Estimation of entropy measures for categorical variables with spatial correlation

Entropy is a measure of heterogeneity widely used in applied sciences, often when data are collected over space. Recently, a number of approaches has been proposed to include spatial information in entropy. The aim of entropy is to synthesize the observed data in a single, interpretable number. In other studies the objective is, instead, to use data for entropy estimation; several proposals can be found in the literature, which basically are corrections of the estimator based on substituting the involved probabilities with proportions. In this case, independence is assumed and spatial correlation is not considered. We propose a path for spatial entropy estimation: instead of intervening on the global entropy estimator, we focus on improving the estimation of its components, i.e. the probabilities, in order to account for spatial effects. Once probabilities are suitably evaluated, estimating entropy is straightforward since it is a deterministic function of the distribution. Following a Bayesian approach, we derive the posterior probabilities of a binomial distribution for categorical variables, accounting for spatial correlation. A posterior distribution for entropy can be obtained, which may be synthesized as wished and displayed as an entropy surface for the area under study.