The Spectral Envelope

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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.