Reyes's I: Measuring Spatial Autocorrelation in Compositions

Compositional observations arise when measurements are recorded as parts of a whole, so that only relative information is meaningful and the natural sample space is the simplex equipped with Aitchison geometry. Despite extensive development of compositional methods, a direct analogue of Moran's $I$ for assessing spatial autocorrelation in areal compositional data has been lacking. We propose Reyes's $I$, a Moran type statistic defined through the Aitchison inner product and norm, which is invariant to scale, to permutations of the parts, and to the choice of the $\operatorname{ilr}$ contrast matrix. Under the randomization assumption, we derive an upper bound, the expected value, and the noncentral second moment, and we describe exact and Monte Carlo permutation procedures for inference. Through simulations covering identical, independent, and spatially correlated compositions under multiple covariance structures and neighborhood definitions, we show that Reyes's $I$ provides stable behavior, competitive calibration, and improved efficiency relative to a naive alternative based on averaging componentwise Moran statistics. We illustrate practical utility by studying the spatial dependence of a composition measuring COVID-19 severity across Colombian departments during January 2021, documenting significant positive autocorrelation early in the month that attenuates over time.