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: A novel composite framework integrating time varying filter-based empirical mode decomposition, fuzzy entropy (FE) theory, singular spectrum analysis (SSA), phase space reconstruction (PSR), compound prediction models adopting kernel-based extreme learning machine (KELM) and convolutional long short-term memory network (ConvLSTM) as well as mutation and hierarchy-based hybrid optimization algorithm, is proposed in this paper.
100 citations
TL;DR: A hybrid empirical model decomposition (EMD) support vector regression (SVR) model designated as EMD-SVR for nonlinear and non-stationary wave prediction is proposed in this paper.
100 citations
TL;DR: The faster learning speed, lesser computational complexity, superior classification accuracy, and short event detection time prove that the proposed HHT-WBELM method can be implemented in the online power quality monitoring system.
Abstract: In this paper, Hilbert Huang transform (HHT) and weighted bidirectional extreme learning machine (WBELM) are integrated to detect and classify power quality events (PQEs) in real time Empirical mode decomposition is used to decompose the nonstationary PQEs into the monocomponent mode of oscillation, known as intrinsic mode functions (IMFs) The efficacious features are extracted by applying the Hilbert transform (HT) on the IMFs An efficient WBELM computational intelligence technique is proposed to recognize the single, as well as multiple PQEs and its performances are compared with the recently developed classifiers such as support vector machine, least-square support vector machine, extreme learning machine, and bidirectional extreme learning machine The recognition architecture of HHT integrated with WBELM (HHT-WBELM) method is tested and compared with the empirical wavelet transform associated with HT and WBELM method, and tunable-Q wavelet transform along with HT and WBELM method The faster learning speed, lesser computational complexity, superior classification accuracy, and short event detection time prove that the proposed HHT-WBELM method can be implemented in the online power quality monitoring system Finally, a hardware prototype is developed based on the digital signal processor to verify the cogency of the proposed method in real time The feasibility of the proposed method is tested and validated by both the simulation and laboratory experiments
100 citations
Cites background from "The empirical mode decomposition an..."
...Each IMF is associated with an equal number of extrema and zerocrossings, and is symmetrical around the local mean [22]....
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TL;DR: A robust and simple automated multivariate empirical wavelet transform (MEWT) algorithm for the decoding of different MI tasks and a robust correlation-based feature selection strategy is applied to largely reduce the system complexity and computational load.
Abstract: The robustness and computational load are the key challenges in motor imagery (MI) based on electroencephalography (EEG) signals to decode for the development of practical brain-computer interface (BCI) systems. In this study, we propose a robust and simple automated multivariate empirical wavelet transform (MEWT) algorithm for the decoding of different MI tasks. The main contributions of this study are four-fold. First, the multiscale principal component analysis method is utilized in the preprocessing module to obtain robustness against noise. Second, a novel automated channel selection strategy is proposed and then is further verified with comprehensive comparisons among three different strategies for decoding channel combination selection. Third, a sub-band alignment method by utilizing MEWT is adopted to obtain joint instantaneous amplitude and frequency components for the first time in MI applications. Four, a robust correlation-based feature selection strategy is applied to largely reduce the system complexity and computational load. Extensive experiments for subject-specific and subject independent cases are conducted with the three-benchmark datasets from BCI competition III to evaluate the performances of the proposed method by employing typical machine-learning classifiers. For subject-specific case, experimental results show that an average sensitivity, specificity and classification accuracy of 98% was achieved by employing multilayer perceptron neural networks, logistic model tree and least-square support vector machine (LS-SVM) classifiers, respectively for three datasets, resulting in an improvement of upto 23.50% in classification accuracy as compared with other existing method. While an average sensitivity, specificity and classification accuracy of 93%, 92.1% and 91.4% was achieved for subject independent case by employing LS-SVM classifier for all datasets with an increase of up to 18.14% relative to other existing methods. Results also show that our proposed algorithm provides a classification accuracy of 100% for subjects with small training size in subject-specific case, and for subject independent case by employing a single source subject. Such satisfactory results demonstrate the great potential of the proposed MEWT algorithm for practical MI EEG signals classification.
100 citations
Cites methods from "The empirical mode decomposition an..."
...Step 2: Hilbert Transform for Instantaneous Components Extraction:Hilbert transform is applied onto each narrow sub-band signal to extract the instantaneous components and the analytical representation is given as below [48],...
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TL;DR: The simulation results show that the proposed hierarchical algorithms have better performance than the overall estimation algorithms without parameter decomposition.
Abstract: This paper studies the modeling of multi-frequency signals based on measured data. With the use of the hierarchical identification principle and the iterative search, several iterative parameter estimation algorithms are derived for the signal models with the known frequencies and the unknown frequencies. For the multi-frequency signals, the hierarchical estimation algorithms are derived by means of parameter decomposition. Through the decomposition, the original optimization problem is transformed into the combination of the nonlinear optimization and the linear optimization problems. The simulation results show that the proposed hierarchical algorithms have better performance than the overall estimation algorithms without parameter decomposition.
99 citations
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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