A novel weighted evidence combination rule based on improved entropy function with a diagnosis application

Abstract

Managing conflict in Dempster–Shafer theory is a popular topic. In this article, we propose a novel weighted evidence combination rule based on improved entropy function. This newly proposed approach can be mainly divided into two steps. First, the initial weight will be determined on the basis of the distance of evidence. Then, this initial weight will be modified using improved entropy function. This new method converges faster when handling high conflicting evidences and greatly reduces uncertainty of decisions, which can be demonstrated by a numerical example where the belief degree is raised up to 0.9939 when five evidences are in conflict, an application in faulty diagnosis where belief degree is increased hugely from 0.8899 to 0.9416 when compared with our previous works, and a real-life medical diagnosis application where the uncertainty of decision is reduced to nearly 0 and the belief degree is raised up to 0.9989.

Keywords

Dempster–Shafer evidence theory information fusion entropy function faulty diagnosis medical diagnosis

Introduction

Information fusion technology is a powerful tool to analyze and handle multi-source uncertain information comprehensively. Dempster–Shafer (D-S) theory of evidence, one kind of information fusion technology, is quite popular with academics to reason with uncertainty information in intelligent systems. D-S theory is presented first in 1967 by Dempster¹ and then improved by Dempster’s student, Shafer.² Compared to traditional probability theory, D-S theory can present and handle uncertainty better. Now this theory has been applied widely and successfully in many fields, such as decision-making,^3–6 target recognition,^7–9 supplier selection,^10,11 reliability analysis,^12,13 and optimization under uncertain environment.^14,15

Though D-S theory has plenty of advantages, there are still several basic and popular problems. Among those problems, one of the most important is that D-S theory will fail to fuse highly conflicting evidences and will cause counter-intuitive results.^16–19 There are two main directions to deal with this kind of problem. One is to pre-process the bodies of evidence^20,21 and the other is to modify the combination rule.^22,23 Generally, in the study of Murphy,²² a simple averaging approach is presented, where all bodies of evidence are seen equally important. However, this idea is not reasonable in practice. In 2004, Deng et al.²³ proposed a weighted combination rule which combines the weight average of the masses for $n - 1$ times to get a better combination result according to combining the weight average of the masses for $n - 1$ times.

In this article, a novel weighted evidence combination rule based on evidence distance and improved entropy function is proposed. This newly proposed approach mainly consists of two steps. First, the initial weight will be determined on the basis of the distance of evidence. Then, this initial weight will be modified using improved entropy function. A numerical example and an application demonstrate the efficiency and effectiveness of this new method.

The remaining article is organized as follows. Section “Preliminaries” starts with preliminaries about D-S theory, evidence distance, and improved belief entropy. In section “The proposed method,” the proposed method is presented. Sections “Experiment” and “Application” give a numerical example and an application in faulty diagnosis. In section “Real-life medical application,” we show a real-life medical diagnosis application. Finally, a comprehensive conclusion is made in section “Conclusion.”

Preliminaries

In this section, some preliminaries are briefly introduced.

Basics of evidence theory

D-S theory of evidence is used for dealing with uncertainty information as an efficient mathematical model in intelligent systems.¹ In 1967, the definition of D-S theory was proposed by Dempster, and then his student Shafer² developed this theory in 1976.

Let $Ω$ be a nonempty finite set and $2^{Ω}$ be the set of all subsets of $Ω$ , denoted as $2^{Ω} = {\emptyset, {θ_{1}}, {θ_{2}}, \dots {θ_{n}},$ ${θ_{1}, θ_{2}}, \dots, {θ_{1}, θ_{2}, \dots, θ_{n}}}$ . In D-S theory,² a basic probability assignment (BPA) is mapping, $2^{Ω} \to [0, 1]$ , that satisfies equations (1) and (2)

\sum_{E \subseteq Ω} m (E) = 1

(1)

m (\emptyset) = 0

(2)

If $m (E) > 0$ , where E is called the focal element, and the set of all the focal elements is called one body of evidences (BOEs). When there are more than one independent BOE, Dempster’s combination rule, equation (3) which is a powerful and important tool in D-S theory, can be used to combine these evidences

m (E) = \frac{\sum_{E_{1}, E_{2} \subseteq Ω, E_{1} \cap E_{2} = E} m_{1} (E_{1}) m_{2} (E_{2})}{1 - K}

(3)

where $K = \sum_{E_{1} \cap E_{2} = \emptyset} m_{1} (E_{1}) m_{2} (E_{2})$ stands for the conflict degree, also called normalization constant. What’s noted is that the combination rule above makes sense only when the conflict degree $m_{\oplus} (\emptyset) \neq 1$ ; otherwise, the rule is not meaningful. Here, we give a specific example about combination rule and show the corresponding results in Table 1.

Table 1.

An example of combination rule.

BOEs	$m (E_{1})$	$m (E_{2})$	$m (E_{3})$
$S_{1}$	0.5	0.2	0.3
$S_{2}$	0.6	0.2	0.2
Combined outcome	0.75	0.1	0.15

Example 1

Suppose that the frame of discernment ${E_{1}, E_{2}, E_{3}}$ is complete and there are two BOEs listed as follows

S_{1} : m_{1} (E_{1}) = 0.5, m_{1} (E_{2}) = 0.2, m_{1} (E_{3}) = 0.3

S_{2} : m_{2} (E_{1}) = 0.6, m_{2} (E_{2}) = 0.2, m_{2} (E_{3}) = 0.2

In the frame of discernment $Ω$ , there are two BOEs, $m_{1}$ and $m_{2}$ . When $m_{1}$ and $m_{2}$ are both reliable, to generate a new BPA, we can apply the following conjunctive rule,²⁴ denoted in equation (4). When only one of them is totally reliable and we are not sure about the other one, then we should apply the disjunctive combination rule to obtain a new BPA as in equation (5)

m_{⊓} (E) = \sum_{E_{1}, E_{2} \subseteq 2^{Ω}, E_{1} \cap E_{2} = E} m_{1} (E_{1}) m_{2} (E_{2}), \forall E \subseteq Ω

(4)

m (E) = \sum_{E_{1}, E_{2} \subseteq 2^{Ω}, E_{1} \cap E_{2} = E} m_{1} (E_{1}) m_{2} (E_{2}), \forall E \subseteq Ω

(5)

Given a proposition $E_{1} \in 2^{Ω}$ , the belief function of $E_{1}$ , denoted as $Bel (E_{1})$ , is defined in equation (6), which represents the total belief that the object is in $E_{1}$ . The plausibility function of $E_{1}$ , $Pl (E_{1})$ , is defined in equation (7), which measures the total belief that can move into $E_{1}$ . In D-S theory, $Bel (E_{1})$ and $Pl (E_{1})$ are called the lower bound function and upper bound function, respectively, denoted as $[Bel (E_{1}), Pl (E_{2})]$ . For any proposition, $Bel (E_{1})$ and $Pl (E_{1})$ satisfy the following relations of equations (8) and (9)

Bel (E_{1}) = \sum_{E_{2} \subseteq E_{1}} m (E_{2})

(6)

The plausibility function of A, denoted as Pl(A), is defined as follows

Pl (E_{1}) = \sum_{E_{1} \cap E_{2} \neq \emptyset} m (E_{2})

(7)

Pl (E_{1}) = 1 - Bel (\bar{E_{1}})

(8)

Pl (E_{1}) \geq Bel (E_{1})

(9)

Here, we give an example about belief function and plausibility function and show its results in Table 2.

Table 2.

An example of Bel and Pl.

Function	${E_{1}}$	${E_{2}}$	${E_{3}}$	${E_{1}, E_{2}}$	${E_{1}, E_{3}}$	${E_{2}, E_{3}}$	${E_{1}, E_{2}, E_{3}}$
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 2

Assume that $Ω = {E_{1}, E_{2}, E_{3}}$ , a BPA is given that $m ({E_{1}}) = 0.4$ , $m ({E_{1}, E_{2}}) = 0.3$ , $m ({E_{2}, E_{3}}) = 0.2$ , and $m ({E_{1}, E_{2}, E_{3}}) = 0.1$ .

But the classical Dempster’s combination rule is not efficient all the time. When BOEs are in high conflict, illogical results will be generated.^25–27

Nowadays, there are mainly two kinds of methodologies. One is to modify the combined rule, and the other is to pre-process evidences. Smets’²⁸ unnormalized combination rule, Dubois and Prade’s²⁹ disjunctive combination rule, and Yager’s²⁰ combination rule belong to the first category. These three alternatives mentioned above are examined and they all proposed a general combination framework. To pre-process data, Murphy’s²² simple average and Deng et al.’s²³ weighted average are popular. In the study of Murphy,²² a simple averaging approach of the primitive BOEs is proposed. And in that case, all BOEs are seen equally important, which is unreasonable in real life. From the study of Deng et al.,²³ we can get a better combination result by combining the weight average of the masses for $n - 1$ times.

Evidence distance

With the wide application of D-S theory, the study about the distance of evidence has attracted more and more interests.³⁰ The dissimilarity measure of evidence can represent the lack of similarity between two BOEs. Performance evaluation,³¹ reliability evaluation,³² conflict evidence combination,²³ target association,³³ 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 et al.’s³⁴ distance, Wen et al.’s³⁵ cosine similarity, Ristic and Semts’³³ transferable belief model (TBM) global distance measure, and Sunberg and Rogers’s³⁶ belief function distance metric. Among those definitions on the distance of evidence, the most frequently used is Jousselme et al.’s³⁴ distance.

Jousselme et al.’s³⁴ distance is identified on the basis of Cuzzolin’s³⁷ geometric interpretation of evidence theory. The power set of the frame of discernment $2^{Ω}$ is regarded as a $2^{N} - linear$ space, and a distance and vectors are defined with the BPA as a particular case of vectors. The Jousselme et al.’s distance is defined as

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

(10)

where $m_{i}, m_{j}$ are two BPAs under the frame of discernment $Ω$ , and D is a $2^{N} \times 2^{N}$ matrix. The element in D is defined as $D (E_{1}, E_{2}) = \frac{| E_{1} \cap E_{2} |}{| E_{1} \cup E_{2} |}, E_{1}, E_{2} \in P (2^{Ω})$ , and |·| represents cardinality. This Jousselme et al.’s³⁴ distance satisfies all four requirements (non-negativity, non-degeneracy, symmetry, and triangle inequality) of a strict distance metric. Jousselme et al.’s distance is an efficient tool to quantify the dissimilarity of two BOEs. An example of Jousselme et al.’s distance is shown below.

Example 3

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

S_{1} : m_{1} (E_{1}) = 0.4 m_{1} (E_{2}) = 0.2 m_{1} (E_{1}, E_{2}) = 0.1 m_{1} (E_{1}, E_{2}, E_{3}) = 0.3

S_{2} : m_{2} (E_{1}) = 0.5 m_{2} (E_{2}) = 0.2 m_{2} (E_{1}, E_{2}) = 0.1 m_{2} (E_{1}, E_{2}, E_{3}) = 0.2

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

\vec{m_{1}} = (\begin{matrix} 0.4 \\ 0.2 \\ 0.1 \\ 0.3 \end{matrix}), \vec{m_{2}} = (\begin{matrix} 0.5 \\ 0.2 \\ 0.1 \\ 0.2 \end{matrix}) and D = (\begin{matrix} 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{matrix})

It follows, $(\vec{m_{1}} - \vec{m_{2}})^{T} = (\begin{matrix} - 0.1 & 0 & 0 & 0.1 \end{matrix})$ and $(\vec{m_{1}} - \vec{m_{2}}) = (\begin{matrix} - 0.1 \\ 0 \\ 0 \\ 0.1 \end{matrix})$

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

Improved belief entropy

Uncertainty is widespread in universe.^38–42 If a probability assignment p is provided, we can apply Shannon’s⁴³ entropy to measure its uncertainty

H = - \sum_{i} p_{i} \log_{2} (p_{i})

(11)

But if a BPA is given, there is no way to measure that uncertainty based on some other main entropies listed in Table 3.

Table 3.

Some main entropies to measure uncertainty.

Proposer	Equation
Dubois and Prade⁴⁴	$N_{DP} (m) = \sum_{A \subseteq X} m (A) \log (\| A \|)$
Yager⁴⁵	$C_{Y} (m) = - \sum A \subseteq Fm (A) \log Pl (A)$
Klir and Ramer⁴⁶	$C_{KR} (m) = - \sum A \subseteq Fm (A) \log \sum_{B \subseteq F} m (B) \frac{\| A \cap B \|}{\| B \|}$
George and Pal⁴⁷	$C_{GP} (m) = - \sum A \subseteq Fm (A) \sum_{B \subseteq F} m (B) (1 - \frac{\| A \cap B \|}{\| A \cup B \|})$
Maluf⁴⁸	$C_{M} (m) = - \sum A \subseteq FPl (A) \log Bel (A)$
Klir and Wierman⁴⁹	$C_{K} (Bel) = - \frac{1}{c} \sum x \subseteq XBel ({x}) \log Bel ({x}) + Pl ({x}) \log Pl ({x}), c = \sum_{x \subseteq X} [Bel ({x}) + Pl ({x})]$

As for such a reason, Deng’s⁵⁰ entropy is presented to measure the uncertainty of BPA, which is a more significant tool to manage uncertainty than Shannon’s⁴³ entropy. Deng’s entropy can deal with the uncertainty represented not only by BPA but also by probability distribution. In other words, Deng’s entropy is the generalization of Shannon’s entropy.^51–54

Deng’s entropy can be denoted as follows

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

(12)

where $F_{i}$ is the proposition in mass function m, and $| F_{i} |$ is the cardinality of $F_{i}$ . Generally, if the belief is only assigned to single elements, then Deng’s⁵⁰ entropy equals Shannon’s entropy, which is denoted as

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})

(13)

However, there is a big shortcoming of Deng’s entropy that it cannot effectively quantify the difference among different BOEs which are assigned by the same mass value. For example, there are two BOEs, S₁ and S₂, as follows

S_{1} : m_{1} (E_{1}, E_{2}) = 0.4 m_{1} (E_{3}, E_{4}) = 0.6

S_{2} : m_{2} (E_{1}, E_{2}) = 0.4 m_{2} (E_{2}, E_{3}) = 0.6

According to equation (12), the uncertainty measures with Deng’s entropy are, respectively, $E_{d} (S_{1}) = 2.5559$ and $E_{d} (S_{2}) = 2.5559$ . However, this result calculated by Deng’s entropy is counter-intuitive, because though these two BOEs have the same mass value, S₁ consists of four targets, denoted as $E_{1}$ , $E_{2}$ , $E_{3}$ , and $E_{4}$ , while S₂ has only three possible targets. And intuitively, what is expected is that S₂ should have less uncertainty than S₁. That means the entropy value of S₁ should be bigger than that of S₂. Therefore, Deng’s entropy cannot quantify this difference and we propose one improved entropy function, which can finish this job very well.

The improved belief entropy is defined as follows

E_{i} = - \sum_{i} m (θ_{i}) \log_{2} (\frac{m (θ_{i})}{2^{| θ_{i} |} - 1} \frac{| θ_{i} |}{| θ |})

(14)

where $| θ_{i} |$ denotes the cardinality of the focal element $θ_{i}$ , $| θ |$ is the total number of element in this BOE, and $\frac{| θ_{i} |}{| θ |}$ is used to represent the uncertain information in a BOE that has been ignored by Deng’s entropy.

Still for the two BOEs, $S_{1}$ and $S_{2}$ above, we calculate the uncertainty by means of the newly proposed entropy function, we can get $E_{i} (S_{1}) = (- 0 . 4^{*} \log_{2} ({\frac{0.4}{2^{2} - 1}}^{*} \frac{2}{4})) + (- 0 . 6^{*} \log_{2} ({\frac{0.6}{2^{2} - 1}}^{*} \frac{2}{4})) = 3.5559$ and $E_{i} (S_{2}) = (- {0.4}^{*} \log_{2} ({\frac{0.4}{2^{2} - 1}}^{*} \frac{2}{3})) + (- {0.6}^{*} \log_{2} ({\frac{0.6}{2^{2} - 1}}^{*} \frac{2}{3})) = 3.1409$ . So, this improved belief entropy can effectively quantify the difference even if the same mass values are assigned on different BOEs.

An example of the comparison of Deng’s entropy and improved entropy function is given and the results are seen in Table 4.

Table 4.

An example of Deng’s entropy.

Source of evidence	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$	$S_{5}$
Deng’s⁵⁰ entropy	3.2061	2.1559	3.2886	1.5850	2.8074
Improved entropy function	3.7986	2.7560	4.0981	3.1699	2.8074

Example 4

Assume that $Ω = {E_{1}, E_{2}, E_{3}}$

S_{1} : m_{1} (E_{1}) = 0.3, m_{1} (E_{1}, E_{2}) = 0.2, m_{1} (E_{1}, E_{2}, E_{3}) = 0.5

S_{2} : m_{2} (E_{2}) = 0.2, m_{2} (E_{2}, E_{3}) = 0.4, m_{2} (E_{3}) = 0.4

S_{3} : m_{3} (E_{1}) = 0.2, m_{3} (E_{2}) = 0.2, m_{3} (E_{1}, E_{3}) = 0.3, m_{3} (E_{1}, E_{2}, E_{3}) = 0.3

S_{4} : m_{4} (E_{1}) = \frac{1}{3}, m_{4} (E_{2}) = \frac{1}{3}, m_{4} (E_{3}) = \frac{1}{3}

S_{5} : m_{5} (E_{1}, E_{2}, E_{3}) = 1

The proposed method

We follow the methods of Wang et al.⁵⁵ Suppose we collect n independent BOEs, denoted as $m_{i}$ , $i = 1, \dots, n$ , and first we pre-process these BOEs using equation (15)

WAM (m) = \sum_{i = 1}^{n} w_{i} \times m_{i}

(15)

where $w_{i}$ is the corresponding weight degree of BOE $m_{i}$ ; and $WAM (m)$ , the short form of weighted average mass, represents the weighted average BPA of n independent BOEs. Based on Murphy,²² we can make full use of classical Dempster’s rule to combine $WAM (m)$ for $n - 1$ times, and then the final combined results can be obtained. However, it is little uneasy for us to find an appropriate weight degree $w_{i}$ of BOE $m_{i}$ . Deng et al.²³ put forward one method based on evidence distance to get the corresponding weight. Here, a novel idea is proposed by us. First, the distance of evidences is utilized to calculate one initial weight. And then we make good use of our proposed improved entropy function to modify this initial weight in order to get a more accurate and reasonable weight. The details of our method are shown below.

Determining initial weight based on distance function

The evidence distance function is shown in equation (10). Intuitively, the less the distance between two BOEs, the more the similarity. So, the similarity measure $SI M_{ij}$ between $m_{i}$ and $m_{j}$ can be defined as

SIM (m_{i}, m_{j}) = 1 - d (m_{i}, m_{j})

(16)

Then, the support degree of a BOE $m_{i} (i = 1, 2, \dots, n)$ , could be defined²³ based on the similarity measure above

SUP (m_{i}) = \sum_{j = 1, j \neq i}^{k} SIM (m_{i}, m_{j})

(17)

Next, we could obtain the credibility degree of $m_{i}$ through equation (18)

CR D_{i} = \frac{SUP (m_{i})}{\sum_{j = 1}^{k} SUP (m_{j})}

(18)

Actually, the credibility degree $CR D_{i} (i = 1, 2, \dots, n)$ is the initial weight of BOE $m_{i} (i = 1, 2, \dots, n)$ we expect because $\sum_{i = 1}^{n} CR D_{i} = 1$ is evident. Next, we will modify this initial weight by means of newly proposed improved belief entropy.

Modifying initial weight based on improved entropy function

Suppose that if one BOE $m_{i}$ is with relatively high credibility degree, then it will have less uncertainty degree than the others and then this BOE should possess more weight because of its good quality. On the contrary, one BOE with low credibility degree and high uncertainty degree should be assigned less weight in order to reduce the bad influence as possible. On the basis of the thinking above, the initial weight can be modified through the following steps:

Step 1. Compute the uncertainty measure denoted as $U (m_{i})$ of each BOE $m_{i} (i = 1, \dots, n)$ based on the improved entropy function in equation (14). After that, normalize $U (m_{i})$ using equation (19) to get the normalized uncertainty measure denoted as $UN (m_{i})$

UN (m_{i}) = \frac{U (m_{i})}{\sum_{i = 1}^{n} U (m_{i})}

(19)

where $UN (m_{i})$ is the normalized uncertainty degree of $m_{i} (i = 1, \dots, n)$ .

Step 2. Modify the initial weight $CRD (m_{i})$ by means of equation (20), denoted as $CRDM (m_{i})$ . And by normalizing $CRDM (m_{i})$ through equation (21), the normalized results are denoted as $CRDMN (m_{i})$

CRDM (m_{i}) = CRD (m_{i}) \times e^{(1 + UN (m_{i}))} (i = 1, 2, \dots, n)

(20)

CRDMN (m_{i}) = \frac{CRDM (m_{i})}{\sum_{i = 1}^{n} CRDM (m_{i})} (i = 1, 2, \dots, n)

(21)

Step 3. So, the weighted averaged BOE denoted as $WAM (m)$ could be acquired in equation (22)

WAM (m) = \sum_{i = 1}^{n} (CRDMN (m_{i}) \times m_{i}) (i = 1, 2, \dots, n)

(22)

Finally, the classical Dempster’s¹ rule is utilized to combine $WAM (m)$ for $n - 1$ times²² and then the final combined results are obtained in favor of making a better decision.

Experiment

In this experiment part, we provide a numerical example to demonstrate the efficiency and effectiveness of this newly proposed novel weighted evidence combination rule.

Example 5

In a multisensor-based automatic target recognition system, suppose that the frame of discernment $Ω = {E_{1}, E_{2}, E_{3}}$ is complete and $E_{1}$ is the real target. From five different sensors, the system collects five BOEs listed as follows

S_{1} : m_{1} (E_{1}) = 0.41, m_{1} (E_{2}) = 0.29, m_{1} (E_{3}) = 0.3

S_{2} : m_{2} (E_{1}) = 0, m_{2} (E_{2}) = 0.9, m_{2} (E_{3}) = 0.1

S_{3} : m_{3} (E_{1}) = 0.58, m_{3} (E_{2}) = 0.07, m_{3} (E_{1}, E_{3}) = 0.35

S_{3} : m_{3} (E_{1}) = 0.58, m_{3} (E_{2}) = 0.07, m_{3} (E_{1}, E_{3}) = 0.35

S_{4} : m_{4} (E_{1}) = 0.55, m_{4} (E_{2}) = 0.1, m_{4} (E_{1}, E_{3}) = 0.35

S_{5} : m_{5} (E_{1}) = 0.6, m_{5} (E_{2}) = 0.1, m_{5} (E_{1}, E_{3}) = 0.3

After calculating through the first step of our proposed method, the credibility degree, that is, the initial weight of each BOE, can be obtained. All results are shown in Tables 5 and 6.

Table 5.

Distance measure between BOEs.

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

Table 6.

Initial weight determined by distance function.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
CRD( $m_{1}$ )	0.5000	0.4284	0.2829	0.2059
CRD( $m_{2}$ )	0.5000	0.2494	0.1365	0.0899
CRD( $m_{3}$ )	–	0.3222	0.2861	0.2321
CRD( $m_{4}$ )	–	–	0.2945	0.2368
CRD( $m_{5}$ )	–	–	–	0.2353

Then, the initial weight generated will be modified by the improved entropy function to get the final weight of each BOE, $CRDMN (m_{i})$ . The results of $U (m_{i})$ and $CRDMN (m_{i})$ are listed in Tables 7 and 8, respectively.

Table 7.

Uncertainty measure of each BOE.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
U( $m_{1}$ )	3.5317	3.5317	3.5317	3.5317
U( $m_{2}$ )	2.0540	2.0540	2.0540	2.0540
U( $m_{3}$ )	–	3.0442	3.0442	3.0442
U( $m_{4}$ )	–	–	3.1264	3.1264
U( $m_{5}$ )	–	–	–	3.0559

Table 8.

Modified weight determined based on improved entropy function.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
CRDMN( $m_{1}$ )	0.4751	0.4131	0.2741	0.1996
CRDMN( $m_{2}$ )	0.5249	0.2658	0.1461	0.0963
CRDMN( $m_{3}$ )	–	0.3211	0.2865	0.2326
CRDMN( $m_{4}$ )	–	–	0.2933	0.2359
CRDMN( $m_{5}$ )	–	–	–	0.2356

Next, by replacing $w_{i}$ of equation (15) with CRDMN, we can get $WAM (m)$ . The results of $WAM (m)$ are shown in Table 9. Finally, we make use of classical combination rule to combine $WAM (m)$ for $n - 1$ (where n is the total number of BOEs), and then the final BPAs will be obtained. The final results are listed in Table 10.

Table 9.

Results of WAM(m).

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
WAM(m)	m( $E_{1}$ ) = 0.1948	m( $E_{1}$ ) = 0.3556	m( $E_{1}$ ) = 0.4399	m( $E_{1}$ ) = 0.4878
	m( $E_{2}$ ) = 0.6102	m( $E_{2}$ ) = 0.3815	m( $E_{2}$ ) = 0.2604	m( $E_{2}$ ) = 0.2080
	m( $E_{3}$ ) = 0.1950	m( $E_{3}$ ) = 0.1505	m( $E_{3}$ ) = 0.0968	m( $E_{3}$ ) = 0.0695
	m( $E_{1}$ , $E_{3}$ ) = 0	m( $E_{1}$ , $E_{3}$ ) = 0.1124	m( $E_{1}$ , $E_{3}$ ) = 0.2029	m( $E_{1}$ , $E_{3}$ ) = 0.2347

Table 10.

Final BPAs calculated by the newly proposed approach.

Item	$m_{1}, m_{2}$	$m_{1}, m_{2}, m_{3}$	$m_{1}, m_{2}, m_{3}, m_{4}$	$m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$
Final BPA	m( $E_{1}$ ) = 0.0846	m( $E_{1}$ ) = 0.5784	m( $E_{1}$ ) = 0.9303	m( $E_{1}$ ) = 0.9939
	m( $E_{2}$ ) = 0.8305	m( $E_{2}$ ) = 0.0253	m( $E_{2}$ ) = 0.0129	m( $E_{2}$ ) = 0.0005
	m( $E_{3}$ ) = 0.0849	m( $E_{3}$ ) = 0.0958	m( $E_{3}$ ) = 0.0351	m( $E_{3}$ ) = 0.0044
	m( $E_{1}$ , $E_{3}$ ) = 0	m( $E_{1}$ , $E_{3}$ ) = 0.0081	m( $E_{1}$ , $E_{3}$ ) = 0.0093	m( $E_{1}$ , $E_{3}$ ) = 0.0012

For a better show of the efficiency and effectiveness, we do a comparison with other different combination rule. The comparison results are shown in Table 11 and Figures 1 –4.

Table 11.

Comparison of different evidence 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’s¹ rule	m( $E_{1}$ ) = 0	m( $E_{1}$ ) = 0	m( $E_{1}$ ) = 0	m( $E_{1}$ ) = 0
	m( $E_{2}$ ) = 0.8969	m( $E_{2}$ ) = 0.6575	m( $E_{2}$ ) = 0.3321	m( $E_{2}$ ) = 0.1422
	m( $E_{3}$ ) = 0.1031	m( $E_{3}$ ) = 0.3425	m( $E_{3}$ ) = 0.6679	m( $E_{3}$ ) = 0.8575
	m( $E_{1}$ ) = 0.0964	m( $E_{1}$ ) = 0.4619	m( $E_{1}$ ) = 0.8362	m( $E_{1}$ ) = 0.9620
Murphy’s²² simple average	m( $E_{2}$ ) = 0.8119	m( $E_{2}$ ) = 0.4497	m( $E_{2}$ ) = 0.1147	m( $E_{2}$ ) = 0.0210
	m( $E_{3}$ ) = 0.0917	m( $E_{3}$ ) = 0.0794	m( $E_{3}$ ) = 0.0410	m( $E_{3}$ ) = 0.0138
	m( $E_{1}$ , $E_{3}$ ) = 0	m( $E_{1}$ , $E_{3}$ ) = 0.0090	m( $E_{1}$ , $E_{3}$ ) = 0.0081	m( $E_{1}$ , $E_{3}$ ) = 0.0032
	m( $E_{1}$ ) = 0.0964	m( $E_{1}$ ) = 0.4974	m( $E_{1}$ ) = 0.9089	m( $E_{1}$ ) = 0.9820
Deng et al.’s²³ weighted average	m( $E_{2}$ ) = 0.8119	m( $E_{2}$ ) = 0.4054	m( $E_{2}$ ) = 0.0444	m( $E_{2}$ ) = 0.0039
	m( $E_{3}$ ) = 0.0917	m( $E_{3}$ ) = 0.0888	m( $E_{3}$ ) = 0.0379	m( $E_{3}$ ) = 0.0107
	m( $E_{1}$ , $E_{3}$ ) = 0	m( $E_{1}$ , $E_{3}$ ) = 0.0084	m( $E_{1}$ , $E_{3}$ ) = 0.0089	m( $E_{1}$ , $E_{3}$ ) = 0.0034
	m( $E_{1}$ ) = 0.0846	m( $E_{1}$ ) = 0.5784	m( $E_{1}$ ) = 0.9303	m( $E_{1}$ ) = 0.9939
Our proposed method	m( $E_{2}$ ) = 0.8305	m( $E_{2}$ ) = 0.0253	m( $E_{2}$ ) = 0.0129	m( $E_{2}$ ) = 0.0005
	m( $E_{3}$ ) = 0.0849	m( $E_{3}$ ) = 0.0958	m( $E_{3}$ ) = 0.0351	m( $E_{3}$ ) = 0.0044
	m( $E_{1}$ , $E_{3}$ ) = 0	m( $E_{1}$ , $E_{3}$ ) = 0.0081	m( $E_{1}$ , $E_{3}$ ) = 0.0093	m( $E_{1}$ , $E_{3}$ ) = 0.0012

Figure 1.

Outcome of m( $E_{1}$ ) based on different combination rules.

Figure 2.

Outcome of m( $E_{2}$ ) based on different combination rules.

Figure 3.

Outcome of m( $E_{3}$ ) based on different combination rules.

Figure 4.

Outcome of m( $E_{1}$ , $E_{3}$ ) based on different combination rules.

As shown in Table 11 and Figures 1 –4, when evidences are in high conflict, the classical Dempster’s combination rule will produce counter-intuitive results which do not reflect the truth and so it is useless in that case. However, although with incremental BOEs, Murphy’s²² simple averaging and Deng et al.’s²³ weighted averaging can give relatively reasonable results, their results are all inferior to those of our newly proposed method. In addition, among these existing methods that we compare, the convergence of ours is best. That is mainly because of using evidence distance function and improved entropy function, the effect of credible evidence is strengthened extremely, and the “bad” evidence has less influence on the final combined results.

Application

Just like our previous work,²⁷ we still apply our new method to the application of fault diagnosis. And the example is the same as that of our previous work.²⁷ Now, a machine has three gears $g_{1}$ , $g_{2}$ , and $g_{3}$ , and there are three kinds of failure fault modes $f_{1}$ , $f_{2}$ , and $f_{3}$ in $g_{1}$ , $g_{2}$ , and $g_{3}$ and combined as fault hypothesis set { $f_{1}$ , $f_{2}$ , $f_{3}$ }. Now, we have three kinds of sensors, $s_{1}$ , $s_{2}$ , and $s_{3}$ , from which we can get the evidence set $e_{1}, e_{2}, e_{3}$ as shown in Table 12.

Table 12.

Prior information of evidences $e_{1}$ , $e_{2}$ , and $e_{3}$ .

Evidence no.	$m (f_{1})$	$m (f_{2})$	$m (f_{2}, f_{3})$	$m (Ω)$	$sufficiency (μ)$	$importance (ν)$
$e_{1}$	0.6	0.1	0.1	0.2	1	1
$e_{2}$	0.05	0.8	0.05	0.1	0.6	0.34
$e_{3}$	0.7	0.1	0.1	0.1	1	1

Similarly, we still consider two kinds of sensor reliability, the static one $R_{i}^{s} = μ_{i} \times ν_{i}$ measured based on the evidence sufficiency and importance index and the dynamic one $R_{i}^{d}$ measured based on $CRDMN (m_{i})$ proposed newly in this work. The final comprehensive reliability $R = R^{s} \times R^{d}$ is used to modify the conflicting evidences, and after fusing evidences, we could get final results with a higher accuracy to address such a fault diagnosis problem than before.

First, we could get the results of static reliability $R^{s}$ of each evidence $e_{1}$ , $e_{2}$ , and $e_{3}$ as listed in Table 13.

Table 13.

Static reliability of evidences $e_{1}$ , $e_{2}$ , and $e_{3}$ .

	$e_{1}$	$e_{2}$	$e_{3}$
$R^{s}$	1	0.204	1

Then, we could get the results of dynamic reliability $R^{d}$ of each evidence $e_{1}$ , $e_{2}$ , and $e_{3}$ as listed in Tables 14 –16.

Table 14.

Initial weight determined in application.

	$e_{1}$	$e_{2}$	$e_{3}$
CRD	0.3971	0.2193	0.3836

Table 15.

Uncertainty measure of each evidence in application.

	$e_{1}$	$e_{2}$	$e_{3}$
U	3.4589	2.7584	3.1225

Table 16.

Modified weight in application.

	$e_{1}$	$e_{2}$	$e_{3}$
CRDMN( $R^{d}$ )	0.4091	0.2096	0.3813

Next, calculate the final comprehensive sensor reliability R and then normalize it which is denoted as RN. The results are shown in Table 17.

Table 17.

Final comprehensive sensor reliability in application.

	$e_{1}$	$e_{2}$	$e_{3}$
R	0.4096	0.0428	0.3813
RN	0.4910	0.0514	0.4576

Finally, modify BPAs with comprehensive sensor reliability RN to get $WAM (m)$ as shown in Table 18, and then using the classical combination rule to combine $WAM (m)$ for two times, we will get the final results as listed in Table 19.

Table 18.

$WAM (m)$ results in application.

	$m (f_{1})$	$m (f_{2})$	$m (f_{2}, f_{3})$	$m (Ω)$
WAM(m)	0.6175	0.0136	0.0974	0.1491

Table 19.

Final results in application.

	$m (f_{1})$	$m (f_{2})$	$m (f_{2}, f_{3})$	$m (Ω)$
Value	0.9416	0.0484	0.0087	0.0013

As seen from Table 19, based on our latest approach, the belief degree of the fault $f_{1}$ is 94.16%, while the fault $f_{2}$ only has a belief degree of 4.84%, and most importantly, the uncertainty of belief $m (Ω)$ reduces to 0.13%. We also compare this newly proposed method with others and the results of comparison are shown in Table 20. As we can see from Table 20, our newly proposed method has significantly more belief degree of fault $f_{1}$ and less uncertainty of belief than Fan and Zuo’s⁵⁶ work and even our previous work,²⁷ which is a great step of progress. That is mainly because the improved entropy is proposed and added to novel combination rule, and both static and dynamic reliability are considered.

Table 20.

Comparison results in application.

	$m (f_{1})$	$m (f_{2})$	$m (f_{2}, f_{3})$	$m (Ω)$
Classical D-S theory¹	0.4519	0.5048	0.0336	0.0096
Fan and Zuo’s⁵⁷ work	0.8119	0.1096	0.0526	0.0259
Our previous work²⁷	0.8899	0.0785	0.0243	0.0073
The newly proposed method	0.9416	0.0484	0.0087	0.0013

Real-life medical application

In this section, we also apply our proposed methodology to one real-life medical diagnosis application, which demonstrates adequately the priority of our approach compared with other recent works. Note that this example of the application section is cited from Li et al.⁵⁷

Suppose that there are a total of four diseases, denoted as $Ω$ = {acute dental abscess, migraine, acute sinusitis, peritonsillar abscess} = ${x_{1}, x_{2}, x_{3}, x_{4}}$ . Suppose that the set of parameters p is given by p = {fever, running nose, weakness, oro-facial pain, nausea vomiting, swelling, trismus, history, physical examination, laboratory investigation} = { $e_{1}$ , $e_{2}$ , $e_{3}$ , $e_{4}$ , $e_{5}$ , $e_{6}$ , $e_{7}$ , $s_{1}$ , $s_{2}$ , $s_{3}$ }. Let A and B be two subsets of p, denoted by A = { $e_{1}$ , $e_{2}$ , $e_{3}$ , $e_{4}$ , $e_{5}$ , $e_{6}$ , $e_{7}$ } and B = { $s_{1}$ , $s_{2}$ , $s_{3}$ }. (F, A) is a fuzzy soft describing symptoms of the diseases, and (G, B) is also a fuzzy soft set describing decision-making tools of the diseases.

Here, we have a patient named Jason. Jason got three symptoms—fever, running noise, and facial pain: What a doctor should do now is to find out the most suitable diagnosis based on Jason’s symptoms: history, physical examination, and laboratory investigation. To deal with such a problem, we consider (F, p) and (G, B). There are four diseases { $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}$ } and nine pairs of parameters $a_{1} = (e_{1}, s_{1})$ , $a_{2} = (e_{1}, s_{2})$ , $a_{3} = (e_{1}, s_{3})$ , $a_{4} = (e_{2}, s_{1})$ , $a_{5} = (e_{2}, s_{2})$ , $a_{6} = (e_{2}, s_{3})$ , $a_{7} = (e_{4}, s_{1})$ , $a_{8} = (e_{4}, s_{2})$ , and $a_{9} = (e_{4}, s_{3})$ , which are the pairs of one symptom and one decision-making tool. Referring to Li et al.’s⁵⁷ work, we know the information structure image matrix, that is, the initial mass function of each pair of parameter shown in Table 21 in our article is also shown in the sixth step of the sixth section in Li et al.’s⁵⁷ study. (How to get these initial mass function is not important to us and we just use these mass to start our proposed method; you can directly look through Li et al.’s⁵⁷ work for more details.)

Table 21.

Initial mass function of nine pairs of symptom and decision-making tool with respect to four diseases.

Diseases	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	$a_{7}$	$a_{8}$	$a_{9}$
$x_{1}$	0.4	0.4	0.3333	0	0	0	0.2143	0.3636	0.2
$x_{2}$	0.1333	0.1333	0.1667	0	0	0	0.2857	0.1364	0.3
$x_{3}$	0.2	0.2	0.25	1	1	1	0.2857	0.1818	0.35
$x_{4}$	0.2667	0.2667	0.25	0	0	0	0.2143	0.3182	0.15

Then, on the basis of our newly proposed method, first we can get the initial weight from $a_{1}$ to $a_{9}$ . The relevant results are shown in Tables 22 and 23.

Table 22.

Distance measures between pairs.

$d (a_{ij})$	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	$a_{7}$	$a_{8}$	$a_{9}$
$a_{1}$	0	0	0.0646	0.6667	0.6667	0.6667	0.1841	0.0465	0.2279
$a_{2}$	0	0	0.0646	0.6667	0.6667	0.6667	0.1841	0.0465	0.2279
$a_{3}$	0.0646	0.0646	0	0.6180	0.6180	0.6180	0.1242	0.0746	0.1666
$a_{4}$	0.6667	0.6667	0.6180	0	0	0	0.5847	0.6788	0.5362
$a_{5}$	0.6667	0.6667	0.6180	0	0	0	0.5847	0.6788	0.5362
$a_{6}$	0.6667	0.6667	0.6180	0	0	0	0.5847	0.6788	0.5362
$a_{7}$	0.1841	0.1841	0.1242	0.5847	0.5847	0.5847	0	0.1819	0.0659
$a_{8}$	0.0465	0.0465	0.0746	0.6788	0.6788	0.6788	0.1819	0	0.2346
$a_{9}$	0.2279	0.2279	0.1666	0.5362	0.5362	0.5362	0.0659	0.2346	0

Table 23.

Initial weight determined in real-life medical application.

	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	$a_{7}$	$a_{8}$	$a_{9}$
$CRD (a_{i})$	0.1204	0.1204	0.1236	0.0976	0.0976	0.0976	0.1209	0.1017	0.1202

Next, the initial weights will be modified by our improved entropy function, which results in final weights of each pair (BOE) $CRDMN (a_{i})$ . Tables 24 and 25 show the relevant results of $U (a_{i})$ and $CRDMN (a_{i})$ , respectively.

Table 24.

Uncertainty measure of each evidence in real-life medical application.

	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	$a_{7}$	$a_{8}$	$a_{9}$
$U (a_{i})$	3.8892	3.8892	3.9592	2	2	2	3.9852	3.8955	3.9261

Table 25.

Modified weight in real-life medical application.

	$a_{1}$	$a_{2}$	$a_{3}$	$a_{4}$	$a_{5}$	$a_{6}$	$a_{7}$	$a_{8}$	$a_{9}$
CRDMN( $a_{i}$ )	0.1225	0.1225	0.1261	0.0932	0.0932	0.0932	0.1234	0.1035	0.1224

Finally, replacing $w_{i}$ of equation (15) with $CRDMN (a_{i})$ , we can get WAM(a), as shown in Table 26. After that, making use of classical combination rule to combine WAM(a) for eight times, the final BPA, that is, the belief measure of each alternative $x_{i}$ will be obtained, as listed in Table 27.

Table 26.

$WAM (m)$ results in real-life medical application.

	$m (x_{1})$	$m (x_{2})$	$m (x_{3})$	$m (x_{4})$	$m (Ω)$
WAM(m)	0.2286	0.1398	0.4570	0.1746	0

Table 27.

Belief measure of each alternative $x_{i}$ in real-life medical application.

	$Bel (x_{1}) / m (x_{1})$	$Bel (x_{2}) / m (x_{2})$	$Bel (x_{3}) / m (x_{3})$	$Bel (x_{4}) / m (x_{4})$
Value	0.0009	0(7.1689e–06)	0.9989	0.0001(6.6195e–05)

As shown in Table 27, the final ranking order of all alternatives is $x_{3} > x_{1} > x_{4} > x_{2}$ . Thus, the optimal choice decision is $x_{3}$ according to the maximum belief measure principle.

Finally, we make a comparison between our newly proposed method and some recent methods in the studies of Wang et al.⁵⁵ and Li et al.⁵⁷ The comparison results shown in Table 28 show that first, this newly proposed approach is as feasible as others; second, our method has better performance with respect to enhancing belief degree and reducing uncertainty of decision, which makes a great contribution to the medical diagnosis field.

Table 28.

Comparison of recent methods in real-life application.

Methods	Ranking order	$Bel (x_{3}) / m (x_{3})$	$m (θ)$	$γ$
Li et al.’s⁵⁷ work	$x_{3} > x_{1} > x_{4} > x_{2}$	0.8349	0.0069	4.716
Wang et al.’s⁵⁵ work	$x_{3} > x_{1} > x_{4} > x_{2}$	0.9906	0.0001	4.716
The present study	$x_{3} > x_{1} > x_{4} > x_{2}$	0.9989	0	4.716

Conclusion

In this article, a novel weighted evidence combination rule based on improved entropy function is presented. This newly proposed approach preserves all the desirable properties of Murphy’s²² and Deng et al.’s²³ work. Besides, through comparison with other methods, an application in fault diagnosis, and a real-life medical diagnosis application, our proposed approach converges faster when handling high conflicting evidences, and so it can help experts do take better and faster decision.

Footnotes

Handling Editor: Pietro Manzoni

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the research and construction of self-regulated learning platform of software technology specialty in higher vocational colleges (no. KJ1502901), and the research and demonstration application based on MOOCS independent learning platform of training support technology for Industry 4 talents (no. GZTG201605).

ORCID iD

Lei Chen

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