On the Reliability Index of the Holo-Hilbert Spectral Analysis

It has been shown that, for full information spectral representation of any time series signal, a spectral form of high-dimensional manifold, known as Holo-Hilbert Spectral Analysis (HHSA), is necessary. The higher-dimensional manifold comes from the additional information of the envelopes of the carriers from a second layer of empirical mode decomposition, implemented through the Ensemble Empirical Mode Decomposition (EEMD) interval approach, which is inherited with some uncertainty. To quantify the reliability of this high-dimensional manifold representation, the traditional confidence limit is impractical, if not impossible. To measure the higher dimensional HHSA reliability, we introduce the Reliability Index (RI), defined as the ratio of the MEAN to STD at any point in the FM-AM space from an ensemble of HHSA (eHolo, for short), which is computed with a wide distribution of noise levels and a large number of ensembles that satisfy the ergodicity hypothesis. In general, the eHolo results in the high carrier frequency region are more reliable, due to the higher degree of freedom of the signals. Using the Length-Of-Day data, we demonstrate the necessity and usefulness of the RI as a measure for reliability. The eHolo also gives us a more robust spectral representation. Based on this study, we suggest that the eHolo and RI should be used for all the HHSA results.