The evolving categories multinomial distribution: introduction with applications to movement ecology and vote transfer

We introduce the evolving categories multinomial (ECM) distribution for multivariate count data taken over time. This distribution models the counts of individuals following iid stochastic dynamics among categories, with the number and identity of the categories also evolving over time. We specify the one-time and two-times marginal distributions of the counts and the first and second order moments. When the total number of individuals is unknown, placing a Poisson prior on it yields a new distribution (ECM-Poisson), whose main properties we also describe. Since likelihoods are intractable or impractical, we propose two estimating functions for parameter estimation: a Gaussian pseudo-likelihood and a pairwise composite likelihood. We show two application scenarios: the inference of movement parameters of animals moving continuously in space-time with irregular survey regions, and the inference of vote transfer in two-rounds elections. We give three illustrations: a simulation study with Ornstein-Uhlenbeck moving individuals, paying special attention to the autocorrelation parameter; the inference of movement and behavior parameters of lesser prairie-chickens; and the estimation of vote transfer in the 2021 Chilean presidential election.