Fourier Methods in the Analysis of nonstationary and nonlinear stochastic processes

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。傅里叶分析是一种公认的分析线性、平稳时间序列的工具。这有几个原因：(i)离散傅里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和线性的，那么谱密度函数的估计可以用来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良好度的手段。然而，人们长期以来一直观察到，一些时间序列模型不能很好地拟合在平稳的线性模型框架内。在很长一段时间内，平稳的假设往往是不现实的。即使在短时间内，线性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缺乏信息和误导性的结论。但与线性模型相比，没有通用的方法来比较非嵌套的非线性模型，检查任何给定模型的充分性等。随着越来越复杂的时间序列模型的引入，开发这样的方法变得越来越重要，研究者解决了这些问题。研究者专注于三个领域，在应用中，非平稳性和非线性可能出现(i)非平稳离散时间随机过程（ii）随机抽样的函数时间序列（iii）非线性，平稳时间序列。这些细节如下。在第一个项目中，研究者利用离散傅里叶变换只解相关二阶平稳时间序列的事实来表征和模拟非平稳行为。在第二个项目中，研究者考虑连续时间序列，它只在离散的，随机抽样的时间点上观察到。这里的重点是功能性时间序列，研究者定义了离散傅里叶变换的修改版本，以测试平稳性并开发拟合优度测试。如上所述，通常线性假设可能太强，在第三个项目中，研究者考虑不一定是线性的平稳时间序列。研究者定义了光谱密度的一种变体，它捕获了时间序列的成对依赖结构。这个变换使我们能够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，研究者检查模型充分性，检验两个时间序列之间的成对依赖是否相等，并通过适当的数据转换测量两个时间序列之间的依赖。对一段时间内观测到的数据（通常称为时间序列）的分析在几个学科中进行研究，包括大气科学、经济学等。由于观测值是随时间变化的，相邻观测值之间通常存在依赖性（依赖性的一个简单度量是相关性）。对这种依赖性的理解和建模使人们能够预测（例如，未来的全球温度）并比较各种不同的时间序列（例如，不同的金融市场）。在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）下，存在丰富的相关结构建模文献。然而，有几个真实的数据例子，没有现实的理由，这些假设应该是正确的，事实上，他们可能是系统的过度简化或根本错误。在这个项目中，研究者开发了统计工具，允许人们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。

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.