AUTOMATIC STATISTICAL TIME-FREQUENCY ANALYSIS

非平稳时间序列（即具有随时间变化的统计特性的时间序列）出现在神经科学研究和临床实践的许多领域。例如，脑电图（EEGs）的频谱特性随着大脑状态的变化而变化，这种变化通常具有重要的临床或科学意义。现有的谱分析方法假定时间序列是平稳随机过程的实现。这些方法可以通过加窗傅里叶变换扩展到非平稳过程，但窗的数量和大小必须主观选择。我们建议发展改进的自动统计方法来分析非平稳多元时间序列。我们将评估这些方法在真实模拟数据和从脑部疾病患者获得的真实多通道脑电图数据中的应用。我们将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)单变量随机过程时变功率谱的估计器；(2)多元随机过程的时变谱密度矩阵、相干和相位谱的估计；(3)时频主成分分析；(4)时频滤波器；(5)循环旋转，以减少由于估计器的二元结构造成的偏差；(6)改进了显示结果的彩色时频图。我们提出的统计方法是基于光滑局部复指数（SLEX）变换，它提供了丰富的正交变换选择。SLEX变换的结构允许我们使用Coifman和Wickerhauser的简单且计算效率高的Best Basis算法来自动选择特定的变换，该变换表示将非平稳时间序列分割成近似平稳的区间。与小波、小波包和余弦包系数不同，SLEX系数包含相位信息，便于估计多元时间序列中时变的相位关系。SLEX变换可以任意接近于锥形傅立叶变换，因此我们提出的使用SLEX变换进行非平稳过程光谱分析的方法将与现有的平稳过程光谱分析方法并行。

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 spectral 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 acquired 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) improved color time-frequency plots for displaying the results. Our 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 simple and 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. SLEX coefficients, unlike wavelet, wavelet packet, and cosine packet coefficients, contain phase information, facilitating the estimation of time-varying phase relationships in multivariate time series. The SLEX transform can be made arbitrarily close to a tapered Fourier transform, so that our proposed methods for spectral analysis of nonstationary processes using the SLEX transform will parallel existing methods for spectral analysis of stationary processes.