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...
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TL;DR: In this paper, an ensemble empirical mode decomposition (EEMD) combined with adaptive thresholding was proposed for seismic denoising, where a signal was decomposed into individual components called intrinsic mode functions (IMFs) and each decomposed signal was then compared with those IMFs resulting from a white noise realization to determine if the original signal contained structural features or white noise only.
Abstract: Random and coherent noise exists in microseismic and seismic data, and suppressing noise is a crucial step in seismic processing. We have developed a novel seismic denoising method, based on ensemble empirical mode decomposition (EEMD) combined with adaptive thresholding. A signal was decomposed into individual components called intrinsic mode functions (IMFs). Each decomposed signal was then compared with those IMFs resulting from a white-noise realization to determine if the original signal contained structural features or white noise only. A thresholding scheme then removed all nonstructured portions. Our scheme is very flexible, and it is applicable in a variety of domains or in a diverse set of data. For instance, it can serve as an alternative for random noise removal by band-pass filtering in the time domain or spatial prediction filtering in the frequency-offset domain to enhance the lateral coherence of seismic sections. We have determined its potential for microseismic and reflection seismic denoising by comparing its performance on synthetic and field data using a variety of methods including band-pass filtering, basis pursuit denoising, frequency-offset deconvolution, and frequency-offset empirical mode decomposition.
135 citations
Cites methods from "The empirical mode decomposition an..."
...Empirical mode decomposition (EMD) developed by Huang et al. (1998) is a powerful signal analysis technique to split nonstationary and nonlinear signal systems, such as seismic data....
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...The sifting procedure ensures the first IMFs contain the detailed components of the input signal; the last one solely describes the signal trend (Huang et al., 1998; Bekara and Van der Baan, 2009; Han and Van der Baan, 2013)....
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TL;DR: In this article, a new signal processing technique, the Hilbert-Huang transform and its marginal spectrum, was used for analysis of vibration signals and fault diagnosis of roller bearings. But the proposed method may provide not only an increase in the spectral resolution but also reliability for the fault detection and diagnosis of Roller bearings.
Abstract: This work presents the application of a new signal processing technique, the Hilbert-Huang transform and its marginal spectrum, in analysis of vibration signals and fault diagnosis of roller bearings. The empirical mode decomposition (EMD), Hilbert-Huang transform (HHT) and marginal spectrum are introduced. First, the vibration signals are separated into several intrinsic mode functions (IMFs) by using EMD. Then the marginal spectrum of each IMF can be obtained. According to the marginal spectrum, the localized fault in a roller bearing can be detected and fault patterns can be identified. The experimental results show that the proposed method may provide not only an increase in the spectral resolution but also reliability for the fault detection and diagnosis of roller bearings.
135 citations
Cites background or methods from "The empirical mode decomposition an..."
...2. An introduction to the Hilbert-Huang Transform [ 17 ]...
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...As pointed out by Huang et al. [ 17 ], the frequency in () h ω has a totally different meaning from the Fourier spectral analysis....
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...Based on these observations, Huang et al. [ 17 ] defined IMF as a class of functions that satisfy two conditions: 1) In the whole data set, the number of extrema and the number of zero-crossings must be either equal or differ at most by one....
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TL;DR: Empirical results demonstrate that the proposed model still work well when the number of decomposition results varies, thus is promising for forecasting crude oil price.
Abstract: Considering the actual demand of crude oil price forecasting, a novel model based on ensemble empirical mode decomposition (EEMD) and long short-term memory (LSTM) is proposed. In practical work, the model trained by historical data will be used in later data. Then the forecasting models based on EEMD need re-execute EEMD to update decomposition results of price series after getting new data. In this process, the decomposition results of same period will not stay entirely identical, and even the number of decomposition results could change Unfortunately, in this case the traditional decomposition-ensemble models trained by historical data break down. To overcome this disadvantage, a method to select same number of proper inputs in different situations of decomposition results is developed. And for extracting feature from selected components more adequately, LSTM is introduced as forecasting method to predict price movement directly. For illustration and verification purposes, the proposed model is used to predict the crude oil spot price of West Texas Intermediate (WTI). Empirical results demonstrate that the proposed model still work well when the number of decomposition results varies, thus is promising for forecasting crude oil price.
135 citations
TL;DR: A forecasting architecture based on a new hybrid decomposition technique (HDT) and an improved flower-pollination algorithm (FPA)-back propagation (BP) neural network prediction algorithm is proposed, which indicates that the proposed model is highly suitable for non-stationary multi-step wind speed forecasting.
135 citations
TL;DR: A novel approach that combines deep learning and data augmentation for EEG classification is proposed that substantially improve the training of neural networks, and both two networks yield relatively higher classification accuracies compared to prevailing approaches.
Abstract: Brain–computer interface provides a new communication bridge between the human mind and devices, depending largely on the accurate classification and identification of non-invasive EEG signals. Recently, the deep learning approaches have been widely used in many fields to extract features and classify various types of data successfully. However, the deep learning approach requires massive data to train its neural networks, and the amount of data impacts greatly on the quality of the classifiers. This paper proposes a novel approach that combines deep learning and data augmentation for EEG classification. We applied the empirical mode decomposition on the EEG frames and mixed their intrinsic mode functions to create new artificial EEG frames, followed by transforming all EEG data into tensors as inputs of the neural network by complex Morlet wavelets. We proposed two neural networks—convolutional neural network and wavelet neural network—to train the weights and classify two classes of motor imagery signals. The wavelet neural network is a new type of neural network using wavelets to replace the convolutional layers. The experimental results show that the artificial EEG frames substantially improve the training of neural networks, and both two networks yield relatively higher classification accuracies compared to prevailing approaches. Meanwhile, we also verified the performance of our new proposed wavelet neural network model in the classification of steady-state visual evoked potentials.
135 citations
Cites background or methods from "The empirical mode decomposition an..."
...The empirical mode decomposition algorithm [31] fits with non-stationary signals that change in the frequency structure within a short period of time [32]–[34]....
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...The algorithm decomposes the original signals into a finite number of functions called intrinsic mode functions (IMFs) [31], each of which represents a non-linear oscillation of the signal....
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References
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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....
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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)....
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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…...
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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....
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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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