Time series on compact spaces, with an application to dynamic modeling of relative abundance data in Ecology

TL;DR

This paper introduces a flexible framework for modeling complex time series data on compact spaces, such as the simplex, with applications to ecological abundance data, unifying Markovian and infinite memory models.

Contribution

It presents a general construction for infinite memory models on compact spaces, including conditions for stationarity and ergodicity, and applies this to ecological abundance data analysis.

Findings

Models encompass Markovian and infinite memory types

Conditions for existence and uniqueness of stationary solutions

Application to ecological abundance data analysis

Abstract

Motivated by the dynamic modeling of relative abundance data in ecology, we introduce a general approach to model stationary Markovian or non Markovian time series on (relatively) compact spaces such as a hypercube, the simplex or a sphere in the 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 details statistical inference in two time series models on the simplex, based on either a Dirichlet or a multivariate logistic-normal conditional distribution. Usefulness of our models to analyze abundance data in ecosystems is also discussed.