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 hybrid forecasting system is proposed in this paper that includes effective data decomposition techniques, recurrent neural network prediction algorithms and error decomposition correction methods, and decomposes the error to correct the previously predicted wind speed.
121 citations
TL;DR: In this paper, a flexible approach for the time-frequency analysis of multicomponent signals involving the use of analytic vectors and demodulation is introduced, and the resulting instantaneous frequency of each component in each tile is not constrained to a set polynomial form in time, and is readily calculated, as is the corresponding Hilbert energy spectrum.
Abstract: In this paper, we introduce a flexible approach for the time-frequency analysis of multicomponent signals involving the use of analytic vectors and demodulation. The demodulated analytic signal is projected onto the time-frequency plane so that, as closely as possible, each component contributes exclusively to a different ‘tile’ in a wavelet packet tiling of the time-frequency plane, and at each time instant, the contribution to each tile definitely comes from no more than one component. A single reverse demodulation is then applied to all projected components. The resulting instantaneous frequency of each component in each tile is not constrained to a set polynomial form in time, and is readily calculated, as is the corresponding Hilbert energy spectrum. Two examples illustrate the method. In order better to understand the effect of additive noise, the approximate variance of the estimated instantaneous frequency in any tile has been formulated by starting with pure noise and studying its evolving covariance structure through each step of the algorithm. The validity and practical utility of the resulting expression for the variance of the estimated instantaneous frequency is demonstrated via a simulation experiment.
121 citations
Cites background or methods from "The empirical mode decomposition an..."
...For any projection indexed by (j, n) the Hilbert spectrum ( Huang et al, 1998;...
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...(d) The Hilbert spectrum and instantaneous frequency For any projection indexed by (j, n), the Hilbert spectrum (Huang et al. 1998; Olhede & Walden 2004a) can be formed by firstly calculating the amplitude and phase sequences given by Aj;n;l Z…...
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...Agglomeration over projections yields the complete Hilbert spectrum (Huang et al. 1998; Olhede & Walden 2004a)....
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...Agglomeration over projections yields the complete Hilbert spectrum ( Huang et al, 1998; Olhede and Walden, 2004a)....
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TL;DR: In this article, the long-term trend of aerosol optical thickness (AOT) over the global oceans has been studied by using a nearly 25-year aerosol record from the Advanced Very High Resolution Radiometer (AVHRR) Pathfinder Atmosphere extended (PATMOS-x) data set.
Abstract: [1] The long-term trend of aerosol optical thickness (AOT) over the global oceans has been studied by using a nearly 25-year aerosol record from the Advanced Very High Resolution Radiometer (AVHRR) Pathfinder Atmosphere extended (PATMOS-x) data set. Both global and regional analyses have been performed to derive the AOT tendencies for monthly, seasonal, and annual mean AOT values at AVHRR 0.63 μm channel (or Channel-1). A linear decadal change of −0.01 is obtained for globally and monthly averaged aerosol optical thickness, τ1, of AVHRR Channel-1. This negative tendency is even more evident for globally and annually averaged τ1 and the magnitude can be up to −0.03/decade. Seasonal patterns in the AOT regional long-term trend are evident. In general, negative tendencies are observed for seasonally averaged τ1 in regions influenced by emissions from industrialized countries and the magnitude can be up to −0.10/decade. Positive tendencies are observed in regions influenced by emissions from fast developing countries and the magnitude can be up to +0.04/decade. For regions heavily influenced by Saharan desert particles, a negative trend with a maximum magnitude of −0.03/decade is detected. However, over regions influenced by smoke from biomass burning, positive tendencies with a maximum magnitude of +0.04/decade are observed. Sensitivity analyses have also been performed to study the effects of radiance calibration, aerosol retrieval algorithm, and spatial resolution of input retrieval radiances on the global aerosol long-term tendencies.
121 citations
TL;DR: In this article, a new method for crack detection in beams based on instantaneous frequency and empirical mode decomposition is proposed, which can be used to improve the effectiveness of vibration-based crack detection techniques.
Abstract: In this paper, a new method for crack detection in beams based on instantaneous frequency and empirical mode decomposition is proposed. The dynamic behaviour of a cantilever beam with a breathing crack under harmonic excitation is investigated both theoretically and experimentally. A simple single-degree-of-freedom system with varying stiffness is employed to simulate the dynamic behaviour of the beam. The time-varying stiffness is modelled using a simple periodic function. Both simulated and experimental response data are analysed by applying empirical mode decomposition and Hilbert transform and the instantaneous frequency of each oscillatory mode is obtained. It is shown that the instantaneous frequency oscillates between frequencies corresponding to the open and closed states revealing the breathing of the crack. The variation of the instantaneous frequency increases with increasing crack depth following a polynomial law and consequently can be used for estimation of crack size. Using the intrinsic modes of the system, the harmonic distortion of the distorted sinusoidal response is calculated. It follows that the harmonic distortion increases with crack depth following definite trends and can be also used as an effective indicator for crack size. The proposed time–frequency approach is superior compared to Fourier analysis and can be used to improve the effectiveness of vibration-based crack detection techniques.
121 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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