Automatic Statistical Time-Frequency Analysis

描述（由申请人提供）：非平稳时间序列（即，时间 具有随时间变化的统计特性的序列）出现在许多领域， 神经科学研究和临床实践。例如，光谱 脑电图（EEG）的特性随大脑状态而变化， 这种变异经常具有重要的临床或科学意义。 现有的谱分析方法假设时间序列是一个实现 一个平稳随机过程。这些方法可以推广到非平稳 处理使用窗口傅立叶变换，但数量和大小的 必须主观地选择窗口。我们建议开发改进的，自动的 非平稳多变量时间序列分析的统计方法。我们 将评估应用于现实模拟数据的方法， 脑功能障碍患者所需的真实的多通道EEG数据。我们 将把我们的自动方法的结果与临床判断进行比较。 我们的具体目标是开发、评估、应用、实施和分发 以下统计方法： (1)一元随机时变功率谱的估计 过程;（2）时变谱密度矩阵的估计， 相干性和多变量随机过程的相谱;（3） 时频主成分分析;（4）时频滤波器;（5） 循环旋转，以减少由于我们的估计的二元结构的偏差; 以及（6）在时间和时间上都是平滑的单变量和多变量过程， 频域 我们首先提出的统计方法是基于平滑局部化 复杂指数（SLEX）变换，它提供了丰富的选择， 正交变换SLEX变换的结构允许我们使用 Coifman和Wickerhauser的计算效率最高的基础算法， 自动选择一个特定的变换，它代表一个分割， 将非平稳时间序列分解为近似平稳的时间间隔。 我们提出的第二种方法（目标6）采取了一种新的途径。 的方法，专注于建模的Periodiograms，我们建议建模的 传递函数直接作为频率和时间的平滑函数 使用平滑样条和使用信号加噪声模型。通过模拟 直接传递函数，我们合金同时平滑的时间和 傅里叶变换中的频率。与周期图不同， 传递函数保留了相位信息，因此 可以直接计算时变互谱、相干性和相位 传递函数。

DESCRIPTION (provided by applicant): Non-stationary time series (i.e., time series with statistical properties that vary over time) arise in many areas of neuroscience research and clinical practice. For example, the spectral properties of electroencephalograms (EEGs) vary with brain state, and frequently this variation is of central clinical or scientific importance. Existing methods of spectra analysis assume that a time series is a realization of a stationary random process. These methods can be extended to non-stationary processes using windowed Fourier transforms, but the number and size of the windows must be chosen subjectively. We propose to develop improved, automatic statistical methods for analysis of non-stationary multivariate time series. We will evaluate the methods in applications to realistic simulated data and to real multi-channel EEG data required from patients with brain disorders. We will compare results from our automatic methods with the clinical judgments. Our specific aims are to develop, evaluate, apply, implement, and distribute the following statistical methods: (1) an estimator of the time-varying power spectrum of a univariate random process; (2) estimators of the time-varying spectral density matrix, coherences, and phase spectra of a multivariate random process; (3) time-frequency principal component analysis; (4) time-frequency filters; (5) cycle-spinning to reduce bias due to the dyadic structure of our estimators; and (6) univariate and multivariate processes that are smooth in both time and frequency domains. Our first proposed statistical methods are based on the Smooth Localized complex Exponential (SLEX) transform, which provides a rich selection of orthogonal transforms. The structure of the SLEX transform allows us to use the computationally efficient Best Basis algorithm of Coifman and Wickerhauser to automatically select a particular transform, which represents a segmentation of a non-stationary time series into approximately stationary intervals. Our second proposed approach (Aim 6) takes a new path. Unlike the traditional approaches that focus on modeling the Periodiograms, we propose to model the transfer function directly as a smooth function in both frequency and time using smoothing splines and to use a signal-plus-noise model. By modeling the transfer function directly, we alloy simultaneous smoothing in time and frequency within the Fourier transformation. Unlike the periodiogram, the transfer function preserves the phase information, and therefore the time-varying cross-spectra, coherence, and phase can be directly calculated from the transfer functions.