  We derive generalized estimators for a number of spatial statistics that have
been used in the analysis of spatially resolved omics data, such as Ripley's K,
H and L functions, clustering index, and degree of clustering, which allow
these statistics to be calculated on data modelled by arbitrary random measures
(RMs). Our estimators generalize those typically used to calculate these
statistics on point process data, allowing them to be calculated on RMs which
assign continuous values to spatial regions, for instance to model protein
intensity. The clustering index (H*) compares Ripley's H function calculated
empirically to its distribution under complete spatial randomness (CSR),
leading us to consider CSR null hypotheses for RMs which are not
point-processes when generalizing this statistic. We thus consider restricted
classes of completely random measures which can be simulated directly (Gamma
processes and Marked Poisson Processes), as well as the general class of all
CSR RMs, for which we derive an exact permutation-based H* estimator. We
establish several properties of the estimators, including bounds on the
accuracy of our general Ripley K estimator, its relationship to a previous
estimator for the cross-correlation measure, and the relationship of our
generalized H* estimator to previous statistics. To test the ability of our
approach to identify spatial patterning, we use Fluorescent In Situ
Hybridization (FISH) and Immunofluorescence (IF) data to probe for mRNA and
protein subcellular localization patterns respectively in polarizing mouse
fibroblasts on micropattened cells. We observe correlated patterns of
clustering over time for corresponding mRNAs and proteins, suggesting a
deterministic effect of mRNA localization on protein localization for several
pairs tested, including one case in which spatial patterning at the mRNA level
has not been previously demonstrated.
