Weighted Evidence Combination Based on Distance of Evidence and Entropy Function

Abstract

Conflict management in Dempster-Shafer theory (D-S theory) is a hot topic in information fusion. In this paper, a new weighted evidence combination on the basis of the distance between evidence and entropy function is presented. The proposed approach is identified as two procedures. First, the weight is determined based on the distance of evidence. Then, the obtained weight value in the first step is modified by making advantage of Deng entropy function. Our proposed method can efficiently cope with high conflicting evidences with better performance of convergence. A numerical example is provided to demonstrate that the proposed method is reasonable and efficient in the end.

1. Introduction

The first emergence of the Dempster-Shafer theory of evidence (D-S theory, also known as evidence theory or theory of belief functions) is identified as an efficient model to reason with uncertainty information in intelligent systems [1]. Dempster's combination rule is the most important tool of D-S theory. This theory was firstly proposed by Dempster in 1967 [1] and has been developed to its present form by his student, Shafer in 1976 [2]. Evidence theory can present and handle uncertainty more preferably than probability theory can [3]. Dempster's combination rule which has some interesting mathematical properties, such as associativity and commutativity, plays an important role in evidence theory [4]. Now the evidence theory is utilized widely in many fields, such as decision making [4–6], supplier selection [7, 8], reliability analysis [9–11], and optimization under uncertain environment [12–14]. Due to the complexity in fault diagnosis [15, 16], sensor data fusion based on evidence theory in these fields is also heavily studied [17, 18].

Although D-S theory has lots of advantages, it will be invalid when highly conflicting evidences are combined and generate counterintuitive results [19–21]. To solve such a problem, two primary methodologies are popular. One is to modify the combined rule [3, 22], and the other is to preprocess the bodies of evidence (BOEs) [23]. There are three popular alternative combination rules that belong to the first type to manage conflict and they are Smets' unnormalized combination rule [24], Dubois and Prade's disjunctive combination rule [25], and Yager's combination rule [3]. The three alternatives mentioned above are all examined and they all proposed a general combination framework. The main work of preprocessing bodies of evidences (BOEs) includes Murphy's simple average in [23], Deng et al.'s weighted average on the basis of distance of evidences in [26], and Han et al.'s modified weighted average in [27]. In [23], a simple averaging approach of the primitive BOEs is proposed, and in that case all BOEs are seen equally important, which is unreasonable in practice. Deng et al. [26] got a better combination result according to combining the weight average of the masses for $n - 1$ times. The approach proposed by Han et al. [27] is a novel weighted evidence combination approach based on the distance of evidence and ambiguity measure (AM) to modify Deng et al.'s work [26].

In this paper, a new uncertainty measure, named Deng entropy, is utilized to address conflicting evidences combination. The numerical example is given to prove the efficiency of the proposed method.

The rest of the paper is organized as follows: Section 2 starts with a brief introduction of the Dempster-Shafer theory, evidence distance, and belief entropy; the proposed method is presented in Section 3; Section 4 gives a numerical example to show the efficiency of the proposed approach; finally, the conclusion is made in Section 5.

2. Preliminaries

In this section, some preliminaries are briefly introduced below.

2.1. Basics of Evidence Theory

Let $2^{Ω}$ be the set of all subsets of Ω, denoted by $2^{Ω} = {⌀, {θ_{1}}, {θ_{2}}, \dots, {θ_{n}}, {θ_{1}, θ_{2}}, \dots, {θ_{1}, θ_{2}, \dots, θ_{n}}}$ . In D-S theory [2], mathematically a basic probability assignment (BPA, also called mass function) is mapping $2^{Ω} \to [0,1]$ that satisfies

\begin{matrix} \sum_{A \subseteq Ω} m (A) = 1, \\ m (⌀) = 0 . \end{matrix}

(1)

m (A) > 0

, A is called a focal element, and the set of all the focal elements is named a body of evidence (BOE). When multiple independent BOEs are available, we can use Dempster's combination rule to obtain the combined evidence as follows:

\begin{matrix} m (A) = \frac{\sum_{B, C \subseteq Ω, B \cap C = A} m_{1} (B) m_{2} (C)}{1 - K}, \end{matrix}

(2)

where

K = \sum_{B \cap C = ⌀} ‍ m_{1} (B) m_{2} (C)

is a normalization constant, called conflict. The combination rule above makes sense only when

m_{\oplus} (⌀) \neq 1

; otherwise, the rule is not meaningful.

Suppose $m_{1}$ and $m_{2}$ are two BOEs belonging to the frame of discernment Ω. If two distinct pieces of evidences $m_{1}$ and $m_{2}$ are both reliable, we can apply the following conjunctive rule [28] to generate a new BPA:

\begin{matrix} m_{⊓} (A) = \sum_{B, C \subseteq Ω, B \cap C = A} m_{1} (B) m_{2} (C), \forall A \subseteq Ω . \end{matrix}

(3)

And if only one of those two BOEs is totally reliable and we are not sure which BOE is totally reliable, we should apply the following disjunctive combination rule [25] to obtain a new BPA as follows:

\begin{matrix} m_{⊔} (A) = \sum_{B, C \subseteq Ω, B \cap C = A} m_{1} (B) m_{2} (C), \forall A \subseteq Ω . \end{matrix}

(4)

Given a proposition $A \in 2^{Ω}$ , the belief function of A, denoted by $B e l (A)$ , is defined as

\begin{matrix} Bel (A) = \sum_{B \subseteq A} m (B) . \end{matrix}

(5)

The plausibility function of A, denoted $P l (A)$ , is defined as follows:

\begin{matrix} Pl (A) = \sum_{A \cap B \neq ⌀} m (B) . \end{matrix}

(6)

$Bel (A)$ represents the total belief that the object is in A and $Pl (A)$ measures the total belief that can move into A. $Bel (A)$ and $Pl (A)$ are called lower bound function and upper bound function, respectively, denoted by $[B e l (A), P l (A)]$ . For any proposition, $B e l (A)$ and $P l (A)$ satisfy the following relations:

\begin{matrix} Pl (A) = 1 - Bel (\bar{A}), \\ Pl (A) \geq Bel (A) . \end{matrix}

(7)

An example of Bel and Pl is given, and the results are shown in Table 1.

Table 1

An example of Bel and Pl.

Function	${a}$	${b}$	${c}$	${a, b}$	${a, c}$	${b, c}$	${a, b, c}$
m	0.4	0	0	0.3	0	0.2	0.1
Bel	0.4	0	0	0.7	0.4	0.2	1
Pl	0.8	0.6	0.3	1	1	0.9	1

Example 1.

Assume $Ω = {a, b, c}$ . and One BOE is given with $m ({a}) = 0.4$ , $m ({a, b}) = 0.3$ , $m ({b, c}) = 0.2$ , and $m ({a, b, c}) = 0.1$ . Bel and Pl can be calculated, and the results are shown in Table 1.

But Dempster's combination rule is not always in force. When BOEs are in high conflict, illogical results will be created [20, 29]. There are some methods to measure the conflict or confusion in a belief function [30]. To preprocess data, such as Murphy's simple average in [23], Deng et al.'s weighted average on the basis of distance of evidences in [26], and Han et al.'s modified weighted average in [27], and to amend the combination rule, such as Smets' unnormalized combination rule [24], Dubois and Prade's disjunctive combination rule [25], and Yager's combination rule [3], are two main methods to fix up highly conflicting evidences combination efficiently. It should be pointed that there are some other combination rules to address the dependence evidence issue, which may be another alternative to handle conflicts [31–34].

2.2. Evidence Distance

With D-S theory applying widely, the study about the distance of evidence has attracted more and more interests [35, 36]. The dissimilarity measure of evidence can represent the lack of similarity between two BOEs. Conflict evidence combination [26], target association [37], and lots of methods of evidence distance are brought up as an appropriate measure of the difference. And several definitions on distance in evidence theory are also proposed, such as Jousselme's distance [38], Wen's cosine similarity [39], Smets' transferable belief model (TBM) global distance measure [37], and Sunberg's belief function distance metric [40]. Among those definitions on the distance of evidence, the most frequently used one is Jousselme's distance [38].

Jousselme's distance [38] is identified on the basis of Cuzzolin's geometric interpretation of evidence theory [41]. The power set of the frame of discernment $2^{Ω}$ is regarded as a $2^{N} - l i n e a r$ space and a distance and vectors are defined with the BPA as a particular case of vectors. Jousselme's distance is defined as

\begin{matrix} d_{i j} = \sqrt{\frac{1}{2} {(\vec{m_{i}} - \vec{m_{j}})}^{T} D (\vec{m_{i}} - \vec{m_{j}})} \end{matrix}

(8)

with

m_{i}, m_{j}

being two BPAs under the frame of discernment Ω, and D is

2^{N} \times 2^{N}

matrix. The element in D is defined as

D (A, B) = | A \cap B | / | A \cup B |

A, B \in P (Ω)

, and

|\cdot|

represents cardinality. This Jousselme distance satisfies all four requirements (nonnegativity, nondegeneracy, symmetry, and triangle inequality) [38] of a strict distance metric. Jousselme's distance is an efficient tool to quantify the dissimilarity of two BOEs. An example of Jousselme's distance is shown below.

Example 2.

Assume there are two BOEs $S_{1}$ and $S_{2}$ :

$S_{1}$ : $m_{1} (A) = 0.4$ , $m_{1} (B) = 0.2$ , $m_{1} (A, B) = 0.1$ , and $m_{1} (A, B, C) = 0.3$ ;

$S_{2}$ : $m_{2} (A) = 0.5$ , $m_{2} (B) = 0.2$ , $m_{2} (A, B) = 0.1$ , and $m_{2} (A, B, C) = 0.2$ .

The values inside the BOE vectors $\vec{m_{1}}$ and $\vec{m_{2}}$ and the distance matrix D are given by

\begin{matrix} \vec{m_{1}} = (\begin{pmatrix} 0.4 \\ 0.2 \\ 0.1 \\ 0.3 \end{pmatrix}), \\ \vec{m_{2}} = (\begin{pmatrix} 0.5 \\ 0.2 \\ 0.1 \\ 0.2 \end{pmatrix}), \\ D = (\begin{pmatrix} 1 & 0 & \frac{1}{2} & \frac{1}{3} \\ 0 & 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{2} & 1 & \frac{2}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} & 1 \end{pmatrix}) . \end{matrix}

(9)

It follows that $(\vec{m_{1}} - \vec{m_{2}})^{T} = (\begin{pmatrix} - 0.1 & 0 & 0 & 0.1 \end{pmatrix})$ and $(\vec{m_{1}} - \vec{m_{2}}) = (\begin{smallmatrix} - 0.1 \\ 0 \\ 0 \\ 0.1 \end{smallmatrix})$ :

\begin{matrix} d_{(m_{1}, m_{2})} = \sqrt{\frac{1}{2} (\begin{pmatrix} - 0.1 & 0 & 0 & 0.1 \end{pmatrix}) (\begin{pmatrix} 1 & 0 & \frac{1}{2} & \frac{1}{3} \\ 0 & 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{2} & 1 & \frac{2}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{2}{3} & 1 \end{pmatrix}) (\begin{pmatrix} - 0.1 \\ 0 \\ 0 \\ 0.1 \end{pmatrix})} = 0.1125 . \end{matrix}

(10)

2.3. Deng Entropy

Uncertainty is widespread in universe [42–45]. If a probability assignment p is provided, we can apply the following Shannon entropy [46] to measure its uncertainty:

\begin{matrix} H = - \sum_{i} p_{i} \log_{2} (p_{i}) . \end{matrix}

(11)

But if BPA is given, there is no way to measure that uncertainty based on some other main entropies listed in Table 2.

Table 2

Some main entropies to measure uncertainty.

Proposer	Equation
Dubois and Prade [54]	$N_{D P} (m) = \sum_{A \subseteq X}^{} m (A) \log (\|A\|)$
Yager [55]	$C_{Y} (m) = - \sum_{A \subseteq F}^{} m (A) \log P l (A)$
Klir and Ramer [30]	$C_{K R} (m) = - \sum_{A \subseteq F}^{} m (A) \log \sum_{B \subseteq F}^{} m (B) \frac{\|A \cap B\|}{\|B\|}$
George and Pal [56]	$C_{G P} (m) = - \sum_{A \subseteq F}^{} m (A) \sum_{B \subseteq F}^{} m (B) (1 - \frac{\|A \cap B\|}{\|A \cup B\|})$
Maluf [57]	$C_{M} (m) = - \sum_{A \subseteq F}^{} P l (A) \log B e l (A)$
Klir and Wierman [58]	$C_{K W} (B e l) = - \frac{1}{c} \sum_{x \subseteq X}^{} B e l (\{x\}) \log B e l (\{x\}) + P l (\{x\}) \log P l (\{x\}), c = \sum_{x \subseteq X}^{} [B e l (\{x\}) + P l (\{x\})]$

As for such a reason, Deng entropy [47] is presented to measure the uncertainty of BPA, which is a more significant tool to manage uncertainty than Shannon entropy [46]. Deng entropy can deal with the uncertainty represented not only by BPA but also by probability distribution. In other words, Deng entropy is the generalization of Shannon entropy [34, 48].

Deng entropy can be denoted as follows:

\begin{matrix} E_{d} = - \sum_{i} m (F_{i}) \log_{2} (\frac{m (F_{i})}{2^{|F_{i}|} - 1}), \end{matrix}

(12)

where

F_{i}

is a proposition in mass function m and

| F_{i} |

is the cardinality of

F_{i}

. In particular, if the belief is only assigned to single elements, the Deng entropy equals Shannon entropy [47]. The following is denoted:

\begin{matrix} E_{d} = - \sum_{i} m (θ_{i}) \log_{2} (\frac{m (θ_{i})}{2^{|θ_{i}|} - 1}) = - \sum_{i} m (θ_{i}) \log_{2} m (θ_{i}) . \end{matrix}

(13)

An example of Deng entropy is given and the results are seen in Table 3.

Table 3

An example of Deng entropy.

Source of evidence	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$	$S_{5}$
Deng entropy [47, 49]	3.2061	2.1559	3.2886	1.5850	2.8074

Example 3.

Assume $Ω = {A, B, C}$ :

$S_{1}$ : $m_{1} (A) = 0.3$ , $m_{1} (A, B) = 0.2$ , $m_{1} (A, B, C) = 0.5$ ;

$S_{2}$ : $m_{2} (B) = 0.2$ , $m_{2} (B, C) = 0.4$ , $m_{2} (C) = 0.4$ ;

$S_{3}$ : $m_{3} (A) = 0.2$ , $m_{3} (B) = 0.2$ , $m_{3} (A, C) = 0.3$ , $m_{3} (A, B, C) = 0.3$ ;

$S_{4}$ : $m_{4} (A) = 1 / 3$ , $m_{4} (B) = 1 / 3$ , $m_{4} (C) = 1 / 3$ ;

$S_{5}$ : $m_{5} (A, B, C) = 1$ .

3. The Proposed Method

The flow graph of our proposed method is shown in Figure 1.

Figure 1

The flow graph of our proposed method.

Supposing that we collect k BOEs $m_{i}$ , $i = 1, \dots, k$ ; the pretreatment [49] of those BOEs can be clarified as

\begin{matrix} M A E (m) = \sum_{i = 1}^{k} w_{i} m_{i}, \end{matrix}

(14)

where

w_{i}

is the corresponding weight degree of BOE

m_{i}

, and

M A E (m)

denoted the weighted average BPA of k pieces of primitive evidences. By making use of classical Dempster's rule to combine

M A E (m)

for

k - 1

times, one can get the final outcome. However, it is a bit difficult to find an appropriate weight

w_{i}

. In our previous work, Deng et al. [26] put forward an evidence combination based on weighted average of evidences. Han et al.'s approach [27] which reposed Deng et al.'s [26] includes two steps: firstly, generate the weights based on the distance of evidence; then, modify the weights generated in the first step by taking advantage of ambiguity measure (AM) [50]. We come up with a new modified weighted evidence combination because of the superior efficiency of Deng entropy [47, 49] compared to AM [50] to measure uncertainty, which is confirmed through Example 4 of the comparison between AM and Deng entropy. For convenience, we still make use of Example 3.

Example 4.

Assume $Ω = {A, B, C}$ :

$S_{1}$ : $m_{1} (A) = 0.3$ , $m_{1} (A, B) = 0.2$ , $m_{1} (A, B, C) = 0.5$ ;

$S_{2}$ : $m_{2} (B) = 0.2$ , $m_{2} (B, C) = 0.4$ , $m_{2} (C) = 0.4$ ;

$S_{3}$ : $m_{3} (A) = 0.2$ , $m_{3} (B) = 0.2$ , $m_{3} (A, C) = 0.3$ , $m_{3} (A, B, C) = 0.3$ ;

$S_{4}$ : $m_{4} (A) = 1 / 3$ , $m_{4} (B) = 1 / 3$ , $m_{4} (C) = 1 / 3$ ;

$S_{5}$ : $m_{5} (A, B, C) = 1$ .

In particular we analyze and compare two BPAs from $S_{4}$ and $S_{5}$ . The BPA of $S_{4}$ , $m_{4} (A) = 1 / 3$ , $m_{4} (B) = 1 / 3$ , $m_{4} (C) = 1 / 3$ , shows that each proposition in system happens with equal probability, which is not the same with the opinion that we know nothing about the system. For example, suppose we will win 8 dollars with frontal side emerging and win 3 dollars with the emergence of opposite side in Game Playing. Then we should bet the frontal side according to average profit. The BPA of $S_{5}$ , $m_{5} (A, B, C) = 1$ , indicates that we know nothing about this system and we do not know how to distribute probability to each proposition. Maybe it will be like $m_{5} (A) = 0.6$ , $m_{5} (B) = 0.1$ , $m_{5} (C) = 0.3$ or like $m_{5} (A) = 0.2$ , $m_{5} (B) = 0.2$ , $m_{5} (C) = 0.6$ and so forth. Everything will be possible. Therefore, the uncertainty of $S_{5}$ should be greater than that of $S_{4}$ . However, as seen from Table 4 and Figure 2, $S_{5}$ 's outcome measured by AM is illogical, but $S_{5}$ 's uncertainty measured by Deng entropy is consistent with logic. So, we have enough confidence to say that Deng entropy is a more efficient tool than AM to measure uncertainty.

Table 4

An example of comparison between AM and Deng entropy.

Uncertainty	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$	$S_{5}$
Uncertainty measured by AM [50]	1.4037	0.9710	1.5395	1.5850	1.5850
Uncertainty measured by Deng entropy [47]	3.2061	2.1559	3.2886	1.5850	2.8074

Figure 2

Uncertainty measured by AM and Deng entropy.

3.1. Determining Weight with Evidence Distance

The distance of evidence is listed in (8). The less the distance between two BOEs is, the more the similarity of those two is. The similarity measure $S i m_{i j}$ between two BOEs can be obtained [51] as

\begin{matrix} S i m (m_{i}, m_{j}) = 1 - d (m_{i}, m_{j}) . \end{matrix}

(15)

The support degree of BOE

m_{i} (i = 1,2, \dots, k)

is defined [26] below:

\begin{matrix} Sup (m_{i}) = \sum_{j = 1, j \neq i}^{k} S i m (m_{i}, m_{j}) . \end{matrix}

(16)

The credibility degree of BOE

m_{i} (i = 1,2, \dots, k)

is obtained [26] as follows:

\begin{matrix} C r d_{i} = \frac{Sup (m_{i})}{\sum_{j = 1}^{k} Sup (m_{j})} . \end{matrix}

(17)

In fact, the credibility degree

C r d_{i}

can be directly regarded as a weight because

\sum_{i = 1}^{k} ‍ C r d_{i} = 1

is evident. Thus, we could make use of

C r d_{i}

to directly replace

w_{i}

in (14) in order to modify original BPAs. As mentioned in [27, 52], the uncertainty degree can also be utilized to construct the weights. A preferable combination result has been gotten by using AM to modify

C r d_{i}

acquired based on the distance of evidence. Due to Deng entropy's better performance of measuring uncertainty than AM, in this paper,

C r d_{i}

is modified by applying Deng entropy as illustrated below.

3.2. Weight Modification on the Basis of Entropy Function

Supposing that one of some bodies of evidence, which have relatively high credibility degree generated in the first step, has less uncertainty degree than the others. We believe that the BOE is more credible and it should possess more weight because of its good quality. On the contrary, if a BOE has both a low credibility degree and a more uncertainty degree, such a BOE is relatively incredible and even causes a wrong result perhaps. So this BOE should possess a less weight.

Based on the thoughts mentioned above, we can modify the weight produced based on the distance of evidence by means of the following steps.

Step 1.

Compute Deng entropy [47] of each BOE $m_{i} (i = 1,2, \dots, k)$ , denoted as $D E (m_{i})$ .

Step 2.

Normalize the obtained $D E (m_{i})$ according to the following equation:

\begin{matrix} D E n (m_{i}) = \frac{D E (m_{i})}{\sum_{i = 1}^{k} D E (m_{i})}, \end{matrix}

(18)

where

D E n (m_{i})

is the normalized uncertainty degree of

m_{i}

Step 3.

Generate the modified weight denoted as follows:

\begin{matrix} C r d m (m_{i}) = C r d (m_{i}) D E n {(m_{i})}^{- Δ C r d (m_{i})} (i = 1,2, \dots, k), \end{matrix}

(19)

where

Δ C r d (m_{i}) = C r d (m_{i}) - (1 / k) \sum_{j = 1}^{k} ‍ C r d (m_{j})

is the difference between the average credibility of all BOEs and a BOE's credibility

C r d (m_{i})

Step 4.

Normalize all $C r d m (m_{i})$ as shown below:

\begin{matrix} C r d m n (m_{i}) = \frac{C r d m n (m_{i})}{\sum_{i = 1}^{k} C r d m n (m_{i})} (i = 1,2, \dots, k) . \end{matrix}

(20)

Step 5.

The weighted averaged BOE denoted $M A E (m)$ is acquired as

\begin{matrix} M A E (m) = \sum_{i = 1}^{k} (C r d m n (m_{i}) m_{i}) (i = 1,2, \dots, k) . \end{matrix}

(21)

If n pieces of evidence are supplied, we can use classical Dempster's rule [1] to combine $M A E (m)$ for $n - 1$ times [23]. After that, we could get a better combined outcome to make a better decision.

4. Experiment

In this section, a numerical example is provided to demonstrate the effectiveness of our proposed method.

Example 5.

In a multisensor-based automatic target recognition system, suppose that the frame of discernment $Ω = {A, B, C}$ is complete and A is the real target. Form five different sensors, the system collects five BOEs listed as follows:

$S_{1}$ : $m_{1} (A) = 0.41$ , $m_{1} (B) = 0.29$ , $m_{1} (C) = 0.3$ ;

$S_{2}$ : $m_{2} (A) = 0$ , $m_{2} (B) = 0.9$ , $m_{2} (C) = 0.1$ ;

$S_{3}$ : $m_{3} (A) = 0.58$ , $m_{3} (B) = 0.07$ , $m_{3} (A, C) = 0.35$ ;

$S_{4}$ : $m_{4} (A) = 0.55$ , $m_{4} (B) = 0.1$ , $m_{4} (A, C) = 0.35$ ;

$S_{5}$ : $m_{5} (A) = 0.6$ , $m_{5} (B) = 0.1$ , $m_{5} (A, C) = 0.3$ .

Through the first step of our proposed method, we can get the credibility degree; that is, we can determine the weight of each BOE based on the distance of evidence. The results are shown in Table 5.

Table 5

The weight determined by the distance of evidence.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
$d (m_{1}, m_{2})$	0.5386	0.5386	0.5386	0.5386
$d (m_{1}, m_{3})$	—	0.3495	0.3495	0.3495
$d (m_{1}, m_{4})$	—	—	0.3257	0.3257
$d (m_{1}, m_{5})$	—	—	—	0.3311
$d (m_{2}, m_{3})$	—	0.8142	0.8142	0.8142
$d (m_{2}, m_{4})$	—	—	0.7850	0.7850
$d (m_{2}, m_{5})$	—	—	—	0.7906
$d (m_{3}, m_{4})$	—	—	0.0300	0.0300
$d (m_{3}, m_{5})$	—	—	—	0.0374
$d (m_{4}, m_{5})$	—	—	—	0.0354
$C r d (m_{1})$	0.5000	0.4284	0.2829	0.2059
$C r d (m_{2})$	0.5000	0.2494	0.1365	0.0899
$C r d (m_{3})$	—	0.3222	0.2861	0.2321
$C r d (m_{4})$	—	—	0.2945	0.2368
$C r d (m_{5})$	—	—	—	0.2353

Then, the weight generated in the first step is modified by using Deng entropy. The final weight of each BOE, denoted by $C r d m n$ , is listed in Table 6.

Table 6

The modified weight determined based on Deng entropy.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
$E d (m_{1})$	1.5664	1.5664	1.5664	1.5664
$E d (m_{2})$	0.4690	0.4690	0.4690	0.4690
$E d (m_{3})$	—	1.8092	1.8092	1.8092
$E d (m_{4})$	—	—	1.8914	1.8914
$E d (m_{5})$	—	—	—	1.7710
$C r d m n (m_{1})$	0.5000	0.4688	0.2936	0.2050
$C r d m n (m_{2})$	0.5000	0.2100	0.1022	0.0653
$C r d m n (m_{3})$	—	0.3211	0.2966	0.2397
$C r d m n (m_{4})$	—	—	0.3076	0.2457
$C r d m n (m_{5})$	—	—	—	0.2443

Next, we can get $M A E (m)$ by means of replacing $w_{i}$ of (14) with $C r d m n$ .

Finally, we make use of classical Dempster's rule [1] to combine $M A E (m)$ shown in Table 7 for $n - 1$ times [23]. After that, the final combined outcome will be obtained shown in Table 8.

Table 7

MAE(m) on the basis of the new modified weight averaging approach.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
MAE(m)	$m (A) = 0.2050$	$m (A) = 0.3785$	$m (A) = 0.4616$	$m (A) = 0.5048$
	$m (B) = 0.5950$	$m (B) = 0.3475$	$m (B) = 0.2286$	$m (B) = 0.1840$
	$m (C) = 0.2000$	$m (C) = 0.1617$	$m (C) = 0.0983$	$m (C) = 0.0680$
	$m (A C) = 0$	$m (A C) = 0.1123$	$m (A C) = 0.2115$	$m (A C) = 0.2432$

Table 8

The final BPA by the proposed method.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
Final BPA	$m (A) = 0.0964$	$m (A) = 0.6495$	$m (A) = 0.9577$	$m (A) = 0.9904$
	$m (B) = 0.8119$	$m (B) = 0.2367$	$m (B) = 0.0129$	$m (B) = 0.0009$
	$m (C) = 0.0917$	$m (C) = 0.1065$	$m (C) = 0.0200$	$m (C) = 0.0068$
	$m (A C) = 0$	$m (A C) = 0.0079$	$m (A C) = 0.0094$	$m (A C) = 0.0019$

We also make advantage of different combination rules to calculate Example 5, and the results are all shown in Table 9 and the comparison is shown in Figures 3–6.

Table 9

Evidence combination outcomes based on different combination rules.

Approach	Combination outcomes
Approach	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
Dempster's rule [1]	$m (A) = 0$	$m (A) = 0$	$m (A) = 0$	$m (A) = 0$
	$m (B) = 0.8969$	$m (B) = 0.6575$	$m (B) = 0.3321$	$m (B) = 0.1422$
	$m (C) = 0.1031$	$m (C) = 0.3425$	$m (C) = 0.6679$	$m (C) = 0.8575$

Murphy's simple average [23]	$m (A) = 0.0964$	$m (A) = 0.4619$	$m (A) = 0.8362$	$m (A) = 0.9620$
	$m (B) = 0.8119$	$m (B) = 0.4497$	$m (B) = 0.1147$	$m (B) = 0.0210$
	$m (C) = 0.0917$	$m (C) = 0.0794$	$m (C) = 0.0410$	$m (C) = 0.0138$
	$m (A C) = 0$	$m (A C) = 0.0090$	$m (A C) = 0.0081$	$m (A C) = 0.0032$

Deng et al.'s weighted average [26]	$m (A) = 0.0964$	$m (A) = 0.4974$	$m (A) = 0.9089$	$m (A) = 0.9820$
	$m (B) = 0.8119$	$m (B) = 0.4054$	$m (B) = 0.0444$	$m (B) = 0.0039$
	$m (C) = 0.0917$	$m (C) = 0.0888$	$m (C) = 0.0379$	$m (C) = 0.0107$
	$m (A C) = 0$	$m (A C) = 0.0084$	$m (A C) = 0.0089$	$m (A C) = 0.0034$

Han et al.'s novel weighted average [27]	$m (A) = 0.0964$	$m (A) = 0.5188$	$m (A) = 0.9246$	$m (A) = 0.9844$
	$m (B) = 0.8119$	$m (B) = 0.3802$	$m (B) = 0.0300$	$m (B) = 0.0023$
	$m (C) = 0.0917$	$m (C) = 0.0926$	$m (C) = 0.0362$	$m (C) = 0.0099$
	$m (A C) = 0$	$m (A C) = 0.0084$	$m (A C) = 0.0092$	$m (A C) = 0.0034$

Our proposed method	$m (A) = 0.0964$	$m (A) = 0.6495$	$m (A) = 0.9577$	$m (A) = 0.9904$
	$m (B) = 0.8119$	$m (B) = 0.2367$	$m (B) = 0.0129$	$m (B) = 0.0009$
	$m (C) = 0.0917$	$m (C) = 0.1065$	$m (C) = 0.0200$	$m (C) = 0.0068$
	$m (A C) = 0$	$m (A C) = 0.0079$	$m (A C) = 0.0094$	$m (A C) = 0.0019$

Figure 3

The outcome of $m (A)$ based on different combination rules.

Figure 4

The outcome of $m (B)$ based on different combination rules.

Figure 5

The outcome of $m (C)$ based on different combination rules.

Figure 6

The outcome of $m (A, C)$ based on different combination rules.

As seen from Table 8 and Figures 3–6, when evidences are in high conflict, classical Dempster's combination rule produces counterintuitive results that do not reflect the truth. With incremental BOEs, although Murphy's simple averaging [23], Deng et al.'s weighted averaging [26], and Han et al.'s novel weight averaging [27] all give reasonable results, their results are all inferior to the outcomes of our proposed approach. Moreover, the performance of convergence of the proposed method is better than any existing method. The main reason for these phenomena mentioned above is that, by making use of the distance of evidence [38] and Deng entropy [47], the effect of credible evidence is strengthened extremely and the “bad” evidence has less effect on the final combined outcomes. The numerical example demonstrates adequately that the proposed method is of efficiency.

5. Conclusion

In this paper, a new modified weighted evidence combination method on the basis of the distance of evidence [35, 51, 53] and Deng entropy [47] is brought up. The proposed method preserves all the desirable properties of Murphy's simple averaging [23], Deng et al.'s weighted averaging [26], and Han et al.'s novel weighted averaging [27]. Comparing all existing methods, the results of our proposed approach converge fastest when handling high conflicting evidences.

Footnotes

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The work is partially supported by National High Technology Research and Development Program of China (863 Program) (Grant no. 2013AA013801), National Natural Science Foundation of China (Grant nos. 61174022, 61573290, and 61503237), and China State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant no. BUAA-VR-14KF-02).

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