Dempster-Shafer evidence theory is widely used in many disciplines since it allows to deal with uncertain information. Several familiar branches of its applications includes statistical learning cuzzolin2008geometric ; huang2014new ; yang2013novel , classification and clustering denoeux1995k ; denoeux2000neural ; masson2008ecm ; liu2013evidential , decision making liu2014PM ; ahn2014use ; yao2014induced , knowledge reasoning kang2012evidential ; denoeux2013maximum , risk assessment and evaluation zhang2013ifsjsp ; Deng2014DAHPSupplier ; yager2014characterizing , and so forth chen2013fuzzy ; deng2014environmental ; zhang2014wei ; yager2014owa ; deng2015parameter . In Dempster-Shafer evidence theory, several key research directions continuingly appeal to researcher’s attention, for example, the combination of multiple evidences murphy2000combining ; lefevre2013preserve ; deng2014improved , conflict management liu2006analyzing ; schubert2011conflict , generation of basic probability assignment (BPA) xu2014nonAppInt ; zhang2014new ; xu2013newKBS , and so on karahan2013persistence ; certa2013multistep ; zhang2014response . Among these points, decision-making based BPA is a crucial issue to be solved, and it has attracted much attention.
A lot of works have been done to construct a reasonable model for the decision making based on the BPA smets1994transferable ; smets2005decision ; merigo2011decision ; nusrat2013descriptive . One widely used model is the transferable belief model (TBM) smets1994transferable , pignistic probabilities are used for decision making in this model. In the TBM, a pignistic probability transformation (PPT) approach has been proposed to bring out probabilities from BPAs. Another well-known probability transformation method is proposed by Barry R. Cobb cobb2006plausibility , which is based on the plausibility function. The main idea of the plausibility transformation method is to assign the uncertain according to the plausibility function with normalization. In cuzzolin2012relative , the semantics and properties of the relative belief transform have been discussed. One method was mentioned namely proportional probability transformation daniel2006transformations . Within the proportional probability transformation, a belief mass assigned to nonsingleton focal element is distributed among ’s elements with respect influenced by the proportion of BPAs assigned to singletons. The proportional probability transformation is influenced by the proportion of BPAs assigned to singletons.
In this paper, a novel probability transformation approach is proposed based on a new entropy measure of BPAs, Deng entropy DengEntropyArXiv . Within the proposed approach, given a BPA, it is expected to find a probability distribution whose Shannon entropy is as close as possible to the entropy of given BPA. The rest of this paper is organized as follows. Section 2 introduces some basic background knowledge. In section 3 the proposed probability transformation approach is presented. Section 4 uses some examples to illustrate the effectiveness of the proposed approach. Conclusion is given in Section 5.
2.1 Dempster-Shafer evidence theory
Dempster-Shafer evidence theory dempster1967upper ; shafer1976mathematical , also called Dempster-Shafer theory or evidence theory, is used to deal with uncertain information. As an effective theory of uncertainty reasoning, Dempster-Shafer theory has an advantage of directly expressing various uncertainties. This theory needs weaker conditions than bayesian theory of probability, so it is often regarded as an extension of the bayesian theory. For completeness of the explanation, a few basic concepts are introduced as follows.
Let be a set of mutually exclusive and collectively exhaustive, indicted by
The set is called frame of discernment. The power set of is indicated by , where
If , is called a proposition.
For a frame of discernment , a mass function is a mapping from to , formally defined by:
which satisfies the following condition:
In Dempster-Shafer theory, a mass function is also called a basic probability assignment (BPA). If , is called a focal element, the union of all focal elements is called the core of the mass function.
For a proposition , the belief function is defined as
The plausibility function is defined as
Obviously, , these functions and are the lower limit function and upper limit function of proposition , respectively.
3 Proposed probability transformation approach based on Deng entropy
In this section, a new measure for the uncertainty of BPA, Deng entropy is introduced first, then a new approach of transforming BPA to probability distribution is proposed based on the concept of Deng entropy.
3.1 Deng entropy
Deng entropy DengEntropyArXiv is a generalized Shannon entropy to measure uncertainty involving in a BPA. Mathematically, Deng entropy can be presented as follows
where, is a proposition in mass function , and is the cardinality of . As shown in the above definition, Deng entropy, formally, is similar with the classical Shannon entropy, but the belief for each proposition is divided by a term which represents the potential number of states in (of course, the empty set is not included).
Specially, Deng entropy can definitely degenerate to the Shannon entropy if the belief is only assigned to single elements. Namely,
3.2 Proposed probability transformation approach
In our view, a primary principle in the transformation process is to minimize the difference of uncertainties involving in the given BPA and obtained probability distribution. In order to implement such optimization transformation, it must be able to calculate the uncertainty of BPA. Exactly, Deng entropy provides a method to measure the uncertainty of BPA as well as probability distribution. Therefore, a novel probability transformation approach based on Deng entropy can be proposed as follows.
Assume the frame of discernment is , given a BPA , a probability distribution associated with is calculated by solving the following optimization problem:
where and are the entropies of BPA and probability distribution , respectively.
4 Numerical examples
In this section, some illustrative examples are given to show the proposed probability transformation approach.
By using the proposed probability transformation approach, a probability distribution is obtained by
we can obtain that
The result shows that the transformed probability distribution has the maximum uncertainty (Shannon entropy) when the given BPA is totally unknown (i.e., ).
Example 2. Given a frame of discernment , there is a BPA:
Due to ; , the associated probability distribution can be calculated by
So, we can get
In this paper, the transformation of BPA to probability distribution has been studied. Based on an idea that minimizing the difference of uncertainties involving in the given BPA and obtained probability distribution, a novel probability transformation approach has been proposed. Finally, several illustrative examples have been given to show the proposed method.
The work is partially supported by China Scholarship Council, National Natural Science Foundation of China (Grant No. 61174022), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20131102130002), R&D Program of China (2012BAH07B01), National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801), the open funding project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No.BUAA-VR-14KF-02), Fundamental Research Funds for the Central Universities (Grant No. XDJK2014D034).
- (1) F. Cuzzolin, A geometric approach to the theory of evidence, Systems, Man, and Cybernetics, Part C: Applications and Reviews, IEEE Transactions on 38 (4) (2008) 522–534.
- (2) S. Huang, X. Su, Y. Hu, S. Mahadevan, Y. Deng, A new decision-making method by incomplete preferences based on evidence distance, Knowledge-Based Systems 56 (2014) 264–272.
- (3) Y. Yang, D. Han, C. Han, F. Cao, A novel approximation of basic probability assignment based on rank-level fusion, Chinese Journal of Aeronautics 26 (4) (2013) 993–999.
- (4) T. Denoeux, A k-nearest neighbor classification rule based on dempster-shafer theory, Systems, Man and Cybernetics, IEEE Transactions on 25 (5) (1995) 804–813.
M.-H. Masson, T. Denoeux, Ecm: An evidential version of the fuzzy c-means algorithm, Pattern Recognition 41 (4) (2008) 1384–1397.
- (7) Z.-g. Liu, Q. Pan, J. Dezert, Evidential classifier for imprecise data based on belief functions, Knowledge-Based Systems 52 (2013) 246–257.
- (8) J. Liu, Y. Li, R. Sadiq, Y. Deng, Quantifying influence of weather indices on pm based on relation map, Stochastic Environmental Research and Risk Assessment 28 (6) (2014) 1323–1331.
- (9) B. S. Ahn, R. R. Yager, The use of ordered weighted averaging method for decision making under uncertainty, International Transactions in Operational Research 21 (2) (2014) 247–262.
- (10) S. Yao, W.-Q. Huang, Induced ordered weighted evidential reasoning approach for multiple attribute decision analysis with uncertainty, International Journal of Intelligent Systems 29 (10) (2014) 906–925.
- (11) B. Kang, Y. Deng, R. Sadiq, S. Mahadevan, Evidential cognitive maps, Knowledge-Based Systems 35 (2012) 77–86.
T. Denoeux, Maximum likelihood estimation from uncertain data in the belief function framework, Knowledge and Data Engineering, IEEE Transactions on 25 (1) (2013) 119–130.
- (13) X. Zhang, Y. Deng, F. T. Chan, P. Xu, S. Mahadevan, Y. Hu, Ifsjsp: A novel methodology for the job-shop scheduling problem based on intuitionistic fuzzy sets, International Journal of Production Research 51 (17) (2013) 5100–5119.
- (14) X. Deng, Y. Hu, Y. Deng, S. Mahadevan, Supplier selection using AHP methodology extended by D numbers, Expert Systems with Applications 41 (1) (2014) 156–167.
- (15) R. R. Yager, N. Alajlan, On characterizing features of owa aggregation operators, Fuzzy Optimization and Decision Making 13 (1) (2014) 1–32.
S. Chen, Y. Deng, J. Wu, Fuzzy sensor fusion based on evidence theory and its application, Applied Artificial Intelligence 27 (3) (2013) 235–248.
- (17) X. Deng, Y. Hu, Y. Deng, S. Mahadevan, Environmental impact assessment based on D numbers, Expert Systems with Applications 41 (2) (2014) 635–643.
- (18) C. Zhang, Y. Hu, F. T. Chan, R. Sadiq, Y. Deng, A new method to determine basic probability assignment using core samples, Knowledge-Based Systems 69 (2014) 140–149.
- (19) R. R. Yager, N. Alajlan, Probabilistically weighted owa aggregation, IEEE Transactions on Fuzzy Systems 22 (1) (2014) 46–56.
- (20) X. Deng, Y. Hu, F. T. Chan, S. Mahadevan, Y. Deng, Parameter estimation based on interval-valued belief structures, European Journal of Operational Research 241 (2) (2015) 579–582.
- (21) C. K. Murphy, Combining belief functions when evidence conflicts, Decision support systems 29 (1) (2000) 1–9.
- (22) E. Lefèvre, Z. Elouedi, How to preserve the conflict as an alarm in the combination of belief functions, Decision Support Systems 56 (2013) 326–333.
- (23) X. Deng, Y. Deng, F. T. Chan, An improved operator of combination with adapted conflict, Annals of Operations Research 223 (1) (2014) 451–459.
- (24) W. Liu, Analyzing the degree of conflict among belief functions, Artificial Intelligence 170 (11) (2006) 909–924.
- (25) J. Schubert, Conflict management in Dempster-Shafer theory using the degree of falsity, International Journal of Approximate Reasoning 52 (3) (2011) 449–460.
- (26) P. Xu, X. Su, S. Mahadevan, C. Li, Y. Deng, A non-parametric method to determine basic probability assignment for classification problems, Applied Intelligence 41 (3) (2014) 681–693.
- (27) C. Zhang, Y. Hu, F. T. Chan, R. Sadiq, Y. Deng, A new method to determine basic probability assignment using core samples, Knowledge-based Systems 69 (2014) 140–149.
- (28) P. Xu, Y. Deng, X. Su, S. Mahadevan, A new method to determine basic probability assignment from training data, Knowledge-Based Systems 46 (2013) 69–80.
- (29) F. Karahan, S. Ozkan, On the persistence of income shocks over the life cycle: Evidence, theory, and implications, Review of Economic Dynamics 16 (3) (2013) 452–476.
- (30) A. Certa, M. Enea, G. M. Galante, C. M. La Fata, A multistep methodology for the evaluation of human resources using the evidence theory, International Journal of Intelligent Systems 28 (11) (2013) 1072–1088.
- (31) Z. Zhang, C. Jiang, X. Han, D. Hu, S. Yu, A response surface approach for structural reliability analysis using evidence theory, Advances in Engineering Software 69 (2014) 37–45.
- (32) P. Smets, R. Kennes, The transferable belief model, Artificial intelligence 66 (2) (1994) 191–234.
- (33) P. Smets, Decision making in the tbm: the necessity of the pignistic transformation, International Journal of Approximate Reasoning 38 (2) (2005) 133–147.
- (34) J. M. Merigó, M. Casanovas, Decision making with dempster-shafer theory using fuzzy induced aggregation operators, in: Recent developments in the ordered weighted averaging operators: Theory and practice, Springer, 2011, pp. 209–228.
- (35) E. Nusrat, K. Yamada, A descriptive decision-making model under uncertainty: combination of dempster-shafer theory and prospect theory, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21 (01) (2013) 79–102.
- (36) B. R. Cobb, P. P. Shenoy, On the plausibility transformation method for translating belief function models to probability models, International Journal of Approximate Reasoning 41 (3) (2006) 314–330.
- (37) F. Cuzzolin, On the relative belief transform, International Journal of Approximate Reasoning 53 (5) (2012) 786–804.
- (38) M. Daniel, On transformations of belief functions to probabilities, International Journal of Intelligent Systems 21 (3) (2006) 261–282.
- (39) Y. Deng, Deng entropy: a generalized shannon entropy to measure uncertainty, arXiv.org (2015) submit/1174300.
- (40) A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The annals of mathematical statistics 38 (2) (1967) 325–339.
- (41) G. Shafer, A mathematical theory of evidence, Princeton university press Princeton, 1976.