Fourier Methods in the Analysis of nonstationary and nonlinear stochastic processes

非平稳和非线性随机过程分析中的傅里叶方法

• 批准号: 1106518
• 负责人: Suhasini Subba Rao
• 金额: $ 12.71万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106518&HistoricalAwards=false
• 关键词: Fourier; Methods; Analysis; nonstationary; nonlinear

The investigator develops new Fourier based methods for analyzing nonstationary and nonlinear time series. Fourier analysis is the well established de facto tool for analyzing linear, stationary time series. There are several reasons for this (i) the discrete Fourier transform asymptotically uncorrelates a stationary time series (ii) if the time series is stationary and linear, then estimates of the spectral density function can be used to identify the underlying linear model (iii) the spectral density function can be used as a means of checking goodness of fit of a linear model. However, it has long been observed that several time series models do not fit well within the stationary, linear model framework. Over long periods of time the assumption of stationarity is often quite unrealistic. Even over short periods of time, the assumption of linearity can be too strong. Applying standard Fourier methods to such data can lead to uninformative and misleading conclusions. But in contrast to linear models, there does not exist universal methods for comparing non-nested, nonlinear models, checking adequacy of any given model, etc. As increasingly complex time series models are introduced, it has become increasingly important to develop such methods, and the investigator addresses these issues. The investigator focuses on three areas where, in applications, nonstationarity and nonlinearity can arise (i) nonstationary discrete time stochastic processes (ii) functional time series with random sampling (iii) nonlinear, stationary time series. These are detailed below. In the first project the investigator exploits the fact that the discrete Fourier transform only decorrelates second order stationary time series to characterize and model nonstationary behavior. In the second project the investigator considers continuous time series, which are only observed at discrete, randomly sampled time points. Here the focus is on functional time series, and the investigator defines a modified version of the discrete Fourier transform to test for stationarity and to develop goodness of fit tests. As mentioned above, often the assumption of linearity can be too strong, and in the third project the investigator considers stationary time series' which are not necessarily linear. The investigator defines a variant of the spectral density which captures the pair-wise dependence structure of a time series. This transformation allows one to understand the dependence structure of the time series on different parts of the domain of the time series. Using this transformation the investigator checks for model adequacy, tests for equality of pair-wise dependence between two time series and measures the dependence between two time series through an appropriate transformations of the data. The analysis of data which is observed over time (usually called a time series) is studied in several disciplines, including the atmospheric sciences, economics etc. As the observations are over time, usually there is dependence (a simple measure of dependence is correlation) between neighboring observations. Understanding and modeling this dependence allows one to forecast (for example, future global temperatures) and compare various different time series (for example, different financial markets). Under the assumption that the time series is stationary (the overall structure does not change over time), and linear (the transition in the times series is smooth), a rich literature on modeling the correlation structure exists. However, there are several real data examples where there are no realistic reasons that these assumptions should hold true, and indeed they could be an oversimplification of the system or simply wrong. In this project, the investigator develops statistical tools which allows one to check whether a time series satisfies the usual assumptions, and if not, how they may violate these assumptions, what impact this may have on standard statistical analysis and how it may effect the conclusions.

研究者开发了新的基于傅立叶的方法来分析非平稳和非线性时间序列。傅立叶分析是分析线性平稳时间序列的公认工具。对此有几个原因：（i）离散傅立叶变换渐近地使平稳时间序列不相关;（ii）如果时间序列是平稳的和线性的，则谱密度函数的估计可以用于识别潜在的线性模型;（iii）谱密度函数可以用作检查线性模型拟合优度的方法。然而，长期以来一直观察到，一些时间序列模型并不适合固定的线性模型框架。在很长一段时间内，平稳性的假设往往是不切实际的。 即使在很短的时间内，线性的假设也可能过于强烈。将标准傅立叶方法应用于此类数据可能会导致无信息和误导性的结论。但是，与线性模型相比，并不存在通用的方法来比较非嵌套，非线性模型，检查任何给定的模型的充分性等，随着越来越复杂的时间序列模型的介绍，它已变得越来越重要，以开发这样的方法，和调查解决这些问题。研究人员专注于三个领域，在应用中，非平稳性和非线性可能会出现（i）非平稳离散时间随机过程（ii）随机采样的功能时间序列（iii）非线性，平稳时间序列。下文将详细介绍这些措施。在第一个项目中，研究人员利用离散傅立叶变换只对二阶平稳时间序列进行去相关来表征和建模非平稳行为的事实。在第二个项目中，研究人员考虑连续时间序列，这只在离散的，随机抽样的时间点观察。这里的重点是功能时间序列，研究人员定义了一个修改后的版本的离散傅立叶变换来测试平稳性，并开发拟合优度测试。如上所述，线性的假设往往过于强烈，在第三个项目中，研究人员认为平稳的时间序列不一定是线性的。研究者定义了一个变量的谱密度，它捕获了时间序列的成对依赖结构。这种转换允许人们理解时间序列在时间序列域的不同部分上的依赖结构。使用这种转换，调查员检查模型的充分性，测试两个时间序列之间的成对依赖的平等性，并通过适当的数据转换来测量两个时间序列之间的依赖性。 随着时间的推移观察到的数据（通常称为时间序列）的分析在几个学科中进行了研究，包括大气科学，经济学等。理解和建模这种依赖性允许人们预测（例如，未来的全球温度）和比较各种不同的时间序列（例如，不同的金融市场）。假设时间序列是平稳的（整体结构不随时间变化）和线性的（时间序列中的过渡是平滑的），存在大量关于相关性结构建模的文献。然而，有几个真实的数据例子，没有现实的理由表明这些假设应该成立，事实上，它们可能是对系统的过度简化或根本错误。在这个项目中，研究人员开发了统计工具，可以检查时间序列是否满足通常的假设，如果不满足，它们如何违反这些假设，这可能对标准统计分析产生什么影响，以及它如何影响结论。