inference of local Gi* using permutations

Hello,

I am currently using the localG () in spdep package, I was wondering if we
can have a conditional permutation-based inference to get the P value for
every Gi*. I saw that a Mote Carlo simulation is used in Moran.MC(), and I
borrowed the following codes from this function and tried to see if I can
do a permutation for localG():

pvals<-matrix(0, nrow = V, ncol = 1)
for (i in 1:V){
 rankresi<-rank(res[i, ])
 xranki <- rankresi[length(res[i, ])]
 diffi <- nsim - xranki
 diffi <- ifelse(diffi > 0, diffi, 0)
 pvali <- punif((diffi + 1)/(nsim + 1))
 pvals[i,]<-pvali
}

Monte Carlo or Hope type tests or permutation bootstraps work like 
analytical permutation in the global case, and are redundant for that 
reason. Monte Carlo (conditional permutation) in the local case requires 
that there is no global autocorrelation or other mis-specifications.

Your code snippet is not an example, and I think is not how permutation is 
actually done. In the local case in ArcGIS, GeoDa, etc., you fix the value 
of interest at i, randomly re-assign all the values to the other 
observations, and recompute the local indicator at i, doing this say 499 
times. Then you move on to the next i, and repeat. In your snippet, we 
cannot see where res is coming from. Is this an n by nsim matrix from nsim 
runs? Might you be needing to compare the sample values with the observed 
value? If you are outputting Z-values of G_i anyway, you may need to step 
back to the actual G_i (here from spdep):

attr(localG(..., return_internals=TRUE), "internals")[,1]

Then

mns <- apply(res, 1, mean)
sds <- apply(res, 1, sd)
permZ <- (obs_localG - mns)/sds

are z values that can be considered as drawn from the normal. However, I 
advise against inference unless the assumptions are met, and p-values must 
be adjusted anyway.

Roger

After running these codes with several different datasets, I found that all
the negative Gi*s have very high P values say 0.999 with 999 permutations,
meaning that there are no significant cold spots. Where is the problem? How
can we do conditional permutation-based inference for localG() with R
spdep? I understand the critics of permutation-based inference for local
indicators, but I just want to explore this. Thank you!

Best

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Roger Bivand
Department of Economics, Norwegian School of Economics,
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