Statistical Methods in the Frequency Domain

摘要：在本文中，我们主要关注与频域统计方法相关的主题。首先，我们提出将平稳时间序列的谱包络方法扩展到非平稳序列的演化谱包络的概念。在另一个项目中，我们将利用基于光滑局部复指数（SLEX）的变换来分析非平稳多时间序列及其主成分。在第三个项目中，我们将考虑在带有协变量的实验设计中收集的时间序列的光谱分析。频谱包络首先被提出作为一种在频域分析平稳分类值时间序列的方法。这项研究的动机是对DNA序列的分析。分析长DNA序列数据的一个常见问题是如何识别分散在整个序列中并被非编码区域分开的编码序列。众所周知，DNA序列是异质的，即使在DNA的短子序列中，也会遇到局部行为。在这个项目中，我们感兴趣的是扩展光谱包络方法来捕捉这些序列的局部行为。为了解决分类值时间序列中的局部行为问题，我们将探索将谱包络与基于二进树的自适应分割（TBAS）方法结合使用来分析局部平稳过程。我们的希望是，这种方法将有助于强调存在于几乎任何长度的分类序列中的任何谐波特征，以快速和自动化的方式。人类基因组计划等项目已经产生了大量的数据。我们相信我们的方法将被证明是一种有用的数据挖掘技术，有助于分析各种基因组计划产生的大量数据。第一个项目侧重于基于傅里叶的方法，第二个项目侧重于其他技术，将提供空间（或时间）和频率定位。我们的目标，一如既往，是为分析大数据集开发计算效率高的算法。在我们最初的研究中，我们将重点关注用于分析分类值非平稳时间序列的SLEX变换，但我们的目标是最终将该技术应用于一般的多个时间序列（及其主成分）。SLEX变换具有特殊的性质，使其成为分析非平稳时间序列的理想方法。SLEX变换是基于在时域和频域都有定域的SLEX基函数。SLEX变换产生时间和频率的分解，并允许在许多正交变换中进行选择。正交性为非平稳时间序列的自动分割提供了高效的计算程序，并有望促进我们对所提出方法的理论要素的研究。正交表示允许存储系数，然后通过非线性阈值等方法处理它们。我们的感觉是，如果数据可以减少到相对较少的有意义的系数，那么这些系数可能在某些类型的二次统计分析中有用。在我们与其他科学家和医生的合作中，我们经常遇到在实验设计中记录几个受试者的时间序列和协变量的设置。分析这些数据缺乏核心的统计程序，我们通常会遇到一些技术，这些技术是由那些几乎没有技术技能或知识来分析相关数据和估计（谱）函数的研究人员以一种特别的方式炮制的。我们在这个项目中的目标是开发一种通用的、用户友好的统计方法，该方法将结合从许多组的几个单位记录的时间序列数据集获得的相关信息，并且也可以测量协变量。我们最初的方法将是利用谱密度估计和广义线性模型之间的关系。

Abstract DMS-0102511 Stoffer & OmbaoIn this proposal, we concentrate on topics relating, in general, to statistical methods in the frequency domain. First, we propose to extend the spectral envelope methodology for stationary time series to the notion of evolutionary spectral envelope for nonstationary series. In another project, we will direct our attention to analyzing nonstationary multiple time series and their principal components using transforms based on smooth localized complex exponentials (SLEX). In a third project, we will consider spectral analysis of time series collected in experimental designs with covariates. The spectral envelope was first proposed as a method to analyze stationary categorical-valued time series in the frequency domain. The motivation for that research was the analysis of DNA sequences. A common problem in analyzing long DNA sequence data is in identifying coding sequences that are dispersed throughout the sequence and separated by regions of noncoding. It is well known that DNA sequences are heterogeneous, and even within short subsequences of DNA, one encounters local behavior. In this project, we are interested in extending the spectral envelope methodology to capture the local behavior of such sequences. To address this problem of local behavior in categorical-valued time series, we will explore using the spectral envelope in conjunction with a dyadic tree-based adaptive segmentation (TBAS) method for analyzing locally stationary processes. Our hope is that this methodology will help emphasize any harmonic feature that exists in a categorical sequence of virtually any length in a quick and automated fashion. Projects such as the human genome project have produced large amounts of data. We believe our methods will prove to be useful as a data mining technique for help in the analysis of the vast quantities of data being produced by various genome projects. While the first project focuses on Fourier based methods, the second project concentrates on other techniques that will give spatial (or time) and frequency localization. Our goal, as always, is to develop computationally efficient algorithms for the analysis of large data sets. In our initial investigations, we will focus on the SLEX transform for analyzing categorical-valued nonstationary time series, but our goal is eventually to apply the technique to multiple time series (and their principal components) in general. The SLEX transform has special properties that make it ideal for analysis of nonstationary time series. The SLEX transform is based on the SLEX basis functions which are localized in both the time and frequency domains. The SLEX transform yields a decomposition in both time and frequency and allows a choice among many orthogonal transforms. Orthogonality leads to computationally efficient procedures for automatic segmentation of nonstationary time series and will hopefully facilitate in our investigation of the theoretical elements of our proposed methodology. An orthogonal representation allows one to store the coefficients and later process them by methods such as nonlinear thresholding. Our feeling is that if the data can be reduced to a relatively small number of meaningful coefficients then these coefficients might be useful in some type of secondary statistical analysis. In our collaborations with other scientists and physicians, we frequently encounter settings where time series, and covariates, are recorded for several subjects in an experimental design. There is an absence of a core of statistical procedures for analyzing such data, and we typically run across techniques that are cooked up in an ad hoc manner by researchers who have little technical skill or knowledge for analyzing correlated data and estimating (spectral) functions. Our goal in this project is to develop a general, user friendly, statistical methodology that will incorporate the relevant information obtained from time series data sets recorded from several units from many groups, and where covariates may also be measured. Our initial approach will be to exploit the relationship between spectral density estimation and generalized linear models.