Stationary Probability Density Function Research Articles

Sequential Monte Carlo with Adaptive Lookahead Support for Improved Importance Sampling

The sequential Monte Carlo, also called the Bayesian particle filter, approximates a posterior probability density function of a latent target state from noisy sensor measurements using a set of Monte Carlo samples. These samples are predicted using an importance density function and then updated using the Bayes’s rule. The updated samples and their corresponding weights provide an estimate of the latent state. The said filtering process is iterated over time for tracking dynamic target states. It is critical to have enough particles in regions of the target state space that contribute to the posterior. The auxiliary and the improved auxiliary particle filters accomplish this by a process that mimics drawing from an importance density that leverages the incoming observation into the sampling step. However these filters are known to fail when the sensor measurements are highly informative and the diffusion over the state transition is large. This paper presents an improvement to the auxiliary particle filter by taking two support points that act as limits in a univariate state space within which particles are samples. The choice of the limits is adaptive. The proposed method is successfully tested using a nonlinear model using simulations.

Stochastic analysis and vibration suppression of a time-delayed system with nonlinear energy sink

Nonlinear energy sink (NES) is of great significance in attenuating structural vibration. Time-delay phenomenon in a mechanical system is inevitable. Different from existing works, the influences of time-delayed factors in a nonlinear oscillator attached with NES are considered in this paper under Gaussian white noise (GWN). Firstly, the time-delayed term is transformed into a linear combination of stiffness and damping terms, and an equivalent system without the time-delayed term is obtained. Stationary probability density function (SPDF) is deduced by leveraging stochastic averaging method. Secondly, good agreement is observed between analytical and numerical results, which verifies the effectiveness of the method. Furthermore, bifurcation conditions for the amplitude of SPDF of the primary system are obtained, indicating that delayed time, feedback gain and damping coefficients can cause stochastic P-bifurcation of the primary system. Finally, the vibration suppression performance of NES is analyzed via energy variation and the time–history diagram of displacement.

Topological detection of phenomenological bifurcations with unreliable kernel density estimates

Phenomenological (P-type) bifurcations are qualitative changes in stochastic dynamical systems whereby the stationary probability density function (PDF) changes its topology. The current state of the art for detecting these bifurcations requires reliable kernel density estimates computed from an ensemble of system realizations. However, in several real world signals such as Big Data, only a single system realization is available—making it impossible to estimate a reliable kernel density. This study presents an approach for detecting P-type bifurcations using unreliable density estimates. The approach creates an ensemble of objects from Topological Data Analysis (TDA) called persistence diagrams from the system’s sole realization and statistically analyzes the resulting set. We compare several methods for replicating the original persistence diagram including Gibbs point process modelling, Pairwise Interaction Point Modelling, and subsampling. We show that for the purpose of predicting a bifurcation, the simple method of subsampling exceeds the other two methods of point process modelling in performance.

Efficient quadratures for high-dimensional Bayesian data assimilation

Bayesian update is a common strategy used to combine (uncertain) model predictions and (noisy) observational data. A computational bottleneck in this data assimilation technique is the evaluation of high-dimensional quadratures involving multivariate probability density functions (PDFs) of system states. We explore “designed quadratures” as a means to reduce the computational cost of Bayesian update of multivariate joint PDFs. A series of numerical experiments demonstrate that our method outperforms stochastic collocation on sparse grids, a popular technique used to perform high-dimensional integration in the context of uncertainty quantification, in terms of both accuracy and computational efficiency.

Bifurcation Regulations Induced by Joint Noise in a Tri-Rhythmic Van Der Pol System

Tri-rhythmical nature has attracted extensive attention from scholars in describing the dynamical behaviors of self-sustained systems. In this paper, we consider a tri-rhythmic van der Pol system and give a bifurcation analysis of a stochastic tri-rhythmic self-sustained system under joint noise perturbation. Based on an approximate approach, we give the stationary probability density function of amplitudes, and we find that the noise and time-delay feedback, regulating the velocity time-delay feedback strength parameter, may not cause transitions among unimodal, bimodal and trimodal in the tri-rhythmic system. More stochastic bifurcations appear by regulating the time delay in this system. The system, surprisingly, undergoes five times of stochastic bifurcations when the time delay is monotonically increased. It is shown that the time delay is more sensitive to the tri-rhythmic system and a much stronger dependence on the system. From a biological point of view, the reaction rate of biological molecules can be enhanced or diminished by the change of the noise intensity or correlation time of Gaussian colored noise. More surprisingly, an increase in displacement feedback will delay the reaction rate; however, the effect of an increase in velocity feedback on the reaction rate depends on the time delay. A detailed research on the parameter space indicates that time delay and colored noise parameters can effectively control the multirhythmicity of the system. Finally, this paper verifies the effectiveness of the theoretical solution through the numerical results of Monte Carlo simulation of the original system. These results may help to further explore forking bifurcations in real-world applications. This research provides new insight into the understanding of nontrivial effects of joint noise on the tri-rhythmic system.

Critical transitions in spatial systems induced by Ornstein–Uhlenbeck noise: spatial mutual information as a precursor

Complex dynamical systems are subject to perturbations across space and time, which can induce a critical transition or tipping in the state of the system. External perturbations are often correlated in time and can interplay with the underlying nonlinearity of the spatial system, affecting the occurrence of critical transitions. Theoretical analysis of the spatial system perturbed by the Ornstein–Uhlenbeck (OU) correlated noise poses challenges beyond the white noise assumptions and is yet to be done. Here, we resort to the mean-field approximation of a spatially extended system perturbed with OU noise and obtain the stationary probability density function deriving the Fokker–Planck equation for the same. This allows us to determine the role of diffusion and noise on the resilience of the spatial system. While the theoretical analysis guides us on the landscape of tipping thresholds of the system, critical transitions customary to a variety of systems, require a priori prediction. Here, we propose a probabilistic information-based indicator—spatial mutual information—that can successfully forecast tippings, complementing the previously developed spatial indicators. Further, validating its reliability on empirical data, we show that spatial mutual information serves as a robust indicator capturing information characteristic to an imminent tipping reaching peaks in its vicinity.

Bifurcation analysis of a tristable system with fractional derivative under colored noise excitation

Stochastic bifurcation has received much attention recently and is of great significance in the research on the dynamics of nonlinear systems. In this paper, we study the bifurcation of a self-sustained tristable system containing fractional derivative under the excitation of two colored noises, where the self-sustained tristable system consists of stable limit cycles and a stable state. The multiple scale method and generalized harmonic function are used to convert the original system into an equal system without obvious time delay and fractional derivative. Then the stationary probability density function (SPDF) of the system is obtained by using the stochastic averaging method to discuss stochastic bifurcation. Based on singularity theory, the equation satisfying amplitude is obtained to derive the corresponding bifurcation diagram. From the bifurcation analysis of the system, it is revealed that fractional order, fractional coefficient, and the intensity and correlation time of colored noise can be utilized as bifurcation parameters, causing peculiar stochastic bifurcation phenomena and regulating the output of the system, as well as the relatively big colored noise intensity and relatively small correlation time facilitate the realization of large magnitude limit cycle. Monte Carlo numerical results verify the validity of the theoretical approach. In addition, appropriately changing the delayed feedback parameter helps to govern the bifurcation range and causes richer bifurcation phenomena. The results of this paper contribute to a better understanding of self-sustained tristable systems with fractional derivative and may provide reference value in resolving the problems of chattering in high-speed aircraft and applications to viscoelastic materials.

Variable step‐size estimation over UDP‐based wireless networks with application to a hydrogen‐powered UAV

AbstractIn this paper, the problem of state estimation for systems over wireless networks using user datagram protocol is focused on. It is known that for such a system, the probability density function of the system state follows a Gaussian mixture model (GMM), and the number of components in this model grows exponentially over time, which makes the computation of optimal estimates infeasible. To compute optimal estimates, based on Kullback‐Leibler divergence, a strategy with variable step‐sizes to truncate and fuse the GMM is proposed. Based on the obtained GMM, a variable step‐size estimator is designed to compute optimal estimates during an estimation cycle. The advantages of the proposed estimator are twofold: (1) its estimation performance is superior to that of existing one‐step fast estimators; (2) its estimation efficiency is much higher than that of the optimal estimator. Finally, trajectory tracking has been proposed for a real‐world hydrogen‐powered unmanned aerial vehicle to show the effectiveness of our methods.

Analysis of the stationary probability density of a generalized and bistable van der Pol system excited by colored noise

The stochastic P-bifurcation behavior of bi-stability in a generalized van der Pol oscillator with the fractional damping under colored noise and thermal excitation is investigated. Firstly, using the principle of minimal mean square error and lin-earization method, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order van der Pol system. Secondly, the system amplitude stationary probability density function is obtained by the stochastic averaging, and then based on the singularity theory, the critical parametric con-ditions for the system amplitude stochastic P-bifurcation are found. Finally, the types of the stationary probability density function of the system amplitude are qualitatively analyzed in each area divided by the transition set curves. The con-sistency between the analytical results and the numerical results acquired from Monte-Carlo simulation also testifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

A Rapid Method for Impact Analysis of Grid-Edge Technologies on Power Distribution Networks

This paper presents a novel rapid estimation method (REM) to perform stochastic impact analysis of grid-edge technologies (GETs) to the power distribution networks. The evolution of network states' probability density functions (PDFs) in terms of GET penetration levels are characterized by the Fokker-Planck equation (FPE). The FPE is numerically solved to compute the PDFs of network states, and a calibration process is also proposed such that the accuracy of the REM is maintained for large-scale distribution networks. The approach is illustrated on a large-scale realistic distribution network using a modified version of the IEEE 8500 feeder, where electric vehicles (EVs) or photovoltaic systems (PVs) are installed at various penetration rates. It is demonstrated from quantitative analyses that the results from our proposed approach have negligible errors comparing with those obtained from Monte Carlo simulations.

Uncertainty propagation in orbital dynamics via Galerkin projection of the Fokker-Planck Equation

The Fokker–Planck equation is a partial differential equation that describes how the probability density function of an object state varies, when subject to deterministic and random forces. The solution to this equation is crucial in many space applications, such as space debris trajectory tracking and prediction, guidance navigation and control under uncertainties, space situational awareness, and mission analysis and planning. However, no general closed-form solutions are known and several methods exist to tackle its solution. In this work, we use a known technique to transform this equation into a set of linear ordinary differential equations in the context of orbital dynamics. In particular, we show the advantages of the applied methodology, which allows to decouple the time and state-dependent components and to retain the entire shape of the probability density function through time, in the presence of both deterministic and stochastic dynamics. With this approach, the probability density function values at future times and for different initial conditions can be computed without added costs, provided that some time-independent integrals are solved offline. We showcase the efficacy and use of this method on some orbital dynamics example, by also leveraging the use of automatic differentiation for efficiently computing the involved derivatives.

A stochastic predator–prey model with distributed delay and Ornstein–Uhlenbeck process: Characterization of stationary distribution, extinction, and probability density function

As the evolution of species relies on not only the current state but also the past information, it is more reasonable and realistic to take delay into an ecological model. This paper deals with a stochastic predator–prey model that considers the distribution delay and assume that the intrinsic growth rate and the death rate in the model are governed by Ornstein–Uhlenbeck process to simulate the random factors in the environment. Based on the existence and uniqueness of the global solution to the model and the boundedness of the order moments of the solution, several conditions are established to analyze the survival of the species. Specifically, a criteria for the existence of the stationary distribution to the stochastic system is established by constructing some suitable Lyapunov functions. And the analytical expression of the probability density function of the model around the quasi‐equilibrium is obtained. Furthermore, the extinction of species in the model is also explored. Finally, numerical simulations are carried out to illustrate the theoretical results obtained in this paper.

Impact of Non-Gaussian Noise and Time Delay on Stability and Stochastic Resonance for a FitzHugh-Nagumo Neural System Subjected to a Multiplicative Periodic Signal

In this paper, we focus on the investigations on the stochastic stability and the stochastic resonance (SR) phenomena for a FitzHugh-Nagumo system with time delay induced by a multiplicative non-Gaussian colored noise and an additive Gaussian colored noise. By use of the fast descent method, the unified colored noise approximation and the two-state theory for the SR, the stationary probability density function (SPDF) and the signal-to-noise ratio (SNR) caused by different noise terms and time delay are explored. The investigation results indicate that the two noise intensities, time delay and the departure parameter from the Gaussian noise can all reduce the probability density around the two stable states and destroy the stability of the neural system; while the two noise correlation times [Formula: see text] and [Formula: see text] can both improve the probability density around both stable states and reinforce the biological stability of the neural system. As regards the SNR, it is found that the two noise intensities and the departure coefficient can all weaken the SR effect, while time delay [Formula: see text] and the correlation time [Formula: see text] of the multiplicative noise will always magnify the SR phenomenon. It is worth to mention that the correlation time [Formula: see text] of the additive noise can stimulate the SR effect, but not alter the maximum of the SNR.

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

Response and bifurcation of fractional Duffing oscillator under recycling noise and time-delayed feedback control are investigated. Firstly, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the Itô differential equations and one-dimensional Markov process are obtained according to stochastic averaging method, and the stochastic stability and stochastic bifurcation of the system are analyzed. Lastly, through joint probability density function diagram and the stationary probability density function diagram, the stochastic bifurcation behavior of system under the different time-delay, fractional order and noise intensity are discussed respectively, the validity of the theory and the occurrence of bifurcation phenomenon are verified.

An Improved Stochastic Averaging on the Piezoelectric Vibrational Harvester Model with Stiffness and Inertia Nonlinearities

In this paper, an improved stochastic averaging method is introduced to solve a monostable piezoelectric vibration energy harvester (VEH) model subject to basal Gaussian white and colored noises. First, the electric voltage is expressed as perturbations to the linear stiffness term and the damping term of the vibrational equation. Second, the original model is simplified into an Itô differential equation by the improved stochastic averaging method. Third, stationary solution of the Fokker–Plank–Kolmogorov function driven from the Itô equation is obtained, so that the stationary probability density functions (PDFs) of the displacement, velocity, and electrical voltage are obtained. With the help of the Monte Carlo, the effectiveness of these theoretical predictions is verified. Finally, the impacts of the nonlinear stiffness coefficient and nonlinear inertia coefficient on the responses are compared. It is found that the stiffness nonlinearity has stronger effect on the outputn electric voltage than the inertia nonlinearity.

Design and analysis of a galloping energy harvester with V-shape spring structure under Gaussian white noise

This paper presents the design of an innovative galloping energy harvester (GEH) that incorporates an elastic structure to reconstructed the traditional GEH. The mathematical model of the GEH is derived based on the Euler–Bernoulli beam theory and Kirchhoff’s law. Due to the nonlinear properties of the model, the partial linearization technique and Fokker–Planck–Kolmogorov equation are used to further explore the joint probability density function (PDF) and the stationary PDF of displacement and velocity. Subsequently, Monte Carlo simulation is used to illustrate the behavior of the theoretical results. The simulation results show that, under different conditions, the stationary PDF exhibits double-peak or single-peak shape, which is caused by the compression or elongation state. Meanwhile, higher wind speeds correspond to larger displacement amplitudes. Then, the stationary PDF of displacement amplitude is obtained using the stochastic average method, and the expressions for mean square voltage and mean output power are derived. Numerical results indicate that when the wind speed is below the critical wind speed for galloping phenomenon, the vibration of the GEH is mainly driven by noise. When the wind speed exceeds the critical wind speed, the GEH is mainly affected by the wind speed. Higher wind speeds correspond to higher mean output power. An optimal electromechanical coupling coefficient and the ratio of the mechanical and electrical can be derived based on the GEH parameters when galloping occurs.

A Markov chain approximation of switched Fokker–Planck equations for a model of on–off intermittency in the postural control during quiet standing

The intermittent on–off switching of feedback control is considered as a major mechanism of postural stabilization during human quiet standing, which can be modeled by switched-type hybrid stochastic delay differential equations with unstable subsystems. Dynamics of the model can also be described by the corresponding switched-type Fokker–Planck (FP) equations. Here, we develop a comprehensive numerical recipe to simulate switched-type FP equations in the case that the probability current is conserved at the switching boundary, as is the case for noise-free models exhibiting C0-continuity for solutions at the boundary, but in a way extendable to cases with discontinuous jumps. Specifically, the FP equations are approximated by a finite state Markov chain model using the finite element method. Then, dynamics of the Markov chain model, including time evolution of probability density function (PDF), stationary PDF, and power spectrum of postural sway are analyzed. We further investigate how the stationary PDF alters as values of important parameters of the model change. Dynamics of the Markov chain model are compared with Monte Carlo-based dynamics of the model, by which the developed numerical recipe is validated. The obtained Markov chain model forms a basis of our future investigations of the intermittent postural control as a Markov decision process.

Fractional stochastic vibration system under recycling noise

The fractional stochastic vibration system is quite different from the traditional one, and its application potential is enormous if the noise can be deployed correctly and the connection between the fractional order and the noise property is unlocked. This article uses a fractional modification of the well-known van der Pol oscillator with multiplicative and additive recycling noises as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Second, the Itô differential equations and One-dimensional Markov process are obtained according to the stochastic averaging method, using Oseledec multiplicative ergodic theorem and maximal Lyapunov exponent to judge local stability, and judging global stability is done by using the singularity theory. Lastly, the stochastic D-bifurcation behavior of the model is analyzed by using the Lyapunov exponent of the dynamical system invariant measure, and the stationary probability density function of the system is solved according to the FPK equation. The results show that the fractional order and noise property can greatly affect the system’s dynamical properties. This paper offers a profound, original, and challenging window for investigating fractional stochastic vibration systems.

Time-dependent probability density function for partial resetting dynamics

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index . For Cauchy and Brownian motions (i.e. ), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.

A Dynamic Behavior Analysis of a Rolling Mill’s Main Drive System with Fractional Derivative and Stochastic Disturbance

Taking the random factors into account, a fractional main drive system of a rolling mill with Gaussian white noise is developed. First, the potential deterministic bifurcation is investigated by a linearized stability analysis. The results indicate that the fractional order changes the system from a stable point to a limit cycle with symmetric phase trajectories. Then, the stochastic response is obtained with the aid of the equivalent transformation of the fractional derivative and stochastic averaging methods. It is found that the joint stationary probability density function appears to have symmetric distribution. Finally, the influence of the fractional order and noise intensity on system dynamics behavior is discussed. The study is beneficial to understand the intrinsic mechanisms of vibration abatement.