Solved – Why is Moran’s I not equal to “-1” in perfectly dispersed point pattern

autocorrelationpattern recognitionrspatial

Is wikipedia wrong…or I don't understand it?

Wikipedia: The white and black squares ("chess pattern") are perfectly dispersed so Moran's I would
be −1. If the white squares were stacked to one half of the board and
the black squares to the other, Moran's I would be close to +1. A
random arrangement of square colors would give Moran's I a value that
is close to 0.

# Example data:
x_coor<-rep(c(1:8), each=8)
y_coor<-rep(c(1:8), length=64)
my.values<-rep(c(1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1), length=64)
rbPal <- colorRampPalette(c("darkorchid","darkorange"))
my.Col <- rbPal(10)[as.numeric(cut(my.values,breaks = 10))]

# plot the point pattern...
plot(y_coor,x_coor,col = my.Col, pch=20, cex=8, xlim=c(0,9),ylim=c(0,9))

So as you can see points are perfectly dispersed

# Distance matrix
my.dists <- as.matrix(dist(cbind(x_coor,y_coor)))
# ...inversed distance matrix
my.dists.inv <- 1/my.dists
# diagonals are "0"
diag(my.dists.inv) <- 0

Moran's I computation library(ape)

Moran.I(my.values, my.dists.inv)
$observed
[1] -0.07775248

$expected
[1] -0.01587302

$sd
[1] 0.01499786

$p.value
[1] 3.693094e-05

Why I get observed = -0.07775248 instead of "-1".

Best Answer

Wikipedia, specifically http://en.wikipedia.org/wiki/Moran's_I as I write, is very wrong on this point.

Although $I$ is a measure of autocorrelation, it is not an exact analogue of any correlation coefficient bounded by $-1$ and $1$. The bounds, unfortunately, are much more complicated.

For a much more careful analysis, see

de Jong, P., Sprenger, C., van Veen, F. 1984. On extreme Values of Moran's $I$ and Geary's $c$. Geographical Analysis 16: 17-24. http://onlinelibrary.wiley.com/doi/10.1111/j.1538-4632.1984.tb00797.x/pdf

I haven't tried to check your calculation.

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