Arbitrary-order Hilbert spectral analysis for time series possessing scaling statistics: a comparison study with detrended fluctuation analysis and wavelet leaders

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter {\mu} = 0.15. We find that the SF underestimates scaling exponent when q > 3. The Hilbert method provides a slight underestimation when q > 5. However, both DFA and WL overestimate the scaling exponents when q > 5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylor’s microscale Reynolds number Relambda \simeq 250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents {\xi}{\theta}(q) quite close to the one for longitudinal velocity.

💡 Research Summary

This paper introduces an extended Hilbert–Huang framework called arbitrary‑order Hilbert spectral analysis (HSA) for characterizing scale‑invariant properties of time series directly in the amplitude‑frequency domain. The method consists of two stages. First, empirical mode decomposition (EMD) adaptively separates the signal into intrinsic mode functions (IMFs) without any a‑priori basis, ensuring each IMF satisfies local extrema and zero‑mean envelope conditions. Second, each IMF undergoes a Hilbert transform to obtain instantaneous amplitude A(t) and phase θ(t), from which the instantaneous frequency ω(t)= (1/2π) dθ/dt is derived. By constructing the joint probability density p(ω, A) of frequency and amplitude, the authors define arbitrary‑order moments L_q(ω)=∫A^q p(ω, A) dA. When the process is scale‑invariant, L_q(ω) follows a power law L_q(ω)∝ω^{−ξ(q)}; the exponent ξ(q)−1 corresponds to the traditional structure‑function exponent ζ(q).

The authors benchmark HSA against three widely used techniques: structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leaders (WL). Synthetic tests include (i) fractional Brownian motion (fBm) and (ii) a multifractal cascade with intermittency parameter μ=0.15. For fBm, all methods recover the expected linear ζ(q) within statistical error. For the multifractal case, SF underestimates ζ(q) for q > 3, DFA and WL overestimate it for q > 5, while HSA shows only a slight under‑estimation for q > 5, yielding the most faithful singularity spectrum among the four.

The practical relevance of HSA is demonstrated on temperature measurements from a turbulent jet (Taylor microscale Reynolds number Re_λ≈250). This passive‑scalar data contain strong ramp‑cliff structures that act as large‑scale, non‑linear features. SF fails to reveal any power‑law scaling because the increment operation filters out the ramp‑cliff contributions. DFA and WL, which rely on Fourier‑type transforms, generate artificial energy flux from low to high frequencies, leading to systematically lower scaling exponents. In contrast, HSA naturally separates the ramp‑cliff components into distinct IMFs, avoiding the need for high‑order harmonics. The resulting scaling exponents ξ_θ(q) closely match those obtained for the longitudinal velocity field, suggesting that the temperature field is less intermittent than previously thought.

Despite its empirical nature—EMD lacks a rigorous theoretical foundation—the authors cite recent advances (e.g., Flandrin et al.) that begin to formalize its statistical properties. They acknowledge current limitations: HSA presently handles only non‑negative moments (q ≥ 0), cannot directly address negative‑order statistics, and its performance depends on stopping criteria and spline interpolation choices. Nevertheless, the extensive numerical and experimental validation demonstrates that HSA provides a more robust and less biased estimation of scaling exponents for non‑linear, non‑stationary signals, especially when large‑scale structures dominate. The paper concludes by recommending further theoretical work on EMD, extension to negative‑order moments, and application of HSA to other complex systems such as climate, physiology, and finance.