1. INTRODUCTION

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. Probability is used as a representation of subjective belief, which is common in a quantitative analysis of events in different applications. Safety assessments should deal with rare events, and thus it is difficult to assess the relative frequencies of these events [1]. The Bayesian approach for the uncertainty analysis is to specify a coherent probability measure as the current state of available knowledge and uses Bayes’ theorem to adjust the probabilities as new evidence, which is unveiled. 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].

Imprecise probability is a generic term for any mathematical model, which measures chance or uncertainty without crisp numerical probabilities. The evidence or Dempster–Shafer (D–S) theory of belief structures is one type of imprecise probabilities, which offers a more flexible representation of uncertainty over the crisp probabilistic approach. Based on the Dempster's work, Shafer [4] proposed the D–S theory as an alternative to Bayesian inference. This theory is a generalization of the Bayesian theory and can robustly deal with incomplete data. It allows the representation of both imprecision and uncertainty [5, 6]. Rather than computing probabilities of propositions, it computes probabilities that evidence supports the propositions or alternatives, which deal with uncertainty reasoning based on incomplete information. The D–S theory tackles the prior probability issue by keeping a track of an explicit probabilistic measure of a possible lack of information. It is suitable for taking into account the disparity among knowledge types [7]. Because it is able to provide a federative framework [8] and combines cumulative evidence for changing prior opinions in the light of new evidence [9].

The D–S theory is well developed and considered as the most appropriate alternative for uncertainty quantification, which deals with lack of knowledge. For example, Jiang et al. [10] proposed a new method to deal with a reliability analysis to solve engineering problems by using the D–S theory under uncertainty. Tang et al. [11] presented an evidential uncertainty quantification method to determine uncertainties with imprecise information, which is included in revealing the material constants of the metal fatigue crack growth model. Yin et al. [12] proposed an efficient algorithm to solve an acoustic problem under uncertainty and applied the Jacobi polynomial for evidence-theory based on an uncertainty analysis. Then, Yin et al. [13] analyzed the response of built-up systems in the mid-frequency range by using the D–S theory and a model of a finite element/statistical energy analysis under uncertainty. Jian et al. [14] evaluated flutter risk and the structural stability of the system, which deals with the D–S theory and uncertainty quantification.

In the probability theory, Shannon entropy is one of the best-known measures of uncertainty for a purely probabilistic system [15]. Based on the Shannon entropy concept, many studies have formulated to handle uncertainty in different fields of science. For example, Thapliyal and Taneja [16] used the concept of the Shannon entropy to evaluate the uncertainty as a measure of inaccuracy associated with distributions of order statistics. Olvera-Guerrero et al. [17] used the non-linear Shannon entropy to measure the uncertainty of a boiling water reactor to assess a system stability and operation.

Gu [18] introduced a new concept of entropy and multi-scale Shannon entropy and then analyzed its application to detect predictive power for the Dow Jones Industrial Average Index (DJIAI). In the fields of component lifetimes of a system, Gupta et al. [19] considered the Shannon entropy as a measure of the uncertainty associated with a residual lifetime distribution and acquired the properties of the residual and the entropy function of order statistics. Tahmasebi et al. [20] proposed a new extension of cumulative residual entropy using the Shannon entropy as a measure of the uncertainty associated with the lifetime of a system. 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. The other applications are fault diagnosis [22], risk [23], etc.

Generally, frameworks, which are considered the uncertainty of the physical events or the behavior of a system, can be categorized into two types of uncertainty. Type one is uncertainty due to stochastic and irreducible variabilities, which is inherent in nature (i.e., aleatory uncertainty), and the other one is resulting from unknown physical phenomena due to a lack of knowledge (i.e., epistemic uncertainty). In Section 2, a brief introduction is provided two uncertainties. 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.

Obviously, in the analysis process of the system performance, uncertainty appears at different steps of analysis and the interaction between these sources of uncertainty cannot be modeled easily. Thus, in order to evaluate and reduce uncertainty, different models and modern mathematical frameworks have been proposed to quantify both aleatory and epistemic uncertainties in systems (e.g., see [24, 25]). However, the existing methods to evaluate the effect of mixed uncertainty using a piece of information and different mathematical frameworks are efficient, and modern frameworks for mathematical representation of uncertainty are still under development.

Therefore, in this paper, the concepts of D–S theory and Shannon entropy are used to handle uncertainty for systems’ failure in each state considering the uncertainty bounds. On the other hand, the objective is to introduce a new technique to calculate uncertainty bounds based on the D–S theory and Shannon entropy. This paper is organized as follows. In Section 2, we introduce the type of uncertainty. Sections 3 and 4 introduce some concepts of the D–S theory and Shannon entropy as a measure of uncertainty. Section 5 describes a heuristic method for the entropy of interval-valued probabilities. It is assumed that there is a system with n different states. Conclusions are summarized in Section 6.

2. TYPES OF UNCERTAINTY

In the field of uncertainty quantification, uncertainty in the governing equations may assume two uncertainties (i.e., aleatory and epistemic). Aleatory uncertainty can be characterized by known probability distributions whilst epistemic uncertainty arises from a lack of knowledge of probabilistic information. It arises from the inherent variation associated with the system under consideration and is irreducible. The sources of aleatory uncertainty are typically represented by using a probabilistic framework. It is referred to inherent uncertainty due to the attribute of intrinsic randomness and probabilistic variables, which is associated with the physical system.

On the other hand, it is a result of a naturally occurring process, variability in the underlying variables or statistical variability, which is stochastic and more collected data in a given model cannot reduce this uncertainty. Thus, stochastic uncertainty in the sequence of possible events is entirely aleatory by nature. Epistemic uncertainty represents any lack of knowledge or information in any phase or activity of the modeling process [26, 27] and it is reducible through the introduction of additional information. Source of this uncertainty is of data completeness and quality. Hence, it can be reduced by more knowledge that is available. Therefore, epistemic uncertainty is reducible as results of an inaccurate scientific understanding of the natural phenomenon being modeled or data quality, completeness or incomplete knowledge of the underlying process.

Frequently, strong statistical information (e.g., probability distribution functions or high-order statistical moments) is not available. Experimental data needed to construct this information are often expensive. Thus no data or only a small collection of data points may be obtainable consequently. In these cases, expert opinion is used to handle epistemic uncertainty in conjunction with the available data to produce weak inferential estimates of parametric characteristics in a form of lower and upper bounds.

Other sources of epistemic uncertainty include limited understanding or misrepresentation of a modeled process, known commonly as a model specification uncertainty. The inclusion of enough additional information about either the model parameters or structure can lead to a reduction in the predicted uncertainty of a model output. Consequently, we can consider epistemic uncertainty as providing (conservative) bounds on an underlying aleatory uncertainty, in which reduction and convergence to the true aleatory uncertainty (or a constant value in some cases) can be obtained given sufficient additional information [28]. To learn more, the reader may find some information about evidence theory [25, 29], possibility theory [30], interval analysis [31], discrete probability distributions [32], fuzzy arithmetic [33] and probability bounds [34, 35]. These theories are different from each other in terms of characterizing the input parameter uncertainty and kind of propagation from a parameter level to model an output level. In this paper, we propose a framework for quantification of epistemic uncertainty. We calculate the upper and lower bounds for uncertainty based on entropy and D–S theory.

3. D–S THEORY OF EVIDENCE

The D–S theory is a mathematical theory of evidence first proposed by Dempster in early 1967 and then extended by Shafter in 1976. It makes inferences based on uncertain knowledge from different information sources. This theory allows strengthening or erosion of beliefs by combining additional sources of evidence, even in the presence of partly contradictory evidence. There are three critical functions in the D–S theory, namely the basic probability assignment function (BPA or m), the belief function (Bel), and the plausibility function (Pl). For a finite set of mutually exclusive and exhaustive propositions (i.e., Ω), a power set 2^Ω is the set of all the subsets of Ω including itself and a null set ∅. The basic probability assignment is a critical variable of evidence theory and does not refer to probability in the classical sense. For any subset A of 2^Ω, the BPA, represented as m(A), defines a mapping of 2^Ω to the interval between 0 and 1. Formally, this description of. can be represented by:

The upper and lower bounds of an interval can be defined from the BPA. This probability interval contains the precise probability of a set of interest in the classical sense and is bounded by two measures, namely Bel and Pl. The lower bound Bel for a set A is defined as the sum of all the BPAs of the proper subsets (B) of the set of interest (A) (B⊆A). The upper bound Pl is the sum of all the BPAs of the sets (B) that intersect the set of interest (A) (B∩A≠ϕ). According to the definition above, the Bel and Pl for a set A can be described by Eqs. (4) and (5), respectively.

It is readily provable that Bel and Pl have the following relation.

4. SHANNON ENTROPY

Shannon [15] presented entropy as a measurement of the uncertainty level of information. The entropy is a function of the probability distribution function. Let x be a nominal attribute on a finite set X=x1,…,xn with probability distribution function p(x). The entropy H(X) of X is defined by:

The Shannon entropy can be taken as a measure of the uncertainty about the realization of a random variable. Entropy has the following properties: (1) by convention, 0 log 0=0 and (2) in experiment, X=x1,…,xn, if pxi=1 and the rest is equal to 0, then Hx=0, implying that there is not any uncertainty in experiment A and a decisive conclusion can be made. On the contrary, if one knows nothing about experimental results beforehand, then pxi=1/ni=1,2,3,…,n, HX reaches its maximum value Hmaxx=log2n. Obviously, the set has a maximal extent of uncertainty in this case. The Shannon entropy is used to measure the uncertainty level of evidence in this study [15].

5. ENTROPY AND INTERVAL-VALUED PROBABILITY

Uncertainty is usually related to lack of knowledge about future events. Thus, a definition of risk can be considered as the amount of lacking information. Therefore, in this approach, we adopt the idea that uncertainty reflects how much we do not know about the future. Entropy is the basic notion in the information theory field. Informally, we can define entropy as the measure of uncertainty of a system that at a given moment can be in one of n states, in which a set of all the possible states is defined, and the probability of some system states is known. We can define entropy formally as follows.

Let us assume that there are n different states, where a system can be in S1,S2,…,Sn. In the following equation, the probability of system S in state s_i is P_i.

If values of all P_i in Eq. (11) are known, then the entropy of a system can be calculated by:

Let us consider how the entropy can be calculated if we are dealing with the interval-valued probability values. Obviously, the entropy itself will be interval-valued. In this section, we introduce a generalized definition of entropy, suitable for interval-valued probabilities. As before, we assume that system can be in n states; however, the probability that a system is in the i-th state is the interval and is equal to Pimin,Pimax. In this case, when probabilities are single-valued rather than interval-valued, it is required that the sum of probabilities is set to 1 as shown below.

If probabilities are interval-valued, then this constraint can be rewritten by:

It is easy to show that Eq. (13) is a special case of Eq. (14), when Pimin=Pimax (where P_i is the probability for each i). Moreover, if we define Pimin=Beli and Pimax=Pli, it can be shown that Eq. (16) holds.

It should be noted that the states with lower probability values are more informative. Thus, we can expect that in order to calculate the upper boundary of entropy H^max, we should use the lower probability bounds Bel_i. We can find the upper boundary of entropy for a system by:

From Eq. (16), it follows that Hmin<Hmax Moreover, as mentioned above, entropy in a system with interval-valued probability is equal to:

The obtained definition of this interval-valued probability is a generalization of the “traditional” entropy of a system with single-valued probabilities (or mass function). Now let us examine whether the additivity feature holds for the generalized entropy in the D–S theory, as defined in Eqs. (17) and (18). If additivity holds, it means whether we have two independent systems (say, X and Y), then entropy based on the D–S theory of a system obtained by joining systems X and Y is equal to the sum of individual generalized entropies in the D–S theory for X and Y. In other words, if additivity holds, then we have:

If we define generalized entropy in the D–S theory as Eqs. (17) and (18), then it can be shown that we have two systems (i.e., X and Y) with states accordingly x1,x2,…,xn and y1,y2,…,ym, with PX∼xi=pi and PY∼yi=rj. Then, lower and upper bounds of the two systems should be summed separately to obtain the corresponding bounds of the summed entropy.

Therefore, we can determine the approximate uncertainty bounds (i.e., lower and upper uncertainty bounds) of two independent systems that should be summed separately to obtain the corresponding bounds of each system. These values from each interval contribute to H^min and H^max are shown in Fig. 1.

Figure 1

Upper and lower bounds of uncertainty.

In addition, the relative entropy is another important concept of entropy. It is known as a measure of distance between two probability distributions. It is evaluated divergence between probabilities or mass function of two systems X(X=x1,x2,…,xn) and (Y=y1,y2,…,ym), where, PX∼xi=pi and PY∼yi=rj.

Let mxi and myi, i=1,2,…,n be two basic mass functions of systems X and Y. If these values are considered as the estimation of exact probabilities under uncertainty, then the relative entropy of measurement mxi with respect to the measure myi is defined by:

Obviously, if and only if xi=myi, HmX∥mY=0. Thus, if belxi and belyi, where i=1,2,…,n, are belief functions of X and Y, then we have:

Furthermore, the lower bound for the relative entropy with respect to plausibility functions plxi and plyi of X and Y is as follows:

6. CONCLUSION

One way to assess the uncertainty between proposed theories under imprecise probabilities is the D–S theory. In probability theory, the best-known measure of uncertainty in a purely probabilistic system is the Shannon's entropy. This paper incorporates information and probabilities for different states of systems considered as an interval-valued probability, and then a heuristic approach is used to generalize entropy in the D–S theory for an uncertainty model. This approach is proposed to approximate upper and lower uncertainties for a presented formal problem. Since it is hard to solve this formal problem directly, the proposed method is a proper method to find an approximate solution. Accordingly, different examples are presented for the simple proposed method. The results show the efficiency of the method to find an approximate solution to quantify uncertainty bounds. Furthermore, the new definition of uncertainty bounds for two independent systems is explained, which is held by the additivity property. Then, quasi-additivity of uncertainty bounds is shown in Fig. 1. Finally, the upper and lower bounds of relative entropy are presented to evaluate the divergence between two mass functions.

ACKNOWLEDGEMENTS

The authors would like to thank the Editor-in-Chief and autonomous reviewers for their constructive comments and helpful suggestions, which greatly improved the presentation of this paper.