Dempster-Shafer evidence theorydempster1967upper ; shafer1976mathematical has attracted more and more attentions recently years. It can handle with uncertain and incomplete information in many fields, such as target recognition, information fusion and decision makingdenoeux2008conjunctive ; dubois1988representation ; heyounewmethod2011 ; han2011weighted1 ; han2011weighted ; zhang2000new ; pan2001some ; he2012new ; deng2011risk ; deng2011new ; deng2011new1 ; deng2010target ; suo2013computational ; tan2012data ; geng2013consensus ; wei2013identifying ; gao2013modified ; kang2012evidential ; chen2013fuzzy . While the evidence are highly conflicting, the Dempster’s combination rule will generate counter-intuitive results, such as the typical conflictive example proposed by Zadehzadeh1986simple
. In the last decade researchers have proposed many approaches to cope with this open issue and certain effort have been obtained. The existing methods can be mainly classified into two categories. The first strategy regards that Dempster’s combination rule is incomplete and modifying the combination rule as alternative, such as Yager’s methodyager1987dempster , Smet’s methodsmets1994transferable ; smets1990combination and Lefevre’s methodlefevre2002belief , etc. The second strategy believes that Dempster’s rule has perfect theoretical foundation and preprocessing the original evidence before combination, such as Haenni’s methodhaenni2002alternatives , Murphy’s methodmurphy2000combining and Deng’s methoddeng2004efficient , etc. We believe that Dempster’s rule is excellent and has been widely applied in recent years. In this paper, preprocessing the original evidence for highly conflicting is adopted. The method of Deng proposeddeng2004efficient in 2004 based on the evidence distance can deal with the conflicting evidence and that the correct sensor can be quickly recognized. The evidence distance of Deng’s method reflects the difference between evidences distance roughly, but can not reflect the degree of difference. In this paper, we propose a new method weighted averaging the evidence, improving Deng’s methoddeng2004efficient . The new method takes both Jousselmejousselme2001new and Hausdorffhausdorff1957set evidence distance into account. Thus, the weights of evidence are more appropriate.
The remainder of this paper is organized as follows. Section 2 presents some preliminaries. The proposed method is presented in section 3. Numerical examples and applications are used to demonstrate the validity of the proposed method in section 4. A short conclusion is drawn in the last section.
In this section, some concepts of Dempster-Shafer evidence theorydempster1967upper ; shafer1976mathematical are briefly recalled. For more information please consult Ref.he2010information . The Dempster-Shafer evidence theory is introduced by Dempster and then developed by Shafer.
In Dempster-Shafer evidence theory, let be the finite set of mutually exclusive and exhaustive elements. It is concerned with the set of all subsets of , which is a powerset of , known as the frame of discernment, denotes as
The mass function of evidence assigns probability to the subset of, also called basic probability assignment(BPA), which satisfies the following conditions
is an empty set and is any subsets of .
Dempster’s combination ruledempster1967upper ; shafer1976mathematical is the first one within the framework of evidence theory which can combine two BPAs and to yield a new BPA . The rule of Dempster’s combination is presented as follows
Where is a normalization constant, namely the conflict coefficient of BPAs.
3 New combination approach
The method of Murphymurphy2000combining purposed regards each BPA as the same role, little relevant to the relationship among the BPAs. In Deng’s weighted methoddeng2004efficient , each BPAs play different roles, that depended on the extent to which they are accredited in system. The similarity of Deng’s method between two BPAs is ascertained by Jousselme distance functionjousselme2001new .
3.1 Two existing evidence distance
The evidence distance proposed by Jousselmejousselme2001new is presented as follows
Let and be two BPAs defined on the same frame of discernment , containing mutually exclusive and exhaustive hypotheses. The metric can be defined as follows
is a similarity matrix, indicates the conflict of focal element in and , where
is the cardinality of subset of the union and , where and may belong to the same BPA or come from different BPAs. indicates the conflict degree between elements and . When two elements have no common object, they are highly conflicting.
Another evidence distance proposed by Sunbergsunberg2013belief is presented as follows
Let and be two BPAs defined on the same frame of discernment , containing mutually exclusive and exhaustive hypotheses. The distance of two BPAs referred to as is defined as follows
Where H(,) is the Hausdorff distancehausdorff1957set between focal elements and . and may belong to the same BPA or come form different BPAs. Positive number is a user-defined tuning parameter. is set to be 1, in this paper, for simplicity. It is defined according to
Where is the distance between two elements of the sets and can be defined as any valid metric distance on the measurement spacehausdorff1957set .
There are five orderable mutually exclusive and exhaustive hypotheses elements: 1, 2, 3, 4 and 5 on the same frame of discernment .
By (4), the Jousselme distance matrix between each elements in a BPA can be obtained as follows
Utilize Hausdorff distance in (6), the Hausdorff distance matrix between each elements in a BPA can be obtained as follows
It is clearly that, the five elements have no object in common. The similarity between each elements are the same value zero in Jousselme distance matrix. In case of this, Jousselme distance matrix can not show the detailed distance of each elements in an orderable system. However, Hausdorff distance matrix can calculate the detail similarity between each orderable elements.
3.2 New combination approach
In this subsection, we purpose an improved combination approach based on Deng’s methoddeng2004efficient . The new method takes advantage of Huasdorff distancehausdorff1957set to update Jousselme distancejousselme2001new .
Let and be two BPAs defined on the same frame of discernment , containing mutually exclusive and exhaustive hypotheses. The distance between and can be defined as
Given there are BPAs in the system, we can calculate the distance between each two BPAs. Thus, the distance matrix is presented as follows
Let be the similarity value between BPA and , thus the can be defined as
It is obvious that while the value of distance between two BPAs are bigger, the similarity of two BPAs are smaller, and vice versa. The similarity function can be represented by a matrix as follows
Let be the support degree of BPA in the system, and the support degree of BPA can be presented as follow
To normalize , the of BPA can be obtained as follows
It is obvious that
indicates the important and credible degree of BPA among all BPAs in the system. It can be regard as the weight of BPA . After obtained the weight of each BPAs, we take advantage of Dempster’s combination ruledempster1967upper ; shafer1976mathematical to yield a new BPA.
The below example is used to demonstrate the detail processes of the new proposed method.
Given there are four BPAs , , and on the same frame of discernment :
Therefore, the new BPA before combination can be obtained as follows
There are four BPAs in this example, we apply Dempster’s combination rule to combine the new BPA three times, the results are presented as follows
4 Numerical examples and Applications
It is known that Dempster-Shafer evidence theorydempster1967upper ; shafer1976mathematical needs less information than Bayes probability to deal with uncertain information. It is often regarded as the extension of Bayes probability.
We utilize the below example to illustrate the effectiveness of the new proposed method.
There are five mass functions on the same frame of discernment, the five BPAs are presented as followsdeng2004efficient
The results of different methods to combine the five BPAs are presented in Table.1. From Table.1, we can see that Dempster’s combination ruledempster1967upper ; shafer1976mathematical can not handel with highly conflicting evidence. Once an element is negatived by any BPAs, no matter how strongly it is supported by other BPAs, its probability will always remain zero.
Murphy’s methodmurphy2000combining regards each evidence plays the same role in the system, considered little relations among evidences. Dengdeng2004efficient improved Murphy’s work and took advantage of an evidence distance as the weight of each evidence. The novel proposed method based on Deng’s method, but utilizes Hausdorff distance to update the distance matrix. Fig.1 indicates that the convergence speed of proposed method is slower than Deng’s method but faster than Murphy’s method, owing to the additional update distance because some sensors may be orderable.
Dempster-Shafer evidence theory is a powerful tool to deal with uncertain and imprecise information in widely fields. However the evidence collected may be multifarious, some of them may be highly conflicting, owing to various noise factors, subjective or objective. The original Dempster combination rule can do nothing for these highly conflicting evidence. Modified methods of Dempster’s combination rule are briefly introduced, and all of them have some drawbacks. The new proposed method inherits all the advantages of Deng’s method. It applies Hausdorff distance to update the Jousselme distance and takes more distance information into account. Numerical examples demonstrate that the new proposed method can discern the correct target, effectively.
- (1) A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The annals of mathematical statistics 38 (2) (1967) 325–339.
- (2) G. Shafer, A mathematical theory of evidence, Vol. 1, Princeton university press Princeton, 1976.
T. Denœux, Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence, Artificial Intelligence 172 (2) (2008) 234–264.
- (4) D. Dubois, H. Prade, Representation and combination of uncertainty with belief functions and possibility measures, Computational Intelligence 4 (3) (1988) 244–264.
- (5) Y. He, L.-F. Hu, X. Guan, Y. Deng, D. Han, A new method of measuring the degree of conflict among general basic probability assignments, Scientia Sinica (Informationis) 41 (8) (2011) 989–997.
D.-Q. Han, C.-Z. Han, Y. Deng, Y. Yang, Weighted combination of conflicting evidence based on evidence variance, Acta Electronica Sinica 39 (3A) (2011) 153–157.
- (7) D.-Q. Han, Y. Deng, C.-Z. Han, Z.-Q. Hou, Weighted evidence combination based on distance of evidence and uncertainty measure, Journal of Infrared and Millimeter Waves 30 (5) (2011) 396–400.
- (8) S.-Y. Zhang, Q. Pan, H.-C. Zhang, A new kind of combination rule of evidence theory, Control and Decision 15 (5) (2000) 540–544.
- (9) Q. Pan, S.-Y. Zhang, Y.-M. Cheng, H.-C. Zhang, Some research on robustness of evidence theory, Acta Automatica Sinica 27 (6) (2001) 798–805.
- (10) Y. He, L. Hu, X. Guan, D. Han, Y. Deng, New conflict representation model in generalized power space, Journal of Systems Engineering and Electronics 23 (1) (2012) 1–9.
- (11) Y. Deng, R. Sadiq, W. Jiang, S. Tesfamariam, Risk analysis in a linguistic environment: a fuzzy evidential reasoning-based approach, Expert Systems with Applications 38 (12) (2011) 15438–15446.
- (12) Y. Deng, F. T. Chan, Y. Wu, D. Wang, A new linguistic mcdm method based on multiple-criterion data fusion, Expert Systems with Applications 38 (6) (2011) 6985–6993.
- (13) Y. Deng, F. T. Chan, A new fuzzy dempster mcdm method and its application in supplier selection, Expert Systems with Applications 38 (8) (2011) 9854–9861.
- (14) Y. Deng, X. Su, D. Wang, Q. Li, Target recognition based on fuzzy dempster data fusion method, Defence Science Journal 60 (5) (2010) 525–530.
- (15) B. Suo, Y. Cheng, C. Zeng, J. Li, Computational intelligence approach for uncertainty quantification using evidence theory, Journal of Systems Engineering and Electronics 24 (2) (2013) 250–260.
- (16) Y. Tan, J. Yang, L. Li, J. Xiong, Data fusion of radar and iff for aircraft identification, Systems Engineering and Electronics, Journal of 23 (5) (2012) 715–722.
- (17) T. Geng, A. Zhang, G. Lu, Consensus intuitionistic fuzzy group decision-making method for aircraft cockpit display and control system evaluation, Journal of Systems Engineering and Electronics 24 (4) (2013) 634–641.
- (18) D. Wei, X. Deng, X. Zhang, Y. Deng, S. Mahadevan, Identifying influential nodes in weighted networks based on evidence theory, Physica A: Statistical Mechanics and its Applications 392 (10) (2013) 2564–2575.
- (19) C. Gao, D. Wei, Y. Hu, S. Mahadevan, Y. Deng, A modified evidential methodology of identifying influential nodes in weighted networks, Physica A: Statistical Mechanics and its Applications 392 (21) (2013) 5490–5500.
- (20) B. Kang, Y. Deng, R. Sadiq, S. Mahadevan, Evidential cognitive maps, Knowledge-Based Systems 35 (2012) 77–86.
- (21) S. Chen, Y. Deng, J. Wu, Fuzzy sensor fusion based on evidence theory and its application, Applied Artificial Intelligence 27 (3) (2013) 235–248.
- (22) L. A. Zadeh, A simple view of the dempster-shafer theory of evidence and its implication for the rule of combination, AI magazine 7 (2) (1986) 85–90.
- (23) R. R. Yager, On the dempster-shafer framework and new combination rules, Information Sciences 41 (2) (1987) 93–137.
- (24) P. Smets, R. Kennes, The transferable belief model, Artificial intelligence 66 (2) (1994) 191–234.
- (25) P. Smets, The combination of evidence in the transferable belief model, Pattern Analysis and Machine Intelligence, IEEE Transactions on 12 (5) (1990) 447–458.
- (26) E. Lefevre, O. Colot, P. Vannoorenberghe, Belief function combination and conflict management, Information Fusion 3 (2) (2002) 149–162.
- (27) R. Haenni, Are alternatives to dempster’s rule of combination real alternatives?: Comments on “about the belief function combination and the conflict management problem”—-lefevre et al, Information Fusion 3 (3) (2002) 237–239.
- (28) C. K. Murphy, Combining belief functions when evidence conflicts, Decision Support Systems 29 (1) (2000) 1–9.
- (29) Y. Deng, W.-K. Shi, Z.-F. Zhu, Efficient combination approach of conflict evidence, Journal of Infrared and Millimeter Waves 23 (1) (2004) 27–32.
- (30) A.-L. Jousselme, D. Grenier, É. Bossé, A new distance between two bodies of evidence, Information Fusion 2 (2) (2001) 91–101.
- (31) F. Hausdorff, Set Theory: Translated from the German by John R. Aumann, Et Al, Vol. 119, AMS Bookstore, 1957.
- (32) Y. He, G. Wang, X. Guan, et al., Information fusion theory with applications, Beijing: Publishing House of Electronics Industry, 2010.
- (33) Z. Sunberg, J. Rogers, A belief function distance metric for orderable sets, Information Fusion 14 (4) (2013) 361–373.
Hongming Mo was born in 1983. He received the B.S.degree from Chongqing Normal University in 2006. He is now an assistant researcher in Sichuan University of Nationalities. His research interests include uncertain information modeling and processing.
Yong Deng was born in 1975. He received the Ph.D.degree from Shanghai Jiaotong University in 2003. He is now a professor in Southwest University. His research interests include uncertain information modeling and processing.