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## MoranI.R (2008-01-14)
## Moran's I Autocorrelation Index
## Copyright 2004 Julien Dutheil, 2007-2008 Emmanuel Paradis
## This file is part of the R-package `ape'.
## See the file ../COPYING for licensing issues.
## code cleaned-up by EP (Dec. 2007)
Moran.I <- function(x, weight, scaled = FALSE, na.rm = FALSE,
alternative = "two.sided")
{
if(dim(weight)[1] != dim(weight)[2])
stop("'weight' must be a square matrix")
n <- length(x)
if(dim(weight)[1] != n)
stop("'weight' must have as many rows as observations in 'x'")
## Expected mean:
ei <- -1/(n - 1)
nas <- is.na(x)
if (any(nas)) {
if (na.rm) {
x <- x[!nas]
n <- length(x)
weight <- weight[!nas, !nas]
} else {
warning("'x' has missing values: maybe you wanted to set na.rm = TRUE?")
return(list(observed = NA, expected = ei, sd = NA, p.value = NA))
}
}
## normalizing the weights:
## Note that we normalize after possibly removing the
## missing data.
ROWSUM <- rowSums(weight)
## the following is useful if an observation has no "neighbour":
ROWSUM[ROWSUM == 0] <- 1
weight <- weight/ROWSUM # ROWSUM is properly recycled
s <- sum(weight)
m <- mean(x)
y <- x - m # centre the x's
cv <- sum(weight * y %o% y)
v <- sum(y^2)
obs <- (n/s) * (cv/v)
## Scaling:
if (scaled) {
i.max <- (n/s) * (sd(rowSums(weight) * y)/sqrt(v/(n - 1)))
obs <- obs/i.max
}
## Expected sd:
S1 <- 0.5 * sum((weight + t(weight))^2)
S2 <- sum((apply(weight, 1, sum) + apply(weight, 2, sum))^2)
## the above is the same than:
##S2 <- 0
##for (i in 1:n)
## S2 <- S2 + (sum(weight[i, ]) + sum(weight[, i]))^2
s.sq <- s^2
k <- (sum(y^4)/n) / (v/n)^2
sdi <- sqrt((n*((n^2 - 3*n + 3)*S1 - n*S2 + 3*s.sq) -
k*(n*(n - 1)*S1 - 2*n*S2 + 6*s.sq))/
((n - 1)*(n - 2)*(n - 3)*s.sq) - 1/((n - 1)^2))
alternative <- match.arg(alternative, c("two.sided", "less", "greater"))
pv <- pnorm(obs, mean = ei, sd = sdi)
if (alternative == "two.sided")
pv <- if (obs <= ei) 2*pv else 2*(1 - pv)
if (alternative == "greater") pv <- 1 - pv
list(observed = obs, expected = ei, sd = sdi, p.value = pv)
}
weight.taxo <- function(x)
{
d <- outer(x, x, "==")
diag(d) <- 0 # implicitly converts 'd' into numeric
d
}
weight.taxo2 <- function(x, y)
{
d <- outer(x, x, "==") & outer(y, y, "!=")
diag(d) <- 0
d
}
correlogram.formula <- function(formula, data = NULL, use = "all.obs")
{
err <- 'formula must be of the form "y1+...+yn ~ x1/x2/../xn"'
use <- match.arg(use, c("all.obs", "complete.obs", "pairwise.complete.obs"))
if (formula[[1]] != "~") stop(err)
lhs <- formula[[2]]
y.nms <- if (length(lhs) > 1)
unlist(strsplit(as.character(as.expression(lhs)), " \\+ "))
else as.character(as.expression(lhs))
rhs <- formula[[3]]
gr.nms <- if (length(rhs) > 1)
rev(unlist(strsplit(as.character(as.expression(rhs)), "/")))
else as.character(as.expression(rhs))
if (is.null(data)) {
## we 'get' the variables in the .GlobalEnv:
y <- as.data.frame(sapply(y.nms, get))
gr <- as.data.frame(sapply(gr.nms, get))
} else {
y <- data[y.nms]
gr <- data[gr.nms]
}
if (use == "all.obs") {
na.fail(y)
na.fail(gr)
}
if (use == "complete.obs") {
sel <- complete.cases(y, gr)
y <- y[sel]
gr <- gr[sel]
}
na.rm <- use == "pairwise.complete.obs"
foo <- function(x, gr, na.rm) {
res <- data.frame(obs = NA, p.values = NA, labels = colnames(gr))
for (i in 1:length(gr)) {
sel <- if (na.rm) !is.na(x) & !is.na(gr[, i]) else TRUE
xx <- x[sel]
g <- gr[sel, i]
w <- if (i > 1) weight.taxo2(g, gr[sel, i - 1]) else weight.taxo(g)
o <- Moran.I(xx, w, scaled = TRUE)
res[i, 1] <- o$observed
res[i, 2] <- o$p.value
}
## We need to specify the two classes; if we specify
## only "correlogram", 'res' is coerced as a list
## (data frames are of class "data.frame" and mode "list")
structure(res, class = c("correlogram", "data.frame"))
}
if (length(y) == 1) foo(y[[1]], gr, na.rm)
else structure(lapply(y, foo, gr = gr, na.rm = na.rm),
names = y.nms, class = "correlogramList")
}
plot.correlogram <-
function(x, legend = TRUE, test.level = 0.05,
col = c("grey", "red"), type = "b", xlab = "",
ylab = "Moran's I", pch = 21, cex = 2, ...)
{
BG <- col[(x$p.values < test.level) + 1]
if (pch > 20 && pch < 26) {
bg <- col
col <- CO <- "black"
} else {
CO <- BG
BG <- bg <- NULL
}
plot(1:length(x$obs), x$obs, type = type, xaxt = "n", xlab = xlab,
ylab = ylab, col = CO, bg = BG, pch = pch, cex = cex, ...)
axis(1, at = 1:length(x$obs), labels = x$labels)
if (legend)
legend("top", legend = paste(c("P >=", "P <"), test.level),
pch = pch, col = col, pt.bg = bg, pt.cex = cex, horiz = TRUE)
}
plot.correlogramList <-
function(x, lattice = TRUE, legend = TRUE,
test.level = 0.05, col = c("grey", "red"),
xlab = "", ylab = "Moran's I",
type = "b", pch = 21, cex = 2, ...)
{
n <- length(x)
obs <- unlist(lapply(x, "[[", "obs"))
pval <- unlist(lapply(x, "[[", "p.values"))
gr <- factor(unlist(lapply(x, "[[", "labels")),
ordered = TRUE, levels = x[[1]]$labels)
vars <- gl(n, nlevels(gr), labels = names(x))
BG <- col[(pval < test.level) + 1]
if (lattice) {
## trellis.par.set(list(plot.symbol=list(pch=19)))
xyplot(obs ~ gr | vars, xlab = xlab, ylab = ylab,
panel = function(x, y) {
panel.lines(x, y, lty = 2)
panel.points(x, y, cex = cex, pch = 19, col = BG)
##lattice::panel.abline(h = 0, lty = 3)
})
} else {
if (pch > 20 && pch < 26) {
bg <- col
CO <- rep("black", length(obs))
col <- "black"
} else {
CO <- BG
BG <- bg <- NULL
}
plot(as.numeric(gr), obs, type = "n", xlab = xlab,
ylab = ylab, xaxt = "n")
for (i in 1:n) {
sel <- as.numeric(vars) == i
lines(as.numeric(gr[sel]), obs[sel], type = type, lty = i,
col = CO[sel], bg = BG[sel], pch = pch, cex = cex, ...)
}
axis(1, at = 1:length(x[[i]]$obs), labels = x[[i]]$labels)
if (legend) {
legend("topright", legend = names(x), lty = 1:n, bty = "n")
legend("top", legend = paste(c("P >=", "P <"), test.level),
pch = pch, col = col, pt.bg = bg, pt.cex = cex, horiz = TRUE)
}
}
}
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