Collaborative Research: The Analysis of Time Series Collected in Experimental Designs

该研究项目有三个基本目标。第一个目标是使用来自多个单元的数据来估计非平稳过程的二阶性质。提出了一种基于小波的随机主题表示功能模型。一个计算效率高的算法估计将开发沿着与理论上的最佳性能的估计。本项目侧重于频域或频谱域，第二个项目将开发时域方法。这个项目的目标是适应结构的时间序列中使用的状态空间模型的混合物。这些混合物的特征在于混合物组分和混合重量中的协变量、治疗和受试者特异性效应。第三个项目侧重于时间和空间组成部分。特别是，该项目通过开发模型和相应的估计器来整合空间信息，以适应时间和空间的数据结构。目标是评估各组之间的差异以及特定主题和协变量对动态结构的影响，研究重点是一个在很大程度上被统计界忽视的领域。 特别是，要解决的问题是如何最好地分析从实验中获得的数据时，在时间和/或空间上顺序采取的数据。 典型的例子包括对具有不同诊断的各种受试者进行EEG的实验，或功能性磁共振成像（fMRI）实验，其中要求具有不同条件的受试者在磁体中执行任务。分析这些数据的困难在于它们通常是不规则的（非均匀或非平稳）。 分析规则时间序列的技术已经很成熟，但需要新的技术来解释时间和/或空间上的不规则性。 事实上，这些数据是在复杂的实验中收集的，这给分析增加了另一层难度。

The research project has three basic goals. The first goal is to estimate the second order properties of a nonstationary process using data from multiple units. A wavelet-based functional model with stochastic subject-specific representation is proposed. A computationally efficient algorithm for estimation will be developed along with theoretical optimal properties of the estimators. While this project is focused on the frequency or spectral domain, the second project will develop methods in the time domain. The goal of this project is to accommodate structural breaks within the time series using mixtures of state-space models. These mixtures are characterized by covariates, treatment and subject-specific effects in both the mixture components and mixing weights. The third project focuses on both temporal and spatial components. In particular, this project integrates spatial information by developing models and corresponding estimators to accommodate data structures in both time and space. The goal is to assess the differences across groups and the subject-specific and covariate effects on the dynamic structure.The research focuses on an area that has been, for the most part, neglected by the statistics community. In particular, the question to be addressed is how to best analyze data obtained from experiments when the data are taken sequentially in time and/or space. Typical examples include experiments where EEGs are taken on various subjects with different diagnoses, or functional magnetic resonance imaging (fMRI) experiments, where subjects with different conditions are requested to perform tasks while in a magnet. The difficulty in analyzing such data is that they are often irregular (nonhomogeneous or nonstationary) in nature. The techniques for analyzing regular time series are well established, but new techniques are needed to account for the irregularities over time and/or space. The fact that these data are collected in complex experiments adds another layer of difficulty onto the analysis.