))))
#> [1] 5
c(l_max(spdep::nb2listw(k5, style="C")))
#> [1] 1
max(Re(eigenw(spdep::nb2listw(k5, style="C"))))
#> [1] 1
ab.listw <- spdep::nb2listw(boston.soi, style="W")
er <- range(eigenw(similar.listw(ab.listw)))
er
#> [1] -0.9708644  1.0000000
res_1 <- lextrW(ab.listw)
c(res_1)
#>   lambda_n   lambda_1 
#> -0.9708644  0.9999991 
if (run) {
B <- as(similar.listw(ab.listw), "CsparseMatrix")
eigs(B, k=1, which="SR")$values
}
#> [1] -0.9708644
if (run) {
eigs(B, k=1, which="LR")$values
}
#> [1] 1
# \dontrun{
ab.listw <- spdep::nb2listw(boston.soi, style="S")
er <- range(eigenw(similar.listw(ab.listw)))
er
#> [1] -0.723495  1.110373
res_1 <- lextrS(ab.listw)
c(res_1)
#>   lambda_n   lambda_1 
#> -0.7230376  1.1103694 
# }
if (run) {
B <- as(similar.listw(ab.listw), "CsparseMatrix")
eigs(B, k=1, which="SR")$values
}
#> [1] -0.723495
if (run) {
eigs(B, k=1, which="LR")$values
}
#> [1] 1.110373