Motivated by the dynamic modeling of relative abundance data in ecology, we introduce a general approach to model time series on the simplex. Our approach is based on a general construction of infinite memory models, called chains with complete connections. Simple conditions ensuring the existence of stationary paths are given for the transition kernel that defines the dynamic. We then study in details two specific examples with a Dirichlet and a multivariate logistic-normal conditional distribution. Inference methods can be based on either likelihood maximization or on some convex criteria that can be used to initialize likelihood optimization. We also give an interpretation of our models in term of additive perturbations on the simplex and relative risk ratios which are useful to analyze abundance data in ecosystems. An illustration concerning the evolution of the distribution of three species of Scandinavian birds is provided.

翻译：受生态学中相对丰度数据动态建模的驱动，我们提出了一种在单纯形上对时间序列进行建模的一般性方法。该方法基于一种称为完全连接链的无限记忆模型的一般构造。我们给出了确保存在平稳路径的简单条件，这些条件针对定义动态过程的转移核。随后，我们详细研究了两个具体示例，分别涉及狄利克雷分布和多元逻辑斯蒂-正态条件分布。推理方法可基于似然最大化或某些可用于初始化似然优化的凸准则。我们还从单纯形上的加法扰动和相对风险比的角度给出了模型解释，这对于分析生态系统中的丰度数据非常有用。最后，我们提供了一个关于三种斯堪的纳维亚鸟类物种分布演变的示例说明。