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...

/pdf/ensemble-empirical-mode-decomposition-a-noise-assisted-data-4htvj7tcuy.pdf

Ensemble empirical mode decomposition: a noise-assisted data analysis method

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.

Variational Mode Decomposition

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.

/pdf/empirical-mode-decomposition-as-a-filter-bank-50wjetwvi7.pdf

Empirical mode decomposition as a filter bank

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.

/pdf/a-new-view-of-nonlinear-water-waves-the-hilbert-spectrum-42def0zzxb.pdf

A new view of nonlinear water waves: the Hilbert spectrum

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.

/pdf/neurophysiological-and-computational-principles-of-cortical-48xnk4ixsg.pdf

Neurophysiological and Computational Principles of Cortical Rhythms in Cognition

The effect of wave breaking on the probability function of surface elevation is examined. The surface elevation limited by wave breaking zeta sub b(t) is first related to the original wave elevation zeta(t) and its second derivative. An approximate, second-order, nonlinear, non-Gaussian model for zeta(t) of arbitrary but moderate bandwidth is presented, and an expression for the probability density function zeta sub b(t) is derived. The results show clearly that the effect of wave breaking on the probability density function of surface elevation is to introduce a secondary hump on the positive side of the probability density function, a phenomenon also observed in wind wave tank experiments.

Probability function of breaking-limited surface elevation

In this paper, the validity of the energy balance equation is verified by statistical analysis. An ensemble of two hundred artificial earthquakes are generated and used as input to a structure on whose floors unanchored bodies are placed. The equation of motion governing the rocking motion of an unanchored body is solved and the probabilities of overturning are obtained. This information is used to show that the energy balance equation proposed by Housner is indeed valid. A few examples are given to demonstrate the use of the energy balance equation.

Energy Balance Equation for Estimating Overturning Potential of an Unanchored Rocking Body Subjected to Earthquake Excitation

Influence of wave-current interactions on fluid force

Using the Stokes waves as a model of nonlinear waves and considering the linear component as a narrow-band Gaussian process, the covariances and spectra of velocity and acceleration components and pressure for points in the vicinity of still water level were derived taking into consideration the effects of free surface fluctuations. The results are compared with those obtained earlier using linear Gaussian waves.

C. C. Tung

Papers

Probability function of breaking-limited surface elevation

Energy Balance Equation for Estimating Overturning Potential of an Unanchored Rocking Body Subjected to Earthquake Excitation

Influence of wave-current interactions on fluid force

Covariances and spectra of the kinematics and dynamics of nonlinear waves

Seismic response of temporary structures