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
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TL;DR: Wang et al. as discussed by the authors investigated the accuracy of a modified EMD-SVM model for monthly streamflow forecasting in the Wei River Basin, which has made an improvement by removing the high frequency (IMF 1 ) based on the conventional EMD SVM model.
224 citations
TL;DR: In this paper, a wavelet-based multifractal analysis and a multiplicative cascading process was used to characterize interaction-dominant dynamics in human cognition, and the results indicated that the major portion of these response series had multiplicative interactions between temporal scales, visible as intermittent periods of large and irregular fluctuations.
Abstract: It has been suggested that human behavior in general and cognitive performance in particular emerge from coordination between multiple temporal scales. In this article, we provide quantitative support for such a theory of interaction-dominant dynamics in human cognition by using wavelet-based multifractal analysis and accompanying multiplicative cascading process on the response series of 4 different cognitive tasks: simple response, word naming, choice decision, and interval estimation. Results indicated that the major portion of these response series had multiplicative interactions between temporal scales, visible as intermittent periods of large and irregular fluctuations (i.e., a multifractal structure). Comparing 2 component-dominant models of 1/f(alpha) fluctuations in cognitive performance with the multiplicative cascading process indicated that the multifractal structure could not be replicated by these component-dominant models. Furthermore, a similar multifractal structure was shown to be present in a model of self-organized criticality in the human nervous system, similar to a spatial extension of the multiplicative cascading process. These results illustrate that a wavelet-based multifractal analysis and the multiplicative cascading process form an appropriate framework to characterize interaction-dominant dynamics in human cognition. This new framework goes beyond the identification of 1/f(alpha) power laws and non-gaussian distributions in response series as used in previous studies. The present article provides quantitative support for a paradigm shift toward interaction-dominant dynamics in human cognition.
222 citations
TL;DR: Variation in vibrissa length predicted motion mean frequencies, including for the highest velocity events, indicating that biomechanics, such as vibrissa resonance, shape signals most likely to drive neural activity.
222 citations
Cites methods from "The empirical mode decomposition an..."
...the micro-motion component, so that “rest” was defined as the average of upper and lower envelopes of the micro-motion component, in a procedure akin to empirical mode decomposition (Huang et al., 1998)....
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01 Sep 2005
TL;DR: This chapter is an introduction to the basic method of the Hilbert-Huang transform, followed by brief descriptions of the recent developments relating to the normalized Hilbert transform, a confidence limit for the Hilbert spectrum, and a statistical significance test for the intrinsic mode function (IMF).
Abstract: The Hilbert-Huang transform (HHT) is an empirically based data-analysis method. Its basis of expansion is adaptive, so that it can produce physically meaningful representations of data from nonlinear and non-stationary processes. The advantage of being adaptive has a price: the difficulty of laying a firm theoretical foundation. This chapter is an introduction to the basic method, which is followed by brief descriptions of the recent developments relating to the normalized Hilbert transform, a confidence limit for the Hilbert spectrum, and a statistical significance test for the intrinsic mode function (IMF). The mathematical problems associated with the HHT are then discussed. These problems include (i) the general method of adaptive data-analysis, (ii) the identification methods of nonlinear systems, (iii) the prediction problems in nonstationary processes, which is intimately related to the end effects in the empirical mode decomposition (EMD), (iv) the spline problems, which center on finding the best spline implementation for the HHT, the convergence of EMD, and two-dimensional EMD, (v) the optimization problem or the best IMF selection and the uniqueness of the EMD decomposition, (vi) the approximation problems involving the fidelity of the Hilbert transform and the true quadrature of the data, and (vii) a list of miscellaneous mathematical questions concerning the HHT.
221 citations
Cites background or methods or result from "The empirical mode decomposition an..."
...As Huang et al. (1998) pointed out, this intra-frequency frequency variation is the hallmark of nonlinear systems....
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...Historically, two different criteria have been used: The first one was used in Huang et al. (1998)....
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...The second complication could come from a highly complicated phase function, as discussed in Huang et al. (1998); then the phase plan will not be a perfect circle....
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...The second complication could come from a highly complicated phase function, as discussed in Huang et al. (1998); then the phase plan will not be a perfect circle. Any deviation from the circle will result in the amplitude being different from unity. Huang and Long (2003) and Huang et al. (2005) conducted detailed comparisons and found the result quite satisfactory....
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...The definition given by Huang et al. (1998, 1999) is hard to quantify....
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01 Jan 2005
TL;DR: Huang's data-driven technique of empirical mode decomposition is given a filter bank interpretation from two complementary perspectives and a stochastic approach shows the spontaneous emergence of an equivalent dyadic filter bank structure when EMD is applied to the versatile class of fractional Gaussian noise processes.
Abstract: Huang’s data-driven technique of empirical mode decomposition (EMD) is given a filter bank interpretation from two complementary perspectives. First, a stochastic approach operating in the frequency domain shows the spontaneous emergence of an equivalent dyadic filter bank structure when EMD is applied to the versatile class of fractional Gaussian noise processes. Second, a similar structure is observed when EMD is operated in the time domain on a deterministic pulse. A detailed statistical analysis of the observed behavior is carried out involving extensive numerical simulations that suggest a number of applications. New EMD-based approaches are used to estimate the scaling exponents in the case of self-similar processes, to perform a fully data-driven spectral analysis, and to denoise-detrend signals that contain noise.
221 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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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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