Statistical Inference for Tree Models with Strong Hierarchical Autocorrelation

当采样单元相互关联时，使用具有树形结构、层次自相关的模型。它们的继承历史由一棵树来建模，树被用来对观测之间的残差相关结构进行参数化。该项目将为这些自相关模型开发一个渐近理论，这些模型产生于树上的Ornstein-Uhlenbeck过程。随着树中提示的数量无限增加，研究人员将确定哪些参数是微遍历的，哪些参数不是微遍历性的。根据参数的微观遍历性和树的拓扑性质，极大似然估计的渐近相合性和收敛速度预计会有很大的变化。当在有界空间区域内的密集位置集合上收集观测时，将在空间统计中的这一渐近框架和填充渐近框架之间建立类比。该项目将完善分层自相关数据的有效样本量的概念，并研究最优抽样设计。这项工作将为开发适当的模型选择工具提供重要的步骤，以检测可能存在的许多Ornstein-Uhlenbeck选择制度，与样本大小相比，具有大量的模型参数。具有层次自相关性的树模型最先出现在进化生物学和生态学中，并与生物物种比较。这些模型现在被用于许多其他领域，从快速进化的病毒的研究到人类语言进化的研究。Ornstein-Uhlenbeck模型被用来检测选择，而不是中性进化，发现选择制度的变化，并确定选择的驱动因素。该项目将为这些模型提供一个统一的统计渐近框架，并将为经验性研究提供最佳做法。将广泛传播计算工具，并为统计学和生物学之间的接口培训提供机会。

Models with tree-structured, hierarchical autocorrelation are used when sampling units are related to each other. Their inheritance history is modeled by a tree, which is used to parametrize the residual correlation structure among observations. The project will develop an asymptotic theory for these autocorrelation models, arising from an Ornstein-Uhlenbeck process along the tree. As the number of tips in the tree grows indefinitely, the investigators will determine which parameters are microergodic and which parameters are not. The asymptotic consistency and the rate of convergence of the maximum likelihood estimator are expected to vary importantly depending on the microergodicity of the parameter and on topological properties of the tree. Analogies will be built between this asymptotic framework and the infill asymptotic framework in spatial statistics, when observations are collected on a dense set of locations within a bounded region of space. The project will refine the concept of effective sample size for hierarchically autocorrelated data and study optimal sampling designs. This work will provide important steps toward developing appropriate model selection tools for the detection of possibly many Ornstein-Uhlenbeck selection regimes, with a large number of model parameters compared to the sample size. Tree models with hierarchical autocorrelation arose first in evolutionary biology and ecology, with the comparison of biological species. These models are now used in many other areas, ranging from the study of rapidly evolving viruses to the study of human language evolution. The Ornstein-Uhlenbeck model is used to detect selection as opposed to neutral evolution, to discover changes in selective regime and to determine driving factors of selection. The project will provide a unified statistical asymptotic framework for these models and will inform best practices for empirical studies. Computational tools will be broadly disseminated, and opportunities will be provided for training at the interface between statistics and biology.