High Dimensional Models for Multivariate Time Series Analysis

该奖学金将侧重于开发高维时间序列分析方法。高维数据的方法论是当前统计学和信号处理领域最重要的研究课题之一，大量数据集激发了基于稀疏性的新统计范式的发展。这些发展主要涉及噪声中的确定性结构，而本程序将把感兴趣的信号建模为随机信号。将观测信号随机建模为时间序列的优点是可以推断出一系列序列的性质，这对于正确理解结构中的不确定性或可变性很重要。传统的时间序列方法局限于平稳过程，其结构在时间上是齐次的。相反，该项目将为非平稳过程类开发理论和方法，这些过程可以在观察的时间过程中经历产生机制的变化。这些过程很重要，因为它们使我们能够模拟可观察量的演变，也使我们能够明确地量化这种演变。非平稳过程在许多应用中被观察到，如地球科学（遥感和卫星观测）、海洋学（漂流和浮子测量）、神经科学（功能性MRI和EEG）和生态学（物种丰度），仅举几个领域。在这样的应用中，单个过程很少引起人们的兴趣，因此我们将开发用于分析多个（或等效的多变量）信号的方法，以量化观察到的过程之间不断发展的相互依赖性。分析非平稳信号的困难在于它们的高度过参数化，如果对多个序列进行推理，这将大大加剧。乍一看，这类问题的可靠估计似乎是不可能的，这是极端过度参数化的结果。稀疏性假设最近被用于相关的过参数化问题的估计。这些方法需要谨慎的扩展和实质性的创新，以涵盖多元和随机信号的情况，我们建议通过这个项目来解决这个问题。开发这种方法的关键是引入新的非平稳过程的稀疏类，建立在高维数据统计的最新发展上。尽管名义上的复杂度很高，但稀疏模型是由一些未知的、更简单的、复杂度更小的结构来描述的。将构建稀疏模型来包含以前不兼容的非平稳过程，从而使我们能够处理缺乏自然分析框架的序列。因此，本提案旨在a)使用稀疏性为单个信号引入新的非平稳过程类别，b)将这些类别扩展到多元过程的丰富家族，用于已知过程的组结构或必须学习的场景，c)对此类过程的可估计性进行理论理解，d)开发一般估计方法以及特定应用的方法。我们期望这项工作对统计数据的影响远远超出时间序列。新形式的稀疏性和方法也将与数学、机器学习和信号处理中的相关问题相关，特别是在定义信号群稀疏性的新形式方面。由于这些方法的发展将使我们能够分析以前无法分析的多个序列，因此这项工作也将具有方法论上的影响，我们打算与我们的合作者一起开发特定于应用程序的方法。

This fellowship will focus on developing methods for high dimensional time series analysis. Methodology for high dimensional data is one of the most important current research topics in statistics and signal processing, where massive data sets have inspired the development of a new statistical paradigm based on sparsity. Such developments have mainly concerned deterministic structure immersed in noise, while this program will model the signal of interest as stochastic. The advantage of modeling an observed signal stochastically as a time series is that one can deduce properties of a population of series, important for the correct understanding of uncertainty or variability in structure.Traditional time series methods are restricted to stationary processes, whose structure is homogeneous in time. The project will instead develop theory and methodology for classes of nonstationary processes, that can experience changes in their generating mechanism over the time course of observation. Such processes are important as they allow us to model the evolution of an observable quantity, and also enable us to quantify this evolution explicitly. Nonstationary processes are observed in a number of applications such as geoscience (remote sensing and satellite observations), oceanography (drifter and float measurements), neuroscience (functional MRI and EEG) and ecology (species abundance) to mention but a few areas. In such applications single processes are rarely of interest, and so we shall develop methods for the analysis of multiple (or equivalently multivariate) signals, to quantify the evolving interdependencies of observed processes.The difficulty in analyzing nonstationary signals is their high degree of overparameterization, that is much exacerbated if inferences are to be made of multiple series. At first glance reliable estimation in such problems seems impossible, as a consequence of the extreme overparameterization. Assumptions on sparsity have recently been used to enable estimation in related overparameterized problems. Such methods need careful extension and substantial innovation to cover the case of multivariate and stochastic signals, that we propose to address via this project. Key to developing such methods is introducing new sparse classes of nonstationary processes, building on recent developments in statistics for high dimensional data. Sparse models despite a nominal degree of high complexity are described by some unknown but simpler structure of smaller complexity. Sparse models will be constructed to contain previously incompatible nonstationary processes, thus enabling us to treat series that lacked a natural analysis framework.This proposal therefore aims to a) introduce new classes of nonstationary processes for single signals using sparsity, b) extend these classes to rich families of multivariate processes for scenarios where either the group structure of the processes is known or has to be learned, c) develop a theoretical understanding of the estimability of such classes of processes and d) develop general estimation methods as well as application specific methodology.We expect this work to impact statistics much beyond time series. New forms of sparsity and methods will also be relevant to related problems in mathematics, machine learning and signal processing, especially in terms of defining new forms of signal group sparsity. The work will also have more than a methodological impact as the development of these methods will allow us to analyze multiple series that previously could not be analyzed, and we intend to develop application specific methods with our collaborators.