Time Series Models on Compact Spaces, With an Application to Dynamic Modeling of Relative Abundance Data in Ecology

Motivated by the dynamic modeling of relative abundance data in ecology, we introduce a general approach for modeling stationary Markovian or non-Markovian time series on (relatively) compact spaces, such as a hypercube, the simplex, or a sphere in a Euclidean space. Our approach is based on a general construction of infinite memory models, called chains with complete connections. The two main ingredients involved in our generic construction are a parametric family of probability distributions on the state space and a map from the state space to the parameter space. Our framework encompasses Markovian models, observation-driven models, and more general infinite memory models. Simple conditions ensuring the existence and uniqueness of a stationary and ergodic path are given. We then study in more detail statistical inference in two time series models on the simplex, based on either a Dirichlet or a multivariate logistic-normal conditional distribution. The usefulness of our models to analyze abundance data in ecosystems is also discussed.