对具有复杂自相关结构的两类数据的统计推断

本项目将对两类具有复杂自相关结构的数据进行统计推断：功能性核磁共振成像（fMRI）数据和分层自相关的生物数据，并重点研究统计推断过程中遇到的高维自协方差矩阵的估计问题。上述两种数据的特点为没有重复观测值且自相关结构复杂：fMRI数据的误差项为非平稳时间序列，分层自相关的生物数据的自相关结构与物种进化树结构有关。在对这两种数据进行统计推断的过程中，估计误差项的自协方差矩阵非常重要，而现有的估计方法对这两类数据并不适用。本项目拟研究高维自协方差矩阵的估计方法并对这两种有代表性的数据做出统计推断。具体的研究内容包括：（1）研究fMRI数据在半参数模型下的统计推断，在非平稳时间序列假设下解决误差自相关性问题。（2）研究具有树状自相关结构的生物数据在线性模型下的统计推断，重点对误差项的自协方差矩阵进行估计，获得矩阵估计的收敛速度，进而论证该估计的最优收敛速度。

This project will conduct statistical inference for two kinds of data with complicated autocorrelation structures: functional magnetic resonance imaging (fMRI) data and biological data with tree-structured autocorrelation; particularly, we will study the high-dimensional autocovariance matrix estimation problem encountered in the process of statistical inference. The two types of data have no replication and the autocorrelation structures are complicated: the error process of fMRI data is a non-stationary time series; the autocorrelation structure of the evolutionary biological data is highly related to the species’ genealogical tree. Estimating the autocovariance matrix is important when analyzing the data, but the existing estimation methods are not applicable here. We aim to develop methods for high-dimensional autocovariance matrix estimation and make statistical inference for the two types of data. In this project, we will address the following issues: (1) perform semiparametric inference for fMRI data and solve the temporal autocorrelation under non-stationary time series assumption; (2) analyze the tree-structured data by linear model, focus on estimating the autocovariance matrix of the error, demonstrate the convergence rate of the estimate and study the optimal convergence rate.