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Evidential reasoning (ER) rule has been widely used in dealing with uncertainty. As an important parameter to measure the inherent property of evidence, the evidence reliability makes the ER rule constitute a generalized reasoning framework. In current research of the ER rule, the evidence reliability tends to be expressed in the form of quantitative value by certain methods or expert knowledge. The single quantitative value lacks the ability to describe the statistical property of reliability, which leads to unreasonable results. In this article, a new ER rule with continuous probability distribution of reliability denoted by ERr-CR is proposed. The combination of two pieces of evidence is discussed in detail, where the reliability is profiled as random variables with specific probability distribution. To characterize the output of ERr-CR, a novel concept of expectation of the expected utility is proposed. In addition, the ERr-CR is expanded to multiple pieces of evidence to show its universality. Further, the basic performances of the ERr-CR are explored to illustrate the rationality. Moreover, a case study of safety assessment of natural gas storage tanks (NGSTs) is conducted to show the potential applications of ERr-CR, which makes the proposed method more practical. [ABSTRACT FROM AUTHOR]

In current studies of safety assessment for complex systems with the evidential reasoning (ER) rule, the evidence reliability is generally given by experts, which makes the observation data by sensors ignored. However, sensors are inevitably affected by such various uncertainties as perturbations in engineering practice, which can reduce their quality and tracking ability. As such, the observation data may become unreliable, and the modeling accuracy of the ER rule is decreased. In this article, a new ER rule-based safety assessment method with sensor reliability for complex systems is proposed, where sensor reliability and perturbation are considered. The coefficient of the variation-based weighting (CVBW) method is employed to obtain sensor weight. The sensor reliability is calculated by static reliability and dynamic reliability, which are determined by experts and the distance-based method, respectively. The perturbation is quantified as a bounded parameter defined as the perturbation factor, which is used to describe uncertainties and aggregate static reliability and dynamic reliability. The performance analysis of safety assessment is conducted to demonstrate the rationality of perturbation and position poor sensors, followed by a safety assessment algorithm. A case study is carried out to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article aims to propose an evidential reasoning (ER) rule considering the parameter uncertainty. As the essential parameters, the evidence weight and reliability make the ER rule constitute a generalized reasoning framework. Theoretically, the weight is affected by subjective cognition, while the reliability mainly reflects objective variation. However, most of the recent researches have focused on the quantitative calculation methods, which make the differences in the property of the two parameters ignored. In this article, a relatively different idea from previous studies is provided, in which multisource uncertainty of parameters is fully considered. On the one hand, the weight is profiled by interval variable with lower and upper bounds. On the other hand, the reliability is modeled by random variable with probability distribution function. Then, a unified inference model for evidence aggregation is developed based on the inference process of the ER rule. In addition, some basic properties of the model are clearly presented to illustrate the rationality of parameter uncertainty. Finally, a practical example is given to show the potential applications of the proposed model. [ABSTRACT FROM AUTHOR]

The belief rule-base (BRB) model is a new intelligent expert system with the characteristics of both expert system and data-driven model. In BRB there are many if-then rules which use belief degrees to express various types of uncertain information, including fuzziness, randomness, and ignorance. As a semi-quantitative modeling tool for complex systems, BRB has the superiorities of dealing both numerical quantitative data and linguistic qualitative knowledge that are derived from heterogeneous sources. Moreover, it is also a white box approach which can provide direct access and transparency to decision makers and stakeholders. Currently, BRB has been widely applied in many fields, such as decision making, reliability evaluation, network security situation awareness, fault diagnosis, and so on. To fully demonstrate the progress of BRB, the original BRB, and some evolution forms are introduced in this article. [ABSTRACT FROM AUTHOR]

Evidential reasoning (ER) rule has been widely used in addressing uncertainty, ignorance, and vagueness information. To explore its performance measure (PM), the perturbation analysis (PA) for the ER rule (ER rule-PA) is conducted, with perturbation taken into consideration. This article aims to analyze the robustness and stability of the ER rule, serving as theoretical basis and technical support for applied research and applications. The combination of two pieces of independent evidence is discussed, and perturbation is added to one piece of evidence. To represent the expected utility of evidence combination under perturbation, perturbation utility is introduced. The novel concept of perturbation coefficient is proposed to characterize the PM of the ER rule (ER rule-PM). The properties of perturbation coefficient are explored to demonstrate the impact of perturbation. The maximum permissible error (MPE) of perturbation coefficient is defined to characterize the acceptability of perturbation. A numerical study is examined to illustrate the implementation process of ER rule-PA. Moreover, a case study of reliability evaluation of aerospace relay is conducted to show the potential applications of ER rule-PA, which makes the proposed method more practical. [ABSTRACT FROM AUTHOR]

The combinatorial explosion problem is a great challenge for belief rule base (BRB) when a complex system has overnumbered attributes and/or referenced values for the attributes. This is because the BRB is conventionally constructed under the conjunctive assumption, conjunctive BRB, which requires covering each possible combination of all referenced values for all attributes. To solve this challenge, this study proposes a generic modeling, inferencing, and optimization approach for BRB under the disjunctive assumption, disjunctive BRB, that can significantly reduce its size. First, a disjunctive BRB is defined based on the mathematical description of the BRB space. The minimum size requirement for a disjunctive BRB is also discussed in comparison to a conjunctive one. Building on this, the generic disjunctive BRB modeling and inferencing procedures are proposed. Furthermore, an improved optimization model with further relaxed restrictions is constructed, and an optimization algorithm is developed in which only the new rule is optimized and its referenced values range is determined by the optimal solution in the former round optimization. The new optimization algorithm is more efficient with fewer variables and a more concise solution space. The results of three case studies confirm that by integrating both experts’ knowledge and historic data, the modeling and inferencing processes can be well understood. Moreover, optimization can further improve the modeling accuracy while it facilitates downsizing the BRB in comparison with previous studies and other approaches. [ABSTRACT FROM AUTHOR]

In current studies of the belief rule base (BRB) model, the attributes are assumed to be fully reliable and the observation data are directly used as input. However, in engineering practice, the observation data may be affected by some disturbance factors, including the quality of the sensors and noise in the environment. Then, the reliability of observation data may be affected and the modeling accuracy of the BRB is therefore influenced. As such, a new BRB model with attribute reliability (BRB-r) is proposed in this paper. In particular, a calculation method of attribute reliability is given based on the statistical method. Moreover, to integrate the attribute reliability into the BRB-r, a new calculation method of matching degree is developed. The model's overall reliability denotes its ability to provide the correct result. When the attributes are unreliable, the overall reliability of the BRB-r is degraded. Thus, a calculation method for the overall reliability of the BRB-r is developed to support decision-making in engineering practice. A case study of the safety assessment of a diesel engine is conducted to demonstrate the efficiency of the proposed BRB-r model. [ABSTRACT FROM AUTHOR]

Nonlinear complex system modeling has drawn attention from diverse fields and many approaches have been developed. Among those approaches, the advantages of the belief rule base (BRB) expert system have been shown for managing multiple types of information under uncertainty and modeling the nonlinearity present in many theoretical and practical complex systems. However, two challenges still need to be addressed. First, BRB needs to be downsized to conserve modeling and computational effort. For this challenge, a new disjunctive assumption is applied, which can significantly downsize BRB while maintaining its completeness. Second, the structure and parameters of BRB need to be jointly optimized. For this challenge, a new Akaike information criterion (AIC)-based optimization objective is derived to represent both modeling accuracy and modeling complexity. Moreover, a joint bi-level optimization model with an AIC-based objective is constructed for the BRB structure and parameters, and a bi-level optimization algorithm is proposed. Three evolutionary algorithms, namely, the genetic algorithm, particle swarm optimization algorithm, and differential evolutionary algorithm, are tested in a comparative fashion to determine the best fit for the optimization engine. The results of two practical case studies show that the joint optimization approach can identify an optimal configuration for both its structure and parameters, which is referred to as the best decision structure in this paper. [ABSTRACT FROM AUTHOR]

It is important to predict the hidden behavior of a complex system. In the existing models for predicting the hidden behavior, the hidden belief rule base (HBRB) is an effective model which can use qualitative knowledge and quantitative data. However, the frame of discernment (FoD) of HBRB which is composed of some states or propositions and the universal set including all states or propositions is not complete. The global ignorance and local ignorance cannot be considered at the same time, which may lead to the inaccurate forecasting results. To solve the problems, a new HBRB model named as PHBRB in which the hidden behavior is described on the FoD of the power set is proposed in this correspondence paper. Furthermore, by using the evidential reasoning rule as the inference tool of PHBRB, a new projection covariance matrix adaption evolution strategy is developed to optimize the parameters of PHBRB so that more accurate prediction results can be obtained. A case study of network security situation prediction is conducted to demonstrate the effectiveness of the newly proposed method. [ABSTRACT FROM AUTHOR]

It is vital to online assess the safety of a complex dynamic system by taking into account the current state, degradation trend, and historical records together. This paper proposes a new safety assessment model with an online algorithm based on the evidential reasoning (ER) approach. It does not only take into account the relative importance of each safety indicator, but also consider the reliability of each indicator. To obtain the integrated safety level, multiple safety indicators are fused at first and the “history,” “current,” and “future” safety states are then integrated. First, a forecasting model based on the third-order Volterra filter is proposed to predict the safety indicators’ information online. Second, an adaptive weighting model is developed to automatically adjust to various conditions and track the characteristics of the dynamic system, and the reliability of each indicator is considered to diminish the influence of inherent disturbance and/or noise. Finally, a safety assessment aggregation scheme based on the ER approach is presented to fuse the history, current, and future safety indicators to obtain the corresponding safety state, and the safety states are then fused synthetically to obtain a comprehensive safety assessment result of the complex dynamic system. A numerical study is examined to demonstrate the implementation and effectiveness. Moreover, a practical example of the inertial navigation system is studied to show the potential applications of the proposed ER-based safety assessment method. [ABSTRACT FROM AUTHOR]

It is vital to assess the lives of newly developed products by using failure data from various testing environments. In the current methods, two steps are generally included. The first step is transforming the failure data under one testing environment into the actual working environment, and the second step is integrating all failure data under the actual working environment into a unified result. However, most available methods cannot use information that includes part failure data and part expert knowledge simultaneously. To resolve the above issue, based on the belief rule base (BRB) and the evidential reasoning (ER) approach, a new BRB-ER-based model is proposed, where the BRB is used to transform the failure data from one testing environment into the actual working environment. The ER approach, which is adopted to aggregate the failure data from different testing environments, is used to assess the life of a product. To conclude, the BRB-ER-based model is applied to represent and integrate asynchronous multisource information. In the proposed model, the initial BRB system is constructed based on experts’ knowledge, which results in uncertainty because of the ambiguous nature of human judgment and calls for training the parameters in the BRB-ER-based model. Therefore, an optimal algorithm that employs the differential evolutionary algorithm is proposed. The proposed model and the optimal algorithm operate in an integrated manner to improve the assessment precision by using both failure data and expert knowledge effectively. A case study in three scenarios and use of the conventional approach is examined to demonstrate the capability and potential applications of the new BRB-ER-based model. [ABSTRACT FROM AUTHOR]

Compared with the observable behavior, it is difficult to predict the hidden behavior of a complex system. In the existing methods for predicting the hidden behavior, a lot of testing data (usually quantitative information) are needed to be sampled. However, some complex engineering systems have the following characteristics: 1) The systems cannot be tested periodically, and the observable information is incomplete; 2) the change process of hidden behavior may be affected by the test; and 3) only part of quantitative information and qualitative knowledge (i.e., semiquantitative information) may be obtained. These characteristics all related to the test are named as testing influence for simplicity. Although a model and a corresponding optimal algorithm for training the model parameters have been proposed to predict the hidden behavior on the basis of semiquantitative information and belief rule base (BRB), the testing influence has not been considered. In order to solve the above problems, a new BRB-based model, which can use the semiquantitative information, is proposed under testing influence in this paper. In the newly proposed forecasting model, there are some parameters of which the initial values are usually assigned by experts and may not be accurate, which can lead to the inaccurate prediction results. As such, an improved optimal algorithm for training the parameters of the forecasting model is further developed on the basis of the expectation-maximization idea and the covariance matrix adaption evolution strategy (CMA-ES). By using the semiquantitative information, the proposed BRB-based model and the improved CMA-ES algorithm can operate together in an integrated manner so as to improve the forecasting precision. A case study is examined to demonstrate the ability and applicability of the newly proposed BRB-based forecasting model and the improved CMA-ES algorithm. [ABSTRACT FROM AUTHOR]

In order to determine the parameters of belief-rule-base (BRB) accurately, several optimization methods have been proposed for training BRB, on the basis of a generic rule-base inference methodology using the evidential reasoning (RIMER) approach. These optimization methods are implemented offline, and such are not suitable for training BRB in a dynamic fashion. In this paper, two recursive algorithms are proposed to update BRB online that can simulate dynamic systems. The main feature of the proposed algorithms is that only partial input and output information is required, which can be incomplete or vague, numerical or judgmental, or mixed. If the internal structure of a BRB is initially decided using expert judgments, domain-specific knowledge and/or commonsense rules, the proposed algorithms can be used to fine-tune the initial BRB online, once input and output datasets become available. Using the proposed algorithms, there is no need to collect a complete set of data before a BRB can be trained, which is necessary if the BRB is used to simulate a dynamic system. A numerical example and a case study are reported to demonstrate the potential of the algorithms for online fault diagnosis. [ABSTRACT FROM PUBLISHER]

Recently, a novel prediction model based on the evidential reasoning (ER) approach is developed to forecast reliability in engineering systems. In order to determine the parameters of the ER-based prediction model, some optimization models have been proposed to train the ER-based prediction model. However, these models are implemented in an offline fashion and thus it is very expensive to train and retrain them when new information is available. This correspondence paper is concerned with developing the recursive algorithms for updating the ER-based prediction model from the probability-based point of view. Using the recursive expectation maximization algorithm, two recursive algorithms are proposed for updating the parameters of the ER-based prediction model under judgmental and numerical outputs, respectively. As such, the proposed algorithms can be used to fine tune the ER-based prediction model online once new information becomes available. We verify the proposed method via a realistic example with missile reliability data. [ABSTRACT FROM PUBLISHER]

In engineering practice, a system's behavior constantly changes over time. To predict the behavior of a complex engineering system, a model can be built and trained using historical data. This paper addresses the forecasting problems with a belief rule base (BRB) to trace and predict system performance in a more interpretable and transparent way. More precisely, it extends the BRB method to handle a system's behavior prediction, and a new prediction model based on BRB is presented, which can model and analyze prediction problems using not only numerical data but human judgmental information as well. The proposed forecasting model includes some unknown parameters that can be manually tuned and trained. To build an effective BRB forecasting model, a multiple-objective optimization model is provided to locally train the BRB prediction model by minimizing the mean square error (MSE). Finally, a practical case study is provided to illustrate the detailed implementation procedures and examine the feasibility of the proposed approach in engineering application. Furthermore, the comparative studies with other state-of-the-art prediction methods are carried out. It is shown that the proposed model is effective and can generate better prediction in terms of accuracy, as well as comprehensibility. [ABSTRACT FROM AUTHOR]