## -----------------------------------------------------------------------------
set.seed(0)
d <- sample(100, 10)
d


## ----autocor1-----------------------------------------------------------------
a <- d[-length(d)]
b <- d[-1]
plot(a, b, xlab='t', ylab='t-1')
cor(a, b)


## ----autocor2-----------------------------------------------------------------
d <- sort(d)
d
a <- d[-length(d)]
b <- d[-1]
plot(a, b, xlab='t', ylab='t-1')
cor(a, b)


## ----acfplot------------------------------------------------------------------
acf(d)


## ----message=FALSE------------------------------------------------------------
library(terra)
p <- vect(system.file("ex/lux.shp", package="terra"))
p <- p[p$NAME_1=="Diekirch", ]
p$value <- c(10, 6, 4, 11, 6) 
as.data.frame(p)


## ----autocor3-----------------------------------------------------------------
par(mai=c(0,0,0,0))
plot(p, col=2:7)
xy <- centroids(p)
points(xy, cex=6, pch=20, col='white')
text(p, 'ID_2', cex=1.5)


## ----message=FALSE------------------------------------------------------------
w <- adjacent(p, symmetrical=TRUE)
class(w)
head(w)


## -----------------------------------------------------------------------------
w


## ----autocor4-----------------------------------------------------------------
plot(p, col='gray', border='blue', lwd=2)
p1 <- xy[w[,1], ]
p2 <- xy[w[,2], ]
lines(p1, p2, col='red', lwd=2)


## -----------------------------------------------------------------------------
wm <- adjacent(p, pairs=FALSE)
wm


## -----------------------------------------------------------------------------
n <- length(p)


## -----------------------------------------------------------------------------
y <- p$value
ybar <- mean(y)


## -----------------------------------------------------------------------------
dy <- y - ybar
g <- expand.grid(dy, dy)
yiyj <- g[,1] * g[,2]


## -----------------------------------------------------------------------------
yi <- rep(dy, each=n)
yj <- rep(dy)
yiyj <- yi * yj


## -----------------------------------------------------------------------------
pm <- matrix(yiyj, ncol=n)


## -----------------------------------------------------------------------------
pmw <- pm * wm
pmw


## -----------------------------------------------------------------------------
spmw <- sum(pmw) 
spmw


## -----------------------------------------------------------------------------
smw <- sum(wm)
sw  <- spmw / smw


## -----------------------------------------------------------------------------
vr <- n / sum(dy^2)


## -----------------------------------------------------------------------------
MI <- vr * sw
MI


## -----------------------------------------------------------------------------
EI <- -1/(n-1)
EI


## -----------------------------------------------------------------------------
ww <-  adjacent(p, "queen", pairs=FALSE)
ww


## -----------------------------------------------------------------------------
ac <- autocor(p$value, ww, "moran")
ac


## -----------------------------------------------------------------------------
m <- sapply(1:99, function(i) {
	autocor(sample(p$value), ww, "moran")
})
hist(m)
pval <- sum(m >= ac) / 100
pval


## -----------------------------------------------------------------------------
n <- length(p)
ms <- cbind(id=rep(1:n, each=n), y=rep(y, each=n), value=as.vector(wm * y))


## -----------------------------------------------------------------------------
ms <- ms[ms[,3] > 0, ]


## -----------------------------------------------------------------------------
ams <- aggregate(ms[,2:3], list(ms[,1]), FUN=mean)
ams <- ams[,-1]
colnames(ams) <- c('y', 'spatially lagged y')
head(ams)


## ----auto5--------------------------------------------------------------------
plot(ams, pch=20, col="red", cex=2)
reg <- lm(ams[,2] ~ ams[,1])
abline(reg, lwd=2)
abline(h=mean(ams[,2]), lt=2)
abline(v=ybar, lt=2)


## ----ngb2---------------------------------------------------------------------
coefficients(reg)[2]