Analyse de l’invariance d’échelle de séries temporelles par la décomposition modale empirique et l’analyse spectrale de Hilbert

Recently, Hilbert-Huang transform (HHT), or Empirical Mode Decomposition (EMD) have become a promising methodology to deal with nonlinear and nonstationary time series [1,2]. This corresponds to a data-driven method with very local ability, both in physical and spectral space [3]. In this work, we propose an extended version of Hilbert spectral analysis, namely arbitrary order Hilbert spectral analysis, to characterize the scale invariant properties in spectral space directly.

The most innovative part of the Hilbert-Huang transform is the Empirical Mode Decomposition, which can separate the original time series into several Intrinsic Mode Functions, called IMF. The starting point of EMD is to consider the time series from real word as multi-component signal. The corresponding characteristic scale is defined as the distance between two successive maxima points (resp. minima points). The intrinsic mode function is proposed to approximate the mono-component signal, which satisfies the following two conditions: (i) the difference between the number of local extrema and the number of zero-crossings must be zero or one; (ii) the running mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero [1,2]. Then the empirical mode decomposition algorithm is proposed to extract IMF modes from a given time series [1,2,16].

After applying EMD algorithm on a given time series, the original signal is then decomposed into several IMF modes, c_i(t). We therefore separate the different scales in physical domain. Then the Hilbert transform is applied on each mode to extract the time-frequency-energy information, amplitude A(t) and instantaneous frequency ω(t) [14,17,18]. The Hilbert spectrum H(ω,t) is thus designed to represent the original time series in time-frequency-energy space. When the marginal integration is taken on H(ω,t) over t, we thus have a marginal Hilbert spectrum h(ω), which is corresponding to the Fourier power spectrum [1,2]. The Hilbert transform is a singular integration, which means that it has a very local ability in spectral space. Hence, the combination of EMD and Hilbert spectral analysis provides a novel powerful time-frequency analysis tool to deal with nonlinear and nonstationary time series [1,2].

The above procedure is also known as Hilbert spectral analysis (HSA). It provides a joint pdf p(ω,A) [17,18]. Thus the Hilbert marginal spectrum can be redefined as the marginal integration of p(ω,A) over A [17,18]. We propose here an arbitrary order Hilbert marginal spectrum Lq(ω) by considering the marginal integration of p(ω,A) over A^q [18]. In case of scale invariance, the arbitrary order spectrum has a power law with scaling exponent N(q). Due to the integration operator, N(q) - 1 corresponds to ζ(q) estimated by the classical structure function analysis [18].

We first verify the arbitrary order HSA method by analyzing synthesized fractional Brownian motion (fBm). We show that the present proposed methodology is very useful to exact the Hurst exponent H when 0<H<1. We then analyze a synthesized multifractal process by considering a discrete cascade with a lognormal model. We compare the scaling exponent ξ(q) estimated by HSA with ζ(q) estimated by the structure function. Graphically, these scaling exponent curves are superposed: this confirms the validity of the present method for monofractal and multifractal processes.

We then consider the periodic effect on both structure function analysis and the arbitrary order HSA. Due to large scale deterministic forcing, such as annual cycle, solar motion, the presence of a periodic component is a very common phenomena in environmental time series. We superpose a sine wave on simulated fBm time series with H = 1/3, which is corresponding to the Hurst number of turbulent velocity, and various intensity I. We perform the structure function analysis and the arbitrary order HSA on these data. By comparing with the results of fBm data without periodic component perturbation, we find that the structure function analysis is strongly influenced by the periodic component. It is found graphically that the periodic effect on the scaling analysis is as large as 2 decades. However, when considering EMD and HSA methods on the same data, we see that the arbitrary order HSA can constrain the periodic effect: the periodic effect on the scaling analysis has an amplitude of about 0.3 decade only.

We finally apply this new method on an experimental homogeneous and isotropic turbulence database, which is characterized by the Reynolds number Re_λ = 720. Both Fourier power spectrum and the Hilbert marginal spectrum predict an over two decades inertial range. We then recover the classical structure function scaling exponents ζ(q) in spectral space.

Résumé

Nous proposons ici une généralisation de l’analyse spectrale de Hilbert qui est effectuée dans le cadre des décompositions modales empiriques. Cette analyse spectrale de Hilbert d’ordre arbitraire permet de caractériser les propriétés d’intermittence des séries temporelles à invariance d’échelle multiple. La méthode est validée en utilisant des simulations de mouvement Brownien fractionnaire, avec différentes valeurs du paramètre H, et avec des simulations lognormales multifractales. Une application est effectuée sur des données « réelles » issues du domaine de la turbulence. La méthode proposée ici fonctionne dans l’espace amplitude-fréquence; cette méthode est la première approche pouvant prendre en compte les exposants d’intermittence dans l’espace des fréquences. Nous montrons également que cette méthode est supérieure à l’approche utilisant les fonctions de structure lorsque la série à analyser présente une invariance d’échelle superposée à une forte composante périodique.