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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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Citations
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Journal ArticleDOI
TL;DR: The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF.
Abstract: A new Ensemble Empirical Mode Decomposition (EEMD) is presented. This new approach consists of sifting an ensemble of white noise-added signal (data) and treats the mean as the final true result. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions in the sifting process, thus making the different scale signals to collate in the proper intrinsic mode functions (IMF) dictated by the dyadic filter banks. As EEMD is a time–space analysis method, the added white noise is averaged out with sufficient number of trials; the only persistent part that survives the averaging process is the component of the signal (original data), which is then treated as the true and more physical meaningful answer. The effect of the added white noise is to provide a uniform reference frame in the time–frequency space; therefore, the added noise collates the portion of the signal of comparable scale in one IMF. With this ensemble mean, one can separate scales naturall...

6,437 citations

Journal ArticleDOI
TL;DR: This work proposes an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently and is a generalization of the classic Wiener filter into multiple, adaptive bands.
Abstract: During the late 1990s, Huang introduced the algorithm called Empirical Mode Decomposition, which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands. EMD is known for limitations like sensitivity to noise and sampling. These limitations could only partially be addressed by more mathematical attempts to this decomposition problem, like synchrosqueezing, empirical wavelets or recursive variational decomposition. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for an ensemble of modes and their respective center frequencies, such that the modes collectively reproduce the input signal, while each being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into multiple, adaptive bands. Our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand. The variational model is efficiently optimized using an alternating direction method of multipliers approach. Preliminary results show attractive performance with respect to existing mode decomposition models. In particular, our proposed model is much more robust to sampling and noise. Finally, we show promising practical decomposition results on a series of artificial and real data.

4,111 citations


Cites background or methods from "The empirical mode decomposition an..."

  • ...Despite the limited mathematical understanding and some obvious shortcomings, the EMDmethod has had significant impact and is widely used in a broad variety of time-frequency analysis applications....

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  • ...Index Terms—AM-FM, augmented Lagrangian, Fourier transform, Hilbert transform, mode decomposition, spectral decomposition, variational problem, Wiener filter....

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Journal ArticleDOI
TL;DR: It turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions, and the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.
Abstract: Empirical mode decomposition (EMD) has recently been pioneered by Huang et al. for adaptively representing nonstationary signals as sums of zero-mean amplitude modulation frequency modulation components. In order to better understand the way EMD behaves in stochastic situations involving broadband noise, we report here on numerical experiments based on fractional Gaussian noise. In such a case, it turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions. It is also pointed out that the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.

2,304 citations

Journal ArticleDOI
TL;DR: In this paper, Hilbert spectral analysis is proposed as an alternative to wavelet analysis, which provides not only a more precise definition of particular events in time-frequency space, but also more physically meaningful interpretations of the underlying dynamic processes.
Abstract: We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

1,945 citations


Cites background or methods from "The empirical mode decomposition an..."

  • ...As discussed in Huang et al (1998a), such a definition gives only a global averaged meaning to the energy and time scales....

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  • ...(7) The procedure is illustrated in Huang et al (1998a)....

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  • ...We will introduce the whole process by starting from the Hilbert transform....

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  • ...31 :4 by C A L IF O R N IA I N ST IT U T E P1: KKK/NBL/spd/mbg P2: NBL/ARY QC: NBL/kba T1: NBL November 24, 1998 12:47 Annual Reviews AR075-12...

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  • ...A degree of statistical stationarity is also defined by Huang et al (1998a)....

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Journal ArticleDOI
Xiao Jing Wang1
TL;DR: A plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention, and implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.
Abstract: Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

1,774 citations

References
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Journal ArticleDOI
TL;DR: In this article, the first detailed basin-wide measurement of the meridional transport of warm surface water in the tropical Pacific during El Nino was obtained using a combination of crossover and collinear difference techniques.
Abstract: Geosat altimeter data are used to obtain the first detailed basin-wide measurement of the meridional transport of warm surface water in the tropical Pacific during El Nino. Using a combination of crossover and collinear difference techniques, continuous sea level time series are constructed on a 2°×1° grid covering the Pacific between 20°N and 20°S for the 4-year period 1985–1989. Zonal integrations of these data are performed over three latitude bands to examine large-scale sea level changes. Comparison with tide gauge data suggests that these zonal averages have an accuracy of better than 1 cm. The Geosat analyses show a clear pattern of water exchange involving principally the equatorial and north equatorial regions. From the onset of the warm event in late 1986 to the mature phase in mid-1987, mean sea level in the equatorial region dropped nearly 5 cm while simultaneously in the north equatorial region it increased by about the same amount. These anomalies, equivalent to 2×104 m3 of upper layer water, persisted for nearly 2 years before both regions returned to normal. A similar pattern and amplitude of north-south water exchange is also observed on the seasonal time scale, consistent with annual variation of the wind stress curl. Water moves northward across 8°N during boreal winter and southward during summer with a net transport of approximately 30 Sv. Thus, in terms of meridional transport, the 1986–1987 El Nino is seen as a low-frequency modulation of the normal seasonal cycle. In contrast to the widely held view that a surplus of equatorial water is required prior to El Nino, no such buildup was observed prior to the 1986–1987 event.

62 citations


"The empirical mode decomposition an..." refers background or methods in this paper

  • ...Because of the importance of the equatorial region in determining the global climate pattern, the altimeter data have been used extensively to study the dynamics of this area (Miller et al. 1988; Miller & Cheney 1990; Zheng et al. 1994, 1995)....

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  • ...Such continuous available data now are derived from both crossover and collinear differences (Miller & Cheney 1990), covering the period from April 1985 Proc....

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Journal ArticleDOI
TL;DR: In this paper, the 2D wavelet functions Arc and Morlet2D are used to decompose a wave field in space b and scale a. As both wavelets are Hardy functions, the transform result is complex, and the phase, φ, is defined over all b.

37 citations


"The empirical mode decomposition an..." refers background in this paper

  • ...Limited applications have also been made to the time–frequency distribution in time series (see, for example, Farge 1992; Long et al. 1993) and two-dimensional images (Spedding et al. 1993)....

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  • ...Limited applications have also been made to the time{frequency distribution in time series (see, for example, Farge 1992; Long et al. 1993) and two-dimensional images ( Spedding et al. 1993 )....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that the observed spectra, coherencies, and dispersions can be described in terms of modulated Stokes-like carrier waves propagating with an angular spread and a lag between amplitude and phase modulations.
Abstract: Observations of ocean surface waves under steady wind conditions show that they do not obey the linear dispersion relation, thus confirming that previously reported results of laboratory experiments also apply to the ocean. It is also shown that mechanically generated surface waves of near-breaking amplitude follow a similar dispersion relation. Observations of coherency of wave signals between two points are also reported, showing the coherency to be higher than in a wave field consisting of independently propagating Fourier components. It is also demonstrated that the observed spectra, coherencies, and dispersions can be described in terms of modulated Stokes-like carrier waves propagating with an angular spread and a lag between amplitude and phase modulations. This is solely proposed as a method of description of observations since there is no present theory for the dynamics of a process where crests are lost due to crest pairing. This violates a crucial premise of modulation theories.

33 citations

Journal ArticleDOI
TL;DR: In this paper, the blocking and trapping of waves and wave packets by inhomogeneous flow fields were studied in a laboratory setting, and evidence for multiple reflections and thus trapping within the current gradient zone near the blocking point was obtained.

21 citations


"The empirical mode decomposition an..." refers background or methods in this paper

  • ...Limited applications have also been made to the time–frequency distribution in time series (see, for example, Farge 1992; Long et al. 1993) and two-dimensional images (Spedding et al. 1993)....

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  • ...Contrary to almost all the previous methods, this new method is intuitive, direct, a posteriori and adaptive, with the basis of the decomposition based on, and derived from, the data....

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Journal ArticleDOI
TL;DR: In this paper, the effects of strong meridional shear of mean flow on propagation of long waves having a wavelength of about 1000 km and period of 25 days observed in the equatorial Pacific and Atlantic Oceans are analyzed.
Abstract: The effects of strong meridional shear of mean flow on propagation of long waves having a wavelength of about 1000 km and period of 25 days observed in the equatorial Pacific and Atlantic Oceans are analyzed in this paper. Information from the weekly multichannel sea surface temperature images derived from the Advanced Very High Resolution Radiometer on board the NOAA series satellites from January 1986 to March 1992 is used for the statistics of the visual parameters of the waves. The characteristic scales of the waves are estimated based on the results by previous investigators and the statistical results of the present study. Solving a potential vorticity equation, the authors obtain a dispersion relation with an additional term depending on the flow shear strength compared with that of the free Rossby wave. The theoretical model provides a good explanation for the variable behavior of long-wave propagation.

20 citations


"The empirical mode decomposition an..." refers background or methods in this paper

  • ...The dynamics of the equatorial Kelvin waves have been discussed extensively by Zheng et al. (1995). They concluded that steady wind cannot excite sustained Kelvin waves, but the variable wind, especially the westerly wind burst, can cause resonant interactions of the excited Kelvin waves with an intrinsic frequency of (1....

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  • ...Because of the importance of the equatorial region in determining the global climate pattern, the altimeter data have been used extensively to study the dynamics of this area (Miller et al. 1988; Miller & Cheney 1990; Zheng et al. 1994, 1995)....

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