Stationary Probability Research Articles

Control of probability flow in Markov chain Monte Carlo-Nonreversibility and lifting.

The Markov chain Monte Carlo (MCMC) method is widely used in various fields as a powerful numerical integration technique for systems with many degrees of freedom. In MCMC methods, probabilistic state transitions can be considered as a random walk in state space, and random walks allow for sampling from complex distributions. However, paradoxically, it is necessary to carefully suppress the randomness of the random walk to improve computational efficiency. By breaking detailed balance, we can create a probability flow in the state space and perform more efficient sampling along this flow. Motivated by this idea, practical and efficient nonreversible MCMC methods have been developed over the past ten years. In particular, the lifting technique, which introduces probability flows in an extended state space, has been applied to various systems and has proven more efficient than conventional reversible updates. We review and discuss several practical approaches to implementing nonreversible MCMC methods, including the shift method in the cumulative distribution and the directed-worm algorithm.

Stochastic dynamics analysis for unilateral vibro-impact systems under combined excitation

Vibro-impact system, as an important type of non-smooth system, exhibits intricately nonlinear characteristics. Inevitably, the vibro-impact system will encounter random excitations, but the conventional methods ar7e not eligible for simultaneous determination of its transient responses and reliabilities. Commonly, existing methods of applying non-smooth transformation tend to ignore the essential non-smooth characteristics of vibro-impact system. To this end, this paper proposes a unified framework based on direct probability integral method (DPIM) to simultaneously determine stochastic dynamic responses and reliabilities of unilateral vibro-impact systems under combined harmonic and random excitation without non-smooth transformation, and captures their complicated dynamical behaviors. Firstly, the impact velocity dependent coefficient of restitution is introduced to establish the motion equation of vibro-impact system. Secondly, the probability density integral equation (PDIE) for the unilateral vibro-impact system is derived from the perspective of probability conservation. Then, the PDIE and governing differential equation of the system is solved in a decoupled and efficient way. Moreover, the first-passage reliability is assessed by introducing extreme value mapping of the stochastic dynamic response. Numerical results of three typical examples using the proposed framework are compared with those using Monte Carlo simulation (MCS), quasi-MCS and from the reference, which highlights the advantages of DPIM in computing the stochastic responses and reliabilities of vibro-impact system under random excitations and random parameters. The stationary probability density functions exhibit periodic fluctuations under combined harmonic and stochastic excitation. Specially, the noise intensity and frequency of harmonic excitation pose the great influence on the reliabilities of systems.

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

Near-Optimal Adaptive Policies for Serving Stochastically Departing Customers

In, “Near-Optimal Adaptive Policies for Serving Stochastically Departing Customers,” Segev considers a multistage stochastic optimization problem originally introduced by Cygan et al. [Cygan M, Englert M, Gupta A, Mucha M, Sankowski P (2013) Catch them if you can: How to serve impatient users. Proc. 4th Innovations Theoretical Comput. Sci. Conf., 485–494], studying how a single server should prioritize stochastically departing customers. In this setting, the objective is to determine an adaptive service policy that maximizes the expected total reward collected along a discrete planning horizon, in the presence of customers who are independently departing between one stage and the next with known stationary probabilities. The paper’s main contribution resides in proposing a quasi-polynomial-time approximation scheme for serving impatient customers. Specifically, letting n be the number of underlying customers, our algorithm identifies in [Formula: see text] time a service policy whose expected reward is within factor [Formula: see text] of the optimal adaptive reward. The method for deriving this approximation scheme synthesizes various stochastic analyses in order to investigate how the adaptive optimum is affected by alterations to several instance parameters, including the reward values, the departure probabilities, and the collection of customers itself.

Assessing extreme significant wave height in China’s coastal waters under climate change

Accurately estimating the return values of significant wave height is essential for marine and coastal infrastructure, particularly as climate change intensifies the frequency and intensity of extreme wave events. Traditional models, which assume stationarity in wave data, often underestimate future risks by neglecting the impacts of climate change on wave dynamics. Combining time series decomposition and recurrence analysis, the research develops a nonstationary framework to predict significant wave height. The stochastic component is modelled using a stationary probability distribution, while the deterministic component is predicted based on sea surface temperature projections from CMIP6 climate scenarios. The model evaluation demonstrates strong predictive capability for both stochastic and deterministic components. Application of the model to China’s coastal waters reveals significant discrepancies between stationary and nonstationary return value estimates. Compared to conventional distribution models, the nonstationary model predicts substantial increases in extreme wave heights. These findings underscore the importance of adopting nonstationary models to more accurately assess future risks posed by extreme wave events in a changing climate.

Inertial amplification as a performance enhancement method for snap-through vibration energy harvester

In recent years, there has been a lot of interest in exploiting the bistable behavior of snap-through systems to harvest energy from vibration sources. The efficient operation of any bistable VEH depends on its ability to exhibit large-amplitude interwell motion. Under weak ambient excitation, bistable VEH performs marginally because of the confinement of motion to a single well. Frequency up-conversion, multi-stability, and adaptive techniques are some of the performance enhancement strategies suggested for bistable-VEH. Considering the VEH's space constraints, the above designs are hard to implement in practical cases. This study introduces an inertial amplification mechanism (IAM) as a simple passive strategy to enhance the performance of a snap-through VEH, a concept not explored in previous studies. The addition of IAM increases the effective mass without increasing the physical mass and thereby enhances energy harvesting, especially from weak ambient excitation sources. The dynamics and performance of the enhanced snap-through VEH are investigated analytically and numerically under harmonic and random excitations. The harmonic balance method (HBM) derives the frequency-amplitude relationship, which shows a hardening behavior and an increase in bandwidth. The effective potential method provides a closed-form expression for the joint probability density function (Joint PDF), which is governed by the Fokker-Planck equation. The joint PDF shows a transition from bimodal to unimodal with an increase in the value of the geometrical parameter. The stochastic averaging method is employed to obtain the stationary probability density function, which defines the long-term dynamics of the VEH. The effects of noise intensity, mass ratio, and inertial amplifier angle on the dynamics are investigated. Finally, the performance of the proposed VEH is compared with a conventional snap-through VEH, an equivalent linear VEH, and a multistable VEH under harmonic and random excitation conditions. The findings suggest that the snap-through VEH with the IAM has advantages over the linear and multistable nonlinear VEH in terms of extracting energy from low-intensity harmonic and random excitation sources. This simple augmentation strategy preserves the original system's bistability, eliminating the need for the complex design of a multistable VEH.

A perspective on conditional spectrum-based determination of response statistics of nonlinear systems

This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system’s displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.

Semi-Markov process-driven maintenance scheduling for Tainter gate system considering multiplelimit states

In extended periods of operation, Taiwan’s reservoir electromechanical systems increasingly require substantial maintenance. This research adopts the semi-Markov process, which accommodates non-exponential distribution of state durations, to formulate optimal maintenance strategies for Tainter gate systems that are noted for their prolonged dormancy and significant operational uncertainties. The methodology initiates with the estimation of failure probabilities across four condition states, analyzing deterioration through the Weibull distribution for both general and latent limit states. The general limit state accounts for time-induced deterioration and effects of dormancy using a Bayesian–Weibull first-order reliability method, while the latent limit state addresses activation failures. Employing the semi-Markov process, an annual transition matrix is computed and combined with failure probabilities to assess the Tainter gates’ system reliability. To identify the most efficient maintenance schedule, a genetic algorithm is applied, targeting the minimization of failure probabilities for both limit states below predefined thresholds and cost reduction. Numerical simulations validate the framework’s efficacy, demonstrating its potential to enhance maintenance planning objectivity and decrease dependence on subjective assessments. The findings highlight the predominance of the general limit state in dictating system failure and underscore the risk Tainter gates face during transition from dormancy to activation, emphasizing the need for thorough monitoring.

Directional switching behavior of swarming systems with social and nonlinear interactions

Coordinated directional switching can occur among members of many mobile biological communities. Some studies show that self-propelled particle models can describe the directional switching behavior well. The key to understanding group movement is to determine the influential factors relevant to directional switching behavior. This paper focuses on the impact of social and nonlinear interactions on the directional switching behavior observed in swarming systems. In which, the nonlinear interaction is represented as a function of a trade-off between the velocity and velocity direction of its neighbors. Based on the framework of dimension reduction theory, the high-dimensional complex model is simplified into a one-dimensional simple model, and the stationary probability density and mean switching time are obtained by theoretical analysis of the one-dimensional model. It can be seen that social and nonlinear interactions play an important role in regulating the directional switching behaviors of swarming systems. Specifically, the increase of group density and nonlinear parameter can inhibit the directional switches. For Erdös-Rényi networks, the large mean degree can suppress the directional switching behavior. For scale-free networks, increasing the degree heterogeneity can reduce the mean switching time. The results reveal the underlying mechanisms by which social and nonlinear interactions influence the directional switching behaviors of swarming systems, and provide a theoretical foundation for the design of bio-inspired devices with specific functions.

Charmenability and stiffness of arithmetic groups

We characterize charmenability among arithmetic groups and deduce dichotomy statements pertaining to normal subgroups, characters, dynamics, representations and associated operator algebras. We do this by studying the stationary dynamics on the space of characters of the amenable radical, and in particular we establish stiffness: any stationary probability measure is invariant. This generalizes a classical result of Furstenberg for dynamics on the torus. Under a higher rank assumption, we show that any action on the space of characters of a finitely generated virtually nilpotent group is stiff.

A stochastic averaging mathematical framework for design and optimization of nonlinear energy harvesters with several electrical DOFs

Energy harvesters for mechanical vibrations are electro-mechanical systems designed to capture ambient dispersed kinetic energy, and to convert it into usable electrical power. The random nature of mechanical vibrations, combined with the intrinsic non-linearity of the harvester, implies that long, time domain Monte-Carlo simulations are required to assess the device performance, making the analysis burdensome when a large parameter space must be explored. Therefore a simplified, albeit approximate, semi-analytical analysis technique is of paramount importance. In this work we present a methodology for the analysis and design of nonlinear piezoelectric energy harvesters for random mechanical vibrations. The methodology is based on the combined application of model order reduction, to project the dynamics onto a lower dimensional space, and of stochastic averaging, to calculate the stationary probability density function of the reduced variables. The probability distribution is used to calculate expectations of the most relevant quantities, like output voltage, harvested power and power efficiency. Based on our previous works, we consider an energy harvester with a matching network, interposed between the harvester and the load, that reduces the impedance mismatch between the two stages. The methodology is applied to the optimization of the matching network, allowing to maximize the global harvested power and the conversion efficiency. We show that the proposed methodology gives accurate predictions of the harvester’s performance, and that it can be used to significantly simplify the analysis, design and optimization of the device.

Online tree-based planning for active spacecraft fault estimation and collision avoidance.

Autonomous robots operating in uncertain or hazardous environments subject to state safety constraints must be able to identify and isolate faulty components in a time-optimal manner. When the underlying fault is ambiguous and intertwined with the robot's state estimation, motion plans that discriminate between simultaneous actuator and sensor faults are necessary. However, the coupled fault mode and physical state uncertainty creates a constrained optimization problem that is challenging to solve with existing methods. We combined belief-space tree search, marginalized filtering, and concentration inequalities in our method, safe fault estimation via active sensing tree search (s-FEAST), a planner that actively diagnoses system faults by selecting actions that give the most informative observations while simultaneously enforcing probabilistic state constraints. We justify this approach with theoretical analysis showing s-FEAST's convergence to optimal policies. Using our robotic spacecraft simulator, we experimentally validated s-FEAST by safely and successfully performing fault estimation while on a collision course with a model comet. These results were further validated through extensive numerical simulations demonstrating s-FEAST's performance.

Stationary distribution and mean extinction time in a generalist prey–predator model driven by Lévy noises

In this study, a delayed generalist prey–predator model incorporating the logistic-like growth and Holling type-III functional response for predator subjected to two Lévy noises is established. Such environmental fluctuations can describe abrupt change in population density. First, we conduct stability analysis and determine the critical value of time delay when Hopf bifurcation takes place. Then, the existence of the global positive solution and stability in distribution of stochastic model are studied. Numerical results are given to illustrate the analytical findings, and mainly dedicated to the consequences of time delay and Lévy noises on the stationary distribution, mean extinction time and reproduction rate of prey–predator populations, via stationary probability distribution function (PDF), mean first passage time (MFPT) and relaxation time, respectively. Our findings indicate that the decreases in time delay, noise intensities (Di,i=1,2), and stability indexes within the range of αi∈[1,2] are more likely to maintain the population in a high density state. Phase transition occurs when the system is in the suitable parameter regime. Additionally, there exist optimal noise intensities that slow down switching of prey species from a stable state to an extinction state. Smaller time delay τ and stability indexes αi inhibit population extinction, whereas smaller skewness parameter β speed up prey extinction. More importantly, two noises can tune the enhanced stability characteristic of the system.

Cauchy universality and random billiards

Motion in bounded domains represents a paradigm in several settings: from billiard dynamics, to random walks in a finite lattice, with applications to relevant physical, ecological, and biological problems. A remarkable universal property, involving the average of return times to the boundary, has been theoretically proposed and experimentally verified in quite different contexts. We discuss here mechanisms that lead to violations of universality, induced by boundary effects and we also emphasize the role played by replacing straight lines with random walks in this framework. We suggest that our analysis should be relevant where nonhomogeneity appears in the stationary probability distribution in bounded domain. Published by the American Physical Society 2024

Probabilistic curvature limit states of corroded circular RC bridge columns: Data-driven models and application to lifetime seismic fragility analyses

Reinforcement corrosion has been recognized as an influential factor in the seismic fragility, both demand and capacity models, of aging reinforced concrete (RC) bridges. For capacity models, accurate and applied prediction tools accounting for aging effects are yet to be well established. Current practices usually perform numerical analyses to obtain time-variant capacity models, which are time-consuming particularly when multi-source structural and environmental uncertainties are considered and sometimes even suffer computational non-convergence. To address these issues, this study leverages a rigorously optimized artificial neural network architecture to develop data-driven models for rapid estimates of probabilistic curvature capacity of corroded circular RC bridge columns with flexural failure modes. An extensive database of multi-level curvature limit states (i.e. slight, moderate, severe, and complete) is created through experimentally validated moment–curvature analyses. A new threshold for the moderate limit state is defined based on the strain of core concrete, rather than cover concrete, to account for the potential full erosion of the cover with drastic corrosion. The data-driven probabilistic capacity models are applied to aid the lifetime seismic fragility assessment of a typical highway bridge, where the spectral acceleration at 1.0 s (Sa-10), peak ground velocity (PGV), and Housner intensity (HI) are found consistently, for the first time, as optimal intensity measures for probabilistic demand modeling of RC columns with different extents of aging effects. For the ease of application, the database and code for the data-driven probabilistic capacity models are accessible at https://bit.ly/3uAa8EY .

Probabilistic ductile deformation limit state prediction of monolithic exterior shear keys based on quantile regression machine learning techniques

Shear strength and ductile deformation capacity of monolithic exterior shear keys play important roles on the seismic behavior of highway bridges. Improving the ductility of the shear key through reasonable design could effectively restrict superstructure displacement and minimize damage to pier columns. However, most existing studies primarily focus on evaluating the shear strength and damage mechanisms of shear keys, with limited attention given to analytical models for assessing ductile deformation capacity. To this end, this study proposes a method that combines quantile regression and machine learning techniques to predict the ductile deformation limit states of shear keys. The proposed numerical modeling method for shear keys is validated using the experimental data on lateral load-displacement curves. On this basis, Latin hypercubic sampling and joint simulation are utilized to establish the ductile deformation limit state database. The ductile deformation limit states of shear keys are characterized by their lateral displacements in three distinct damage states. The main influencing factors of each limit states are investigated by correlation analysis. Three quantile regression machine learning models are trained and constructed to conduct point and interval prediction for the ductile deformation limit states. All three models, particularly the Gaussian process quantile regression model, are proven to demonstrate exceptional prediction accuracy and uncertainty quantification. Experimental data validation confirms that these models outperform the prediction accuracy of existing simplified models.

LN: A Flexible Algorithmic Framework for Layered Queueing Network Analysis

Layered queueing networks (LQNs) are an extension of ordinary queueing networks useful to model simultaneous resource possession and stochastic call graphs in distributed systems. Existing computational algorithms for LQNs have primarily focused on mean-value analysis. However, other solution paradigms, such as normalizing constant analysis and mean-field approximation, can improve the computation of LQN mean and transient performance metrics, state probabilities, and response time distributions. Motivated by this observation, we propose the first LQN meta-solver, called LN, that allows for the dynamic selection of the performance analysis paradigm to be iteratively applied to the submodels arising from layer decomposition. We report experiments where this added flexibility helps us to reduce the LQN solution errors. We also demonstrate that the meta-solver approach eases the integration of LQNs with other formalisms, such as caching models, enabling the analysis of more general classes of layered stochastic networks. Additionally, to support the accurate evaluation of the LQN submodels, we develop novel algorithms for homogeneous queueing networks consisting of an infinite server node and a set of identical queueing stations. In particular, we propose an exact method of moment algorithms, integration techniques for normalizing constants, and a fast non-iterative mean-value analysis technique.

Minimum magnitude boundaries in probabilistic seismic hazard analysis: an insight from structural engineering

In order to systematically advance our understanding of the minimum magnitude limit (Mmin) in the probabilistic seismic hazard analysis (PSHA) calculations, a novel and useful approach utilising a broad range of Single-Degree-of-Freedom oscillators and hazard conditions is being developed and tested. We have determined the most reasonable Mmin value for a variety of structures by examining the impact of Mmin on the mean annual frequency (MAF) of various limit states (LSs) (including the collapse capacity). The originality of the suggested methodology in the current work, known as the MAF saturation strategy, is the recommended Mmin, which is the cut-off value at which lesser magnitude events do add to the hazard but do not significantly change the MAF. The current work is the first to offer the MAF saturation strategy methodology, which searches for the cut-off magnitude at which the MAF value essentially remains constant even when smaller values of this cut-off are utilised as Mmin for hazard assessments. Therefore, given a series of carefully chosen ground motions in each oscillator instance, an incremental dynamic analysis is carried out (by applying the Hunt and Fill algorithm), and the appropriate LS (including the collapse capacity defined as global instability) points are calculated. Thus, the relationship between the distribution of LSs and the Engineering Demand Parameter and intensity measure is found. A simple point source hazard curve is convoluted with this distribution, yielding the structure-specific MAF. In order to find the cut-off lower magnitude (Mmin), this convolution is repeated for several Mmin values. This cut-off is defined as the point at which, when lower values are utilised as Mmin in the PSHA computation, the MAF’s values do not change considerably (with a five per cent threshold). The acquired data were thoroughly discussed in relation to various structural features and seismic input factors. The primary findings showed that each of the structures under consideration requires a Mmin value in the range of 4–4.3. Put otherwise, the suggestions seen in technical literature, which range from 4.5 to 5, are not cautious, at least not when it comes to probabilistic structural limit state frequency. The derived Mmin value is mostly controlled by the natural period of the structure and is largely unaffected by other structural characteristics like ductility, damping ratio and overstrength factor.

Twinning quality sensors in wastewater treatment process via optimized echo state network-based soft sensors

The presence of a large amount of quality-related but hard-to-measure variables usually makes effective monitoring of industrial processes difficult, and even impossible. Soft computing techniques and digital twins can revolutionize standard approaches to solve this issue, even though industrial data are frustrated by nonlinearity, disturbances, and dynamic behaviors. In this light, this paper proposes a data-driven soft sensor with the help of the Fourier amplitude sensitivity test (FAST) and improved probabilistic regularized echo state network (IPRESN), then to twin the quality hardware sensors. Within this framework, the initial step is to rely on the FAST method to measure the collaborated importance of multiple process variables with respect to each quality variable. This quantification facilitates the selection of auxiliary variables for the training of IPRESN. Furthermore, a probabilistic regularization method is learned to improve the robust performance of the echo state network (ESN) by re-designing a new objective function to minimize the mean and variance of the modeling error. Additionally, the whale optimization algorithm (WOA) is improved and then used to optimize the critical hyper-parameters of ESN, thus mitigating the risks associated with suboptimal hyperparameter selection. The effectiveness of the proposed method is verified through a Benchmark Simulation Model 2 (BSM2) and a full-scale wastewater treatment plant data set. The findings show the potential of the proposed method in facilitating digital twin implementations for hardware sensors.