Probability Boxes Research Articles

Characterization and propagation of uncertainties associated with limited data using a hierarchical parametric probability box

AbstractUncertainty of random variables is commonly characterized from measurement data. In practice, data might be insufficient in order to obtain an accurate probability model. In this work, we assume that the type of distribution of the considered random variable is known a priori, and use a hierarchical parametric probability box (p‐box) – which is a set of distributions whose parameters are uncertain – to account for the limited data. Using Bayes' rule, knowledge about the variability of these parameters is updated. Propagation of these uncertainties through a computational model usually suffers from computational burden. A surrogate model approximating the computational model is constructed to reduce computational cost.

A probability‐box approach on uncertain correlation lengths by stochastic finite element method

AbstractIn order to regard mixed aleatory and epistemically uncertain random fields within stochastic finite element method, a probability box approach using stochastic collocation method is introduced. The influence of an interval‐valued correlation length on the output is investigated.

Global sensitivity analysis in the context of imprecise probabilities (p-boxes) using sparse polynomial chaos expansions

Global sensitivity analysis aims at determining which uncertain input parameters of a computational model primarily drives the output quantities of interest. Sobol’ indices are now routinely applied in this context when the input parameters are modelled by classical probability theory using random variables. In many practical applications however, input parameters are affected by both aleatory and epistemic uncertainty. In this respect, imprecise probability theory have become popular in the last decade. In this paper, we consider that the uncertain input parameters are modelled by parametric probability boxes (p-boxes). We propose interval-valued (so-called imprecise) Sobol’ indices as an extension of their classical definition. An original algorithm based on the concepts of augmented space, isoprobabilistic transforms and sparse polynomial chaos expansions is devised to allow for the computation of these imprecise Sobol’ indices at extremely low cost. In particular, phantoms points are introduced to build an experimental design in the augmented space (necessary for the calibration of the sparse PCE) which leads to a smart reuse of runs of the original computational model. The approach is applied to three analytical and engineering examples which illustrate the efficiency of the algorithms against brute-force double-loop Monte Carlo simulation.

A General Methodology for Uncertainty Quantification in Engineering Analyses Using a Credible Probability Box

Uncertainty quantification (UQ) is gaining in maturity and importance in engineering analysis. While historical engineering analysis and design methods have relied heavily on safety factors (SF) with built-in conservatism, modern approaches require detailed assessment of reliability to provide optimized and balanced designs. This paper presents methodologies that support the transition toward this type of approach. Fundamental concepts are described for UQ in general engineering analysis. These include consideration of the sources of uncertainty and their categorization. Of particular importance are the categorization of aleatory and epistemic uncertainties and their separate propagation through an UQ analysis. This familiar concept is referred to here as a “two-dimensional” approach, and it provides for the assessment of both the probability of a predicted occurrence and the credibility in that prediction. Unique to the approach presented here is the adaptation of the concept of a bounding probability box to that of a credible probability box. This requires estimates for probability distributions related to all uncertainties both aleatory and epistemic. The propagation of these distributions through the uncertainty analysis provides for the assessment of probability related to the system response, along with a quantification of credibility in that prediction. Details of a generalized methodology for UQ in this framework are presented, and approaches for interpreting results are described. Illustrative examples are presented.

Sensitivity Analysis for Systems under Epistemic Uncertainty with Probability Bounds Analysis

It is of paramount important to identify and rank the influence of components on the performance of interest. System sensitivity analysis provides a quantitative tool for accessing the importance of components within a specific system configuration. In practice, however, due to lack of information, there exist epistemic uncertainty within the components distribution parameters, which makes it is hard to estimate the reliability of the corresponding system. In this paper, survival signature is adopted to evaluate the system performance, and the area value of the probability box is introduced to reflect the epistemic uncertainty of the system. Also, in order to find out which component or components set is more sensitive to the system, the probability bounds analysis which bases on pinching method is used. Two case studies are presented to show the applicability of the approaches.

Terahertz spectral unmixing based method for identifying gastric cancer

At present, many researchers are exploring biological tissue inspection using terahertz time-domain spectroscopy (THz-TDS) techniques. In this study, based on a modified hard modeling factor analysis method, terahertz spectral unmixing was applied to investigate the relationships between the absorption spectra in THz-TDS and certain biomarkers of gastric cancer in order to systematically identify gastric cancer. A probability distribution and box plot were used to extract the distinctive peaks that indicate carcinogenesis, and the corresponding weight distributions were used to discriminate the tissue types. The results of this work indicate that terahertz techniques have the potential to detect different levels of cancer, including benign tumors and polyps.

A Fresh Look at Z-numbers – Relationships with Belief Functions and p-boxes

ABSTRACTThis paper proposes a new approach to the notion of Z-number, i.e., a pair of fuzzy sets modelling a probability-qualified fuzzy statement. Originally, a Z-number is viewed as the fuzzy set of probability functions stemming from the flexible restriction of the probability of a fuzzy event by a fuzzy probability. This representation leads to complex calculations and does not reduce to the original fuzzy event when the attached probability is 1. Simpler representations are proposed, that avoid these pitfalls. We note that when both fuzzy sets forming the Z-number are crisp, the generated set of probabilities is representable by a special kind of belief function that corresponds to a probability box (p-box). Two proposals are made to generalise this approach when the two sets are fuzzy. One approach considers a Z-number as a weighted family of crisp Z-numbers, obtained by independent cuts of the two fuzzy sets. In the alternative approach, a Z-number can be turned into a pair of possibility distributions forming a generalized p-box. In that case, the probability of each cut of the fuzzy event is upper and lower bounded by two probability values. Then computation with Z-numbers come down to uncertainty propagation with random intervals.

Uncertain Power Flow Analysis Based on Evidence Theory and Affine Arithmetic

A novel uncertain power flow analysis method is proposed in this letter. It utilizes evidence theory to fuse probabilistic, possibilistic, and interval uncertainties. Besides, the extended affine arithmetic is applied to handle variable dependencies existed in power flow. The final analysis results are represented as probability boxes (p-boxes). The comparisons with Monte Carlo simulation are given to demonstrate the validity of the proposed method.

Bayesian networks and probability boxes to model uncertainty in unavailability assessment

In this paper the problem of uncertainty in assessing unavailability of Safety Instrumented Systems (SIS) is treated. The characteristic parameters of the SIS including Common Cause Failure (CCF) factors are replaced by probability densities families (p-boxes) allowing reliability experts to express their uncertainty on the statement of values probabilities. We show how the imprecision is modelled and propagated in a Bayesian networks which induces uncertainty about the failure probability on demand of the SIS and its Safety Integrity Level (SIL). We will demonstrate how the uncertainty in the values of some characteristic parameters causes significant variations in the level risk.

Bayesian networks and probability boxes to model uncertainty in unavailability assessment

Bayesian networks and probability boxes to model uncertainty in unavailability assessment

Assessment of power grid vulnerabilities accounting for stochastic loads and model imprecision

Vulnerability and robustness are major concerns for future power grids. Malicious attacks and extreme weather conditions have the potential to trigger multiple components outages, cascading failures and large blackouts. Robust contingency identification procedures are necessary to improve power grids resilience and identify critical scenarios. This paper proposes a framework for advanced uncertainty quantification and vulnerability assessment of power grids. The framework allows critical failure scenarios to be identified and overcomes the limitations of current approaches by explicitly considering aleatory and epistemic sources of uncertainty modelled using probability boxes. The different effects of stochastic fluctuation of the power demand, imprecision in power grid parameters and uncertainty in the selection of the vulnerability model have been quantified. Spectral graph metrics for vulnerability are computed using different weights and are compared to power-flow-based cascading indices in ranking N-1 line failures and random N-k lines attacks. A rank correlation test is proposed for further comparison of the vulnerability metrics. The IEEE 24 nodes reliability test power network is selected as a representative case study and a detailed discussion of the results and findings is presented.

Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory

Abstract Random set theory is a general framework which comprises uncertainty in the form of probability boxes, possibility distributions, cumulative distribution functions, Dempster-Shafer structures or intervals; in addition, the dependence between the input variables can be expressed using copulas. In this paper, the lower and upper bounds on the probability of failure are calculated by means of random set theory. In order to accelerate the calculation, a well-known and efficient probability-based reliability method known as subset simulation is employed. This method is especially useful for finding small failure probabilities in both low- and high-dimensional spaces, disjoint failure domains and nonlinear limit state functions. The proposed methodology represents a drastic reduction of the computational labor implied by plain Monte Carlo simulation for problems defined with a mixture of representations for the input variables, while delivering similar results. Numerical examples illustrate the efficiency of the proposed approach.

Structural reliability analysis based on probability and probability box hybrid model

In many structural reliability analysis problems, probability approach is often used to quantify the uncertainty, while it needs a great amount of information to construct precise distributions of the uncertain parameters. However, in many practical engineering applications, distributions of some uncertain variables may not be precisely known due to lack of sufficient sample data. Hence, a complex hybrid reliability problem will be caused when the random and non-precise probability variables both exist in a same structure. In this paper, a new hybrid reliability analysis method is developed based on probability and probability box (p-box) models. Random distributions are used to deal with the uncertain parameters with sufficient information, while the probability box models are employed to deal with the non-precise probability variables. Due to the existence of the p-box parameters, a limit-state band will be resulted and the corresponding reliability index will belong to an interval instead of a fixed value. According to the interval analysis, the hybrid reliability model based on random and probability box variables is constructed and the complex nesting optimization problem will be involved in this hybrid reliability analysis. In order to obtain the minimal and maximal reliability index, the corresponding solution strategy is developed, in which the intergeneration projection genetic algorithm (IP-GA) with fine global convergence performance is employed as inner and outer optimization solver. Four numerical examples are investigated to demonstrate the effectiveness of the present method.

Microgrid Energy Management Combining Sensitivities, Interval and Probabilistic Uncertainties of Renewable Generation and Loads

This paper proposes an optimal energy management approach combining sensitivities, interval, and probabilistic uncertainties of wind and solar power sources and loads in microgrid. Affine arithmetic (AA) is used to model the interval uncertainties and sensitivities in nodal power injections. However, all the elements in the interval solutions of AA-optimal power flow may not be significant in view of the probabilistic nature of statistical data. So, those elements which are significant with a desired confidence level are boxed using probability boxes obtained by deriving best fitting discrete state probability distribution functions (PDFs) for load and renewable power injections. Thus, the original hard bounded affine intervals are made soft bounded using the derived joint PDFs, forming new less conservative and more feasible intervals of cost and power flow variables. The minimization of the operational cost is taken care of by stochastic weight tradeoff particle swarm optimization. The method is tested in CIGRE LV benchmark microgrid with fuel cell, microturbine, diesel generator, wind, and solar power sources.

Tighter bounds on the probability of failure than those provided by random set theory

Random set theory is a generalization of Dempster-Shafer evidence theory, that employs an infinite number of focal elements. It can be used for the estimation of the bounds of the probability of failure of structural systems when there is both aleatory and epistemic uncertainty in the representation of the input variables. Indeed, this framework allows to model basic variables as cumulative distribution functions, distribution-free probability boxes, possibility distributions or families of intervals provided by experts, while representing the dependence of the implied variables by means of copulas. This paper reviews another method, which poses the calculation of the bounds of the probability of failure as a reliability-based-design-optimization problem. It is proved theoretically and by means of numerical experiments, that the latter method provides tighter bounds on the probability of failure than those estimated by random set theory. We also theoretically show some interesting relationships between the random set-based method and the optimization approach.

Construction of families of probability boxes and corresponding membership functions at different fractiles

AbstractUncertainty comes in many forms in the real world and is an unavoidable component of human life. Generally, two types of uncertainties arise, namely, aleatory and epistemic uncertainty. Probability is a well established mathematical tool to handle aleatory uncertainty and fuzzy set theory is a tool to handle epistemic uncertainty. However, in certain situations, parameters of probability distributions may be tainted with epistemic uncertainty; and so, representation of parameters of probability distributions may be treated as fuzzy numbers (may be of different shapes). A probability box (P‐box) can be constructed when parameters are not precisely known. In this paper, an attempt has been made to construct families of P‐boxes when parameters of probability distributions are bell shaped or normal fuzzy numbers; and from these families of P‐boxes, membership functions are generated at different fractiles for different alpha levels.

Definition and Implementation of a Method for Uncertainty Aggregation in Component-Based System Simulation Models

Component-based system simulation models are used throughout all development phases for design and verification of both physical systems and control software, not least in the aeronautical industry. However, the application of structured methods for uncertainty quantification (UQ) of system simulation models is rarely seen. To enable dimensionality reduction of a UQ problem and to thereby make UQ more feasible for industry-grade system simulation models, this paper describes a pragmatic method for uncertainty aggregation. The central idea of the proposed aggregation method is to integrate information obtained during common practice component-level validation directly into the components, and to utilize this information in model-level UQ. A generic component output uncertainty description has been defined and implemented in a Modelica library for modeling and simulation (M&S) of aircraft vehicle systems. An example is provided on how to characterize and quantify a component's aggregated output uncertainty based on the component-level bench test measurement data. Furthermore, the industrial applicability of the uncertainty aggregation method is demonstrated in an approximate UQ of an aircraft liquid cooling system simulation model. For cases when the concept of thorough UQ resulting in probability boxes is not feasible, the demonstrated approximate UQ using aggregated uncertainties is considered to be a pragmatic alternative fairly in reach for the common M&S practitioner within the area of system simulation.

An efficient reliability analysis approach for structure based on probability and probability box models

In this paper, a new reliability analysis method for engineering structures is developed based on probability and probability box (p-box) models. Random variable distributions are used to deal with the uncertain parameters with sufficient information, while the p-box models are employed to deal with the uncertain-but-bounded variables. Due to the existence of the p-box parameters, a limit-state band will result and a complex nesting optimization problem will be involved in this reliability analysis. To reduce the computational burden, an efficient decoupling strategy is developed to solve the nesting optimization problem. Through interval analysis for the probability transformation process, the complex nesting optimization problem can be transformed to a single-layer optimization model. Then, the optimum solution and corresponding reliability index can be obtained by introducing a sequential quadratic programming (SQP) method. Four engineering numerical examples are investigated to demonstrate the effectiveness of the present method.

Interval reliability analysis under the specification of statistical information on the input variables

Structural reliability analysis should often face the problem that there is uncertainty about the probabilistic specification of the input random variables implied in a specific problem. In the poorest information case only the knowledge of intervals of fluctuation is available, while the opposite situation is that of a full specification of the joint distribution function of the variables. Parametric or non parametric probability boxes, arising from basic statistical tools such as confidence intervals or Kolmogorov–Smirnov distance, are intermediate situations. In any case, random or fixed intervals emerge for describing eitheistribution parameters under epistemic uncertainty. A crude Monte Carlo simulation for assessing the corresponding interval of the failure probability requires the solution of a double optimization problem for each probability level of the limit state function. This, obviously, implies a prohibitive computational labor. In this paper a method for drastically simplifying this task is proposed. The method exploits the ordering statistics representation property of the reliability plot, which is shown to approximately obey an orthogonal hyperbolic pattern. Accordingly a two-level FORM approach with relaxed accuracy constraints is used to derive the polar vectors for building two plots, one for the input variable space and the second to the epistemic uncertainty space. Using a variety of examples, it is demonstrated that the extrema of the failure probability are contained amongst the samples located in specific sectors of the upper level plot as indicated by the hyperbolae. This means that, after solving the two-level FORM problem, it suffices to calculate the failure probability (or the reliability index) for a small number of mean samples thus selected. Obviously, the method yields the same reliability interval estimates as the crude two-level Monte Carlo because its samples are underlying in it.

Model validation based on random set theory

This paper considers the model validation under epistemic uncertainty in model inputs. The random set theory is used to quantify the uncertainty of model prediction. With the probability box obtained by the random set theory, a pignistic probability transformation is applied to construct a single probability distribution to be the prior distribution for the model prediction. Then a posterior probability distribution is updated based on the experimental observations in Bayesian principle. The Bayes factor derived from the ratio between the posterior and the prior probability distributions is used as the validation metric to quantify the extent to which the experimental observations support the model. A thermal conduction example and an aerospace bolted joint example are presented to illustrate the proposed method. It is shown that the method presented in this paper provides a convenient mechanism to consider different types of uncertainty during the model validation.