Uncertainty measure in data fusion applications is a hot topic; quite a few methods have been proposed to measure the degree of uncertainty in DempsterShafer framework. However, the existing methods pay little attention to the scale of the frame of discernment (FOD), which means a loss of information. Due to this reason, the existing methods cannot measure the difference of uncertain degree among different FODs. In this paper, an improved belief entropy is proposed in DempsterShafer framework. The proposed belief entropy takes into consideration more available information in the body of evidence (BOE), including the uncertain information modeled by the mass function, the cardinality of the proposition, and the scale of the FOD. The improved belief entropy is a new method for uncertainty measure in DempsterShafer framework. Based on the new belief entropy, a decisionmaking approach is designed. The validity of the new belief entropy is verified according to some numerical examples and the proposed decisionmaking approach.
Decisionmaking in the uncertain environment is common in real world applications, such as civil engineering [
DempsterShafer evidence theory [
In the probabilistic framework, Shannon entropy [
Lately, another uncertainty measure named Deng entropy [
In order to verify the validity of the improved belief entropy, a decisionmaking approach in target identification is designed based on the new belief entropy. In the proposed method, the uncertain degree of the sensor data is measured by the new belief entropy; then the uncertain degree will be used as the relative weight of each sensor report modeled as the body of evidence (BOE); after that, the BPAs of the BOE will be modified by the weight value; finally, the decisionmaking is based on the fusion results of applying Dempster’s rule of combination to the modified BPAs.
The rest of this paper is organized as follows. In Section
In this section, some preliminaries are briefly introduced, including DempsterShafer evidence theory, Shannon entropy, and some typical uncertainty measures in DempsterShafer framework.
Let
A
If
A
A BPA
In DempsterShafer evidence theory, two independent mass functions, denoted as
As an uncertainty measure of information volume in a system or process, Shannon entropy plays a central role in information theory. Shannon entropy indicates that the information volume of each piece of information is directly connected to its uncertainty degree.
Shannon entropy, as the information entropy, is defined as follows [
In DempsterShafer framework, some uncertainty measures for the BOE are presented, as is shown in Table
Uncertainty measures in DempsterShafer framework.
Uncertainty measure  Definition 

Hohle’s confusion measure [ 

Yager’s dissonance measure [ 

Dubois & Prade’s weighted Hartley entropy [ 

Klir & Ramer’s discord measure [ 

Klir & Parviz’s strife measure [ 

George & Pal’s total conflict measure [ 

Lately, another belief entropy, named Deng entropy, is presented to measure the uncertainty in the BOE. Deng entropy, denoted as
In DempsterShafer framework, the uncertain information modeled in the BOE includes the mass function and the FOD. However, the existing uncertainty measures only focus on the mass function [
Consider a target identification problem; assume that two reliable sensors report the detection results independently. The results are represented by BOEs shown as follows:
Recalling (
The limitation of Deng entropy also exists in Dubois & Prade’s weighted Hartley entropy [
The results calculated by Deng entropy and the weighted Hartley entropy are counterintuitive. Although the two BOEs have the same mass value, the FOD of the first BOE
To address this issue, an improved belief entropy is proposed.
In DempsterShafer framework, the improved belief entropy is proposed as follows:
In detail, compared with the confusion measure [
With the new belief entropy, recall the issue in Example
It can be concluded that both the weighted Hartley entropy and Deng entropy cannot measure the different uncertain degree between these two BOEs, while the new belief entropy can effectively measure the difference by taking into consideration more available information of the BOE. According to Table
Calculation results in Example
BOEs  Weight Hartley entropy [ 
Deng entropy [ 
Improved belief entropy 


1  2.5559  2.1952 

1  2.5559  2.0750 
In order to show the rationality and merit of the proposed belief entropy, some numerical examples are presented in this section. In Section
Consider a target identification problem; if the target reported by the sensor is
Shannon entropy
It is obvious that the uncertainty degree for a certain event is zero. So the values of Shannon entropy, Deng entropy, and the improved belief entropy are all zero.
Consider the mass functions
Shannon entropy
According to Examples
In order to test the efficiency and merit of the new belief entropy, recall the numerical example in [
Consider the mass functions
Deng entropy
The uncertain degree in Example
It seems that the uncertain degree measured by Dubois & Prade’s weighted Hartley entropy, Deng entropy, and the modified belief entropy is rising significantly with the increasing of the element number in proposition
Improved belief entropy with a variable element number in
Cases  Deng entropy  Improved belief entropy 


2.6623  2.5180 

3.9303  3.7090 

4.9082  4.6100 

5.7878  5.4127 

6.6256  6.1736 

7.4441  6.9151 

8.2532  7.6473 

9.0578  8.3749 

9.8600  9.1002 

10.6612  9.8244 

11.4617  10.5480 

12.2620  11.2714 

13.0622  11.9946 

13.8622  12.7177 
Comparison among different uncertainty measures.
In this section, a decisionmaking approach in sensor data fusion is presented. After uncertainty measure with the improved belief entropy, the modified BOEs are fused with Dempster’s rule of combination. Finally, decisionmaking is based on the fused results.
In target recognition, sometimes, decisionmaking is based on reports from sensors. Consider the problem in [
With the incoming data of sensors, which is the target? The target recognition system needs to make a decision based on the fusion results of the sensor data. Intuitively,
Before further addressing the BOEs reported by sensors, the quality of this information is quantified with the proposed belief entropy. Recalling (
Intuitively, the second BOE
In real applications, for example, air battle, the realtime requirement is highly concerned, so decisionmaking in target recognition needs to be finished in real time. If decisionmaking needs to be finished in real time, then data fusion needs to be processed instantly with the upcoming sensor report. The procedures for decisionmaking based on the improved belief entropy are designed in Figure
The flowchart of the proposed approach for decisionmaking.
Evidence from sensor report is modeled as the BOE.
An example of this step is given in Section
Uncertainty measure of each BOE with the improved belief entropy.
Generally, the more dispersive the mass value is assigned among the power set, the bigger the new belief entropy of the BOE will be. An illustrative example of this step is shown in Section
Relative weight of the incoming BOEs is calculated based on the improved uncertainty measure.
A big entropy value corresponds to a big information volume. It is commonly accepted that the bigger the entropy is, the higher the uncertain degree will be. The relative weight of each BOE is defined as the relative weight of the new belief entropy. For the
The relative weight of each BOE in Section
The modified BPAs are derived for sensor data fusion.
For a proposition
For example, with (
The weighted BPAs are fused with Dempster’s rule of combination in (
For example, the modified BPAs in Step
Realtime decisionmaking based on the fused results.
For example, taking into consideration the target recognition problem expressed in Section
Based on the aforementioned six steps, the decisionmaking results corresponding to two, three, four, and five BOEs are shown in Table
Decisionmaking results based on sensor data fusion.
BOEs  Fusion results  Recognition result 



Uncertain; no proposition has a belief over 60.00% 






































In order to test the efficiency of the proposed method, the fusion results are compared with the other methods in [
Fusion results with different combination rules.






Dempster’s rule [ 





















Yager’s rule [ 


























Murphy’s rule [ 





















Deng et al.’s method [ 





















Zhang et al.’s method [ 





















Yuan et al.’s method [ 





















The proposed method 


















Fusion results of two BOEs.
Fusion results of three BOEs.
Fusion results of four BOEs.
Fusion results of five BOEs.
Figure
Figure
Figure
The performance of different methods shown in Figure
It seems that the performance of the proposed method is similar to Yuan et al.’s method [
A few reasons contribute to the effectiveness of the proposed decisionmaking approach. First of all, the sensor data is preprocessed properly before applying the combination rules. This is very important in sensor data fusion especially if there is conflict evidence. Secondly, the weights of BOEs are calculated based on the proposed belief entropy. The effectiveness and superiority of the new belief entropy verified in Section
In this paper, an improved belief entropy is proposed. The new belief entropy improves the performance of Deng entropy and some other uncertainty measures in DempsterShafer framework. The new belief entropy considers the uncertain information consisted in not only the mass function and the cardinality of the proposition, but also the scale of the FOD and the relative scale of each proposition with respect to the FOD. Numerical examples show that the new belief entropy can quantify the uncertain degree of the BOE more accurately than the other uncertainty measures.
A new decisionmaking approach is presented in this paper and applied to a case study. The new uncertainty measurebased decisionmaking approach shows the efficiency and merit of the new belief entropy. In the following researches, the proposed belief entropy will be used to solve more problems related to uncertain information processing in real world applications.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work is partially supported by National Natural Science Foundation of China (Grant no. 61671384), Natural Science Basic Research Plan in Shaanxi Province of China (Grant no. 2016JM6018), the Fund of SAST (Grant no. SAST2016083), and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University.