The Spectral Envelope

光谱包络是近年来引入的一种在光谱域分析分类时间序列的方法，并已成功地应用于DNA序列分析。在发展范畴时间序列的谱包络理论和方法的过程中，人们发现这个概念有许多有趣的扩展。尽管有许多方向可以追求这一想法，但本研究至少将重点放在将光谱包络线的概念扩展到实值时间序列的分析上。在这里，主要目标将是获得实值（单变量和多变量）时间序列的最优变换，这些时间序列可能是非线性的，非高斯的，或具有一些未预料到的依赖结构，并包络（或覆盖）时间序列的任何此类变换的频谱。近年来，光谱包络作为一种分析时间或空间（时间序列）上连续观察到的定性数据的方法被引入，并已成功地应用于DNA序列分析。在为这些数据发展光谱包络的理论和方法时，人们发现这个概念有许多有趣的扩展。尽管有许多方向可以追求这一想法，但本研究至少将重点放在将光谱包络线的概念扩展到定量时间序列数据的分析上。在这里，主要目标将是获得时间序列的最优变换，这些时间序列可能是非线性的，非正态的，或者具有一些不可预料的依赖结构，并发现这些数据的任何有趣的循环（重复）属性。

The spectral envelope was recently introduced as a method for analyzing categorical time series in the spectral domain and has been successfully applied in analyzing DNA sequences. In developing the theory and methodology of the spectral envelope for categorical time series, it was discovered that there are many interesting extensions to the concept. Although there are numerous directions in which to pursue this idea, this research will focus, at least, on extending the concept of the spectral envelope to the analysis of real-valued time series. Here, the main goals will be to obtain optimal transformations of real-valued (univariate and multivariate) time series that may be nonlinear, non-Gaussian, or have some unsuspected dependence structure, and to envelop (or blanket) the spectrum of any such transformation of the time series. The spectral envelope was recently introduced as a method for analyzing qualitative data observed sequentially in time or space (time series) and has been successfully applied in analyzing DNA sequences. In developing the theory and methodology of the spectral envelope for such data, it was discovered that there are many interesting extensions to the concept. Although there are numerous directions in which to pursue this idea, this research will focus, at least, on extending the concept of the spectral envelope to the analysis of quantitative time series data. Here, the main goals will be to obtain optimal transformations of time series that may be nonlinear, non-Normal, or have some unsuspected dependence structure, and to discover any interesting cyclic (repetitive) properties of these data.