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Journal ArticleDOI

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

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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Journal ArticleDOI
TL;DR: In this paper, a wavelet-based approach for estimating evolutionary power spectral density (EPSD) functions of vector-valued nonstationary processes is presented, based on the assumption that the time-varying rate of the modulation function of the process is slow as compared to the localized wavelet functions.

91 citations

Journal ArticleDOI
TL;DR: The results indicate that the proposed EEMD–MKNN model has a higher forecast precision than EMD–KNN, KNN method and ARIMA, and has high predictive precision for short-term forecasting.
Abstract: In this paper, we propose a new two-stage methodology that combines the ensemble empirical mode decomposition (EEMD) with multidimensional k -nearest neighbor model (MKNN) in order to forecast the closing price and high price of the stocks simultaneously. The modified algorithm of k -nearest neighbors (KNN) has an increasingly wide application in the prediction of all fields. Empirical mode decomposition (EMD) decomposes a nonlinear and non-stationary signal into a series of intrinsic mode functions (IMFs), however, it cannot reveal characteristic information of the signal with much accuracy as a result of mode mixing. So ensemble empirical mode decomposition (EEMD), an improved method of EMD, is presented to resolve the weaknesses of EMD by adding white noise to the original data. With EEMD, the components with true physical meaning can be extracted from the time series. Utilizing the advantage of EEMD and MKNN, the new proposed ensemble empirical mode decomposition combined with multidimensional k -nearest neighbor model (EEMD–MKNN) has high predictive precision for short-term forecasting. Moreover, we extend this methodology to the case of two-dimensions to forecast the closing price and high price of the four stocks (NAS, S&P500, DJI and STI stock indices) at the same time. The results indicate that the proposed EEMD–MKNN model has a higher forecast precision than EMD–KNN, KNN method and ARIMA.

91 citations

Journal ArticleDOI
TL;DR: It is demonstrated that DFA analysis for the cleaned Mauna Loa record yields scaling comparable to a mid-latitude station, and the EMD algorithm helps to identify dominating frequencies in the time series, which allows to separate fluctuations from the remaining background.
Abstract: Correlations for daily data of total ozone column are investigated by detrended fluctuation analysis (DFA). The removal of annual periodicity does not result in a background-free signal for the tropical station Mauna Loa. In order to identify the remaining quasiperiodic constituent, the relatively new method of empirical mode decomposition (EMD) is tested. We found that the so-called intrinsic mode functions do not represent real signal components of the ozone time series, their amplitude modulation is very sensitive to local changes such as random data removal or smoothing. Tests on synthetic data further corroborate the limitations of decomposing quasiperiodic signals from noise with EMD. Nevertheless the EMD algorithm helps to identify dominating frequencies in the time series, which allows to separate fluctuations from the remaining background. We demonstrate that DFA analysis for the cleaned Mauna Loa record yields scaling comparable to a mid-latitude station.

91 citations

Journal ArticleDOI
TL;DR: Through the numerical examples and simulations, the SSD method is shown to be able to accurately retrieve different components concealed in the data, minimizing at the same time the generation of spurious components.
Abstract: This study introduces singular spectrum decomposition (SSD), a new adaptive method for decomposing nonlinear and nonstationary time series in narrow-banded components. The method takes its origin from singular spectrum analysis (SSA), a nonparametric spectral estimation method used for analysis and prediction of time series. Unlike SSA, SSD is a decomposition method in which the choice of fundamental parameters has been completely automated. This is achieved by focusing on the frequency content of the signal. In particular, this holds for the choice of the window length used to generate the trajectory matrix of the data and for the selection of its principal components for the reconstruction of a specific component series. Moreover, a new definition of the trajectory matrix with respect to the standard SSA allows the oscillatory content in the data to be enhanced and guarantees decrease of energy of the residual. Through the numerical examples and simulations, the SSD method is shown to be able to accurately retrieve different components concealed in the data, minimizing at the same time the generation of spurious components. Applications on time series from both the biological and the physical domain are also presented highlighting the capability of SSD to yield physically meaningful components.

91 citations

Journal ArticleDOI
TL;DR: In this article, a local rub-impact fault diagnosis method based on EMD (empirical mode decomposition) is proposed for rotor system with local rub impact fault, which can be applied to the rotor systems efficiently.

91 citations

References
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Journal ArticleDOI
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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Journal ArticleDOI
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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