The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis
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.
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...
Citations
More filters
TL;DR: In this paper, a demodulation technique based on improved local mean decomposition (LMD) is proposed for mechanical diagnoses, and a method of boundary process and a strategy for determining the step size of moving average are presented to improve the LMD algorithm.
Abstract: Demodulation is an available method for mechanical diagnoses, and a demodulation technique based on improved local mean decomposition (LMD) is proposed in this paper. A method of boundary process and a strategy for determining the step size of moving average are presented to improve the LMD algorithm. Instantaneous amplitude (IA) and instantaneous frequency (IF) of the signal can be computed independently of Hilbert transform using the improved LMD method. A well-constructed description of the derived IA and IF is given in the form of instantaneous time–frequency spectrum (ITFS) which preserves both the time and frequency information simultaneously. Results of three synthetic signals indicate that this proposed method is the best demodulation approach to extracting the all-round carrier and modulated components as well as the accurate IF, compared with Hilbert–Huang transform and stationary wavelet transform. The validity of the technique is then demonstrated on a real rotor system of a gas turbine with rub-impact fault. Due to the opposite friction during operation, the transient fluctuations of the IF of the fundamental harmonic component are successfully identified in the ITFS. In addition, we find that the proposed technique is more effective and sensitive than other methods in detecting sub-harmonics and FM components contained in the rub-impact signals. Thus the present method is powerful in the analysis of modulated signals and is an effective tool for the detection of rub-impact faults.
115 citations
TL;DR: Numerical results show the benefits of the proposed approach for mode reconstruction in comparison to similar techniques that do not make use of demodulation, and numerical investigations show that the proposed Approach sharpens the TF representation on which it is built.
Abstract: In this paper, we present a novel technique for the retrieval of the modes of a multicomponent signal using a time-frequency (TF) representation of the signal. Our approach is based on a novel ridge extraction method that takes into account the fact that the TF representation is both discrete in time and frequency, followed by a demodulation procedure. Numerical results show the benefits of the proposed approach for mode reconstruction in comparison to similar techniques that do not make use of demodulation. Furthermore, numerical investigations show that the proposed approach sharpens the TF representation on which it is built.
115 citations
Additional excerpts
...January 13, 2017 DRAFT 21...
[...]
TL;DR: The proposed TFR based on the improved eigenvalue decomposition of Hankel matrix and Hilbert transform has achieved classification accuracy 100% for the studied EEG database and gives good performance in terms of Renyi entropy measure.
Abstract: Time–frequency representation (TFR) is useful for non-stationary signal analysis as it provides information about the time-varying frequency components. This study proposes a novel TFR based on the improved eigenvalue decomposition of Hankel matrix and Hilbert transform (IEVDHM–HT). In the proposed method, first the authors decompose non-stationary signals using the IEVDHM with suitably defined criterion for eigenvalue selection, requirement of number of iterations, and new component merging criteria. Furthermore, the HT is applied on extracted components in order to obtain the TFR of non-stationary signals. The performance of proposed TFR has been evaluated on synthetic signals in clean and white noise environment with different signal-to-noise ratios. The proposed method gives good performance in terms of Renyi entropy measure in comparison with other existing methods. Application of the proposed TFR is also shown for the classification of epileptic seizure electroencephalogram (EEG) signals. The least-square support vector machine (LS-SVM) with radial basis function kernel is used for classification of seizure and seizure-free EEG signals obtained from the publicly available database by the University of Bonn, Germany. The proposed method has achieved classification accuracy 100% for the studied EEG database.
115 citations
Cites background or methods from "The empirical mode decomposition an..."
...The arrangement of amplitude envelope and instantaneous frequency functions of the IMFs provide HHT-based TFR [14]....
[...]
...The MMSC satisfied by such type of mono-component signals, which have more than one eigenvalue pairs, is based on conditions given as follows [14]....
[...]
...The HHT [14] makes use of empirical mode decomposition (EMD) and the HT....
[...]
...The HT is applied on each decomposed component obtained in the previous step that provides additional information about instantaneous amplitude and instantaneous frequency [14]....
[...]
...The implementation of condition 2 has been performed as given in [14] with same interpolation function....
[...]
TL;DR: In this paper, a novel method for fault diagnosis of rolling bearings based on multifractal detrended fluctuation analysis (MF-DFA) and Mahalanobis distance criterion (MDC) was proposed.
115 citations
TL;DR: The proposed multi-stage MEMD-ACO-RF model outperformed models that were compared during the testing phase and has shown the prospects of an accurate forecasting tool for monthly solar radiation forecasting.
114 citations
References
More filters
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.
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.
16,554 citations
"The empirical mode decomposition an..." refers background in this paper
...(ii) Lorenz equation The famous Lorenz equation (Lorenz 1963) was proposed initially to study deterministic non-periodic flow....
[...]
Book•
01 Jan 1974
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.
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.
8,808 citations
Book•
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.
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.
6,693 citations
"The empirical mode decomposition an..." refers background in this paper
...A brief tutorial on the Hilbert transform with the emphasis on its physical interpretation can be found in Bendat & Piersol (1986)....
[...]
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…...
[...]
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.
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
5,806 citations
"The empirical mode decomposition an..." refers background in this paper
...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....
[...]
...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)....
[...]