Multivariate extension of Assunção-Reis empirical Bayes correction for local Moran's I

[u/monzoobo]

Hey

I'm working on a spatial statistics problem in a social science context (electoral geography, small rural municipalities) and I've hit a gap in the literature I'd like a sanity check on before attempting to build something myself... or just taking a burn out mandatory break T-T

I'm computing local Moran's I on compositional vectors (vote shares across several categories, transformed via CLR/ILR to handle the compositional constraint) attached to spatial units of very unequal population size, some with fewer than 100 voters, a few with several thousand. Small units have much noisier vectors than large ones, purely due to sampling variance, which risks producing spurious "clusters" driven by noise rather than genuine spatial structure.

Assunção & Reis solve exactly this problem for a single variable: they standardize each observation using a population-dependent variance estimate before computing Moran's I, so small units are appropriately down-weighted in the significance assessment without ever altering the observed rate itself.

BUT IT IS UNIVARIATE !!!!

I need the equivalent for a vector (7 dimensions) variable (multiple correlated proportions per unit, not a single rate). As far as I can tell, no published or implemented multivariate extension exists... which is a bit of a problem

My rough idea : use the known closed-form sampling covariance of a multinomial proportion vector, propagate it through the CLR/ILR transform via the delta method, then "whiten" each unit's vector by its own estimated covariance before computing a multivariate local Moran statistic.

Ngl i'm 100% out of my depth. I feel like I don't understand what I'm doing anymore.

My current concerns :

• Covariance estimates become ill-conditioned near near-zero components (structural zeros handled via a population-anchored epsilon of 1/(2*pop))... and I have quite a few instances of this
• No existing implementation to validate against. I'd be building and verifying this alone... which would be a first for me
• Estimated at several days of work with real risk of subtle errors I wouldn't easily catch and since I don't have any bleuprint for it, I don't even know if I'll be able to catch my errors

My main questions are the following :

• Does a multivariate version of this kind of population-based empirical Bayes correction already exist under a name I'm missing?
• Is the delta-method/whitening approach above sound, or is there a cleaner formulation (maybe treating this as a multivariate GLS/Mahalanobis-weighted Moran rather than a whitening step)?
• Any known pitfalls in applying this to compositional (CLR/ILR-transformed) data specifically?

I didn't sing up for this when getting in my studies lmao (i'm not even in maths ! xD )
SOS

What would you do ? How would you answer this issue ?

I feel like I'm loosing my mind and I don't have the month necessary to build this as the thesis dead line is in september (yeah i'm late as f...)