On Holo-Hilbert spectral analysis: a full informational spectral representation for nonlinear and non-stationary data

Abstract

The Holo-Hilbert spectral analysis (HHSA) method is introduced to cure the deficiencies of traditional spectral analysis and to give a full informational representation of nonlinear and non-stationary data. It uses a nested empirical mode decomposition and Hilbert–Huang transform (HHT) approach to identify intrinsic amplitude and frequency modulations often present in nonlinear systems. Comparisons are first made with traditional spectrum analysis, which usually achieved its results through convolutional integral transforms based on additive expansions of an a priori determined basis, mostly under linear and stationary assumptions. Thus, for non-stationary processes, the best one could do historically was to use the time–frequency representations, in which the amplitude (or energy density) variation is still represented in terms of time. For nonlinear processes, the data can have both amplitude and frequency modulations (intra-mode and inter-mode) generated by two different mechanisms: linear additive or nonlinear multiplicative processes. As all existing spectral analysis methods are based on additive expansions, either a priori or adaptive, none of them could possibly represent the multiplicative processes. While the earlier adaptive HHT spectral analysis approach could accommodate the intra-wave nonlinearity quite remarkably, it remained that any inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase-lock modulations were left untreated. To resolve the multiplicative processes issue, additional dimensions in the spectrum result are needed to account for the variations in both the amplitude and frequency modulations simultaneously. HHSA accommodates all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions. The Holo prefix in HHSA denotes a multiple dimensional representation with both additive and multiplicative capabilities.

On the intrinsic timescales of temporal variability in measurements of the surface solar radiation

“…The short-term intrinsic periodicities in the solar proxies appear to be indicative of "randomly distributed dynamical processes in the solar atmosphere" that are closely related to the 11-year solar activity and therefore, unsurprisingly, the high-frequency modes are found to be modulated by this latter cycle (Kolotkov et al, 2016). But this phenomenon is not limited to solar activity signals and has also been identified in surface air temperature records (Paluš, 2014), and time series of the sea level (Liu et al, 2007), and may indicate crossscale nonlinear couplings (Paluš, 2014;Huang et al, 2016).…”

On the temporal variability of the surface solar radiation by means of spectral representations

“…A slightly slanted plateau between 2 days and 2-3 months can also be identified, whose power declines towards lower scales. Preliminary investigations suggest that this high-frequency region is noise-like and modulated by the yearly cycle, and is an indication of cross-scale coupling of phase-amplitude (Paluš, 2014;Huang et al, 2016); further research is needed for a definitive answer. If this plateau is noise-like, it would impact numerical modelling of the SSI in the context of forecasting, as it would imply that, for this specific site at Carpentras, there exists a deterministic yearly component, a gap in the spectrum at medium scales, and a stochastic component modulated by the highly energetic yearly cycle.…”

“…Any user who accepts the BSRN data release guidelines (http://www.bsrn.awi.de/en/data/conditions_of_ data_release) may ask Gert König-Langlo (Gert.KoenigLanglo@awi.de) (König-Langlo et al, 2015) to obtain an account to download these datasets.…”

“…The measuring station is part of the Baseline Surface Radiation Network (BSRN, König-Langlo et al, 2015). BSRN is a worldwide radiometric network providing accurate readings of the SSI at 1 min temporal resolution and with an uncertainty requirement at 5 W m −2 for SSI (Ohmura et al, 1998).…”

Adaptive data analysis: theory and applications

“…The authors for the articles in this theme issue stem from many different fields, including applied mathematics (with articles on new theoretical developments by Cicone et al [1]; Daubechies et al [2]; Hou & Shi [3]; Joliffe & Cadima [4]; and Mallat [5]), physics (Meignen et al [6]), physiology (Yeh et al [7]), general science and data analysis (Huang et al [8]), meteorology and climate (Qiao et al [9] and Wu et al [10]) and biomedical research (Hemakom et al [11]). …”