Covariances among join-count spatial autocorrelation measures

Spatial distributions of biological variables are often well-characterized with pairwise measures of spatial autocorrelation. In this article, the probability theory for products and covariances of join-count spatial autocorrelation measures are developed for spatial distributions of multiple nominal (e.g. species or genotypes) types. This more fully describes the joint distributions of pairwise measures in spatial distributions of multiple (i.e. more than two) types. An example is given on how the covariances can be used for finding standard errors of weighted averages of join-counts in spatial autocorrelation analysis of more than two types, as is typical for genetic data for multiallelic loci