Amplitude Probability Density Function Research Articles

Spontaneous emergence of rogue waves in partially coherent waves: A quantitative experimental comparison between hydrodynamics and optics.

Rogue waves are extreme and rare fluctuations of the wave field that have been discussed in many physical systems. Their presence substantially influences the statistical properties of a partially coherent wave field, i.e., a wave field characterized by a finite band spectrum with random Fourier phases. Their understanding is fundamental for the design of ships and offshore platforms. In many meteorological conditions waves in the ocean are characterized by the so-called Joint North Sea Wave Project (JONSWAP) spectrum. Here we compare two unique experimental results: the first one has been performed in a 270 m wave tank and the other in optical fibers. In both cases, waves characterized by a JONSWAP spectrum and random Fourier phases have been launched at the input of the experimental device. The quantitative comparison, based on an appropriate scaling of the two experiments, shows a very good agreement between the statistics in hydrodynamics and optics. Spontaneous emergence of heavy tails in the probability density function of the wave amplitude is observed in both systems. The results demonstrate the universal features of rogue waves and provide a fundamental and explicit bridge between two important fields of research. Numerical simulations are also compared with experimental results.

Neuro-Spike Communications With Multiple Synapses Under Inter-Neuron Interference

The nervous system is a complex intra-body communication system, called neuro-spike communications, for transferring vital information throughout the body. This system consists of unreliable neural components like axon and synapses. The nervous system can deal with the unreliability of its components by making use of multiple synapses. In this paper, we analyze the performance of neuro-synaptic channels consisting of multiple synapses. It is assumed that synaptic channels are subject to synaptic noise, random vesicle release, and inter-neuron interference. The optimal detection for two cases of multiple cooperative synapses and multiple interfering synapses is investigated. The closed-form expressions of the probability density function of variable quantal amplitude are derived that is used to calculate the probability of detection error.

Bifurcation Regulations Governed by Delay Self-Control Feedback in a Stochastic Birhythmic System

We aim to investigate bifurcation behaviors in a stochastic birhythmic van der Pol (BVDP) system subjected to delay self-control feedback. First, the harmonic approximation is adopted to drive the delay self-control feedback to state variables without delay. Then, Fokker–Planck–Kolmogorov (FPK) equation and stationary probability density function (SPDF) for amplitude are obtained by applying stochastic averaging method. Finally, dynamical scenarios of the change of delay self-control feedback as well as noise that markedly influence bifurcation performance are observed. It is found that: the big feedback strength and delay will suppress the large amplitude limit cycle (LC) while the relatively big noise strength facilitates the large amplitude LC, which imply the proposed regulation strategies are feasible. Interestingly enough, the inner LC is never destroyed due to noise. Furthermore, the validity of analytical results was verified by Monte Carlo simulation of the dynamics.

Probabilistic distribution and stochastic P-bifurcation of a hybrid energy harvester under colored noise

In this manuscript, a hybrid energy harvesting system combining piezoelectric and electromagnetic transduction and subjected to colored noise is investigated. By using the stochastic averaging method, the stationary probability density functions of amplitudes are obtained and reveal interesting dynamics related to the long term behavior of the device. From stationary probability densities, we discuss the stochastic bifurcation through the qualitative change which shows that noise intensity, correlation time and other system parameters can be treated as bifurcation parameters. Numerical simulations are made for a comparison with analytical findings. The Mean first passage time (MFPT) is numerical provided in the purpose to investigate the system stability. By computing the Mean residence time (TMR), we explore the stochastic resonance phenomenon; we show how it is related to the correlation time of colored noise and high output power.

Robust identification of harmonic oscillator parameters using the adjoint Fokker-Planck equation.

We present a model-based output-only method for identifying from time series the parameters governing the dynamics of stochastically forced oscillators. In this context, suitable models of the oscillator's damping and stiffness properties are postulated, guided by physical understanding of the oscillatory phenomena. The temporal dynamics and the probability density function of the oscillation amplitude are described by a Langevin equation and its associated Fokker-Planck equation, respectively. One method consists in fitting the postulated analytical drift and diffusion coefficients with their estimated values, obtained from data processing by taking the short-time limit of the first two transition moments. However, this limit estimation loses robustness in some situations-for instance when the data are band-pass filtered to isolate the spectral contents of the oscillatory phenomena of interest. In this paper, we use a robust alternative where the adjoint Fokker-Planck equation is solved to compute Kramers-Moyal coefficients exactly, and an iterative optimization yields the parameters that best fit the observed statistics simultaneously in a wide range of amplitudes and time scales. The method is illustrated with a stochastic Van der Pol oscillator serving as a prototypical model of thermoacoustic instabilities in practical combustors, where system identification is highly relevant to control.

Amplitude stochastic response of Rayleigh beams to randomly moving loads

We consider the problem of the nonlinear response of a Rayleigh beam to the passage of a train of forces moving with stochastic velocity. The Fourier transform and the theory of residues is used to estimate the mean-square amplitude of the beam, while the stochastic averaging method gives the stationary probability density function of the oscillations amplitude. The analysis shows that the effect of the load random velocities is highly nonlinear, leading to a nonmonotonic behavior of the mean amplitude versus the intensity of the stochastic term and of the load weight. The analytic approach is also checked with numerical simulations. The effect of loads number on the system response is numerically investigated.

Non-Gaussian amplitude PDF modeling of ultrasound images based on a novel generalized Cauchy-Rayleigh mixture

In this paper, a mixture of generalized Cauchy distribution and Rayleigh distribution that possesses a closed-form expression is proposed for modeling the heavy-tailed Rayleigh (HTR) distribution. This new approach is developed for analytically modeling the amplitude distribution of ultrasound images based on the HTR distribution. HTR as a non-Gaussian distribution is basically the amplitude probability density function (PDF) of the complex isotropic symmetric α-stable (S α S) distribution which appears in the envelope distribution of ultrasonic images. Analytic expression for HTR distribution is a momentous consideration in signal processing with stable random variables. Furthermore, we introduce a mixture ratio estimator based on the energy of amplitude PDF which contains both α and γ parameters. For a quantitative assessment, we compare the accuracy and computational complexity of the proposed mixture with other approximations of HTR distribution through several numeral simulations on synthetic random samples. Experimental results obtained from the Kolmogorov-Smirnov (K-S) distance and Kullback-Leibler (K-L) divergence as the goodness-of-fit tests on real ultrasound images reveal the favor of the new mixture model.

Statistical characterization of high-to-medium frequency mesoscale gravity waves by lidar-measured vertical winds and temperatures in the MLT

We present the first statistical study of gravity waves with periods of 0.3–2.5h that are persistent and dominant in the vertical winds measured with the University of Colorado STAR Na Doppler lidar in Boulder, CO (40.1°N, 105.2°W). The probability density functions of the wave amplitudes in temperature and vertical wind, ratios of these two amplitudes, phase differences between them, and vertical wavelengths are derived directly from the observations. The intrinsic period and horizontal wavelength of each wave are inferred from its vertical wavelength, amplitude ratio, and a designated eddy viscosity by applying the gravity wave polarization and dispersion relations. The amplitude ratios are positively correlated with the ground-based periods with a coefficient of ~0.76. The phase differences between the vertical winds and temperatures (φW−φT) follow a Gaussian distribution with 84.2±26.7°, which has a much larger standard deviation than that predicted for non-dissipative waves (~3.3°). The deviations of the observed phase differences from their predicted values for non-dissipative waves may indicate wave dissipation. The shorter-vertical-wavelength waves tend to have larger phase difference deviations, implying that the dissipative effects are more significant for shorter waves. The majority of these waves have the vertical wavelengths ranging from 5 to 40km with a mean and standard deviation of ~18.6 and 7.2km, respectively. For waves with similar periods, multiple peaks in the vertical wavelengths are identified frequently and the ones peaking in the vertical wind are statistically longer than those peaking in the temperature. The horizontal wavelengths range mostly from 50 to 500km with a mean and median of ~180 and 125km, respectively. Therefore, these waves are mesoscale waves with high-to-medium frequencies. Since they have recently become resolvable in high-resolution general circulation models (GCMs), this statistical study provides an important and timely reference for them.

Quantitative diagnosis method based on model of tissue structure change for liver fibrosis

To realize a quantitative diagnosis method of liver fibrosis, we have been developing a modeling method for the probability density function of the echo amplitude. We developed a scatterer distribution model of diseased livers considering the liver lobule structure, because computer simulations are effective for obtaining information on the continuous stage of diseases. We examined relationship between tissue characterization and ultrasonic B-mode image which is simulated by this scatterer distribution model. From these analyses, we proposed a multi-Rayleigh model with three Rayleigh distributions, corresponding to the distribution of the echo amplitude from hypoechoic, normal, and fibrous tissues. We showed quantitatively that the proposed model can model the amplitude distribution of liver fibrosis echo data adequately. We also found that fibrous indices can be estimated stably using the proposed model. To evaluate the liver fibrosis more accurately, each pixel's amplitude value in the clinical echo ima...

Response and Reliability of a Giant Magnetostrictive Film Actuator Subject to Axial Random Excitation

A giant magnetostrictive thin film (GMF) actuator designed as a simply supported beam under axial white noise excitation is studied. Firstly, a non-dimensional stochastic difference equation governing the motion is simplified by the modified stochastic averaging method. Subsequently, the stationary probability density function of amplitude and the joint probability density function of displacement and velocity were obtained respectively. Finally, the reliability was studied by the backward Kolmogorov (BK) equation to guarantee the safety of the actuator. Numerical simulations are carried out to verify the theoretical analysis.

Investigation on the probabilistic distribution of mooring line tension for fatigue damage assessment

This paper is to investigate the applicable probability density functions (PDFs) for the short- and long-term mooring line tension amplitudes to be used for accurately assessing the mooring line fatigue damage with the spectral method. Statistical characteristics of the mooring line tensions were investigated, in which kurtosis and skewness were computed and compared to validate the non-normality of the line tension response. Several PDFs, including Rayleigh, GEV, and Gamma distributions, are used to express the short-term statistical distribution of the mooring line tension amplitudes. Results indicate that the Gamma distribution can give much better fitting than the Rayleigh and GEV distributions. In order to identify the long-term distribution of the tension amplitudes, the Gamma distribution is modified by bringing in a location parameter. Compared with the generally used Weibull distribution, the modified Gamma distribution can characterize the long-term PDF of the tension amplitudes much better. Moreover, the modified Gamma distribution can give much more accurate assessment of the mooring line fatigue damage than the Weibull distribution.

Skin-friction behavior in the transitionally-rough regime

Results of an experimental investigation of the skin friction for a range of rough surfaces in fully-developed, turbulent channel flow are presented. The fifteen surfaces were generated by grit blasting with various types and sizes of media. The Reynolds number (Rem) for the experiments based on the bulk mean velocity and the channel height spanned from 10,000–300,000. The shape of the roughness function (ΔU+) for all of these surfaces shows self-similarity with an effective roughness height (keff+) indicating that a scale does exist to collapse results throughout the entire roughness function. The roughness function displays inflectional behavior in the transitionally-rough regime in a similar manner to the results of Nikuradse for uniform sand. The equivalent sand grain roughness height can be predicted using krms and the skewness (Sk) of the probability density function of the roughness amplitude. A long-wavelength filtering procedure based on the Taylor microscale (λ) of the roughness is presented, and it is recommended that roughness profiles be filtered at a scale of ∼ 100λ.

Stability evaluation of parameter estimation of multi-Rayleigh model for ultrasound B-mode image of liver fibrosis

A diagnosis of liver fibrosis using an ultrasound B-mode image has the advantages of real-time observation and noninvasive properties. In our previous study, a multi-Rayleigh model was proposed to express a probability density function (PDF) of echo amplitudes from a fibrotic liver. From the multi-Rayleigh model, fibrosis parameters, such as the amount of fibrotic tissue and its progressive ratio, can be extracted. To quantitatively evaluate liver fibrosis using the multi-Rayleigh model, it is important to evaluate the stability of the estimation method of multi-Rayleigh model parameters. In this study, a numerical simulation using random variables following the multi-Rayleigh model was performed and the estimation stability of the parameters of the multi-Rayleigh model with two components was examined. From the simulation results, it was found that estimation becomes unstable under a certain condition owing to statistical variations of moments, which are inputs in the estimation algorithm. The instability of estimated parameters could be evaluated by focusing on changes in moments upon changes in multi-Rayleigh model parameters. It was indicated that we can evaluate the reliability of the estimated parameters of the multi-Rayleigh model only from the estimated values.

Bifurcations Induced in a Bistable Oscillator via Joint Noises and Time Delay

In this paper, noise-induced and delay-induced bifurcations in a bistable Duffing–van der Pol (DVP) oscillator under time delay and joint noises are discussed theoretically and numerically. Based on the qualitative changes of the plane phase, delay-induced bifurcations are investigated in the deterministic case. However, in the stochastic case, the response of the system is a stochastic non-Markovian process owing to the existence of noise and time delay. Then, methods have been employed to derive the stationary probability density function (PDF) of the amplitude of the response. Accordingly, stochastic P-bifurcations can be observed with the variations in the qualitative behavior of the stationary PDF for amplitude. Furthermore, results from both theoretical analyses and numerical simulations best demonstrate the appearance of noise-induced and delay-induced bifurcations, which are in good agreement.

Probability density function formalism for optical coherence tomography signal analysis: a controlled phantom study.

The distribution of backscattered intensities as described by the probability density function (PDF) of tissue-scattered light contains information that may be useful for tissue assessment and diagnosis, including characterization of its pathology. In this Letter, we examine the PDF description of the light scattering statistics in a well characterized tissue-like particulate medium using optical coherence tomography (OCT). It is shown that for low scatterer density, the governing statistics depart considerably from a Gaussian description and follow the K distribution for both OCT amplitude and intensity. The PDF formalism is shown to be independent of the scatterer flow conditions; this is expected from theory, and suggests robustness and motion independence of the OCT amplitude (and OCT intensity) PDF metrics in the context of potential biomedical applications.

Incremental Dynamic Analysis: A Nonlinear Stochastic Dynamics Perspective

AbstractAn efficient stochastic incremental dynamic analysis (IDA) methodology for nonlinear/hysteretic oscillators is developed by resorting to nonlinear stochastic dynamics concepts and tools such as stochastic averaging and statistical linearization. Specifically, modeling the excitation as a nonstationary stochastic process possessing an evolutionary power spectrum (EPS), an approximate closed-form expression is derived for the parameterized oscillator response amplitude probability density function (PDF). In this regard, an IDA surface is determined providing the PDF of the engineering demand parameter (EDP) for a given intensity measure (IM) value. In contrast to a computationally expensive Monte Carlo simulation (MCS) based determination of the IDA surface, the methodology developed herein determines the EDP PDF at minimal computational cost. Further, an approximate closed-form expression is derived for the parameterized nonlinear oscillator response EPS as well; thus, a conceptually novel IDA surf...

Probabilistic characteristics of noisy Van der Pol type oscillator with nonlinear damping

The exact Fokker–Planck equation for the joint probability distribution of amplitude and phase of a Van der Pol oscillator perturbed by both additive and multiplicative noise sources with arbitrary nonlinear damping is first derived by the method of functional splitting of averages. We truncate this equation in the usual manner using the smallness of the damping parameter and obtain a general expression for the stationary probability density function of oscillation amplitude, which is valid for any nonlinearity in the feedback loop of the oscillator. We analyze the dependence of this stationary solution on system parameters and intensities of noise sources for two different situations: (i) Van der Pol generator with soft and hard excitation regimes; (ii) Van der Pol oscillator with negative nonlinear damping. As shown, in the first case the probability distribution of amplitude demonstrates one characteristic maximum, which indicates the presence of a stable limit cycle in the system. The non-monotonic dependence of stationary probability density function on oscillation frequency is also detected. In the second case we examine separately three situations: linear oscillator with two noise sources, nonlinear oscillator with additive noise and nonlinear oscillator with frequency fluctuations. For the last two situations, noise-induced transitions in the system under consideration are found.

An approximate stochastic dynamics approach for nonlinear structural system performance-based multi-objective optimum design

A novel approach for structural system optimal design considering life cycle cost is developed. Specifically, a performance-based multi-objective design optimization framework for nonlinear/hysteretic multi-degree-of-freedom (MDOF) structural systems subject to evolutionary stochastic excitation is formulated. In the core of the stochastic structural analysis component of the proposed framework lies an efficient approximate dimension reduction technique based on the concepts of statistical linearization and of stochastic averaging for determining the non-stationary system response amplitude probability density functions (PDFs); thus, computationally intensive Monte Carlo simulations are circumvented. Note that the approach can readily handle stochastic excitations of arbitrary non-separable evolutionary power spectral density (EPSD) forms that exhibit strong variability in both the intensity and the frequency content. Further, approximate closed-form expressions are derived for the non-stationary inter-story drift ratio amplitude PDFs corresponding to each and every DOF. In this regard, considering appropriately defined damage measures structural system related fragility curves are determined at a low computational cost as well. Finally, the structural system design optimization problem is formulated as a multi-objective one to be solved by a genetic algorithm based approach. A building structure comprising the versatile Bouc-Wen (hysteretic) model serves as a numerical example for demonstrating the efficiency of the proposed methodology.

Performance Bounds for Diversity Receptions Over Arbitrarily Correlated Nakagami-<inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula> Fading Channels

Prior performance analyses of diversity receptions over arbitrarily correlated Nakagami- $m$ fading channels use power correlation matrix to define correlation among branches. However, the power correlation matrix cannot uniquely determine the joint probability density function of the branch amplitudes as well as the system performance. In this paper, we derive asymptotically tight and closed-form upper and lower bounds for the error rates and the outage probabilities of maximum-ratio combining, equal-gain combining, and selection combining over arbitrarily correlated Nakagami- $m$ fading channels. The correlation between the Nakagami- $m$ random variables is defined through the correlation matrix of the accompanying Gaussian random variables, which can uniquely determine the joint probability density function of the Nakagami- $m$ variables. Using the analytical results and Monte Carlo simulation, we show that the performance of diversity reception systems over Nakagami- $m$ fading channels cannot be uniquely determined by the power correlation matrix of the branches. Furthermore, we also study the factors that determine the asymptotic performance of the diversity receptions.

A constant phase approach for the frequency response of stochastic linear oscillators

When studying a mechanical structure, evaluation of its frequency response function (FRF) over a given frequency range is one of the main interests. Computational cost aside, evaluating FRFs presents no methodological difficulty in the deterministic case. Doing this when the model includes some uncertain parameters may however be more difficult as multimodality and discontinuity can arise around resonances. Indeed, even for a single degree of freedom system, it can be shown that usual methods of the probabilistic frame such as generalized Polynomial Chaos may fail to properly describe the probability density function of the response amplitude. This study proposes another approach which involves a shift in the usual quantities used to draw FRFs. Instead of computing the stochastic response for a given excitation frequency, this work adopts a constant response phase point of view. For each phase value of the oscillator response, the uncertainty over some parameters is propagated to the corresponding uncertain amplitudes and excitation frequencies. This provides much smoother variations of the involved quantities which are much easier to describe using a simple Polynomial Chaos approach. Both analytical and numerical results will be exposed for a single degree of freedom oscillator whose stiffness follows a uniform law.