Time Series Models on Compact Spaces, With an Application to Dynamic Modeling of Relative Abundance Data in Ecology
提出一种在紧空间（如超立方体、单纯形或球面）上对平稳马尔可夫或非马尔可夫时间序列建模的通用方法，基于完全连接链构造，并给出存在唯一平稳遍历路径的条件，最后以单纯形上的狄利克雷和多元逻辑斯蒂正态模型为例讨论统计推断及其在生态丰度数据分析中的应用。
ABSTRACT 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.