To solve the invalidation problem of DempsterShafer theory of evidence (DS) with high conflict in multisensor data fusion, this paper presents a novel combination approach of conflict evidence with different weighting factors using a new probabilistic dissimilarity measure. Firstly, an improved probabilistic transformation function is proposed to map basic belief assignments (BBAs) to probabilities. Then, a new dissimilarity measure integrating fuzzy nearness and introduced correlation coefficient is proposed to characterize not only the difference between basic belief functions (BBAs) but also the divergence degree of the hypothesis that two BBAs support. Finally, the weighting factors used to reassign conflicts on BBAs are developed and Dempster’s rule is chosen to combine the discounted sources. Simple numerical examples are employed to demonstrate the merit of the proposed method. Through analysis and comparison of the results, the new combination approach can effectively solve the problem of conflict management with better convergence performance and robustness.
Multisensor data fusion is a technology that combines information from several sources to form a unified picture [
For a few years, a variety of combination methods have been proposed to achieve effective data fusion on high degree of conflicting sources of evidence. By studying them, the overall methods can be summarized into two main categories. The first is to improve the rules of combination [
Therefore, the dissimilarity measure between two sources of evidence plays a crucial role in the discounting method. Jousselme et al. [
In this study, a novel combination approach of conflict evidence is proposed. The novel dissimilarity measure is defined through integrating the fuzzy nearness and correlation coefficient by Hamacher Tconorm rule [
The rest of this paper is organized as follows. In Section
The frame of discernment, denoted by
Suppose two bodies of evidence
Note that there are two limitations in applying DS evidence theory. One is that the counterintuitive results can be generated when high conflicting evidence is infused using Dempster’s rule as shown in classical Zadeh’s example [
Assume
According to (
Consider two equal
According to (
When working in the probabilistic framework, the focal elements are singletons and exclusive, and the degree of the conflict becomes easier to compute regardless of the intrinsic relationship between BBAs. Probabilistic transformation is a useful tool to map BBAs to probabilities. A classical transformation is the pignistic transformation [
The fundamental goal of our approach is to allocate reasonable weighting factors to the evidence and make a much better combination. The derivation of the weights of the sources is based on the widely welladopted principle that
The degree of conflict between BBAs has been measured in many works, including conflict coefficient
By utilizing the information contained in the belief function and plausibility function of the propositions in the DS, a new method for transforming BBA into probability is defined as
The improved probabilistic transform function satisfies
If
If
If
Let the BBA over the same frame of discernment
In Example
The results of probabilistic transformation in Example







0.5333  0.3333  0.1333  0.4213 

0.5435  0.2913  0.1652  0.4291 

0.5000  0.3571  0.1429  0.4310 

0.5615  0.2956  0.1429  0.4179 
Fuzzy set theory is specially designed to provide a language with syntax and semantics to translate qualitative judgments into numerical reasoning and to capture subjective and vague uncertainty. In this theory, fuzzy nearness is used to measure the level of similarity between two objects. In this work, we use the fuzzy nearness [
Assume we have got a sequence of
Let us consider the frame
One gets
The dissimilarity between
However, the fuzzy nearness is not stable. If
A conflict between two BBAs can be interpreted qualitatively as the fact that one source strongly supports one hypothesis and the other strongly supports another hypothesis, and the two hypotheses are not compatible (their intersection is empty) [
Let
In Example
However, according to the definition of the correlation coefficient, it only considers the elements that two sources of evidence strongly support and ignores the other elements of BBAs. Actually, the fuzzy nearness and correlation coefficient are complementary and they separately capture different aspects of the dissimilarity of BBAs.
Taking into account both of them in the elaboration of a new measure of dissimilarity seems therefore a natural way to capture two aspects of the dissimilarity of BBAs. Consider the analysis in [
Based on the improved probabilistic transformation, the new dissimilarity measure denoted by
In order to verify the effectiveness of the new dissimilarity measure, Example
Let
Comparisons of different conflict measure methods when subset
From Figure
Suppose the number of sources of evidence is
The support degree of all sources of evidence to evidence
The credibility of evidence can be calculated by the following formula:
The weight of evidence is defined by
Let
A typical architecture of the proposed combination method is shown in Figure
The typical architecture of the proposed combination scheme.
The classic Zadeh’s example is used in this part to illustrate that the proposed approach can solve the invalidation problem of DS combination rule with high conflict. The combined results are tabulated and are listed in Table
Combination results of
Methods 





DS [ 
0 

0  0 
Our results 

0.0919 

0.1699 
From Table
The new source of evidence







0.9  0.1  0  0 
DS [ 
0 

0  0 
Our results 

0.0716 

0.1217 
In the real application of decisionmaking support systems, the interference of surroundings or the aberrant measurement of sensors always leads to the varying of the collected belief functions within a certain range. Therefore, the robustness of the combination method directly affects the synthesis results.
Let us consider three simple Bayesian BBAs over the frame
From Table
Four sources of evidence in Example


 


0  0.9  0.1 

0.6  0.25  0.15 

0.75  0.15  0.1 

0.7  0.2  0.1 
Combination results of different methods in Example
Methods 







DS [ 
0.0000  0.0000  0.9574  0.9677  0.0426  0.0323 
Yager [ 
0.5700  0.5320  0.1478  0.197  0.0775  0.0775 
Murphy [ 
0.5235  0.4674  0.4674  0.5235  0.0091  0.0091 
Y. Deng [ 
0.7264  0.6823  0.2502  0.2968  0.0234  0.0209 
Liu [ 
0.8332  0.7958  0.1454  0.1829  0.0214  0.0213 
Our results  0.8837  0.8513  0.0931  0.1159  0.0232  0.0327 
The fusion results in Table
In this section, a synthetic numerical example of a simulation of the multisensor based automatic target recognition system is employed to analyze the effectiveness of the proposed approach of combination.
Let the frame of target discernment be
In Table
The collected sources of evidence of the target recognition system.





 


0.8  0.1  0  0  0  0.1 

0.5  0.2  0.1  0.2  0  0 

0  0.9  0.1  0  0  0 

0.5  0.1  0.1  0.1  0  0.2 

0.6  0.1  0  0  0.1  0.2 
Combination results of different methods of the target recognition system.
Methods 





DS [ 
















Yager [ 



























Murphy [ 




























Y. Deng [ 





















Liu [ 




























Our results 

























The belief assignment allocated to target
As can be observed in Table
In this paper, a new method has been proposed to combine conflict sources of evidence with different weighting factors. The merit of the new method proposed in this work lies in the elaboration of an efficient probability transformation and a comprehensively probabilisticbased dissimilarity measure which can be used for the determination of the weighting factors of the sources involved in the fusion process. Through aforementioned analysis and comparison, the proposed approach can effectively solve the counterintuitive behaviors of the classical DS rule in combining highly conflicting sources. Furthermore, it can make the right decision with better robustness and effectiveness performance for the decisionmaking support system or target detection system.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported in part by the Natural Science Foundation of China under Grant no. 61370097.