**2. Theoretical Background**

#### *2.1. D–S Evidence Theory*

The D–S evidence theory is a method of uncertainty reasoning, proposed by Dempster in 1967 and later improved and developed by Shafer [28]. The D–S evidence method can produce a probability interval to an uncertain event by fusing multiple evidences with known probability distribution. As an indeterminate reasoning method, D–S evidence theory uses weaker conditions than Bayesian, and has the ability to quantify unknown and uncertainty [29]. The evidence theory contains three important functions: basic probability assignment function, belief function, and plausibility function. The basic probability assignment function is the probability distribution of all possible faults in each state, the belief function is the lower bound of the probability of the fault event, and the plausibility function is the upper bound of the probability of the fault event. The belief function and the plausibility function can be obtained by calculating the sum of the basic probability assignment function, and the final decision is made after combining multiple evidences from different sources.

The D–S evidence theory consists of the following parts [30].

• Frame of discernment:

A variety of possible mutually exclusive hypothesis *Xi*(*i* = 1, 2, ··· ,*s*) of a question constitute a finite and non-empty set, which is called the frame of discernment, denoted as Ω = {*X*1, *X*2, ··· , *Xs*}.

• Basic probability assignment (BPA) function:

BPA function is also known as the mass function. Suppose *H* is a subset of Ω, if function *m*(*H*) satisfying

$$\begin{array}{c} (\mathbf{a}) \ m(\boldsymbol{\phi}) = 0 \\ (\mathbf{b}) \ \sum m(H) = 1 \\ (\mathbf{c}) \ m(H) > 0 \end{array}$$

then, function *m*(*H*) is called the basic probability assignment of *H* on Ω.

• Belief function:

In the frame of discernment, the belief function represents the sum of the basic probability assignment functions of all subsets of *H*. The expression of the belief function is as follows:

$$bel(H) = \sum\_{Y/Y \subseteq H} m(Y) \tag{1}$$

• Plausibility function:

In the frame of discernment, the plausibility function represents the degrees of belief for not denying *H*, which is the sum of the basic probability assignments of all the subsets intersecting *H*. The expression of the plausibility function is as follows:

$$pl(H) = \sum\_{S/S \cap H \neq \mathcal{D}} m(S) \tag{2}$$
