The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

doi:10.1098/RSPA.1998.0193

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The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

Journal Article•DOI•

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

Norden E. Huang¹, Zheng Shen², Steven R. Long³, Man-Li C. Wu¹, Hsing H. Shih⁴, Quanan Zheng⁵, Nai-Chyuan Yen⁶, C. C. Tung⁷, Henry H. Liu⁸ - Show less +5 more•Institutions (8)

Goddard Space Flight Center¹, Johns Hopkins University², Wallops Flight Facility³, National Ocean Service⁴, University of Delaware⁵, United States Naval Research Laboratory⁶, North Carolina State University⁷, Naval Surface Warfare Center⁸

08 Mar 1998-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (The Royal Society)-Vol. 454, Iss: 1971, pp 903-995

TL;DR: In this paper, a new method for analysing nonlinear and nonstationary data has been developed, which is the key part of the method is the empirical mode decomposition method with which any complicated data set can be decoded.

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Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

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Journal Article•DOI•

Novel Particle Swarm Optimization-Based Variational Mode Decomposition Method for the Fault Diagnosis of Complex Rotating Machinery

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Xian-Bo Wang, Zhixin Yang, Xiaoan Yan

01 Feb 2018-IEEE-ASME Transactions on Mechatronics

TL;DR: In this article, a particle swarm optimization-based variational mode decomposition method was proposed for fault detection in rotating machinery, which adopts the minimum mean envelope entropy to optimize the parameters (α$ and K$ ) in the existing variational decomposition.

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Abstract: The vibration signals of faulty rotating machinery are typically nonstationary, nonlinear, and mixed with abundant compounded background noise. To extract the potential excitations from the observed rotating machinery, signal demodulation and time–frequency analysis are indispensable. This work proposes a novel particle swarm optimization-based variational mode decomposition method, which adopts the minimum mean envelope entropy to optimize the parameters ( $\alpha$ and $K$ ) in the existing variational mode decomposition. The proposed fault-detection framework separated the observed vibration signals into a series of intrinsic modes. A certain number of the intrinsic modes are then selected by means of the Hilbert transform-based square envelope spectral kurtosis. Subsequently, in this study, the feature representations were reconstructed via the selected intrinsic modes; then, the envelope spectra of the real faulty conditions were generated in the rotating machinery. To verify the performance of the proposed method, a testbed platform of a gearbox with a combination of different faults was implemented. The experimental results demonstrated that the proposed method represented the patterns of the fault frequency more explicitly than the available empirical mode decomposition, the local mean decomposition, and the wavelet package transform method.

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181 citations

Additional excerpts

...5(a) demonstrates the SNR distributions using PSOVMD, EMD, LMD, and WPT under the condition of varied σ....
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...The EMD removes the mean of the envelopes as a low-pass centerline in order to isolate the high-frequency oscillations as the mode and recursively extracts the low-pass centerline [20]....
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...In addition, the conclusion presented in [23] revealed that the VMD outperformed the EMD in terms of tone detection, separation, and noise robustness....
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...The proposed combinational method, referred to as PSO-based VMD (PSO-VMD), improves the original VMD method to be with self-adaptive property similar to the EMD and LMD methods....
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...The EMD generates a series of complete and almost orthogonal intrinsic mode functions (IMFs) [21], which represent the intrinsic oscillation modes embedded in the signal....
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Journal Article•DOI•

Near-Surface Turbulence in the Presence of Breaking Waves

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Johannes Gemmrich¹, David M. Farmer²•Institutions (2)

Fisheries and Oceans Canada¹, University of Rhode Island²

01 May 2004-Journal of Physical Oceanography

TL;DR: In this paper, a three-axis pulse-to-pulse coherent acoustic Doppler profiler and acoustic resonators were used to reveal the turbulence and bubble field beneath breaking waves in the open ocean at wind speeds up to 14 m s−1.

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Abstract: Observations with a three-axis pulse-to-pulse coherent acoustic Doppler profiler and acoustic resonators reveal the turbulence and bubble field beneath breaking waves in the open ocean at wind speeds up to 14 m s−1. About 55%–80% of velocity wavenumber spectra, calculated with Hilbert spectral analysis based on empirical mode decomposition, are consistent with an inertial subrange. Time series of turbulent kinetic energy dissipation at approximately 1 m beneath the free surface and 1-Hz sampling rate are obtained. High turbulence levels with dissipation rates more than four orders larger than the background dissipation are linked to wave breaking. Initial dissipation levels beneath breaking waves yield the Hinze scale of the maximum bubble size aH ≅ 2 × 10−3 m. Turbulence induced by discrete breaking events was observed to decay as e ∝ tn, where n = −4.3 is close to the theoretical value for isotropic turbulence (−17/4). In the crest region above the mean waterline, dissipation increases as e(z) ...

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181 citations

Journal Article•DOI•

Lithium-ion batteries remaining useful life prediction based on a mixture of empirical mode decomposition and ARIMA model

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Yapeng Zhou¹, Miaohua Huang¹•Institutions (1)

Wuhan University of Technology¹

01 Oct 2016-Microelectronics Reliability

TL;DR: A novel approach which combines empirical mode decomposition (EMD) and autoregressive integrated moving average (ARIMA) model is proposed for RUL prognostic, which gives a more satisfying and accurate prediction result.

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181 citations

Cites background from "The empirical mode decomposition an..."

...in 1998 [27], which can convert a group of time series into locally narrow band components, named intrinsic mode functions (IMFs)....
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Journal Article•DOI•

Forecasting tourism demand based on empirical mode decomposition and neural network

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Chun-Fu Chen¹, Ming-Cheng Lai¹, Ching-Chiang Yeh¹•Institutions (1)

National Taipei University of Business¹

01 Feb 2012-Knowledge Based Systems

TL;DR: Experimental results show that the proposed model outperforms the single BPN model without EMD preprocessing and the traditional autoregressive integrated moving average (ARIMA) models.

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Abstract: Due to the fluctuation and complexity of the tourism industry, it is difficult to capture its non-stationary property and accurately describe its moving tendency. In this study, a novel forecasting model based on empirical mode decomposition (EMD) and neural network is proposed to predict tourism demand (i.e. the number of arrivals). The proposed approach first uses EMD, which can adaptively decompose the complicated raw data into a finite set of intrinsic mode functions (IMFs) and a residue, which have simpler frequency components and higher correlations. The IMF components and residue are than modeled and forecasted using back-propagation neural network (BPN) and the final forecasting value can be obtained by the sum of these prediction results. In order to evaluate the performance of the proposed approach, the majority of international visitors to Taiwan are used as illustrative examples. Experimental results show that the proposed model outperforms the single BPN model without EMD preprocessing and the traditional autoregressive integrated moving average (ARIMA) models.

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180 citations

Cites background or methods from "The empirical mode decomposition an..."

...The detail algorithm for EMD is shown as follows [15,8,14,35]:...
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...[15], is perfectly suitable for nonlinear and non-stationary signal analysis, which adaptively represents the local characteristic of the given signal [15]....
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...The second condition modifies classical global requirement to a local one; it is necessary so that the instantaneous frequency will not have the unwanted fluctuations induced by asymmetric wave forms [15]....
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...By using EMD, any complicated signal can be decomposed into a finite and often small number of Intrinsic Mode Functions (IMFs) [15], which have simpler frequency components and stronger correlations, thus are easier and more accurate to forecast....
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...[15], is a form of adaptive time series decomposition technique using the Hilbert–Huang transform (HHT) for nonlinear and non-stationary time series data....
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Journal Article•DOI•

Damage detection in structures under traveling loads by Hilbert–Huang transform

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N. Roveri¹, Antonio Carcaterra¹•Institutions (1)

Sapienza University of Rome¹

01 Apr 2012-Mechanical Systems and Signal Processing

TL;DR: In this article, a novel HHT-based method for damage detection of bridge structures under a traveling load is proposed, which uses a single point measurement and is able to identify the presence and the location of the damage along the beam.

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180 citations

Cites background or methods from "The empirical mode decomposition an..."

...In general, they can detect natural frequency modifications induced by damage, but are not able to locate it [7–9]....
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...The wavelet transform is an adaptive window Fourier method [14], handling nonstationary signals only for linear systems [7]....
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...According to properties (1) and (2) in the last section, to generate the family of IMFs the so-called sifting process is used [7]:...
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...More generally, the proposed, HHT-based, technique should still reveal the damage location when the moving load produces nonlinear and nonstationary data, a task that conventional time–frequency techniques, such as wavelet transform or STFT, can hardly handled [7]....
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...[7] developed an innovative time–frequency technique, known as Hilbert–Huang transform (HHT), able to analyze nonlinear systems and/or nonstationary data....
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References

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Deterministic nonperiodic flow

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Edward N. Lorenz¹•Institutions (1)

Massachusetts Institute of Technology¹

01 Mar 1963-Journal of the Atmospheric Sciences

TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.

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Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

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...(ii) Lorenz equation The famous Lorenz equation (Lorenz 1963) was proposed initially to study deterministic non-periodic flow....
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Linear and Nonlinear Waves

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TL;DR: In this paper, a general overview of the nonlinear theory of water wave dynamics is presented, including the Wave Equation, the Wave Hierarchies, and the Variational Method of Wave Dispersion.

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Abstract: Introduction and General Outline. HYPERBOLIC WAVES. Waves and First Order Equations. Specific Problems. Burger's Equation. Hyperbolic Systems. Gas Dynamics. The Wave Equation. Shock Dynamics. The Propagation of Weak Shocks. Wave Hierarchies. DISPERSIVE WAVES. Linear Dispersive Waves. Wave Patterns. Water Waves. Nonlinear Dispersion and the Variational Method. Group Velocities, Instability, and Higher Order Dispersion. Applications of the Nonlinear Theory. Exact Solutions: Interacting Solitary Waves. References. Index.

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RANDOM DATA Analysis and Measurement Procedures

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Julius S. Bendat, Allan G. Piersol

01 Jan 1971

TL;DR: A revised and expanded edition of this classic reference/text, covering the latest techniques for the analysis and measurement of stationary and nonstationary random data passing through physical systems, is presented in this article.

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Abstract: From the Publisher: A revised and expanded edition of this classic reference/text, covering the latest techniques for the analysis and measurement of stationary and nonstationary random data passing through physical systems. With more than 100,000 copies in print and six foreign translations, the first edition standardized the methodology in this field. This new edition covers all new procedures developed since 1971 and extends the application of random data analysis to aerospace and automotive research; digital data analysis; dynamic test programs; fluid turbulence analysis; industrial noise control; oceanographic data analysis; system identification problems; and many other fields. Includes new formulas for statistical error analysis of desired estimates, new examples and problem sets.

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...A brief tutorial on the Hilbert transform with the emphasis on its physical interpretation can be found in Bendat & Piersol (1986)....
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Theory of communication

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Dennis Gabor

01 Jan 1946

5,910 citations

"The empirical mode decomposition an..." refers methods in this paper

...In order to obtain meaningful instantaneous frequency, restrictive conditions have to be imposed on the data as discussed by Gabor (1946), Bedrosian (1963) and, more recently, Boashash (1992): for any function to have a meaningful instantaneous frequency, the real part of its Fourier transform has…...
[...]

Journal Article•DOI•

Mathematical analysis of random noise

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S. O. Rice

01 Jul 1944-Bell System Technical Journal

TL;DR: In this paper, the authors used the representations of the noise currents given in Section 2.8 to derive some statistical properties of I(t) and its zeros and maxima.

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Abstract: In this section we use the representations of the noise currents given in section 2.8 to derive some statistical properties of I(t). The first six sections are concerned with the probability distribution of I(t) and of its zeros and maxima. Sections 3.7 and 3.8 are concerned with the statistical properties of the envelope of I(t). Fluctuations of integrals involving I2(t) are discussed in section 3.9. The probability distribution of a sine wave plus a noise current is given in 3.10 and in 3.11 an alternative method of deriving the results of Part III is mentioned. Prof. Uhlenbeck has pointed out that much of the material in this Part is closely connected with the theory of Markoff processes. Also S. Chandrasekhar has written a review of a class of physical problems which is related, in a general way, to the present subject.22

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5,806 citations

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...In general, if more quantitative results are desired, the original skeleton presentation is better; if more qualitative results are desired, the smoothed presentation is better....
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...Therefore, the parameter, ν, defined as N21 −N20 = 1 π2 m4m0 −m22 m2m0 = 1 π2 ν2, (3.7) offers a standard bandwidth measure (see, for example, Rice 1944a, b, 1945a, b; Longuet-Higgins 1957)....
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