Optimal transformations and the spectral envelope for real-valued time series

The concept of a spectral envelope for exploring the periodic nature of real-valued time series is introduced. This concept follows naturally from the data-dependent approach proposed by Stoffer et al. (1993) for spectral analysis and scaling of categorical processes. Here, the notion of the spectral envelope is applied in the context of transformations of a time series, and a data-dependent approach for selecting optimal transformations is proposed. These transformations help emphasize periodicities that may exist in the real-valued process. The definition of the spectral envelope is also extended to include multivariate time series. Several examples are used to illustrate the application of this methodology and asymptotic properties of the procedure are established.