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High Dimensional Models for Multivariate Time Series Analysis

用于多元时间序列分析的高维模型

基本信息

批准号：
EP/I005250/1
负责人：
Sofia Olhede
金额：
$ 126.15万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI005250%2F1
关键词：
Dimensional Models Multivariate Time Series

项目摘要

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.

该奖学金将专注于开发高维时间序列分析的方法。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，海量数据集激发了基于稀疏性的新统计范式的发展。这样的发展主要涉及沉浸在噪声中的确定性结构，而这个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导出序列总体的性质，这对于正确理解结构中的不确定性或变异性非常重要。传统的时间序列方法仅限于平稳过程，其结构在时间上是均匀的。相反，该项目将为非平稳过程的类别开发理论和方法，这些过程可以在观察的时间过程中经历其生成机制的变化。这样的过程很重要，因为它们允许我们对可观测量的演变进行建模，也使我们能够明确地量化这种演变。在一些应用中观察到非平稳过程，例如地球科学(遥感和卫星观测)、海洋学(漂浮物和浮游物测量)、神经科学(功能核磁共振和脑电波)和生态学(物种丰富度)。在这类应用中，很少涉及单个过程，因此我们将开发多个(或等价的多变量)信号的分析方法，以量化观察过程的演变的相互依赖关系。分析非平稳信号的困难在于其高度的过度参数化，如果要由多个序列进行推断，则会加剧这一困难。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。关于稀疏性的假设最近被用来在相关的过参数问题中进行估计。这些方法需要仔细的扩展和实质性的创新，以涵盖多变量和随机信号的情况，我们建议通过这个项目来解决这些问题。开发这种方法的关键是引入新的稀疏类非平稳过程，建立在高维数据统计的最新发展基础上。稀疏模型尽管名义上具有很高的复杂性，但用一些未知但更简单的较小复杂性的结构来描述。稀疏模型将被构建为包含先前不相容的非平稳过程，从而使我们能够处理缺乏自然分析框架的序列。因此，本建议的目的是a)利用稀疏性为单信号引入新的非平稳过程类，b)在已知或必须学习过程的组结构的情况下将这些类扩展到丰富的多变量过程族，c)发展对这类过程的可估计性的理论理解，d)发展一般的估计方法以及应用特定的方法。我们期望这项工作对统计的影响远远超出时间序列。稀疏性的新形式和方法也将与数学、机器学习和信号处理中的相关问题相关，特别是在定义信号组稀疏性的新形式方面。这项工作还将产生不仅仅是方法论上的影响，因为这些方法的发展将使我们能够分析以前无法分析的多个系列，我们打算与我们的合作者一起开发特定于应用的方法。