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

physical review. e, statistical, nonlinear and soft matter physics

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