Stationary Probability Distribution Research Articles

Topological phase transition in cavity optomechanical system with periodical modulation

We investigate the topological phase transition and the enhanced topological effect in a cavity optomechanical system with periodical modulation. By calculating the steady-state equations of the system, the steady-state conditions of cavity fields and the restricted conditions of effective optomechanical couplings are demonstrated. It is found that the cavity optomechanical system can be modulated to different topological Su–Schrieffer–Heeger (SSH) phases via designing the optomechanical couplings legitimately. Meanwhile, combining the effective optomechanical couplings and the probability distributions of gap states, we reveal the topological phase transition between trivial SSH phase and nontrivial SSH phase via adjusting the decay rates of cavity fields. Moreover, we find that the enhanced topological effect of gap states can be achieved by enlarging the size of system and adjusting the decay rates of cavity fields.

Water quality evolution mechanism modeling and health risk assessment based on stochastic hybrid dynamic systems

Water quality assessment analysis is an important technical means for water pollution prevention and control. In this research area, mechanism models and real observation data of water quality evolution are always used to perform water quality assessment. However, the existing water quality evolution mechanism modeling researches commonly use a single time-invariant model to model the water quality evolution process. It is inappropriate to directly describe the complex behavior of long-term water quality evolution with the existing models, since the evolution process contain different feature states, and the water quality evolution characteristics under these states are different. In addition, the existing water quality assessment methods are mostly methods for directly processing and calculating the observation data of water quality. This makes the existing methods difficult to effectively compensate for the contingency and randomness in the water quality evolution process, which leads to deviations and errors in performing the water quality assessment. Considering these deficiencies, this paper proposes a water quality evolution mechanism modeling and health risk assessment method based on stochastic hybrid dynamic systems (SHDS). Firstly, a hybrid water quality evolution mechanism (H-WQEM) model is established based on SHDS, and a hybrid improved fruit fly optimization algorithm (H-IFFOA) is proposed to identify the unknown parameters of the H-WQEM model. Then, an improved interacting multiple model extended Kalman filter algorithm (IIMM-EKF) is employed to estimate the probability distribution of the hybrid state of the H-WQEM model, including the probability distribution of different feature states of water bodies and the probability distribution of water quality indexes under these states. Finally, the health degree of water quality is proposed as an indicator to achieve a quantitative assessment of the water health risk status. Real observation data from a monitoring station at Baiyangwan in China is used to validate the effectiveness of the proposed method. The results show that the method can effectively describe the complex water quality evolution process, and reasonably assess the water quality health risk status.

Feedback control of stochastic gene switches using PIDE models

Achieving control of gene regulatory circuits is one of the goals of synthetic biology, as a way to regulate cellular functions for useful purposes (in biomedical, environmental or industrial applications). The inherent stochastic nature of gene expression makes it challenging to control the behavior of gene regulatory networks, and increasing efforts are being devoted in the field to address different control problems.In this work, we combine the efficient modeling of stochastic gene regulatory networks by means of Partial Integro-Differential Equations with feedback control, in order to keep protein levels at the target (pre-defined) stationary probability distribution. In particular, we achieve the closedloop stabilization of bi-modal toggle-switches in the stochastic regime within the region of low probability (around the minimum located between the two modes of the uncontrolled system).

A PDE-BASED ADAPTIVE KERNEL METHOD FOR SOLVING OPTIMAL FILTERING PROBLEMS

In this paper, we introduce an adaptive kernel method for solving the optimal filtering problem. The computational framework that we adopt is the Bayesian filter, in which we recursively generate an optimal estimate for the state of a target stochastic dynamical system based on partial noisy observational data. The mathematical model that we use to formulate the propagation of the state dynamics is the Fokker-Planck equation, and we introduce an operator decomposition method to efficiently solve the Fokker-Planck equation. An adaptive kernel method is introduced to adaptively construct Gaussian kernels to approximate the probability distribution of the target state. Bayesian inference is applied to incorporate the observational data into the state model simulation. Numerical experiments have been carried out to validate the performance of our kernel method.

Dual Control by Reinforcement Learning Using Deep Hyperstate Transition Models

In dual control, the manipulated variables are used to both regulate the system and identify unknown parameters. The joint probability distribution of the system state and the parameters is known as the hyperstate. The paper proposes a method to perform dual control using a deep reinforcement learning algorithm in combination with a neural network model trained to represent hyperstate transitions. The hyperstate is compactly represented as the parameters of a mixture model that is fitted to Monte Carlo samples of the hyperstate. The representation is used to train a hyperstate transition model, which is used by a standard reinforcement learning algorithm to find a dual control policy. The method is evaluated on a simple nonlinear system, which illustrates a situation where probing is needed, but it can also scale to high-dimensional systems. The method is demonstrated to be able to learn a probing technique that reduces the uncertainty of the hyperstate, resulting in improved control performance.

Assistant decision-making model for colour matching of outer packing

In order to solve the problem of low matching accuracy in traditional methods, a decision-making model of colour matching of outer packaging is designed. The support vector machine algorithm is used to determine the finite dataset in two-dimensional space, and the similar data is fused to extract the colour matching data feature of the product outer packaging. The state characteristics and probability distribution of feature data are determined by dynamic Bayesian network, and the data features are preprocessed. First particle swarm optimisation algorithm is used to determine the particle velocity and the best position, then the optimal scheme of colour matching of the outer packaging is obtained, and finally, the auxiliary decision model of the colour matching of the outer packaging is completed. Experimental results show that the proposed method can effectively improve the matching accuracy of the colour matching of the outer packaging, with the highest of 95%.

Global-to-Local Design for Self-Organized Task Allocation in Swarms

Programming robot swarms is hard because system requirements are formulated at the swarm level (i.e., globally) while control rules need to be coded at the individual robot level (i.e., locally). Connecting global to local levels or vice versa through mathematical modeling to predict the system behavior is generally assumed to be the grand challenge of swarm robotics. We propose to approach this problem by programming directly at the swarm level. Key to this solution is the use of heterogeneous swarms that combine appropriate subsets of agents whose hard-coded agent behaviors have known global effects. Our novel global-to-local design methodology allows to compose heterogeneous swarms for the example application of self-organized task allocation. We define a large but finite number of local agent controllers and focus on the global dynamics of behaviorally heterogeneous swarms. The user inputs the desired global task allocation for the swarm as a stationary probability distribution of agents allocated over tasks. We provide a generic method that implements the desired swarm behavior by mathematically deriving appropriate compositions of heterogeneous swarms that approximate these global user requirements. We investigate our methodology over several task allocation scenarios and validate our results with multiagent simulations. The proposed global-to-local design methodology is not limited to task allocation problems and can pave the way to formal approaches to design other swarm behaviors.

State Prediction of Human-in-the-Loop Multi-rotor System with Stochastic Human Behavior Model

Reachability analysis is a widely used method to analyze the safety of a Human-in-the-Loop Cyber Physical System (HiLCPS). It allows the HiLCPS to respond against an imminent threat in advance by predicting reachable states of the system. However, it could lead to an unnecessarily conservative reachable set if the prediction only relies on the system dynamics without explicitly considering human behavior, and thus the risk might be overestimated. To avoid the conservativeness, we present a state probability distribution function (pdf) prediction method which takes into account a stochastic human behavior model represented as a Gaussian Mixture Model (GMM). In this paper, we focus on the multi-rotor controlled by a human operator in a near-collision situation. The stochastic human behavior model is trained using experimental data to represent the human operators’ evasive maneuver. Then, we can retrieve a human control input pdf from the trained stochastic human behavior model using the Gaussian Mixture Regression (GMR). The proposed algorithm predicts the multi-rotor's future state pdf by propagating the pdf of the retrieved human control input according to the given dynamics, which yields closed-loop analysis of the HiLCPS. Human subject experiment results are provided to demonstrate the effectiveness of the proposed algorithm.

Stochastic Modeling in Stock Market Trend Prediction

n Modern centuries a lot of predicting techniques take been proposed and applied for the stock market movement prediction. In this paper, the pattern examinations of the financial exchange forecast are introduced by utilizing Hidden Markov Model with the one day distinction in close incentive for a particular period. The likelihood esteems π gives the pattern level of the stock costs which is determined for all the notice arrangement and stowed away successions. It supports for decision makers to make decisions in case of indecision on the basis of the proportion of probability values found from the steady state probability distribution.

Quantum information approach to high energy interactions.

High energy hadron interactions are commonly described by using a probabilistic parton model that ignores quantum entanglement present in the light-cone wave functions. Here, we argue that since a high energy interaction samples an instant snapshot of the hadron wave function, the phases of different Fock state wave functions cannot be measured-therefore the light-cone density matrix has to be traced over these unobservable phases. Performing this trace with the corresponding [Formula: see text] Haar integration measure leads to 'Haar scrambling' of the density matrix, and to the emergence of entanglement entropy. This entanglement entropy is determined by the Fock state probability distribution, and is thus directly related to the parton structure functions. As proposed earlier, at large rapidity [Formula: see text] the hadron state becomes maximally entangled, and the entanglement entropy is [Formula: see text] according to QCD evolution equations. When the phases of Fock state components are controlled, for example in spin asymmetry measurements, the Haar average cannot be performed, and the probabilistic parton description breaks down. This article is part of the theme issue 'Quantum technologies in particle physics'.

Research and application of stochastic resonance in quad-stable potential system

To solve the problem of low weak signal enhancement performance in the quad-stable system, a new quad-stable potential stochastic resonance (QSR) is proposed. Firstly, under the condition of adiabatic approximation theory, the stationary probability distribution (SPD), the mean first passage time (MFPT), the work (W), and the power spectrum amplification factor (SAF) are derived, and the impacts of system parameters on them are also extensively analyzed. Secondly, numerical simulations are performed to compare QSR with the classical Tri-stable stochastic resonance (CTSR) by using the genetic algorithm (GA) and the fourth-order Runge–Kutta algorithm. It shows that the signal-to-noise ratio (SNR) and mean signal-to-noise increase (MSNRI) of QSR are higher than CTSR, which indicates that QSR has superior noise immunity than CTSR. Finally, the two systems are applied in the detection of real bearing faults. The experimental results show that QSR is superior to CTSR, which provides a better theoretical significance and reference value for practical engineering application.

METHOD OF IDENTIFICATION OF OBJECT STATES ACCORDING TO THE RESULTS OF FUZZY MEASUREMENTS OF CONTROLLED PARAMETERS

The subject of consideration is the task of identifying the states of an object based on the results of fuzzy measurements of a set of controlled parameters. The fuzziness of the initial data of the task further complicates it due to the resulting inequality of the controlled parameters. The aim of the study is to develop a method of identifying the states of a fuzzy object using a fuzzy mechanism of logical output taking into account possible differences in the level of information content of its controlled parameters. The method of obtaining the desired result is based on the modification of the known mathematical apparatus for building an expert system of artificial intelligence by solving two subtasks. The first is the development of a method for assessing the informativity of controlled parameters. The second is the development of a method for constructing a mechanism for logical inference of the relative state of an object based on the results of measuring controlled parameters, which provides identification. In the first problem, a method is proposed for estimating the informativity of parameters, free from the known disadvantages of the traditional Kulbak informativity measure. In implementing the method, it is assumed that the range of possible values for each parameter is divided into subbands in accordance with possible states of the object. For each of these states, the function of belonging to the fuzzy values of the corresponding parameter is defined. At the same time, the correct problem of estimating the informativity of a parameter is solved for cases when this parameter is measured accurately or determined fuzzily by its belonging function. The fundamental difference between the proposed logical output mechanism and the traditional one is the refusal to use the production rule base, which ensures the practical independence of the computational procedure from the dimension of the task. To solve the main problem of identifying states, a non-productive approach is proposed, the computational complexity of which practically does not depend on the dimension of the problem (the product of the number of possible states Results.per the number of controlled parameters). The logic output mechanism generates a probability distribution of the system states. In this case, a set of functions of belonging of each parameter to the range of its possible values for each of the states of the object is used, as well as a set of functions of belonging to fuzzy measurement results of each parameter. Conclusions. Thus, a method of identifying the state of fuzzy objects with a fuzzy non-productive output mechanism is proposed, the complexity of which does not depend on the dimension of the task.

Covariance Steering of Discrete-Time Stochastic Linear Systems Based on Wasserstein Distance Terminal Cost

We consider a class of stochastic optimal control problems for discrete-time linear systems whose objective is the characterization of control policies that will steer the probability distribution of the terminal state of the system close to a desired Gaussian distribution. In our problem formulation, the closeness between the terminal state distribution and the desired (goal) distribution is measured in terms of the squared Wasserstein distance which is associated with a corresponding terminal cost term. We recast the stochastic optimal control problem as a finite-dimensional nonlinear program whose performance index can be expressed as the difference of two convex functions. This representation of the performance index allows us to find local minimizers of the original nonlinear program via the so-called convex-concave procedure [1]. Finally, we present non-trivial numerical simulations to demonstrate the efficacy of the proposed technique by comparing it with sequential quadratic programming methods in terms of computation time.

Probabilistic Seismic Demand Models and Life-Cycle Fragility Estimates for High-Rise Buildings

This paper presents a probabilistic framework for estimating the time-variant seismic fragility of high-rise buildings under carbonation-induced corrosion with a multiple damage model. A 42-story steel frame RC core tube (SFRCT) building was selected as a representative high-rise building to illustrate the proposed method. The main novelties of this study are that (1) a Bayesian updating method is used to determine the posterior probability distributions of the unknown parameters in the time-varying demand models of the maximum interstory drift and the residual drift at the top floor, (2) an analytical derivation method is used to compute the probability distribution of the corrosion initiation time in reinforcements, (3) the uncertainties associated with the corrosion initiation time and unknown parameters in demand models are incorporated into fragility estimates based on total probability theory, and (4) the elliptical copula and Archimedean copula are used to develop bivariate fragility estimates that consider the correlations among the limit states associated with maximum drifts and residual drifts, respectively. The results indicate that the seismic demand and fragility estimates of this high-rise building increase as the corrosion initiation time, which highlights the necessity of accounting for the uncertainty in the corrosion initiation time. In addition, the Gaussian copula and the Clayton copula (special cases of the elliptical copula and the Archimedean copula, respectively) produce similar joint probability distributions for the two-dimensional limit states; however, the Clayton copula can be used in a more general condition. The proposed methodology is applicable for evaluating the seismic life-cycle fragility of both existing and new high-rise buildings under multicriteria performance indicators.

Statistical Analysis of Cropland Area in Canada using the Autoregressive Hidden Markov Time Series Model

Crop production and other agricultural activities are as old as human existence and becoming increasingly intensive, spatially concentrated and specialized. However, diversification in economic activities and recent development in technology in many developed countries have led to significant increase in land use. Thereby, resulting to huge reduction in the total land area available for agricultural activities especially crop production. This study examines the distribution of cropland area in Canada in relation to three contributing factors using the Autoregressive Hidden Markov time series Model (AR-HMM) due to the limitations of the ordinary Autoregressive model in the accuracy of its parameter estimation. Expectation-Maximization (E-M) algorithm method was used to estimate the model parameters so as to investigate the effects of the factors on cropland distribution using secondary data from Food and Agriculture Organisation (FAO). Jarque-Bera and D'Agostino normality tests were carried out to examine the normality of the series. Augmented Dickey Fuller (ADF) and the KPSS tests established the stationarity of the series. The ideal stationary probability distribution for transition was at AR (3)-HMM with the minimum Bayesian Information Criterion (BIC) of 16270.62. The prior transition states for the HMM are 0.462, 0.260 and 0.278 respectively. In conclusion, this study suggests that deforestation and other land use activities as a result of commercial and technological advancements should be minimized to ensure more available cropland area.

Reliability modelling for linear and circular [formula omitted]-out-of-[formula omitted]: F systems with shared components

Reliability analysis for k-out-of-n systems has received much attention in the field of reliability due to its practical importance. In this work, models for linear and circular k-out-of-n: F systems with shared components, consisting of m subsystems, are discussed. The finite Markov chain imbedding approach cannot be used directly to derive reliability of the considered systems because of the common area existing between adjacent subsystems. For this reason, we first derive the probability that the number of failed components is a given value in the m common areas. Then, all disjoint cases, each of which is regarded as a series system consisting of m subsystems with different initial state probability distributions, are obtained, with which the nith step transition rate matrix is calculated by using the Markov chain technique, for i=1,2,…,m. Finally, the reliability of the system is obtained by summing the reliabilities of all cases along with the finite Markov chain imbedding approach. Some numerical examples are presented to illustrate the efficiency of the proposed reliability evaluation method.

Study of invariance of nonextensive statistics under the uniform energy spectrum translation

The general formalisms of the q-dual statistics, the Boltzmann–Gibbs statistics, and three versions of the Tsallis statistics known as Tsallis-1, Tsallis-2, and Tsallis-3 statistics have been considered in the canonical ensemble. We have rigorously proved that the probability distribution of the Tsallis-1 statistics is invariant under the uniform energy spectrum translation at a fixed temperature. This invariance demonstrates that the formalism of the Tsallis-1 statistics is consistent with the fundamentals of the equilibrium statistical mechanics. The same results we have obtained for the probability distributions of the Tsallis-3 statistics, Boltzmann–Gibbs statistics, and q-dual statistics. However, we have found that the probability distribution of the Tsallis-2 statistics, the expectation values of which are not consistent with the normalization condition of probabilities, is indeed not invariant under the overall shift in energy as expected.

Steady state of overdamped particles in the non-conservative force field of a simple non-linear model of optical trap

Optically trapped particles are often subject to a non-conservative scattering force arising from radiation pressure. In this paper, we present an exact solution for the steady state statistics of an overdamped Brownian particle subjected to a commonly used force field model for an optical trap. The model is the simplest of its kind that takes into account non-conservative forces. In particular, we present the exact results for certain marginals of the full three-dimensional steady state probability distribution, in addition to results for the toroidal probability currents that are present in the steady state, as well as for the circulation of these currents. Our analytical results are confirmed by numerical solution of the steady state Fokker–Planck equation.

Optimal inspection and maintenance planning for deteriorating structural components through dynamic Bayesian networks and Markov decision processes

Civil and maritime engineering systems, among others, from bridges to offshore platforms and wind turbines, must be efficiently managed, as they are exposed to deterioration mechanisms throughout their operational life, such as fatigue and/or corrosion. Identifying optimal inspection and maintenance policies demands the solution of a complex sequential decision-making problem under uncertainty, with the main objective of efficiently controlling the risk associated with structural failures. Addressing this complexity, risk-based inspection planning methodologies, supported often by dynamic Bayesian networks, evaluate a set of pre-defined heuristic decision rules to reasonably simplify the decision problem. However, the resulting policies may be compromised by the limited space considered in the definition of the decision rules. Avoiding this limitation, Partially Observable Markov Decision Processes (POMDPs) provide a principled mathematical methodology for stochastic optimal control under uncertain action outcomes and observations, in which the optimal actions are prescribed as a function of the entire, dynamically updated, state probability distribution. In this paper, we combine dynamic Bayesian networks with POMDPs in a joint framework for optimal inspection and maintenance planning, and we provide the relevant formulation for developing both infinite and finite horizon POMDPs in a structural reliability context. The proposed methodology is implemented and tested for the case of a structural component subject to fatigue deterioration, demonstrating the capability of state-of-the-art point-based POMDP solvers of solving the underlying planning stochastic optimization problem. Within the numerical experiments, POMDP and heuristic-based policies are thoroughly compared, and results showcase that POMDPs achieve substantially lower costs as compared to their counterparts, even for traditional problem settings.

Construction and Application of Media Corpus Based on Big Data

Aiming at the problems of poor data quality and low application rate in the construction of existing media corpus, this paper proposes the construction and application research of media corpus based on big data. Media corpus data are collected, the data are divided into four categories, the heuristic data item column sorting algorithm is introduced to sort all collection processes, the minimum value of data item collection rate is determined, on this basis, the maximum value of quantity in media corpus is determined, and data collection is realized in media corpus data through sliding window. Then, the state characteristics and probability distribution of feature data are determined by dynamic Bayesian network, the relationship between the state variables and dimensions of media corpus data is determined, and the media corpus data state is processed by component to complete the preprocessing of media corpus data; finally, through the application research of storage and encryption of the designed database through big data technology, the storage structure data and encryption secret key are designed to realize the construction and application of media corpus. The experimental results show that the data quality of the media corpus constructed by the proposed method is high, and its application rate has been improved to a certain extent.