Uncertain Random Variables Research Articles

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.

Arc entropy of uncertain random variables and its applications

Due to the complexity of the real world, randomness and uncertainty are ubiquitous and interconnected in the real world. In order to measure the research objects that contain both randomness and uncertainty in practical problems, and extend the entropy theory of uncertain random variables, this paper introduces the arc entropy of uncertain random variables and the arc entropy of their functions. On this basis, the mathematical properties of arc entropy and two key formulas for calculating arc entropy are also studied and derived. Finally, two types of the mean variance entropy model with the risk and diversification are established, and the corresponding applications to rare book selection for the rare book market are also introduced.

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.

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.

Partial Gini Coefficient for Uncertain Random Variables with Application to Portfolio Selection

The partial Gini coefficient measures the strength of dispersion for uncertain random variables, while controlling for the effects of all random variables. Similarly to variance, the partial Gini coefficient plays an important role in uncertain random portfolio selection problems, as a risk measure to find the optimal proportions for securities. We first define the partial Gini coefficient as a risk measure in uncertain random environments. Then, we obtain a computational formula for computing the partial Gini coefficient of uncertain random variables. Moreover, we apply the partial Gini coefficient to characterize risk of investment and investigate a mean-partial Gini model with uncertain random returns. To display the performance of the mean-partial Gini portfolio selection model, some computational examples are provided. To compare the mean-partial Gini model with the traditional mean-variance model using performance ratio and diversification indices, we apply Wilcoxon non-parametric tests for related samples.

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.

Further results on laws of large numbers for uncertain random variables

Further results on laws of large numbers for uncertain random variables

Reliability analysis of rock slopes considering the uncertainty of joint spatial distributions

To reasonably and effectively evaluate the stability of jointed rock slopes, this study proposes a reliability analysis method that considers the uncertainty of the joint spatial distribution. First, the discrete fracture network (DFN) model is used to characterize the variability of joint geometric parameters, and a random variable of joint location is assumed to be a random integer that follows a uniform distribution in the random number library. Then, Latin hypercube sampling is used to sample uncertain random variables to generate random numbers, and construct the corresponding DFN model. Finally, the safety factor (Fs) of each numerical model under sampling is calculated by strength reduction method, and the failure probability of jointed rock slope is obtained by statistical analysis of Fs. The comparative analysis results demonstrate that considering the uncertainty of joint spatial distribution is crucial for accurately analyzing the reliability of rock slopes. This method comprehensively considers the main internal factors that affect the reliability of jointed rock slopes, including the uncertainty of rock strength parameters and variability of geometrical parameter and the location randomness of joints, making the results of the evaluation of the reliability of rock slopes more precise and reasonable.

Renyi entropy of uncertain random variables and its application to portfolio selection

Renyi entropy of uncertain random variables and its application to portfolio selection

Reliability analysis of uncertain random systems based on uncertain differential equation

Typical failure modes of degradation and shock processes have been applied which are named as soft failures and hard failures, respectively. This paper discusses the reliability of uncertain random systems where randomness is regarded as objective indeterminacy with enough sample data, and uncertainty is referred to epistemic indeterminacy with insufficient sample data. Considering that the combination of internal and external factors, the degradation process is modelled by an uncertain differential equation, and external shocks are driven by an uncertain random renewal process where state variables are uncertain random variables. Chance measure is applied to define the reliability index which is distinguished from traditional reliability assessment methods that utilize probability measure. Three types of reliability models possessed with independent failures are presented, where the general reliability index formulas are provided. The analysis results of Caohai Nature Reserve show that uncertain random modelling method provides an effective approach to system reliability assessment.

Uncertain random portfolio selection with different mental accounts based on mixed data

Due to the complexity of security markets, securities with massive effective data, invalid data and insufficient data may exist at the same time. When there are massive effective data, the security returns are regarded as random variables, or they can be regarded as uncertain variables when the data are insufficient or invalid. The common portfolio selection problem assumes that the investors put money into one mental account for management. Considering that the investors invest money in separate accounts in reality, this paper discusses a portfolio selection problem with different mental accounts under uncertain random environment. Firstly, we formulate an uncertain random model for the portfolio optimization problem with random risky securities and uncertain risky securities. The chance distributions of uncertain random variables are derived when the variables obey different distributions. On this basis, two equivalent forms of the uncertain random portfolio model based on mental accounts are presented. Finally, numerical simulations with two and three mental accounts are conducted to analyze the reality and practicability of the established models.

Uncertain random portfolio selection with high order moments

<p style='text-indent:20px;'>In financial phenomena, investors are confronted with a hybrid uncertainty, where random and uncertain returns can occur simultaneously. This paper studies the kurtosis of an uncertain random variable and its application in portfolio selection. We are aware that the kurtosis measures the extreme values in each tail of the distribution. However, the notion of kurtosis for uncertain random variables has not been properly established. Thus, we introduce the concept of kurtosis of an uncertain random variable and extract some significant properties. Next, by invoking chance distribution we derive the kurtosis of three particular uncertain random variables, as well as the variance. Then, we present an uncertain random mean-variance-skewness-kurtosis portfolio selection model with their corresponding variations to meet various investors' requirements. Finally, two numerical examples are presented along with a comparative study to explain and illustrate the main results.</p>

Semi entropy of uncertain random variables and its application to portfolio selection

Semi entropy of uncertain random variables and its application to portfolio selection

Portfolio Optimization of Uncertain Random Returns based on Partial Exponential Entropy

Partial entropy is a device to measure uncertainty of an uncertain random variable. In order to characterize indeterminacy of uncertain random variables, the concept of partial exponential entropy for uncertain random variables is presented. By invoking inverse uncertainty distribution, we derive a formula for computing partial exponential entropy for uncertain random variables. As an application of partial exponential entropy, portfolio selection problems for uncertain random returns are optimized via partial entropy-mean models. For better understanding, several examples are provided.

On Some Laws of Large Numbers for Uncertain Random Variables

Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.

Sine Entropy of Uncertain Random Variables

Entropy is usually used to measure the uncertainty of uncertain random variables. It has been defined by logarithmic entropy with chance theory. However, this logarithmic entropy sometimes fails to measure the uncertainty of some uncertain random variables. In order to solve this problem, this paper proposes two types of entropy for uncertain random variables: sine entropy and partial sine entropy, and studies some of their properties. Some important properties of sine entropy and partial sine entropy, such as translation invariance and positive linearity, are obtained. In addition, the calculation formulas of sine entropy and partial sine entropy of uncertain random variables are given.

Cross Entropy Chance Distribution Model of Uncertain Random Shortest Path Problem

The shortest path problem (SPP) is one of the most typical and basic optimization problems in network theory for decades, and it covers a series of practical application problems, such as urban planning, logistics transportation, engineering and power grid strain analysis, etc. The circumstance where the weight of arcs in a network contains both randomness and uncertainty is considered, and the case of the weights of arcs with uncertain random variables is focused on in this paper. Here, we introduced a new model of the SPP which is based on the new definition of uncertain random variables cross entropy, and the newly established model can be used to find the path with the closest chance distribution to the ideal SP. The efficiency of this model is also evaluated in the final part.

Portfolio optimization in real financial markets with both uncertainty and randomness

Financial market is a complex system full of unknown and indeterminacy. It is well known that uncertainty and randomness are two basic types of indeterminacy. Hence, the complexity of real financial markets may lead to various types of security returns. They are usually assumed as random variables when there are enough historical data. If there is a lack of available data, they can be considered as uncertain variables. However, uncertainty and randomness often exist simultaneously. In this paper, we consider a portfolio optimization problem in real financial markets with both uncertainty and randomness. First, the skewnesses for three kinds of uncertain random variables are derived. Then, in an uncertain random environment, considering different risk preferences, a mean-variance-skewness model for the portfolio optimization problem is proposed. In addition, we use the normalization method to eliminate the impact of investment returns and risks of different units. Finally, numerical simulations are carried out to show that the proposed model is realistic and applicable.

Tsallis entropy of uncertain random variables and its application

Tsallis entropy is a flexible extension of Shanon (logarithm) entropy. Since entropy measures indeterminacy of an uncertain random variable, this paper proposes the concept of partial Tsallis entropy for uncertain random variables as a flexible devise in chance theory. An approach for calculating partial Tsallis entropy for uncertain random variables, based on Monte Carlo simulation, is provided. As an application in finance, partial Tsallis entropy is invoked to optimize portfolio selection of uncertain random returns via crow search algorithm.

New approach for uncertain random multi-objective programming problems based on CESD criterion

To overcome the defects that the traditional approach for multi-objective programming under uncertain random environment (URMOP) neglects the randomness and uncertainty of the problem and the volatility of the results, a new approach is proposed based on expected value-standard deviation value criterion (C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ESD</inf> criterion). Firstly, the effective solution to the URMOP problem is defined; then, by applying sequence relationship between the uncertain random variables, the UR-MOP problem is transformed into a single-objective programming (SOP) under uncertain random environment (URSOP), which are transformed into a deterministic counterpart based on the C <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ESD</inf> criterion. Then the validity of the new approach is proved that the optimal solution to the SOP problem is also efficient for the URMOP problem; finally, a numerical example and a case application are presented to show the effectiveness of the new approach.