# localC: Compute Local Geary statistic In spdep: Spatial Dependence: Weighting Schemes, Statistics

 localC R Documentation

## Compute Local Geary statistic

### Description

The Local Geary is a local adaptation of Geary's C statistic of spatial autocorrelation. The Local Geary uses squared differences to measure dissimilarity unlike the Local Moran. Low values of the Local Geary indicate positive spatial autocorrelation and large refers to negative spatial autocorrelation.

Inference for the Local Geary is based on a permutation approach which compares the observed value to the reference distribution under spatial randomness. localC_perm() returns a pseudo p-value. This is not an analytical p-value and is based on the number of permutations and as such should be used with care.

### Usage

localC(x, ..., zero.policy=NULL)

## Default S3 method:
localC(x, listw, ..., zero.policy=attr(listw, "zero.policy"))

## S3 method for class 'formula'
localC(formula, data, listw, ..., zero.policy=attr(listw, "zero.policy"))

## S3 method for class 'list'
localC(x, listw, ..., zero.policy=attr(listw, "zero.policy"))

## S3 method for class 'matrix'
localC(x, listw, ..., zero.policy=attr(listw, "zero.policy"))

## S3 method for class 'data.frame'
localC(x, listw, ..., zero.policy=attr(listw, "zero.policy"))

localC_perm(x, ..., zero.policy=NULL, iseed=NULL, no_repeat_in_row=FALSE)

## Default S3 method:
localC_perm(x, listw, nsim = 499, alternative = "two.sided", ...,
zero.policy=attr(listw, "zero.policy"), iseed=NULL, no_repeat_in_row=FALSE)

## S3 method for class 'formula'
localC_perm(formula, data, listw, nsim = 499,
alternative = "two.sided", ..., zero.policy=attr(listw, "zero.policy"), iseed=NULL,
no_repeat_in_row=FALSE)


### Arguments

 x a numeric vector, numeric matrix, or list. See details for more. formula A one-sided formula determining which variables to be used. listw a listw object created for example by nb2listw. data Used when a formula is provided. A matrix or data frame containing the variables in the formula formula. nsim The number of simulations to be used for permutation test. alternative A character defining the alternative hypothesis. Must be one of "two.sided", "less" or "greater". ... other arguments passed to methods. zero.policy default attr(listw, "zero.policy") as set when listw was created, if attribute not set, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA. iseed default NULL, used to set the seed for possible parallel RNGs no_repeat_in_row default FALSE, if TRUE, sample conditionally in each row without replacements to avoid duplicate values, https://github.com/r-spatial/spdep/issues/124

### Details

The Local Geary can be extended to a multivariate context. When x is a numeric vector, the univariate Local Geary will be calculated. To calculate the multivariate Local Moran provide either a list or a matrix. When x is a list, each element must be a numeric vector of the same length and of the same length as the neighbours in listw. In the case that x is a matrix the number of rows must be the same as the length of the neighbours in listw.

While not required in the univariate context, the standardized Local Geary is calculated. The multivariate Local Geary is always standardized.

The univariate Local Geary is calculated as c_i = \sum_j w_{ij}(x_i - x_j)^2 and the multivariate Local Geary is calculated as c_{k,i} = \sum_{v=1}^{k} c_{v,i} as described in Anselin (2019).

### Value

A numeric vector containing Local Geary statistic with attribute pseudo-p when localC_perm() is used. pseudo-p is an 8 column matrix containing

 E.Ci expectation of the Local Geary statistic based on permutation sample Var.Ci variance of Local Geary based on permutation sample Z.Ci standard deviate of Local Geary based on permutation sample Pr() p-value of Local Geary statistic using pnorm() using standard deviates based on permutation sample means and standard deviations Pr() Sim rank() and punif() of observed statistic rank for [0, 1] p-values using alternative= Pr(folded) Sim the simulation folded [0, 0.5] range ranked p-value (based on https://github.com/pysal/esda/blob/4a63e0b5df1e754b17b5f1205b8cadcbecc5e061/esda/crand.py#L211-L213) Skewness the output of e1071::skewness() for the permutation samples underlying the standard deviates Kurtosis the output of e1071::kurtosis() for the permutation samples underlying the standard deviates

### Author(s)

Josiah Parry josiah.parry@gmail.com and Roger Bivand

### References

Anselin, L. (1995), Local Indicators of Spatial Association—LISA. Geographical Analysis, 27: 93-115. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.1538-4632.1995.tb00338.x")}

Anselin, L. (2019), A Local Indicator of Multivariate Spatial Association: Extending Geary's c. Geogr Anal, 51: 133-150. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/gean.12164")}

### Examples

orig <- spData::africa.rook.nb
listw <- nb2listw(orig)
x <- spData::afcon$totcon (A <- localC(x, listw)) listw1 <- nb2listw(droplinks(sym.attr.nb(orig), 3, sym=TRUE), zero.policy=TRUE) (A1 <- localC(x, listw1, zero.policy=FALSE)) (A2 <- localC(x, listw1, zero.policy=TRUE)) run <- FALSE if (require(rgeoda, quietly=TRUE)) run <- TRUE if (run) { W <- create_weights(as.numeric(length(x))) for (i in 1:length(listw$neighbours)) {
set_neighbors_with_weights(W, i, listw$neighbours[[i]], listw$weights[[i]])
update_weights(W)
}
set.seed(1)
B <- local_geary(W, data.frame(x))
all.equal(A, lisa_values(B))
}
if (run) {
set.seed(1)
C <- localC_perm(x, listw, nsim = 499, conditional=TRUE,
alternative="two.sided")
cor(ifelse(lisa_pvalues(B) < 0.5, lisa_pvalues(B), 1-lisa_pvalues(B)),
attr(C, "pseudo-p")[,6])
}
# pseudo-p values probably wrongly folded https://github.com/GeoDaCenter/rgeoda/issues/28
## Not run:
tmap_ok <- FALSE
if (require(tmap, quietly=TRUE)) tmap_ok <- TRUE
if (run) {
# doi: 10.1111/gean.12164
guerry_path <- system.file("extdata", "Guerry.shp", package = "rgeoda")
cor(st_drop_geometry(g)) #(Tab. 1)
lw <- nb2listw(poly2nb(g))
moran(g$Crm_prs, lw, n=nrow(g), S0=Szero(lw))$I
moran(g$Crm_prp, lw, n=nrow(g), S0=Szero(lw))$I
moran(g$Litercy, lw, n=nrow(g), S0=Szero(lw))$I
moran(g$Donatns, lw, n=nrow(g), S0=Szero(lw))$I
moran(g$Infants, lw, n=nrow(g), S0=Szero(lw))$I
moran(g$Suicids, lw, n=nrow(g), S0=Szero(lw))$I
}
if (run) {
o <- prcomp(st_drop_geometry(g), scale.=TRUE)
cor(st_drop_geometry(g), o$x[,1:2])^2 #(Tab. 2) } if (run) { g$PC1 <- o$x[, "PC1"] brks <- c(min(g$PC1), natural_breaks(k=6, g["PC1"]), max(g$PC1)) if (tmap_ok) tm_shape(g) + tm_fill("PC1", breaks=brks, midpoint=0) + tm_borders() # Fig. 1 else pplot(g["PC1"], breaks=brks) } if (run) { g$PC2 <- -1*o$x[, "PC2"] # eigenvalue sign arbitrary brks <- c(min(g$PC2), natural_breaks(k=6, g["PC2"]), max(g$PC2)) if (tmap_ok) tm_shape(g) + tm_fill("PC2", breaks=brks, midpoint=0) + tm_borders() # Fig. 2 else plot(g["PC2"], breaks=brks) } if (run) { w <- queen_weights(g) lm_PC1 <- local_moran(w, g["PC1"], significance_cutoff=0.01, permutations=99999) g$lm_PC1 <- factor(lisa_clusters(lm_PC1), levels=0:4,
labels=lisa_labels(lm_PC1)[1:5])
is.na(g$lm_PC1) <- g$lm_PC1 == "Not significant"
g$lm_PC1 <- droplevels(g$lm_PC1)
if (tmap_ok) tm_shape(g) + tm_fill("lm_PC1", textNA="Insignificant",
colorNA="gray95") + tm_borders() # Fig. 3
else plot(g["lm_PC1"])
}
if (run) {
set.seed(1)
lm_PC1_spdep <- localmoran_perm(g$PC1, lw, nsim=9999) q <- attr(lm_PC1_spdep, "quadr")$pysal
g$lm_PC1_spdep <- q is.na(g$lm_PC1_spdep) <- lm_PC1_spdep[,6] > 0.02 # note folded p-values
g$lm_PC1_spdep <- droplevels(g$lm_PC1_spdep)
if (tmap_ok) tm_shape(g) + tm_fill("lm_PC1_spdep", textNA="Insignificant",
colorNA="gray95") + tm_borders() # rep. Fig. 3
else plot(g["lm_PC1_spdep"])
}
if (run) {
lg_PC1 <- local_g(w, g["PC1"], significance_cutoff=0.01,
permutations=99999)
g$lg_PC1 <- factor(lisa_clusters(lg_PC1), levels=0:2, labels=lisa_labels(lg_PC1)[0:3]) is.na(g$lg_PC1) <- g$lg_PC1 == "Not significant" g$lg_PC1 <- droplevels(g$lg_PC1) if (tmap_ok) tm_shape(g) + tm_fill("lg_PC1", textNA="Insignificant", colorNA="gray95") + tm_borders() # Fig. 4 (wrong) else plot(g["lg_PC1"]) g$lg_PC1a <- cut(g$PC1, c(-Inf, mean(g$PC1), Inf), labels=c("Low", "High"))
is.na(g$lg_PC1a) <- lisa_pvalues(lg_PC1) >= 0.01 g$lg_PC1a <- droplevels(g$lg_PC1a) if (tmap_ok) tm_shape(g) + tm_fill("lg_PC1", textNA="Insignificant", colorNA="gray95") + tm_borders() # Fig. 4 (guess) else plot(g["lg_PC1"]) } if (run) { lc_PC1 <- local_geary(w, g["PC1"], significance_cutoff=0.01, permutations=99999) g$lc_PC1 <- factor(lisa_clusters(lc_PC1), levels=0:4,
labels=lisa_labels(lc_PC1)[1:5])
is.na(g$lc_PC1) <- g$lc_PC1 == "Not significant"
g$lc_PC1 <- droplevels(g$lc_PC1)
if (tmap_ok) tm_shape(g) + tm_fill("lc_PC1", textNA="Insignificant",
colorNA="gray95") + tm_borders() # Fig. 5
else plot(g["lc_PC1"])
}
if (run) {
set.seed(1)
system.time(lc_PC1_spdep <- localC_perm(g$PC1, lw, nsim=9999, alternative="two.sided")) } if (run) { if (require(parallel, quietly=TRUE)) { ncpus <- max(2L, detectCores(logical=FALSE), na.rm = TRUE)-1L # test with single core if (ncpus > 1L) ncpus <- 1L cores <- get.coresOption() set.coresOption(ncpus) system.time(lmc_PC1_spdep1 <- localC_perm(g$PC1, lw, nsim=9999,
alternative="two.sided", iseed=1))
set.coresOption(cores)
}
}
if (run) {
g$lc_PC1_spdep <- attr(lc_PC1_spdep, "cluster") is.na(g$lc_PC1_spdep) <- attr(lc_PC1_spdep, "pseudo-p")[,6] > 0.01
g$lc_PC1_spdep <- droplevels(g$lc_PC1_spdep)
if (tmap_ok) tm_shape(g) + tm_fill("lc_PC1_spdep", textNA="Insignificant",
colorNA="gray95") + tm_borders() # rep. Fig. 5
else plot(g["lc_PC1_spdep"])
}
if (run) {
g$both_PC1 <- interaction(g$lc_PC1, g$lm_PC1) g$both_PC1 <- droplevels(g$both_PC1) if (tmap_ok) tm_shape(g) + tm_fill("both_PC1", textNA="Insignificant", colorNA="gray95") + tm_borders() # Fig. 6 else plot(g["both_PC1"]) } if (run) { lc005_PC1 <- local_geary(w, g["PC1"], significance_cutoff=0.005, permutations=99999) g$lc005_PC1 <- factor(lisa_clusters(lc005_PC1), levels=0:4,
labels=lisa_labels(lc005_PC1)[1:5])
is.na(g$lc005_PC1) <- g$lc005_PC1 == "Not significant"
g$lc005_PC1 <- droplevels(g$lc005_PC1)
if (tmap_ok) tm_shape(g) + tm_fill("lc005_PC1", textNA="Insignificant",
colorNA="gray95") + tm_borders() # Fig. 7
else plot(g["lc005_PC1"])
}
if (run) {
g$lc005_PC1_spdep <- attr(lc_PC1_spdep, "cluster") is.na(g$lc005_PC1_spdep) <- attr(lc_PC1_spdep, "pseudo-p")[,6] > 0.005
g$lc005_PC1_spdep <- droplevels(g$lc005_PC1_spdep)
if (tmap_ok) tm_shape(g) + tm_fill("lc005_PC1_spdep", textNA="Insignificant",
colorNA="gray95") + tm_borders() # rep. Fig. 7
else plot(g["lc005_PC1_spdep"])
}
if (run) {
lc001_PC1 <- local_geary(w, g["PC1"], significance_cutoff=0.001,
permutations=99999)
g$lc001_PC1 <- factor(lisa_clusters(lc001_PC1), levels=0:4, labels=lisa_labels(lc001_PC1)[1:5]) is.na(g$lc001_PC1) <- g$lc001_PC1 == "Not significant" g$lc001_PC1 <- droplevels(g$lc001_PC1) if (tmap_ok) tm_shape(g) + tm_fill("lc001_PC1", textNA="Insignificant", colorNA="gray95") + tm_borders() # Fig. 8 else plot(g["lc001_PC1"]) if (run) { g$lc001_PC1_spdep <- attr(lc_PC1_spdep, "cluster")
is.na(g$lc001_PC1_spdep) <- attr(lc_PC1_spdep, "pseudo-p")[,6] > 0.001 g$lc001_PC1_spdep <- droplevels(g$lc001_PC1_spdep) if (tmap_ok) tm_shape(g) + tm_fill("lc001_PC1_spdep", textNA="Insignificant", colorNA="gray95") + tm_borders() # rep. Fig. 8 else plot(g["lc001_PC1_spdep"]) } } if (run) { lc_PC2 <- local_geary(w, g["PC2"], significance_cutoff=0.01, permutations=99999) g$lc_PC2 <- factor(lisa_clusters(lc_PC2), levels=0:4,
labels=lisa_labels(lc_PC2)[1:5])
is.na(g$lc_PC2) <- g$lc_PC2 == "Not significant"
g$lc_PC2 <- droplevels(g$lc_PC2)
if (tmap_ok) tm_shape(g) + tm_fill("lc_PC2", textNA="Insignificant",
colorNA="gray95") + tm_borders() # Fig. 9
else plot(g["lc_PC2"])
}
if (run) {
lmc_PC <- local_multigeary(w, g[c("PC1","PC2")], significance_cutoff=0.00247,
permutations=99999)
g$lmc_PC <- factor(lisa_clusters(lmc_PC), levels=0:1, labels=lisa_labels(lmc_PC)[1:2]) is.na(g$lmc_PC) <- g$lmc_PC == "Not significant" g$lmc_PC <- droplevels(g$lmc_PC) table(interaction((p.adjust(lisa_pvalues(lmc_PC), "fdr") < 0.01), g$lmc_PC))
}
if (run) {
if (tmap_ok) tm_shape(g) + tm_fill("lmc_PC", textNA="Insignificant",
colorNA="gray95") + tm_borders() # Fig. 10
else plot(g["lmc_PC"])
}
if (run) {
set.seed(1)
lmc_PC_spdep <- localC_perm(g[c("PC1","PC2")], lw, nsim=9999, alternative="two.sided")
all.equal(lisa_values(lmc_PC), c(lmc_PC_spdep))
}
if (run) {
cor(attr(lmc_PC_spdep, "pseudo-p")[,6], lisa_pvalues(lmc_PC))
}
if (run) {
g$lmc_PC_spdep <- attr(lmc_PC_spdep, "cluster") is.na(g$lmc_PC_spdep) <- p.adjust(attr(lmc_PC_spdep, "pseudo-p")[,6], "fdr") > 0.01
g$lmc_PC_spdep <- droplevels(g$lmc_PC_spdep)
if (tmap_ok) tm_shape(g) + tm_fill("lmc_PC_spdep", textNA="Insignificant",
colorNA="gray95") + tm_borders() # rep. Fig. 10
else plot(g["lmc_PC_spdep"])
}
if (run) {
lmc_vars <- local_multigeary(w, st_drop_geometry(g)[, 1:6],
significance_cutoff=0.00247, permutations=99999)
g$lmc_vars <- factor(lisa_clusters(lmc_vars), levels=0:1, labels=lisa_labels(lmc_vars)[1:2]) is.na(g$lmc_vars) <- g$lmc_vars == "Not significant" g$lmc_vars <- droplevels(g$lmc_vars) table(interaction((p.adjust(lisa_pvalues(lmc_vars), "fdr") < 0.01), g$lmc_vars))
}
if (run) {
if (tmap_ok) tm_shape(g) + tm_fill("lmc_vars", textNA="Insignificant",
colorNA="gray95") + tm_borders() # Fig. 11
else plot(g["lmc_vars"])
}
if (run) {
set.seed(1)
system.time(lmc_vars_spdep <- localC_perm(st_drop_geometry(g)[, 1:6], lw,
nsim=9999, alternative="two.sided"))
}
if (run) {
all.equal(lisa_values(lmc_vars), c(lmc_vars_spdep))
}
if (run) {
cor(attr(lmc_vars_spdep, "pseudo-p")[,6], lisa_pvalues(lmc_vars))
}
if (run) {
if (require(parallel, quietly=TRUE)) {
ncpus <- max(2L, detectCores(logical=FALSE), na.rm = TRUE)-1L
# test with single core
if (ncpus > 1L) ncpus <- 1L
cores <- get.coresOption()
set.coresOption(ncpus)
system.time(lmc_vars_spdep1 <- localC_perm(st_drop_geometry(g)[, 1:6], lw,
nsim=9999, alternative="two.sided", iseed=1))
set.coresOption(cores)
}
}
if (run) {
all.equal(lisa_values(lmc_vars), c(lmc_vars_spdep1))
}
if (run) {
cor(attr(lmc_vars_spdep1, "pseudo-p")[,6], lisa_pvalues(lmc_vars))
}
if (run) {
g$lmc_vars_spdep <- attr(lmc_vars_spdep1, "cluster") is.na(g$lmc_vars_spdep) <- p.adjust(attr(lmc_vars_spdep1, "pseudo-p")[,6], "fdr") > 0.01
g$lmc_vars_spdep <- droplevels(g$lmc_vars_spdep)
if (tmap_ok) tm_shape(g) + tm_fill("lmc_vars_spdep", textNA="Insignificant",
colorNA="gray95") + tm_borders() # rep. Fig. 11
else plot(g["lmc_vars_spdep"])
}

## End(Not run)
## Not run:
library(reticulate)
use_python("/usr/bin/python", required = TRUE)
gp <- import("geopandas")
ps <- import("libpysal")
W <- listw2mat(listw)
w <- ps$weights$full2W(W, rownames(W))
w$transform <- "R" esda <- import("esda") lM <- esda$Moran_Local(x, w)
all.equal(unname(localmoran(x, listw, mlvar=FALSE)[,1]), c(lM$Is)) # confirm x and w the same lC <- esda$Geary_Local(connectivity=w)$fit(scale(x)) # np$std missing ddof=1
n <- length(x)
D0 <- spdep:::geary.intern((x - mean(x)) / sqrt(var(x)*(n-1)/n), listw, n=n)
# lC components probably wrongly ordered https://github.com/pysal/esda/issues/192
o <- match(round(D0, 6), round(lC$localG, 6)) all.equal(c(lC$localG)[o], D0)
# simulation order not retained
lC\$p_sim[o]
attr(C, "pseudo-p")[,6]

## End(Not run)


spdep documentation built on June 22, 2024, 11:23 a.m.