Chance Theory Research Articles

Uncertain random programming models in the framework of U-S chance theory and their applications

Uncertain random programming models in the framework of U-S chance theory and their applications

Uncertain random variables and laws of large numbers under U-C chance space

Uncertainty theory was founded by Baoding Liu for modeling belief degrees about human uncertainty. And chance theory was pioneered by Yuhan Liu based on probability theory and uncertainty theory. In many cases, imprecise risk situation in subjective or objective setting and human uncertainty simultaneously appear in a complex system. To measure uncertain random events in this system, this paper combines uncertainty space with convex non-additive probability space into a new kind of two-dimensional chance space called U-C chance space. Moreover, a new framework for uncertain random variables combining uncertainty theory with convex non-additive probability theory is provided. For applications of this new framework, it can be applied to characterize some kinds of phenomena which possess the characteristics of both imprecise risk situations and belief degrees about human uncertainty in situations of great events, such as financial crisis, major natural disaster, etc. The main contribution of this paper is to derive the types of Kolmogorov and Marcinkiewicz-Zygmund LLNs for uncertain random variables satisfying some conditions under U-C chance space, where the convex non-additive probability is totally monotone. Furthermore, several examples are stated and explained.

A Redundancy Allocation Model for Uncertainty Random Water Supply System in Water Saving Management Contract Projects

Water saving management contract (WSMC) projects provide advanced technology and management for a water supply system to achieve water conservation and set redundant components to ensure water supply reliability. Project managers focus on the reliability optimization problem and require redundancy allocation strategies of the above system. This paper presents an optimization method by dealing with the lifetime of the whole water supply system. Assuming the lifetimes of advanced components are uncertain variables and the old ones are random variables, a reliability optimization model of water supply systems is established based on chance theory, and the redundancy allocation solutions are obtained by an optimization toolkit. A WSMC case in Shenzhen, China is studied and the results show that the reliability of the water supply system has been in a high state based on the allocation strategy. This study provides theoretical support for improving water-saving safety and popularizing the WSMC service mechanism.

Uncertain stochastic hybrid zero-sum games based on forward uncertain difference equations and backward stochastic difference equations

We investigate the interplay between forward uncertain difference equations and backward stochastic difference equations, which belong to distinct mathematical frameworks: uncertainty theory and probability theory, respectively. By establishing the connections between uncertain expectation, probabilistic expectation, and chance expectation, we develop the chance theory framework for analyzing uncertain stochastic hybrid zero-sum games constrained by both forward uncertain difference equations and backward stochastic difference equations. Through an equivalent transformation, we convert these games into deterministic difference equation solving problems. By backward-solving the difference equations and applying mathematical induction, analytical expressions for the equilibrium solutions of two types of uncertain stochastic hybrid zero-sum games are obtained. Furthermore, we explore solution methods for zero-sum games with different operational orders of uncertain and probabilistic expectations and highlight the fundamental differences through numerical examples.

Optimal control and zero‐sum game subject to differential equations with Liu processes and random matrices

AbstractThis paper presents a differential equation including both random matrices and a Liu process. Then we demonstrate that the solution to this equation exists and is unique. Under the framework of chance theory, problems of optimal control and two‐person zero‐sum game subject to differential equations are considered. An equation of optimality is provided for solving a problem of optimal control. Then equilibrium equations are proposed to identify the saddle‐point of a two‐person zero‐sum game problem. As an extension, we generalize the obtained results to the problems subject to differential equations including both random matrices and multiple Liu processes. Finally, we utilize the acquired theoretical results to analyze a portfolio selection game problem.

The Role of Pre-Socratics in Ṣadrā's Philosophy

Abstract: This article analyzes Ṣadrā's (d. 1045/1635–1636) use of historical sources on pre-Socratics in his discussions devoted to the two correlated problems of the origination of the world and the theory of chance. His overall attitude to pre-Socratics is that of an Illuminationist philosopher. This is facilitated by the fact that the sources he consults present Neoplatonic interpretations of a number of pre-Socratic views that he finds easy to reconcile with his own doctrines. In his discussion on the material world's origination, Ṣadrā sees them as precursors of his own doctrine of substantial motion. However, when it comes to chance Ṣadrā has to address Ibn Sina's refutation of the views of Democritus and Empedocles. Ṣadrā's discussion is based on Rāzī's summary treatment of Ibn Sina's critique but intertwined with assertions (which trace back to Suhrawardī) that pre-Socratic views were misunderstood or even fabricated.

Uncertain random accelerated degradation modelling and statistical analysis with aleatory and epistemic uncertainties from multiple dimensions

Accelerated degradation testing (ADT) is commonly employed for degradation analysis and reliability evaluations. Due to limitations in practical ADTs, aleatory and epistemic uncertainties are simultaneously embodied in time, unit, and stress dimensions but they are not considered comprehensively and integrated scientifically currently. To address such problems, we focus on the most common situation in ADTs that test time is relatively long (reflecting aleatory uncertainties in time dimension) while the numbers of test units and stress levels are quite small (reflecting epistemic uncertainties in unit and stress dimensions), and establish an uncertain random accelerated degradation model (URADM). In the URADM, aleatory and epistemic uncertainties are quantified using probability theory and uncertainty theory, respectively, and they are integrated using chance theory. Then, first passage time and reliability evaluations are derived. Next, a three-step uncertain random statistical method is presented for parameter estimations, where aleatory and epistemic uncertainties are divided carefully and quantified separately. A simulation study and a practical case are conducted to show the effectiveness of the URADM. Results reveal that the URADM can not only predict deterministic degradation trends with high accuracy as existing methods do, but contribute degradation boundaries well covering ADT data with narrower boundaries and higher stability.

Reliability importance analysis of uncertain random [formula omitted]-out-of-[formula omitted] systems with multiple states

This paper analyzes the importance of components within a multiple-state uncertain random k-out-of-n system, which is called ur-m-k-out-of-n system for short. This system simultaneously contains randomness and uncertainty. The states of components exhibit polymorphism and are defined as uncertain random variables. The component importance analysis is both a prerequisite and a foundation for system reliability optimization. For this reason, the marginal reliability importance (MRI) and joint reliability importance (JRI) are proposed to examine the importance of an individual component or couple components of ur-m-k-out-of-n systems using chance theory. Some theorems are given to calculate MRI and JRI. Moreover, this paper gives several numerical examples to illustrate how to calculate MRI and JRI.

On the choice problem in two-dimensional Ellsberg experiment with randomness and uncertainty

ABSTRACT The classical Ellsberg experiment provides decision-makers with a choice problem in which the likelihood of winning the gambling is uncertain. In this paper, we consider a two-dimensional Ellsberg experiment in which the likelihood of winning the gambling and the income after that are both uncertain, establish a mathematical model to characterize the corresponding Ellsberg experiment and solve the choice problem based on probability theory, uncertainty theory, and chance theory.

Computing the reliability of mixed uncertain random [formula omitted]-out-of-[formula omitted] systems with multiple possible states

This paper presents an uncertain random multi-state k-out-of-n system (urm-k-out-of-n system, for short), where both random components and uncertain components coexist. States of the system and components display multiplicity and are defined as uncertain random variables. Since reliability is a crucial index to reflect system performance, this paper proves some formulas to calculate the reliability of urm-k-out-of-n systems in accordance with chance theory, which have two special cases, random systems and uncertain systems. Later, a mixed algorithm for multi-valued decision diagram (MDD) and binary search is designed to assess the reliability index. Compared to the general algorithm, it operates in significantly less time when the system becomes complex.

Mean-CVaR Portfolio Optimization Models based on Chance Theory

The indeterminacy of financial markets leads investors to face different types of security returns. Usually, security returns are assumed to be random variables when sufficient transaction data are available. If data are missing, they can be regarded as uncertain variables. However, uncertainty and randomness coexist. In this situation, chance theory is the main tool to deal with this complex phenomenon. This paper investigates the conditional value at risk (CVaR) of uncertain random variables and its application to portfolio selection. First, we define the CVaR of uncertain random variables and discuss some of its mathematical properties. Then, we propose an uncertain random simulation to approximate the CVaR. Next, we define the inverse function of the CVaR of uncertain random variables, as well as a computational procedure. As an application in finance, we establish uncertain random mean-CVaR portfolio selection models. We also perform a numerical example to illustrate the applicability of the proposed models. Finally, we numerically compare the mean-CVaR models with the mean-variance models with respect to the optimal investment strategy.

Linear quadratic control for multiple time‐delayed uncertain random systems

AbstractThis study focuses on time‐delayed linear quadratic control problems (LQCPs). Firstly, a general time‐delayed control problem for a linear time‐varying system is investigated. This problem is equivalent to an optimization problem without time delays, and corresponding recurrence equations are proposed accordingly. Furthermore, based on recurrence equations and chance theory, two different types of LQCPs are researched, where the control weighting matrices are positive definite and indefinite, respectively. Exact solutions for the optimal results of these LQCPs are provided. Finally, numerical examples are provided to demonstrate the findings.

UNPRINCIPLED

AbstractIt is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one’s credence function, when conditionalized on the chance-making facts, should coincide with the objective chance function. It is a shortcoming of a theory of chance if it implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped.

Optimal control problems subject to uncertain random discrete-time noncausal systems

Uncertain random discrete-time noncausal systems (URDTNSs) and optimal control problems (OCPs) are investigated under the framework of chance theory with chance expectation in this paper. We convert OCPs subject to URDTNSs into conditional equivalent uncertain random OCPs employing an algebraic transformation method. Then recurrence equations are proposed accordingly to transform the uncertain random OCPs into solving problems for deterministic difference equations. The OCPs having linear and quadratic objective functions along with three kinds of URDTNSs involving linear, quadratic, and cubic control terms are studied. As a generalization of the main findings, we study two types of OCPs for time-delayed URDTNSs and give a unified framework for solving such problems. Meanwhile, analytical expressions of optimal solutions for these kinds of OCPs are acquired. Furthermore, numerical examples are offered to highlight the usefulness of our results.

Study on Structural Reliability Analysis Method Based on Chance Theory

Many factors influence structural reliability in practice engineering. Some factors can be measured to obtain lots of data, but others are difficult to acquire statistical data. In view of this situation, a new structural reliability analysis method is proposed using chance theory, which is composed of probability theory and uncertainty theory and can reflect random uncertainty and cognitive uncertainty simultaneously. The performance function of a structural mechanical element is defined, and when it is a random uncertain variable, the chance distribution is established. Then the calculated method of failure measures and reliability measures for the structural mechanical element is put forward. Furthermore, considering the series system and parallel system, the performance function of the structural system is proposed, and the calculated method of failure measure is determined by theoretical proof. The results can provide a new approach to analyzing structural reliability under the uncertain circumstance of lack of statistical data.

Uncertain stochastic hybrid differential game system with V-n jumps: Saddle point equilibrium strategies and application to advertising duopoly game

This paper focuses on the saddle point equilibrium strategies for uncertain stochastic hybrid differential game systems under chance theory. We consider systems with dynamics described by both an uncertain differential equation with V-n jumps (UDEV) and a stochastic differential equation (SDE), with linear and quadratic performance index functions. Based on chance theory and the equation of optimality of uncertain stochastic hybrid optimal control model, the saddle equilibrium equation for general uncertain stochastic hybrid differential game system is established. Furthermore, three classes of differential game models are discussed by using equilibrium equation. Finally, an advertising duopoly game problem as an application is provided, the corresponding optimal profit difference value and advertising strategies are analyzed.

MATHEMATICAL MODEL FOR CALCULATION OF RECREATIONAL LOAD ON GREEN OPEN SPACE IN CITY

The article presents a description of the rank model for calculating the recreational load on the territory of green open space. The object of our research is the demand for open green spaces in the city for local residents. Our model is based on the theory of chances. Within the framework of research need imagine that all territories (including territories of location green open space and territories of location residential district) are taken as material points. This means that the internal structure of the allocated territory can be ignored within the framework of this model. The attitude of the urban population towards green spaces that are close to their place of residence is important to study in different ways. Our model includes two parts: in the first part of this model the recreational load is considered as a mathematical expectation of the number of visitors; in the second part, the recreational load is considered as a Poisson random variable with a time-dependent parameter. Decision makers can use this model for substantiation of urban planning standards, for developing urban areas and for landscaping.

Hybrid uncertainty quantification of dependent competing failure process with chance theory

Dependent competing failure process (DCFP) has been widely studied in recent years. When evaluating the reliability of DCFP system, the aleatory uncertainty due to randomness and the epistemic uncertainty due to lack of data and knowledge should be considered. In this paper, the hybrid uncertainty of DCFP system is analyzed and quantified with chance theory. The uncertainty random renewal reward process and the arithmetic Liu process are used to describe the shock and degradation process. The quantification of uncertainty propagation is realized by decoupling DCFP from the physical correlation aspect. Reliability of DCFP system considering two kinds of unilateral dependencies and mutual dependency is modeled and a combinational algorithm is proposed to solve it. Finally, the DCFP effect in control system of an aircraft is studied as a case, aleatory uncertainty and epistemic uncertainty are modeled, and the system reliability is evaluated. Meanwhile, the effects of key parameters on system reliability are discussed. Results show that the chance theory is an appropriate method to qualify the hybrid uncertainty for engineering systems when data is insufficient. The mutual dependency of soft failure and hard failure will reduce the system reliability. The smaller the parameter of hard failure, the greater the influence of mutual dependency on the results.

Uncertain random enhanced index tracking for portfolio selection with parameter estimation and hypothesis test

The enhanced index tracking is an effective method for portfolio optimization that focuses on imitating the behavior of a special benchmark and achieving an excess return. In addition, uncertainty and randomness are two intrinsic attributes of real world systems. In this paper, we study an uncertain random enhanced index tracking for portfolio optimization with parameter estimation and hypothesis test based on chance theory, which combines uncertainty theory and probability theory. First, we formulate an uncertain random mean–variance enhanced index tracking model including both random risky securities and uncertain risky securities. Then the presented model is transformed into a quadratic programming problem and the analytical solutions are derived for some special cases. Furthermore, the uncertain hypothesis test is applied to examine the correctness of the unknown parameters solved by uncertain parameter estimation. Finally, three numerical simulations are presented for showing the applicability of the formulated models and the effectiveness of the enhanced index tracking strategies.

Regularity Index of Uncertain Random Graph

A graph containing some edges with probability measures and other edges with uncertain measures is referred to as an uncertain random graph. Numerous real-world problems in social networks and transportation networks can be boiled down to optimization problems in uncertain random graphs. Actually, information in optimization problems in uncertain random graphs is always asymmetric. Regularization is a common optimization problem in graph theory, and the regularity index is a fundamentally measurable indicator of graphs. Therefore, this paper investigates the regularity index of an uncertain random graph within the framework of chance theory and information asymmetry theory. The concepts of k-regularity index and regularity index of the uncertain random graph are first presented on the basis of the chance theory. Then, in order to compute the k-regularity index and the regularity index of the uncertain random graph, a simple and straightforward calculating approach is presented and discussed. Furthermore, we discuss the relationship between the regularity index and the k-regularity index of the uncertain random graph. Additionally, an adjacency matrix-based algorithm that can compute the k-regularity index of the uncertain random graph is provided. Some specific examples are given to illustrate the proposed method and algorithm. Finally, we conclude by highlighting some potential applications of uncertain random graphs in social networks and transportation networks, as well as the future vision of its combination with symmetry.