Incomplete Probability Information Research Articles

Testing out-of-sample portfolio performance using second-order stochastic dominance constrained optimization approach

Second-order Stochastic Dominance (SSD) criterion can be used to support portfolio decision making under risk and uncertainty. In this paper, we develop novel robust SSD criteria to capture the strength of dominance and portfolio optimization models utilizing these criteria to identify portfolios whose in-sample SSD dominance over a given benchmark is likely to hold also out-of-sample. The developed models can incorporate incomplete probability information by allowing a set of feasible state probabilities. We also show that these portfolio optimization models can be formulated as linear programming problems. We report results from applying these SSD-based portfolio optimization models with different sets of state probabilities in an empirical application, with a focus on evaluating the out-of-sample portfolio performance of the optimized portfolios.

Plastic deformation-based rolling bearing reliability and sensitivity analysis under incomplete probability information

It is currently difficult to adequately characterize the probability distribution information of rolling bearings, and research on the reliability of rolling bearings based on plastic deformation is lacking. To address this problem, the Latin hypercube design, BP neural networks, and higher-order moment method are combined, fully considering the randomness of the load, geometry, and material parameters of the rolling bearing. The dynamics model of the rolling bearing was constructed by using the Hertz contact theory and Jones model, combined with a Latin hypercube design to obtain neural network training samples. Based on the rational construction of the neural network structure, the mapping relationship between the equivalent stress and the design variables is obtained. The state function of the rolling bearing is constructed according to the stress–strength interference model, and then, the reliability index and reliability of the rolling bearing are obtained using the higher-order moment method, followed by a reliability sensitivity analysis. Finally, the proposed method is applied to the reliability analysis of a certain type of angular contact ball bearing, and the calculation results are compared with the Monte Carlo method results to demonstrate the correctness and effectiveness of the proposed method.

The Choquet integral-based Shapley function for n-person cooperative games with probabilistic hesitant fuzzy coalitions

The use of probabilistic hesitant fuzzy elements (PHFEs) to represent the alliance status in cooperative games can describe the player's participation levels and associated probability. This paper proposes an effective Shapley function for probabilistic hesitant fuzzy cooperative games based on generalized Choquet integral. We first define the probabilistic hesitant fuzzy coalition of cooperative games. After that, a supplementary model of incomplete probability information is established. Without losing any probability information, an adjustment algorithm is developed to derive the consistent probability of coalitions. Then the characteristic function of probabilistic hesitant fuzzy cooperative games is constructed based on generalized Choquet integral. The Choquet integral-based Shapley function is presented according to the proposed characteristic function. Some desirable properties of Shapley function are discussed in detail. The CIG algorithm is established for generating consistent imputation of cooperative games. Finally, the proposed algorithm is applied to the cooperative production of alloy materials.

Copula-based methods for global sensitivity analysis with correlated random variables and stochastic processes under incomplete probability information

Global sensitivity analysis (GSA) plays an important role in uncertainty analysis and quantification. Conventional GSA for structures requires tackling two main challenges: (1) the incomplete probability information of inputs and (2) the effects caused by the static/dynamic correlation of random variables or stochastic processes. In this paper, two kinds of novel copula-based methods for variance-based GSA are proposed to address these challenges. Based on the known samples, the proposed methods can choose the optimal copula function to construct the joint distribution of inputs, and compute the global sensitivity indices combined with Monte Carlo (MC) simulation. Time-variant copula function is used to generate the samples of time series which are both auto-correlated and cross-correlated, and the proposed methods are extended to develop time-variant GSA of dynamic structures with correlated random variables and stochastic processes. Four engineering examples are given to illustrate the good applicability and capability of the proposed methods for the dependent model functions under incomplete probability information.

A Vine Copula-Based Global Sensitivity Analysis Method for Structures with Multidimensional Dependent Variables

For multidimensional dependent cases with incomplete probability information of random variables, global sensitivity analysis (GSA) theory is not yet mature. The joint probability density function (PDF) of multidimensional variables is usually unknown, meaning that the samples of multivariate variables cannot be easily obtained. Vine copula can decompose the joint PDF of multidimensional variables into the continuous product of marginal PDF and several bivariate copula functions. Based on Vine copula, multidimensional dependent problems can be transformed into two-dimensional dependent problems. A novel Vine copula-based approach for analyzing variance-based sensitivity measures is proposed, which can estimate the main and total sensitivity indices of dependent input variables. Five considered test cases and engineering examples show that the proposed methods are accurate and applicable.

Probabilistic Linguistic Z Number Decision-Making Method for Multiple Attribute Group Decision-Making Problems with Heterogeneous Relationships and Incomplete Probability Information

Probabilistic Linguistic Z Number Decision-Making Method for Multiple Attribute Group Decision-Making Problems with Heterogeneous Relationships and Incomplete Probability Information

Time-dependent reliability and optimal design of scraper chains based on fretting wear process

PurposeThis paper aims to present a dynamic reliability model of scraper chains based on the fretting wear process and propose a reasonable structural optimization method.Design/methodology/approachFirst, the dynamic tension of the scraper chain is modeled by considering the polygon effect of the scraper conveyor. Then, the numerical wear model of the scraper chain is established based on the tangential and radial fretting wear modes. The scraper chain wear process is introduced based on the diameter wear rate. Furthermore, the time-dependent reliability of scraper chains based on the fretting wear process is addressed by the third-moment saddlepoint approximation (TMSA) method. Finally, the scraper chain is optimized based on the reliability optimization design theory.FindingsThere is a correlation between the wear and the dynamic tension of the scraper conveyor. The unit sliding distance of fretting wear is affected by the dynamic tension of the scraper conveyor. The reliability estimation of the scraper chain with incomplete probability information is achieved by using the TMSA for the method needs only the first three statistical moments of the state variable. From the perspective of the chain drive system, the reliability-based optimal design of the scraper chain can effectively extend its service life and reduce its linear density.Originality/valueThe innovation of the work is that the physical model of the scraper chain wear is established based on the dynamic analysis of the scraper conveyor. And based on the physical model of wear, the time-dependent reliability and optimal design of scraper chains are carried out.

Reliability Analysis of 3D Rockfill Dam Slope Stability Based on the Copula Function

Abstract The evaluation of slope stability reliability under the condition of incomplete probability information is one of the most popular issues in geotechnical engineering. The key difficulty in...

Sealing reliability and reliability sensitivity analysis of the piston–cylinder bore friction pair in a hydraulic pump under incomplete probability information

Incomplete probability information makes it impossible to determine the probability distribution of a random variable accurately. Such a lack of probability information is manifested by the fact that the reliability index of engineering products can only be estimated qualitatively and approximately based on the readily available data. In this paper, the sealing reliability of the piston–cylinder bore friction pair in a hydraulic pump is designed and its sensitivity is analyzed under incomplete probability information. A reliability design model that incorporates both the motion and load effects is established. A method for designing the sealing reliability and for analyzing its sensitivity under incomplete probability information is proposed for the piston–cylinder bore friction pair, and the trends of the change in the reliability with the design parameters are determined by a simulation. The influence of the changes in the design parameters on the reliability is clarified. The reliability design problem of the piston–cylinder bore friction pair under the dual effects of motion and load is solved using the available probability information. The reliability of the piston–cylinder bore friction pair is designed and its sensitivity is analyzed using limited probability information through a computer program.

Double-quantified linguistic variable

Regarding the natural language representation, there are two main methods, namely, the linguistic variable and multi-granularity linguistic term set. Although these two techniques are big achievements in the natural language representation researches, there are still some challenges: 1) the linguistic variable cannot handle the probability information, and 2) the multi-granularity linguistic term set cannot handle the negative words and incomplete probability information. To bridge these research gaps, this study proposes the double-quantified linguistic variable to quantify the natural language from two aspects: membership degree and probability. Firstly, we compare the linguistic variable with the multi-granularity linguistic term set and identify the differences and shortages of them. Based on these analyses, the double-quantified linguistic variable as a quintuple is introduced and every part of the double-quantified linguistic variable is explained in detail. After that, the basic operations and comparison method of the double-quantified linguistic variables are given. This study ends with some concluding remarks and future research directions.

Portfolio diversification based on stochastic dominance under incomplete probability information

Identifying efficient portfolio diversification strategies subject to stochastic dominance (SD) criteria usually assumes that the state-space of future asset returns can be captured by a fixed sample of equally probable historical returns. This paper relaxes this assumption by developing SD criteria under incomplete information on state probabilities. Specifically, we identify portfolios that dominate a given benchmark for any state probabilities in a given set. The proposed approach is applied to analyze if industrial diversification can be utilized to outperform the market portfolio. The results from this application demonstrate that the use of set-valued state probabilities can help to improve out-of-sample performance of SD-based portfolio optimization.

A non-parametric copula approach to dependence modelling of shear strength parameters and its implications for geotechnical reliability under incomplete probability information

This paper aims to evaluate the potential of the non-parametric copula approach to dependence modelling of shear strength parameters and address the problem of copula selection for geotechnical reliability under incomplete probability information. The optimal non-parametric copula is determined by truncating the basis functions in the kernel according to the Akaike information criterion and Bayesian information criterion. The fitting of copulas to the same dataset shows that the non-parametric approach generally gives better copula models. As the non-parametric copula maximizes its relative entropy (Kullback–Leibler divergence) with respect to the independence copula subject to given constraints, the approximated copula should be the best model according to the principle of maximum entropy. The interpretation of the incomplete probability information as constraints on the expectation of a basis function is later provided, and the non-parametric copula approach is applied to solve two geotechnical reliability problems under incomplete probability information. By measuring the relative entropy of the copulas, the best reliability result and reliability-based design under incomplete probability information can be selected. The comparison between the parametric and non-parametric copula shows that the non-parametric copula almost always encodes the minimum dependence information in the context of given probability information.

Oparta na niezawodności systemu optymalizacja konstrukcji łańcucha przenośnikowego zgrzebłowego o zależnych przyczynach uszkodzeń

With the incomplete probability information, the joint probability distribution modelling and system reliability-based optimization design of structural systems with failure interactions are challenging problems in the domain of reliability. This article is designed to propose a system reliability-based optimization design method for optimizing the scraper chain with multiple failure modes. Firstly, the common failure modes of the scraper chain are analysed. For each failure mode, a reliability model for the failure of scraper chains is obtained. Secondly, aiming at the joint failure probability modelling problem, a method for estimating the failure probability of the scraper chain based on system reliability is proposed. The reliability of scraper chains is calculated by the stochastic perturbation technique and the four-moment method. And then, the optimization design problem is discussed based on system reliability. And the optimal model is established. Finally, the effectiveness of the method is verified by the illustrative example of scraper chains. The proposed joint failure probability estimation method and design optimization are shown in the example. The results obtained can provide a reference for the optimal design of the scraper chain.

Modeling face reliability in tunneling: A copula approach

This research develops a copula-enabled approach for modeling and assessing the reliability of the dependent system with incomplete probability information. A bivariate framework consisting of the supporting pressure and the ground settlement is proposed to estimate the excavation face reliability in tunneling under limited observations. The impacts of the selection of copula functions, the variation of the failure criteria, and the size of the measured data and Monte Carlo simulation sampling on the reliability estimation result are explored. A realistic tunnel case in the Wuhan metro system, China, is used to demonstrate the applicability and effectiveness of the developed approach. Results indicate that: (1) the failure probability will lead to an overestimated result, which is more than twice of the reference value (that is without considering the negative dependence); (2) the commonly used Gaussian copula function is likely to underestimate the failure probability and lead to less conservative designs; (3) a setting size of 105 samples is reasonable in the case to achieve the required quality in estimation; and (4) the failure probability gradually rises along with an increase of the limit value of the supporting pressure and decreases along with an increase of the limit value of the ground settlement.

Reliability analysis for high-lift device based on Copula function and Bayesian updating

In order to improve the reliability analysis accuracy of the aircraft high-lift, an approach based on the Copula function theory and Bayesian updating is proposed. Considering the influence of the random variables’ correlation in the process of updating, choosing the reasonable prior joint distribution and likelihood function is crucial. Under the condition of the incomplete probability information, the analytic expressions of the prior joint distribution and likelihood function of the correlated random variables are derived through the Copula function. Then, the posterior joint distribution is obtained by Bayesian updating. The reliability of the lifting device is calculated based on the posterior distribution. The case analysis shows that the reliability results based on the proposed approach are more accurate and more coincident with the factual situation than the reliability analysis results based on the independence assumption of random variables.

Response–Surface–Based Embankment Reliability under Incomplete Probability Information

Response–Surface–Based Embankment Reliability under Incomplete Probability Information

Towards reliability evaluation involving correlated multivariates under incomplete probability information: A reconstructed joint probability distribution for isoprobabilistic transformation

Reliability evaluation under incomplete probability information (prescribed marginal distributions and correlation coefficients) is a challenging task. The widely used Nataf transformation inherently assumes a normal copula for dependence modeling, which can be inappropriate in some cases. This paper aims to provide a more general isoprobabilistic transformation method for reliability evaluations under incomplete probability information. To this end, the joint probability distribution is represented using the pair-copula decomposition approach, which is highly flexible in dependence modeling. The desired pair-copula parameters are retrieved from the incomplete probability information by a simulation-based method. Finally, based on the reconstructed joint probability distribution, the Rosenblatt’s transformation is adopted for the subsequent reliability evaluation. The proposed method is illustrated in a tunnel excavation reliability problem. Several dependence structures characterized by different pair-copulas are investigated to provide insights into the effect of copula selection on reliability results.

Subset simulation for non-Gaussian dependent random variables given incomplete probability information

Reliability analysis under incomplete probability information is a challenging task. This paper focuses on the extension of the subset simulation (SS), an efficient reliability approach, to the modeling of any dependent random variables under incomplete probability information. The Nataf transformation is commonly adopted to generate correlated random variables given marginal distributions and correlations; however, it inherently assumes a Gaussian dependence structure. In contrast, the Rosenblatt transformation can be used to generate correlated random variables with any dependence structure; however, the joint probability information must be known. To remove the limitation, the vine copula approach, which is highly flexible in dependence modeling, is used to reconstruct the joint probability information from the prescribed marginal distributions and correlations. The copula parameters in the vine structure are retrieved using an efficient approximation method. Three copula cases including the Gaussian and non-Gaussian dependence structures are investigated by applying the proposed method to a numerical example. The failure probabilities and the effects of the uncertain parameters correspond to different cases are compared, which aims to provide insights into the impact of the dependence structure on the SS results when only the incomplete probability information is given.

Stochastic response surface method for reliability problems involving correlated multivariates with non-Gaussian dependence structure: Analysis under incomplete probability information

This paper aims to provide a stochastic response surface method (SRSM) that can consider non-Gaussian dependent random variables under incomplete probability information. The Rosenblatt transformation is adopted to map the random variables from the original space into the mutually independent standard normal space for the stochastic surrogate model development. The multivariate joint distribution is reconstructed by the pair-copula decomposition approach, in which the pair-copula parameters are retrieved from the incomplete probability information. The proposed method is illustrated in a tunnel excavation example. Three different dependence structures characterized by normal copulas, Frank copulas, and hybrid copulas are respectively investigated to demonstrate the effect of dependence structure on the reliability results. The results show that the widely used Nataf transformation is actually a special case of the proposed method if all pair-copulas are normal copulas. The effect of conditioning order is also examined. This study provides a new insight into the SRSM-based reliability analysis from the copula viewpoint and extends the application of SRSM under incomplete probability information.

Bivariate distribution of shear strength parameters using copulas and its impact on geotechnical system reliability

The objective of this paper is to investigate the effect of copulas for constructing the bivariate distribution of shear strength parameters on system reliability of geotechnical structures. First, the bivariate distribution of shear strength parameters is constructed using copulas. Second, the implementation procedure of system reliability analysis using direct Monte Carlo simulation (MCS) is developed. Finally, the system reliability of a retaining wall and a rock wedge slope is presented to explore the effect of copula selection on geotechnical system reliability. The results indicate that the system reliability of geotechnical structures under incomplete probability information could not be determined uniquely because the bivariate distribution of cohesion and friction angle with given marginal distributions and correlation coefficient could not be determined uniquely. The copulas for modeling dependence structure between cohesion and friction angle have a significant influence on the system reliability of geotechnical structures. Such an influence includes two phases separately. The first phase is that the dependence structure between shear strength parameters characterized by copulas affects the reliability of single failure mode, depending on the marginal distributions, dependence structure between shear strength parameters, and reliability level of each failure mode. The second phase is that the reliability of each failure mode influences on system reliability, only depending on reliability level of each failure mode and correlations among various failure modes. It is important to distinguish between the effect of copula selection on reliability of each failure mode and that on geotechnical system reliability.