Amplitude Probability Density Function Research Articles

Delay-induced stochastic bifurcations in a bistable system under white noise.

In this paper, the effects of noise and time delay on stochastic bifurcations are investigated theoretically and numerically in a time-delayed Duffing-Van der Pol oscillator subjected to white noise. Due to the time delay, the random response is not Markovian. Thereby, approximate methods have been adopted to obtain the Fokker-Planck-Kolmogorov equation and the stationary probability density function for amplitude of the response. Based on the knowledge that stochastic bifurcation is characterized by the qualitative properties of the steady-state probability distribution, it is found that time delay and feedback intensity as well as noise intensity will induce the appearance of stochastic P-bifurcation. Besides, results demonstrated that the effects of the strength of the delayed displacement feedback on stochastic bifurcation are accompanied by the sensitive dependence on time delay. Furthermore, the results from numerical simulations best confirm the effectiveness of the theoretical analyses.

Analysis of X-Band SAR Sea-Clutter Distributions at Different Grazing Angles

Modeling sea clutter is a difficult problem due to the interaction of the sea surface characteristics (wind speed and wind direction), the geometry of acquisition (grazing angle and azimuth angle), and radar parameters (frequency, polarization, and resolution). In synthetic aperture radar (SAR) imagery, the effect of coherent averaging and motion from the sea will also influence the statistics. An accurate description of the sea surface amplitude probability density function is important for robust target detection. The goal of this paper is to identify a common distribution which matches two different airborne SAR X-band data sets obtained from different locales and at different grazing angles. The first data set was collected by the French Aerospace Laboratory (ONERA) SETHI radar off the coast of France, at low grazing angles (3° and 10°). The second is the Ingara medium grazing angle (15°-45°) sea clutter data set collected by the Defence Science and Technology Organisation off the coast of Australia. In this paper, a number of recently developed distributions are considered, including the K+Rayleigh and Pareto+noise, both of which account for thermal noise. To measure the effectiveness of the model fit, the Bhattacharyya distance and the threshold error are computed with particular attention given to the tail region, where the threshold is determined in a detection scenario.

Integrable turbulence and formation of rogue waves

In the framework of the focusing nonlinear Schrödinger equation we study numerically the nonlinear stage of the modulation instability (MI) of the condensate. The development of the MI leads to the formation of ‘integrable turbulence’ (Zakharov 2009 Stud. Appl. Math. 122 219–34). We study the time evolution of its major characteristics averaged across realizations of initial data—the condensate solution seeded by small random noise with fixed statistical properties.We observe that the system asymptotically approaches to the stationary integrable turbulence, however this is a long process. During this process momenta, as well as kinetic and potential energies, oscillate around their asymptotic values. The amplitudes of these oscillations decay with time t as t−3/2, the phases contain the nonlinear phase shift that decays as t−1/2, and the frequency of the oscillations is equal to the double maximum growth rate of the MI. The evolution of wave-action spectrum is also oscillatory, and characterized by formation of the power-law region ∼|k|−α in the small vicinity of the zeroth harmonic k = 0 with exponent α close to 2/3. The corresponding modes form ‘quasi-condensate’, that acquires very significant wave action and macroscopic potential energy.The probability density function of wave amplitudes asymptotically approaches the Rayleigh distribution in an oscillatory way. Nevertheless, in the beginning of the nonlinear stage the MI slightly increases the occurrence of rogue waves. This takes place at the moments of potential energy modulus minima, where the PDF acquires ‘fat tales’ and the probability of rogue waves occurrence is by about two times larger than in the asymptotic stationary state.Presented facts need a theoretical explanation.

Probability image of tissue characteristics for liver fibrosis using multi-Rayleigh model with removal of nonspeckle signals

We have been developing a quantitative diagnostic method for liver fibrosis using an ultrasound image. In our previous study, we proposed a multi-Rayleigh model to express a probability density function of the echo amplitude from liver fibrosis and proposed a probability imaging method of tissue characteristics on the basis of the multi-Rayleigh model. In an evaluation using the multi-Rayleigh model, we found that a modeling error of the multi-Rayleigh model was increased by the effect of nonspeckle signals. In this paper, we proposed a method of removing nonspeckle signals using the modeling error of the multi-Rayleigh model and evaluated the probability image of tissue characteristics after removing the nonspeckle signals. By removing nonspeckle signals, the modeling error of the multi-Rayleigh model was decreased. A correct probability image of tissue characteristics was obtained by removing nonspeckle signals. We concluded that the removal of nonspeckle signals is important for evaluating liver fibrosis quantitatively.

Influence of fatigue–life data modelling on the estimated reliability of a structure subjected to a constant-amplitude loading

This article describes how a selected material’s fatigue–life-curve model influences the calculated reliability of a structure subjected to a dynamic loading. A uni-axially loaded structural beam with a fully-reversal constant loading amplitude was considered. The reliability for a certain number of cycles-to-failure was calculated as a cross-section of the probability distributions representing the load–amplitude scatter and the scatter of the material’s fatigue–life curve. The probability density function (PDF) of the loading amplitude was modelled by a uniform and a Gaussian PDF. The scattered fatigue–life curve was modelled by a conditional two-parametric Weibull’s PDF. Its parameters were estimated using two procedures: (i) a two-phase procedure and (ii) a direct procedure. Following the two-phase procedure a conditional PDF of the number of cycles-to-failure was estimated first and then converted into a corresponding conditional PDF of the stress amplitudes. In the direct procedure the conditional PDF of the stress amplitudes was modelled directly from the fatigue–life data. The two procedures were tested on 12 sets of simulated fatigue–life data and a set of experimental fatigue–life data. The two fatigue–life-curve models for the experimental data set were applied for calculating the reliability for the selected structural beam.

Softening Duffing Oscillator Reliability Assessment Subject to Evolutionary Stochastic Excitation

AbstractAn approximate analytical technique for assessing the reliability of a softening Duffing oscillator subject to evolutionary stochastic excitation is developed. Specifically, relying on a stochastic averaging treatment of the problem, the oscillator time-dependent survival probability is determined in a computationally efficient manner. In comparison with previous techniques that neglect the potential unbounded response behavior of the oscillator when the stiffness element acquires negative values, the technique developed in this paper readily takes this aspect into account by introducing a special form for the oscillator nonstationary response amplitude probability density function (PDF). A significant advantage of the technique relates to the fact that it can readily handle cases of stochastic excitations that exhibit variability in both the intensity and the frequency content. Numerical examples include a softening Duffing oscillator under evolutionary earthquake excitation and a softening Duffi...

Intermittency in generalized NLS equation with focusing six-wave interactions

We study numerically the statistics of waves for generalized one-dimensional Nonlinear Schrödinger (NLS) equation that takes into account focusing six-wave interactions, dumping and pumping terms. We demonstrate the universal behavior of this system for the region of parameters when six-wave interactions term affects significantly only the largest waves. In particular, in the statistically steady state of this system the probability density function (PDF) of wave amplitudes turns out to be strongly non-Rayleigh one for large waves, with characteristic “fat tail” decaying with amplitude |Ψ| close to ∝exp⁡(−γ|Ψ|), where γ>0 is constant. The corresponding non-Rayleigh addition to the PDF indicates strong intermittency, vanishes in the absence of six-wave interactions, and increases with six-wave coupling coefficient.

Comprehensive comparisons of geodesic acoustic mode characteristics and dynamics between Tore Supra experiments and gyrokinetic simulations

In a dedicated collisionality scan in Tore Supra, the geodesic acoustic mode (GAM) is detected and identified with the Doppler backscattering technique. Observations are compared to the results of a simulation with the gyrokinetic code GYSELA. We found that the GAM frequency in experiments is lower than predicted by simulation and theory. Moreover, the disagreement is higher in the low collisionality scenario. Bursts of non harmonic GAM oscillations have been characterized with filtering techniques, such as the Hilbert-Huang transform. When comparing this dynamical behaviour between experiments and simulation, the probability density function of GAM amplitude and the burst autocorrelation time are found to be remarkably similar. In the simulation, where the radial profile of GAM frequency is continuous, we observed a phenomenon of radial phase mixing of the GAM oscillations, which could influence the burst autocorrelation time.

Polynomial chaos expansion method in estimating probability distribution of rotor-shaft dynamic responses

Abstract The main purpose of the study is an assessment of computational efficiency of selected numerical methods for estimation of vibrational response statistics of a large multi-bearing turbo-generator rotor-shaft system. The effective estimation of the probability distribution of structural responses is essential for robust design optimization and reliability analysis of such systems. The analyzed scatter of responses is caused by random residual unbalances as well as random stiffness and damping parameters of the journal bearings. A proper representation of these uncertain parameters leads to multidimensional stochastic models. Three estimation techniques are compared: Monte Carlo sampling, Latin hypercube sampling and the sparse polynomial chaos expansion method. Based on the estimated values of the first four statistical moments the probability density function of the maximal vibration amplitude is evaluated by the maximal entropy principle method. The method is inherently suited for an accurate representation of the probability density functions with an exponential behavior, which appears to be characteristic for the investigated rotor-shaft responses. Performing multiple numerical tests for a range of sample sizes it was found that the sparse polynomial chaos method provides the best balance between the accuracy and computational effectiveness in estimating the unknown probability distribution of the maximal vibration amplitude

SAR probability density function estimation using a generalized form of K-distribution

In this paper, generalized forms of amplitude and intensity K-distributions are proposed to model multilook synthetic aperture radar (SAR) data. The approach is based upon a product model in which the amplitude and intensity distributions of the SAR backscatter component are assumed to be Laguerre orthogonal expansions of the traditional clutter distribution. The SAR intensity and amplitude probability density functions (pdfs) are derived and turn out to be the weighted combination of a series of K-distributions. Several theoretical and implementation issues regarding the proposed generalization are discussed. Comparative experimental results using real SAR data indicates that these generalized forms of K may allow better estimation of the SAR pdf than that of the traditional K distribution.

The peak response distributions of structure–DVA systems with nonlinear damping

Dynamic vibration absorbers (DVAs) with nonlinear damping are often modelled using a power-law equivalent viscous damping relationship. There is currently not a method available to predict the peak response of this type of nonlinear DVA without resorting to computationally expensive nonlinear simulations. Since the peak response of the DVA is required during the design process, it is advantageous to have a simplified method to estimate the peak response.In this study, statistical linearization is employed to represent the nonlinear damping as amplitude-dependent viscous damping and predict the rms response of the structure–DVA system. Subsequently, statistical nonlinearization is used to describe the probability density function of the DVA response amplitude. A probability density function is developed, which enables the peak response expected during an interval of time (e.g. 1-h) to be estimated from the rms response of the structure–DVA system. Higher power-law damping exponents are shown to result in smaller peak factors.Results of nonlinear simulations reveal that the model can estimate the peak structural and DVA responses with acceptable accuracy. A plot is developed to show the peak factors for nonlinear DVAs as a function of the number of system cycles for several power-law damping exponents. This plot can be used to estimate the peak response of a nonlinear DVA as a function of its rms response.

Mean Stress Effect Correction in Frequency-domain Methods for Fatigue Life Assessment

Two fatigue life calculation methods are presented. One defined in the time domain and the second one defined in the frequency domain – both supplemented with a mean stress effect correction. The method is verified on the basis of own results for the S355JR steel. The authors analyze six models for the designation of the probability density function (PDF) of stress amplitudes used in the calculation process. The results are presented in the form of probability distributions before and after PSD transformation and the calculated fatigue life's are being compared with the experimental ones in fatigue comparison graphs.

Noise Propagation in Multiple-Input ADC-Based Measurement Systems

Abstract In this paper, the complete statistical characterization of the amplitude spectrum at the output of a multiple-input ADC-based measurement system is derived under the assumption of input channels with different noise levels. In practical applications the input channels correspond to the spatial components of a vector field (e.g., magnetic/electric field). Each output spectral line represents the amplitude of the vector field at a specific frequency. Such amplitude is a random variable depending on the noise levels (internal and external noise) of the input channels. Closed form analytical solution for the probability density function of the vector field amplitude is not available in the mathematical literature under the hypothesis of different noise levels. Therefore, an analytical expression for the probability density function is derived on the basis of a Laguerre series expansion. The impact of the kind of time window, the sampling frequency, and the number of samples is clearly derived and put into evidence. Approximate analytical expressions for the mean value and the variance of the vector field amplitude are also provided. Analytical results are validated by means of numerical simulations.

English

Amplitude and phase statistics of SAR complex interferogram are significant in the study of interferometry and polarimetry. To reduce statistical variations, multi-look processing is adopted by averaging spatially the complex interferogram. In this study, we derive and validate three kinds of probability density functions (PDFs) of multi-look interferogram for different surface feature scenes. For simple homogeneous areas with the gamma distribution intensity, a concise product-form interferometry phase PDF is derived, which is equivalent to a conventional Gauss hypergeometric PDF. For complicated areas with the K and G0 distributions intensity, two new interferometry amplitude PDFs named as Gamma-K and Gamma-G are proposed, and their phase PDFs are approximately preserved. Finally three typical areas including grass, mountain, and city are picked out from a pair of RADARSAT-2 SAR images and studied. Experimental results indicate good agreement between the computed histograms and the theoretical distributions. The results obtained can be applied to the feature classification of polarisation SAR data and the estimation of decorrelation effect of interferometric SAR. Science Journal, Vol. 64, No. 6, November 2014, pp.564-570, DOI:http://dx.doi.org/10.14429/dsj.64.4747

When is respiratory management necessary for partial breast intensity modulated radiotherapy: A respiratory amplitude escalation treatment planning study

PurposeThe impact of typical respiratory motion amplitudes (∼2mm) on partial breast irradiation (PBI) is minimal; however, some patients have larger respiratory amplitudes that may negatively affect dose homogeneity. Here we determine at what amplitude respiratory management may be required to maintain plan quality. Methods and MaterialsTen patients were planned with PBI IMRT. Respiratory motion (2–20mm amplitude) probability density functions were convolved with static plan fluence to estimate the delivered dose. Evaluation metrics included target coverage, ipsilateral breast hotspot, homogeneity, and uniformity indices. ResultsDegradation of dose homogeneity was the limiting factor in reduction of plan quality due to respiratory motion, not loss of coverage. Hotspot increases were observed even at typical motion amplitudes. At 2 and 5mm, 2/10 plans had a hotspot greater than 107% and at 10mm this increased to 5/10 plans. Target coverage was only compromised at larger amplitudes: 5/10 plans did not meet coverage criteria at 15mm amplitude and no plans met minimum coverage at 20mm. ConclusionsWe recommend that if respiratory amplitude is greater than 10mm, respiratory management or alternative radiotherapy should be considered due to an increase in the hotspot in the ipsilateral breast and a decrease in dose homogeneity.

Amplitude scintillation effects on SAR

Space-based low-frequency synthetic aperture radar (SAR) is affected by the ionosphere, which induces both phase and amplitude fluctuations, known as scintillation, into the radar signal. This paper describes the effect of amplitude scintillation on SAR imagery. The two-way amplitude and intensity probability density functions (pdf) for both monostatic and bistatic SAR are derived from the one-way Nakagami-m distribution. The moments are then used to determine the SAR radiometric calibration error and image contrast from the one-way S 4 index. It is also shown that monostatic SAR experiences an S 4-dependent radar cross-section (RCS) enhancement that is not experienced by bistatic SAR. The anisotropy of the ionospheric irregularities strongly affects the degree to which amplitude scintillation will be visible in SAR imagery. The description of anisotropic effects is reviewed and extended to cover SAR. The variation over the Earth is illustrated, showing that a sun-synchronous satellite will experience the strongest effect near Brazil. Two PALSAR images of the same area of Brazilian rainforest are compared, one of which shows azimuthal streaking, corresponding to an amplitude modulation of ± 1 dB. The one-way S 4 index is determined from this imagery using both the RCS enhancement and image contrast measures of S 4, which produce similar results.

Evaluation of muscle force classification using shape analysis of the sEMG probability density function: a simulation study.

In this work, we propose to classify, by simulation, the shape variability (or non-Gaussianity) of the surface electromyogram (sEMG) amplitude probability density function (PDF), according to contraction level, using high-order statistics (HOS) and a recent functional formalism, the core shape modeling (CSM). According to recent studies, based on simulated and/or experimental conditions, the sEMG PDF shape seems to be modified by many factors as: contraction level, fatigue state, muscle anatomy, used instrumentation, and also motor control parameters. For sensitivity evaluation against these several sources (physiological, instrumental, and neural control) of variability, a large-scale simulation (25 muscle anatomies, ten parameter configurations, three electrode arrangements) is performed, by using a recent sEMG-force model and parallel computing, to classify sEMG data from three contraction levels (20, 50, and 80% MVC). A shape clustering algorithm is then launched using five combinations of HOS parameters, the CSM method and compared to amplitude clustering with classical indicators [average rectified value (ARV) and root mean square (RMS)]. From the results screening, it appears that the CSM method obtains, using Laplacian electrode arrangement, the highest classification scores, after ARV and RMS approaches, and followed by one HOS combination. However, when some critical confounding parameters are changed, these scores decrease. These simulation results demonstrate that the shape screening of the sEMG amplitude PDF is a complex task which needs both efficient shape analysis methods and specific signal recording protocol to be properly used for tracking neural drive and muscle activation strategies with varying force contraction in complement to classical amplitude estimators.

Simulation of non-Gaussian stochastic processes by amplitude modulation and phase reconstruction

Stochastic processes are used to represent phenomena in many diverse fields. Numerical simulation method is widely applied for the solution to stochastic problems of complex structures when alternative analytical methods are not applicable. In some practical applications the stochastic processes show non-Gaussian properties. When the stochastic processes deviate significantly from Gaussian, techniques for their accurate simulation must be available. The various existing simulation methods of non-Gaussian stochastic processes generally can only simulate super-Gaussian stochastic processes with the high-peak characteristics. And these methodologies are usually complicated and time consuming, not sufficiently intuitive. By revealing the inherent coupling effect of the phase and amplitude part of discrete Fourier representation of random time series on the non-Gaussian features (such as skewness and kurtosis) through theoretical analysis and simulation experiments, this paper presents a novel approach for the simulation of non-Gaussian stochastic processes with the prescribed amplitude probability density function (PDF) and power spectral density (PSD) by amplitude modulation and phase reconstruction. As compared to previous spectral representation method using phase modulation to obtain a non-Gaussian amplitude distribution, this non-Gaussian phase reconstruction strategy is more straightforward and efficient, capable of simulating both super-Gaussian and sub-Gaussian stochastic processes. Another attractive feature of the method is that the whole process can be implemented efficiently using the Fast Fourier Transform. Cases studies demonstrate the efficiency and accuracy of the proposed algorithm.

Statistical modeling and signal reconstruction processing method of EMF for slurry flow measurement

According to the experimental data, an amplitude probability density function (PDF) model of the slurry flow signal is built up for electromagnetic flowmeter (EMF) by the method combining statistical analysis with numerical fitting, in order to reveal the effect of slurry noise on the flow signal and describe the features of slurry flow signal. Based on this model, a signal reconstruction processing algorithm is proposed to deal with the output signal of EMF sensor for realizing the slurry flow measurement. At the same time, the high-low voltage switching mode based square-wave excitation method is presented for EMF so as to reduce the slurry noise interferences. A slurry-type EMF transmitter is developed with a DSP chip – TMS320F28335, to implement the signal processing algorithm and control function. Finally water flow calibrations and slurry flow experiments are conducted to verify the reliability and stability of the method and system. Experimental results show that its measurement accuracy of water flow is better than 0.5%, and its steady-state volatility of paper slurry is less than 3%, and its dynamic response time is less than 4s.

Nonlinear Analysis of Stochastic Heave Motion for Truss Spar

Applying the Path Integral Solution Method, the stochastic heave motion of Truss Spar platform was studied in this paper. Taking a platform for example, probability density function of stochastic heave motion amplitude was calculated under the conditions of different wave excitations. The results show that, the changing process of probability density function in the time domain has an obvious periodicity. As the increasing of external excitation, the heave motion amplitude and speed both increases. Meanwhile, as the increasing of excitation amplitude, the probability density function of heave motion has a more pronounced periodicity.