Poisson White Noise Research Articles

A novel method based on augmented Markov vector process for the time-variant extreme value distribution of stochastic dynamical systems enforced by Poisson white noise

The probability density function (PDF) of the time-variant extreme value process for structural responses is of great importance. Poisson white noise excitation occurs widely in practical engineering problems. The extreme value distribution of the response of systems excited by Poisson white noise processes is still not yet readily available. For this purpose, in the present paper, a novel method based on the augmented Markov vector process for the PDF of the time-variant extreme value process for a Poisson white noise driven dynamical system is proposed. Specifically, the augmented Markov vector (AMV) process is constructed by combining the extreme value process and its underlying response process. Then the joint probability density of the AMV can be evaluated by solving the Chapman-Kolmogorov Equation, e.g., via the path integral solution (PIS). Further, the PDF of the time-variant extreme value process is obtained, and can be used, say, to estimate the dynamic reliability of a stochastic system. For the purpose of illustration and verification, several numerical examples are studied and compared with Monte Carlo solution. Problems to be further studied are also discussed.

Diffusion caused by two noises—active and thermal

The diffusion of colloids inside an active system—e.g. within a living cell or the dynamics of active particles itself (e.g. self-propelled particles) can be modeled through overdamped Langevin equation which contains an additional noise term apart from the usual white Gaussian noise, originating from the thermal environment. The second noise is referred to as ‘active noise’ as it arises from activity such as chemical reactions. The probability distribution function (PDF or the propagator) in space-time along with moments provides essential information for understanding their dynamical behavior. Here we employ the phase-space path integral method to obtain the propagator, thereby moments and PDF for some possible models for such noise. At first, we discuss the diffusion of a free particle driven by active noise. We consider four different possible models for active noise, to capture the possible traits of such systems. We show that the PDF for systems driven by noises other than Gaussian noise largely deviates from normal distribution at short to intermediate time scales as a manifestation of out-of-equilibrium state, albeit converges to Gaussian distribution after a long time as a consequence of the central limit theorem. We extend our work to the case of a particle trapped in a harmonic potential and show that the system attains steady state at long time limit. Also, at short time scales, the nature of distribution is different for different noises, e.g. for particle driven by dichotomous noise, the probability is mostly concentrated near the boundaries whereas a long exponential tail is observed for a particle driven by Poissonian white noise.

Power shrinkage—curvelet domain image denoising using a new scale-dependent shrinkage function

Image processing and analysis algorithms are at the heart of applications in various scientific fields, such as medical diagnosis, military imaging, and astronomy. However, images are typically exposed to noise contamination during their acquisition and transmission. In this paper, we explore recent advancements in image denoising using curvelet domain shrinkage and present a novel scale-dependent shrinkage function, which we call power shrinkage, to enhance restored image quality. Experimental results confirm our proposed method to perform better than classical thresholding and to outperform recent state-of-the-art approaches in denoising different types of noises including speckle, Poisson and additive white Gaussian noise.

Exact stationary solutions of the Kolmogorov–Feller equation in a bounded domain

We present the first detailed analysis of the statistical properties of jump processes bounded by a saturation function and driven by Poisson white noise, being a random sequence of delta pulses. The Kolmogorov–Feller equation for the probability density function (PDF) of such processes is derived and its stationary solutions are found analytically in the case of the symmetric uniform distribution of pulse sizes. Surprisingly, these solutions can exhibit very complex behavior arising from both the boundedness of pulses and processes. We show that all features of the stationary PDF (number of branches, their form, extreme values probability, etc.) are completely determined by the ratio of the saturation function width to the half-width of the pulse-size distribution. We verify all theoretical results by direct numerical simulations.

Influence of Poisson White Noise on the Response Statistics of Nonlinear System and Its Applications to Bearing Fault Diagnosis

In view of complex noise background in engineering practices, this paper presents a rescaled method to detect failure features of bearing structure in the Poisson white noise background. To realize the scale transformation of the fault signal with Poisson white noise, a general scale transformation (GST) method is introduced based on the second-order underdamped nonlinear system. The signal features are successfully extracted through the proposed rescaled method in the simulated and experimental cases. We focus on the influence of Poisson white noise parameters and damping coefficient on the response of nonlinear system. The impulse arrival rate and noise intensity have opposite effects on the realization of stochastic resonance (SR) and the extraction of bearing fault features. Poisson white noise with smaller impulse arrival rate or larger noise intensity is easier to induce SR to extract bearing fault features. The optimal matching between the nonlinear system and the input signal is formed by the optimization algorithm, which greatly improves the extraction efficiency of fault features. Compared with the normalized scale transformation (NST) method, the GST has significant advantages in recognizing the bearing structure failure. The differences and connections between Poisson white noise and Gaussian white noise are discussed in the rescaled system excited by the experiment signal. This paper might provide several practical values for recognizing bearing fault mode in the Poisson white noise.

Non-stationary response of MDOF dynamical systems under combined Gaussian and Poisson white noises by the generalized cell mapping method

A block matrix analysis procedure of the generalized cell mapping (GCM) method is proposed in this paper. The proposed method solves the storage problem in the response analysis of nonlinear stochastic dynamical system with the GCM method, and makes it possible to compute the non-stationary and stationary probability density functions (PDFs) of multi-degree-of-freedom (MDOF) nonlinear systems without using supercomputing. Two examples of two-degree-of-freedom systems under external or parametric excitations of combined Gaussian and Poisson white noises are presented to demonstrate the efficiency of the proposed method. Monte Carlo (MC) simulations are used to verify the accuracy of the solutions obtained with the proposed method.

Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

A generalized compatible cell mapping (CCM) method is proposed in this paper to take advantages of the simple cell mapping (SCM) method, the generalized cell mapping (GCM) method together with a subdivision procedure. A coarse cell partition is first used to obtain a covering set of the global attractor. Then, a finer global attractor is obtained by the subdivision process. The probabilistic response of stochastic dynamic systems is obtained by the sparse matrix analysis algorithm applied to the covering set of the global attractor. Because the computational domain is the covering set of the global attractor rather than the whole state space, the numerical efficiency of the proposed method can be greatly improved as compared to the GCM. A three-dimensional and a four-dimensional dynamical system under Poisson white noise excitation are studied to demonstrate the effectiveness of the proposed method for the probabilistic response analysis. Monte Carlo simulations show a good agreement with the proposed method.

Statistics of bounded processes driven by Poisson white noise

We study the statistical properties of jump processes in a bounded domain that are driven by Poisson white noise. We derive the corresponding Kolmogorov–Feller equation and provide a general representation for its stationary solutions. Exact stationary solutions of this equation are found and analyzed in two particular cases. All our analytical findings are confirmed by numerical simulations.

Continuous Finite-Dimensional Locally Optimal Filtering of Jump Diffusions

This paper considers continuous-time state estimation for a dynamic system with Gaussian and Poisson white noise. We design a finite-dimensional filter with a lower computational cost than the extended Kalman–Bucy filter: its order is the dimension of the estimated part of the system state vector. The nonlinear structure of the filter is selected using the mean squared optimal unbiased estimation for each infinitesimal period of time. We develop an algorithm to find the structural functions of the filter that is based on the Kolmogorov–Feller equation for the density function. A numerical method to calculate them a priori by sequential Monte Carlo trials is presented, which requires the histograms of the desired functions. Due to its cumbersome form, some numerical and analytical approximations of the suggested filter are also given. They structurally coincide with the corresponding nonlinear extensions of the Kalman–Bucy filter but have a considerably smaller order and calculable parameters.

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping under combined harmonic and Poisson white noise parametric excitations is investigated. A stochastic averaging method and the Khasminskii’s procedure are applied to evaluate the largest Lyapunov exponent. Then the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are verified by those from the Monte Carlo simulation of the original system.

Stochastic stability of viscoelastic systems under Gaussian and Poisson white noise excitations

As the use of viscoelastic materials becomes increasingly popular, stability of viscoelastic structures under random loads becomes increasingly important. This paper aims at studying the asymptotic stability of viscoelastic systems under Gaussian and Poisson white noise excitations with Lyapunov functions. The viscoelastic force is approximated as equivalent stiffness and damping terms. A stochastic differential equation is set up to represent randomly excited viscoelastic systems, from which a Lyapunov function is determined by intuition. The time derivative of this Lyapunov function is then obtained by stochastic averaging. Approximate conditions are derived for asymptotic Lyapunov stability with probability one of the viscoelastic system. Validity and utility of this approach are illustrated by a Duffing-type oscillator possessing viscoelastic forces, and the influence of different parameters on the stability region is delineated.

Multivariate integer-valued time series with flexible autocovariances and their application to major hurricane counts

This paper examines a bivariate count time series with some curious statistical features: Saffir–Simpson Category 3 and stronger annual hurricane counts in the North Atlantic and eastern Pacific Ocean Basins. As land and ocean temperatures on our planet warm, an intense climatological debate has arisen over whether hurricanes are becoming more numerous, or whether the strengths of the individual storms are increasing. Recent literature concludes that an increase in hurricane counts occurred in the Atlantic Basin circa 1994. This increase persisted through 2012; moreover, the 1994–2012 period was one of relative inactivity in the eastern Pacific Basin. When Atlantic activity eased in 2013, heavy activity in the eastern Pacific Basin commenced. When examined statistically, a Poisson white noise model for the annual severe hurricane counts is difficult to resoundingly reject. Yet, decadal cycles (longer term dependence) in the hurricane counts is plausible. This paper takes a statistical look at the issue, developing a stationary multivariate count time series model with Poisson marginal distributions and a flexible autocovariance structure. Our auto- and cross-correlations can be negative and have long-range dependence; features that most previous count models cannot achieve in tandem. Our model is new in the literature and is based on categorizing and super-positioning multivariate Gaussian time series. We derive the autocovariance function of the model and propose a method to estimate model parameters. In the end, we conclude that severe hurricane counts are indeed negatively correlated across the two ocean basins. Some evidence for long-range dependence is also presented; however, with only a 49-year record, this issue cannot be definitively judged without additional data.

Stochastic Dynamics of a Time-Delayed Ecosystem Driven by Poisson White Noise Excitation.

We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson white noise. The stochastic averaging method and the perturbation method are applied to calculate the approximate stationary probability density functions for both predator and prey populations. The influences of system parameters and the Poisson white noises are investigated in detail based on the approximate stationary probability density functions. It is found that, increasing time delay parameter as well as the mean arrival rate and the variance of the amplitude of the Poisson white noise will enhance the fluctuations of the prey and predator population. While the larger value of self-competition parameter will reduce the fluctuation of the system. Furthermore, the results from Monte Carlo simulation are also obtained to show the effectiveness of the results from averaging method.

Stochastic Responses of Lightly Nonlinear Vibroimpact System with Inelastic Impact Subjected to External Poisson White Noise Excitation

A procedure for analyzing stationary responses of lightly nonlinear vibroimpact system with inelastic impact subjected to external Poisson white noise excitation is proposed. First, the original vibroimpact system is transformed to a new system without velocity jump in terms of the Zhuravlev nonsmooth coordinate transformation and the Dirac delta function. Second, the averaged generalized Fokker-Planck-Kolmogorov (FPK) equation for transformed system under parametric excitation of Poisson white noise is derived by stochastic averaging method. Third, the averaged generalized FPK equation is solved by using the perturbation technique and inverse transformation of the Zhuravlev nonsmooth coordinate transformation to obtain the approximately stationary solutions for response probability density functions of original vibroimpact system. Last, analytical and numerical results for two typical lightly nonlinear vibroimpact systems are presented to assess the effectiveness of the proposed method. It is found that they are in good agreement and the proposed method is quite effective.

An Analysis of a Nonlinear System Excited by Combined Gaussian and Poisson White Noises Using Complex Fractional Moments

An Analysis of a Nonlinear System Excited by Combined Gaussian and Poisson White Noises Using Complex Fractional Moments

An asymptotic method for quasi-integrable Hamiltonian system with multi-time-delayed feedback controls under combined Gaussian and Poisson white noises

In the present paper, we consider an approximate approach for predicting the responses of the quasi-integrable Hamiltonian system with multi-time-delayed feedback control under combined Gaussian and Poisson white noise excitations. Two-step approximation is taken here to obtain the responses of such system. First, based on the property of the system solution, the time-delayed system state variables are approximated by using the system state variables without time delay. After this approximation, the system is converted to the one without time delay but with delay time as parameters. Then, stochastic averaging method for quasi-integrable Hamiltonian system under combined Gaussian and Poisson white noises is applied to simplify the converted system to obtain the averaged stochastic integro-differential equations and generalized Fokker–Planck–Kolmogorov equations for both non-resonant and resonant cases. Finally, two examples are worked out to show the detailed procedure of proposed method for the illustrative purpose. And the influences of the time delay on the responses of the systems are also discussed. In addition, the validity of the results obtained by present method is verified by Monte Carlo simulation.

Stochastic sensitivity analysis of nonautonomous nonlinear systems subjected to Poisson white noise

The stochastic sensitivity function (SSF) method is extended to estimate the stationary probability distribution around periodic attractors of nonautonomous nonlinear dynamical systems subjected to Poisson white noise in this paper. After deriving the stochastic sensitivity functions of period-N cycle of mapping systems based on the characteristic of Poisson process, non-autonomous dynamical systems around periodic attractors are converted to mapping systems by constructing a stroboscopic map, and then the stochastic sensitivity functions of periodic attractors of nonautonomous nonlinear systems can be obtained by adopting the results of mapping systems. It is found that the stochastic sensitivity functions depend on the product of noise intensity and the arrival rate of Poisson processes. To illustrate the validity of the proposed method, a Henon map driven by Poisson processes and a Mathieu–Duffing oscillator under Poisson white noise are studied.

The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise

We propose a method to find an approximate theoretical solution to the mean first exit time (MFET) of a one-dimensional bistable kinetic system subjected to additive Poisson white noise, by extending an earlier method used to solve stationary probability density function. Based on the Dynkin formula and the properties of Markov processes, the equation of the mean first exit time is obtained. It is an infinite-order partial differential equation that is rather difficult to solve theoretically. Hence, using the non-Gaussian property of Poisson white noise to truncate the infinite-order equation for the mean first exit time, the analytical solution to the mean first exit time is derived by combining perturbation techniques with Laplace integral method. Monte Carlo simulations for the bistable system are applied to verify the validity of our approximate theoretical solution, which shows a good agreement with the analytical results.

A theoretical analysis for mean exit time of a Bi-stable system under combined Gaussian and Poisson white noise excitations

In this work, an approximate method is proposed to simultaneously compute an analytic solution for the mean exit time statistics in a bi-stable system with combined Gaussian and Poisson white noise excitations. Firstly, the method employs a perturbation scheme coupled with the Laplace integral method to estimate the mean exit time analytically. Then, a bi-stable system under both external and parametric excitations of combined Gaussian and Poisson white noises are investigated illustratively with the proposed technique, and effects of noise intensity and random magnitude of impulses on the mean exit time are discussed respectively. Finally, the effectiveness of the theoretical results is verified by means of Monte Carlo simulation, and good agreement will be observed.

Stochastic averaging of quasi partially integrable and resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for quasi partially integrable and resonant Hamiltonian systems excited by the combined Gaussian and Poisson white noises is proposed. The averaged stochastic integro-differential equations (SIDEs) and generalized Fokker-Planck-Kolmogorov (GFPK) equation for the quasi partially integrable Hamiltonian system with r(1<r<n) independent first integrals in involution and β(1≤β≤r−2) resonant relations are derived. The dimension of averaged SIDEs and the GFPK equation is equal to the number of the independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. A 4-degree-of-freedom (DOF) quasi partially integrable and resonant Hamiltonian system under combined Gaussian and Poisson white noise excitations are calculated as an example to illustrate the application of the proposed method. Different resonant cases and non-resonant case are worked out for the purpose of comparison. The Monte Carlo simulations are also carried out to verify the effectiveness and accuracy of the proposed stochastic averaging method.