Belief reliability evaluation with uncertain right censored time‐to‐failure data under small sample situation

Uncertain right censoring which describes the situation that the censoring settings of test units are unrelated to their failure time often occurs in the life testing. Besides, in the reliability evaluations using the collected time-to-failure data from the practical life testing, the small-sample situation is common since the test resources are usually limited. Under such situation, the large-sample based probability theory is not appropriate anymore to evaluate reliability. To address such problem, based on a new theory of reliability-belief reliability theory, the reliability evaluation is conducted to cope with the limited censored time-to-failure data under uncertain right censoring, where the uncertain measure is employed. Firstly, a brief introduction to the belief reliability theory is conducted. Then, the lifetime model and belief reliability evaluation under the uncertain measure are presented, and the corresponding statistics method for unknown parameter estimation is provided with objective measures. Finally, a simulation study is used to illustrate the proposed method. The results show that the proposed method can achieve more accurate and concrete reliability evaluation results with limited right-censored time-to-failure data, compared to the probability reliability method.

Field experimentation on constructed systems demands consideration of many mechanisms of epistemic and aleatory uncertainties as well as human errors and subjectivity. This is especially true in operational modal analysis (OMA) applications that aim to identify the dynamic properties of a structure. Although statistics and probability theory are sufficient for quantifying aleatory uncertainty and bounding the resulting errors in OMA results, there is much debate as to whether the same tools may also be used to quantify epistemic uncertainty. This study explored a framework for better understanding the distinctions and impacts of these two types of uncertainties in OMA and how human errors and subjectivity may be classified. A physical laboratory model was designed to simulate four key sources of epistemic uncertainty that represented the primary test variables: structural complexity (changing boundary conditions, nonlinearity), ambient excitation characteristics (magnitude, directionality, and ban...

The potential applications and challenges of uncertain data mining have recently attracted interests from researchers. Most uncertain data mining algorithms consider aleatory (random) uncertainty of data, i.e. these algorithms require that exact probability distributions or confidence values are attached to uncertain data. However, knowledge about uncertainty may be incomplete in the case of epistemic (incomplete) uncertainty of data, i.e. probabilities of uncertain data may be imprecise, coarse, or missing in some applications. The paper focuses on uncertain data which miss probabilities, specially, value-uncertain discrete objects which miss probabilities (for short uncertain objects). On the other hand, classification is one of the most important tasks in data mining. But, to the best of our knowledge, there is no method to learn Naïve Bayesian classifier from uncertain objects. So the paper studies Naïve Bayesian classification of uncertain objects. Firstly, the paper defines interval probabilities of uncertain objects from probabilistic cardinality point of view, and bridges the gap between uncertain objects and the theory of interval probability by proving that interval probabilities are F-probabilities. Secondly, based on the theory of interval probability, the paper defines conditional interval probabilities including the intuitive concept and the canonical concept, and the conditional independence of the intuitive concept. Further, the paper gives a formula to effectively compute the intuitive concept. Thirdly, the paper presents a Naïve Bayesian classifier with interval probability parameters which can handle both uncertain objects and certain objects. Finally, experiments with uncertain objects based on UCI data show satisfactory performances.

Multi-state systems (MSSs) are common in real-word applications, in which a system and/or its components exhibit multiple stochastically-dependent states or performance levels. Such characteristic poses challenges to the reliability evaluation of MSSs. Multi-valued decision diagrams (MDDs) have been developed to address the reliability analysis of MSSs under the assumption that the failure-time parameters of system components are deterministic. However, due to epistemic uncertainty, it is often difficult or impossible to obtain the determinate values of the component parameters. Therefore, this paper aims to address the MDD-based reliability evaluation of MSSs with epistemic uncertainty by incorporating the interval theory and fuzzy set theory. The proposed methods are verified through a detailed case study of a high-speed train bogie system. The results show that the proposed methods can obtain practical reliability evaluation results reflecting the condition of epistemic uncertainty.

An approach to combining unreliable pieces of evidence and their propagation in a system response analysis

The objective of the study described in this paper was to develop an efficient uncertainty quantification framework capable of analyzing uncertainty in integrated spacecraft system models. Specifically, this paper discusses the capabilities of the developed framework and the res ults when applied to the multidisciplinary analysis of a reusable launch veh icle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The Second-Order Probability Theory utilizing a stochastic response surface obta ined with Point-Collocation NonIntrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. This study has also demonstrated the feasibility of the developed uncertainty quantifica tion framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems. I. Introduction ncertainties are generally ubiquitous in the analys is and design of highly complex engineering systems. Uncertainties can arise from the lack of k nowledge in physical modeling (epistemic uncertainty), inherent variations in the systems (a leatory uncertainty), and numerical errors in the computational procedures used for analysis. It is i mportant to account for all of these uncertainties in applications such as robust and reliable design of multi-disciplinary aerospace systems. A reusable la unch vehicle (RLV) is a highly complex aerospace system, which represents a cost viable option for access t o space missions due to its reusability aspect. Since an RLV system is composed of various subsystems which must work together to serve an over-arching p urpose or mission objective/constraint, a multidisciplinary approach for the analysis and des ign of RLV systems is required. Uncertainties are generally present in the models used in each discipline of an RLV multidisciplinary analysis framework. It is important to account for all of these uncertainties for accurate and reliabl e estimates of the RLV system performance. The primary purpose of this paper will be to demonstrate an efficient approach for uncertainty quantification in the multidiscipl inary analysis of a RLV, which has a mixture of aleatory (inherent) and epistemic input uncertainties. Uncertainty quantification will be performed on various output variables of interest f or an RLV system such as the weight and vehicle crossrange. For the propagation of mixed (aleatory-epist emic) uncertainty, Second-Order Probability Theory utilizing Point-Collocation Non-Intrusive Polynomia l Chaos (NIPC) 1 will be used. 2-3 In general, the NIPC methods, which are based on the spectral representa tion of uncertainty, are computationally more effic ient than traditional Monte Carlo methods for moderate number of uncertain variables and can give highly 1 Aerospace Engineer, 4020 Long Beach Blvd, Long Beach, CA, 90807, Member AIAA

Accelerated degradation testing (ADT) aids the reliability and lifetime evaluations for highly reliable products. In engineering applications, the number of test items is generally small due to finance or testing resource constraints, which leads to the rare knowledge to evaluate reliability and lifetime. Consequently, the epistemic uncertainty is embedded in ADT data and the large-sample based probability theory is no longer appropriate. In this paper, we introduce the uncertainty theory, which is a theory different from the probability theory, to account for such uncertainty due to small samples and build up a framework of ADT modeling. In this framework, an uncertain accelerated degradation model is first proposed based on the arithmetic Liu process. Then, the uncertain statistics for parameter estimations are presented correspondingly, which is completely constructed on objectively observed ADT data. An application case and a simulation case are used to illustrate the proposed methodology. With further comparisons to the Wiener process based accelerated degradation model (WADM) and the Bayesian-WADM, the sensitivities of these models to sample sizes are explored and the results show that the proposed model is superior to the other two probability-based models under the small sample size.

Epistemic uncertainty exists in system availability evaluation due to the lack of data and information. To address it, this article proposes a series of definitions of the uncertainty theory-based availability, called belief availability, expanding the scope of belief reliability by introducing uncertainty-measured logistics and maintenance into belief reliability. Based on an uncertain alternating renewal process, we construct a belief availability model for repairable systems subject to the epistemic uncertainty. From the model, we derive formulas of several belief availability metrics, including belief availability (inherent, achieved, and operational), delay time ratio, maintenance time ratio, and belief failure frequency. We find an interesting property that the states order or the initial state in the model will not influence these metrics. A case study about the oxygen generation system (OGS) on the international space station was conducted to analyze the impact of its working, logistics, and maintenance time on the OGS belief operational availability. The results show a potential application in belief availability tradeoff between the OGS's availability-related parameters. In addition, we compared the proposed availability with probability one based on the time distributions of the OGS states, illustrating our method can effectively reduce the deviation of availability evaluation with insufficient data.

A general sampling problem can be described by an Ellsberg urn, which is a mathematical model that assumes that balls are randomly drawn from an urn with an uncertain numbers of colored balls. This means that the Ellsberg urn is essentially an intricate model with simultaneous randomness and epistemic uncertainty, and this is the core problem discussed in this paper. Since practical sampling is usually processed in an intricate environment, the solution for an equivalent mathematical problem is necessary. Suppose an Ellsberg urn contains three unknown numbers of colored balls (i.e., a two-degrees-of-freedom Ellsberg urn), and three balls are randomly drawn from the urn. Compared to the published papers, this paper first constructs a chance space with two-dimensional uncertainty space and three-dimensional probability space to rigorously calculate the color distributions for those drawn balls by uncertainty theory, probability theory, and chance theory. Moreover, it is interesting to find that all cases of the drawn balls are symmetric in such a specific situation of a sample problem with epistemic uncertainty.

In this note, it is proved that Liu’s uncertain measure and Choquet capacity differ from each other. Specifically, over any locally compact separable metric space such as \(\mathbb {R}^n\), a Liu’s uncertain measure is generally not a Choquet capacity. Further, Liu’s uncertain measure is not an ordinary probability measure in the sense of Kolmogorov. Notice that Choquet theory is consistent with Kolmogorov’s probability theory, as a direct generalization of the latter by allowing the random elements to take closed sets of the underlying space as the values.

Uncertainties, including aleatory and epistemic uncertainties, always exist in multidisciplinary system. Due to the discontinuous nature of epistemic uncertainty and the complex coupled relation among subsystems, the computational efficiency of reliability-based multidisciplinary design optimization (RBMDO) with mixed aleatory and epistemic uncertainties is extremely low. A novel RBMDO procedure is presented in this paper based on combined probability theory and evidence theory (ET) to deal with hybrid-uncertainties and improve the computational efficiency. Firstly, based on Bayes method, a novel method to define the probability density function of the aleatory variables is proposed. Secondly, the conventional equivalent normal method (J-C method) is modified to reliability analysis with hybrid-uncertainties. Finally, a novel RBMDO procedure is suggested by integrating the modified J-C method into the frame of sequence optimization and reliability analysis (SORA). Numerical examples and engineering example are applied to demonstrate the performance of the proposed method. The examples show the excellence of the RBMDO method both in computational efficiency and accuracy. The proposed method provides a practical and effective reliability design method for multidisciplinary system.

The ability to describe variables in a health risk model through probability theory enables us to estimate human health risk. These types of risk assessment are interpreted as probabilistic risk assessment (PRA). Generally, PRA requires specific estimate of the parameters of the probability density of the input variables. In all circumstances, such estimates of the parameters may not be available due to the lack of knowledge or information. Such types of variables are treated as uncertain variables. These types of information are often termed uncertainty which are interpreted through fuzzy theory. The ability to describe uncertainty through fuzzy set theory enables us to process both random variable and fuzzy variable in a single framework. The method of processing aleatory and epistemic uncertainties into a same framework is coined as hybrid method. In this paper, we are going to talk about such type of hybrid methodology for human health risk assessment. Risk assessment on human health through different pathways of exposure has been attempted many a times combining Monte Carlo analysis and extension principle of fuzzy set theory. The emergence of credibility theory enables transforming fuzzy variable into credibility distribution function which can be used in those hybrid analyses. Hence, an attempt, for the first time, has been made to combine probability theory and credibility theory to estimate risk in human health exposure. This method of risk assessment in the presence of credibility theory and probability theory is identified as probabilistic-credibility method (PCM). The results obtained are then interpreted through probability theory, unlike the other hybrid methodology where the results are interpreted in terms of possibility theory. The results obtained are then compared with probability-fuzzy risk assessment (PFRA) method. Generally, decision under hybrid methodology is made on the index of optimism. An optimistic decision maker estimates from the $$\alpha$$ -cut at 1, whereas a pessimistic decision maker estimates from the $$\alpha$$ -cut at 0. The PCM is an optimistic approach as the decision is always made at $$\alpha$$ =1.

A marginal regression approach for correlated censored survival data has become a widely used statistical method. Examples of this approach in survival analysis include from the early work by Wei et al. (J Am Stat Assoc 84:1065-1073, 1989) to more recent work by Spiekerman and Lin (J Am Stat Assoc 93:1164-1175, 1998). This approach is particularly useful if a covariate's population average effect is of primary interest and the correlation structure is not of interest or cannot be appropriately specified due to lack of sufficient information. In this paper, we consider a semiparametric marginal proportional hazard mixture cure model for clustered survival data with a surviving or "cure" fraction. Unlike the clustered data in previous work, the latent binary cure statuses of patients in one cluster tend to be correlated in addition to the possible correlated failure times among the patients in the cluster who are not cured. The complexity of specifying appropriate correlation structures for the data becomes even worse if the potential correlation between cure statuses and the failure times in the cluster has to be considered, and thus a marginal regression approach is particularly attractive. We formulate a semiparametric marginal proportional hazards mixture cure model. Estimates are obtained using an EM algorithm and expressions for the variance-covariance are derived using sandwich estimators. Simulation studies are conducted to assess finite sample properties of the proposed model. The marginal model is applied to a multi-institutional study of local recurrences of tonsil cancer patients who received radiation therapy. It reveals new findings that are not available from previous analyses of this study that ignored the potential correlation between patients within the same institution.

In the South African legal system of fact finding and proof the relevance of an evidentiary fact is not governed by the rules of the law of evidence but by a set of extra-legal principles based on the logic of inferential reasoning and probability theory. However, there is no definitive legal definition, or practical test, of what constitutes relevance in a post-constitutional South African curial context, except for an ambiguous pre-1961 reference to a "blend of common sense, judicial experience and logic, laying outside the law". This article critically evaluates the relationship between relevance and admissibility in the adversarial adjudicative process, with particular reference to the peculiarities of the South African legal system, in which the procedural framework of the fact-finding process has been subjected to a post-apartheid constitutional democracy. In addition, this article provides an interpretative synthesis of prevailing international scholarship in the field, develops a functional three-legged practical relevance test for ease of application by all legal practitioners in the courtroom and provides a uniquely different possible statutory definition of relevance and admissibility.

Probability theory is well suited to treat uncertainties when their origin comes only from the natural variability of components' failure (aleatory uncertainty). On the other hand, if the uncertainties are due to incompleteness, imprecision or ignorance of the reliability data (epistemic uncertainty), several theories can be used. Previous studies have shown that the Transferable Belief Model (TBM) which is an interpretation of the Dempster Shafer theory is a well suited framework to treat both types of uncertainties. Nevertheless, past experiences have proven that the computational cost of the TBM based model grows exponentially with the size of the system. To overcome this situation, generalized expressions were defined so as to have a more efficient way to study the systems' reliability of series-parallel and parallel-series configurations. Through the use of these expressions, there is no need to go by the TBM operations (conjunctive rule, extensions, etc.) to evaluate the overall system's reliability.