Metric for spatial auto cross correlation?

I am trying to find anomalies in time series data with geolocations. For different time intervals, e.g. for some week or for some day or for 6 consecutive months, I want to learn whether there is spatial auto cross correlation (whether time series of locations which are close to each other are often correlated).

It seems that Moran's I is unsuitable for my case because it does not support cross-correlation (aka correlation of time series), it can only work with one value per location.

Bonus points if the algorithm can also find correlation with small time lag.

I will also cluster my locations by geographic coordinates or by theme and then use the algorithm to check whether clustering makes sense - whether time series inside of clusters are correlated.

Doing this in a time-space context is another barrel of monkeys (that I am not sure about), but it sounds to me like wavelet coherence (or wavelet covariance) might be worth looking into.

Wavelet coherence is to wavelet covariance as correlation is to covariance. Wavelet coherence is wavelet covariance that has been normalized (mainly for the purpose of avoiding statistically significant peaks in the cross wavelet spectrum that are due to noise). Such a method would allow you to look at cross correlation of two time series.

If you are an R user, look into the package "biwavelet" (and analogous packages). There seems to have been a surge of new packages for this sort of thing in the last several years.

Most of the empirical applications of wavelet covariance and coherence analysis have been in climate science...and matlab seems to be popular in those realms, so packages for matlab might also be useful.

Another option would be bivariate Moran's I, but I would think that wavelet coherence would be better because it sounds like it is peaks in the local spectrum that would be of interest to you...bivariate Moran's I is a global analysis (and thus would not tell you about anomalous positions).

Hope this helps!

CDA

Edit: PS--You can also do a wavelet coherence analysis for your spatial data...but both time and space is where it would get tricky.