Abstract
This paper presents a new structure as a simple method at two uncertainties (i.e., aleatory and epistemic) that result from variabilities inherent in nature and a lack of knowledge. Aleatory and epistemic uncertainties use the concept of the entropy and Dempster-Shafer (D-S) theory, respectively. Accordingly, we propose the generalized Shannon entropy in the D-S theory as a measure of uncertainty. This theory has been originated in the work of Dempster on the use of probabilities with upper and lower bounds. We describe the framework of our approach to assess upper and lower uncertainty bounds for each state of a system. In this process, the uncertainty bound is calculated with the generalized Shannon entropy in the D-S theory in different states of these systems. The probabilities of each state are interval values. In the current study, the effect of epistemic uncertainty is considered between events with respect to the non-probabilistic method (e.g., D-S theory) and the aleatory uncertainty is evaluated by using an entropy index over probability distributions through interval-valued bounds. Therefore, identification of total uncertainties shows the efficiency of uncertainty quantification.
Highlights
Several mathematical frameworks can be used to evaluate an uncertainty analysis
Probability is used as a representation of subjective belief, which is common in a quantitative analysis of events in different applications
Though Bayesian inference can be employed to determine the probability of decisions correctness based on prior information, it has some disadvantages, namely (1) the knowledge required to generate the prior probability distributions may not be available, (2) instabilities may occur when conflicting data are presented and/or the number of unknown propositions is large compared to the known propositions [2], (3) available information should be characterized by a specific distribution or an exact assertion of the truth of a proposition for the decision maker, and (4) Bayesian inference offers a few opportunities to express incomplete information or partial belief [3]
Summary
Several mathematical frameworks can be used to evaluate an uncertainty analysis. Probability theory is the most traditional representation of uncertainty, which is familiar to non-mathematicians. Toomaj and Doostparast [21] introduced a new concept of a stochastic order for comparing mixed systems and used cumulative residual entropy, which measures the residual uncertainty of a random variable. They applied these concepts to evaluate component lifetimes of competing systems using the Shannon entropy, in order to handle uncertainty. For adequate data existing in probability theory, numerous uncertain models (e.g., various types of entropies) are used as the most appropriate evaluation for aleatory uncertainty quantification. Non-probabilistic methods based on interval specifications or alternative mathematical frameworks (e.g., D–S theory) are proposed for possible better representations of epistemic uncertainty.
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