Since first proposed by Dempster dempster1967upper , and then developed by Shafer shafer1976mathematical , the Dempster-Shafer theory of evidence, which is also called Dempster-Shafer theory or evidence theory, has been paid much attentions for a long time and continually attracted growing interests. Even as a theory of reasoning under the uncertain environment, Dempster-Shafer theory has an advantage of directly expressing the “uncertainty ”by assigning the probability to the subsets of the set composed of multiple objects, rather than to each of the individual objects, so it has been widely used in many fields bloch1996some ; srivastava2003applications ; cuzzolin2008geometric ; masson2008ecm ; denoeux2011maximum ; Dengnew2011 ; denoeux2013maximum ; yang2013discounted ; yang2013novel ; wei2013identifying ; liu2013evidential ; deng2014supplier .
Due to improve the Dempster-Shafer theory of evidence, many studies have been devoted for combination rule of evidence yager1996aggregation ; gebhardt1998parallel ; yang2013discounted ; lefevre2013preserve ; yang2013evidential , confliction problem yager1987dempster ; lefevre2002belief ; liu2006analyzing ; schubert2011conflict ; tchamova2012behavior , generation of mass function bastian2010universal ; cappellari2012systematic ; liu2013belief ; burger2013randomly ; liu2014credal , uncertain measure of evidence klir1991generalized ; bachmann2010uncertainty ; bronevich2010measures ; baker2012measuring , and so on couso2010independence ; limbourg2010uncertainty ; jirouvsek2011compositional ; luo2012agent ; karahan2013persistence ; mao2014model ; zhang2014response . One open issue of evidence theory is the decision making based on the basic probability assignments, many works have been done to construct a reasonable model for the decision making smets1994transferable ; smets2005decision ; cobb2006plausibility ; daniel2006transformations ; merigo2011decision ; nusrat2013descriptive .
In the transferable belief model (TBM) smets1994transferable , pignistic probabilities are used for decision making. The transferable belief model is presented to represent quantified beliefs based on belief functions. TBM was constructed by two levels. The credal level where beliefs are entertained and quantified by belief functions. The pignistic level where beliefs can be used to make decisions and are quantified by probability functions. The main idea of the pignistic probability transformation is to transform the multi-elements subsets into singleton subsets by an average method. Though the pignistic probability transformation is widely used, it can not describe the unknown for the multi-elements subsets. Hense, a generalization of the pignistic probability transformation called multiscale probability transformation of basic probability assignment is proposed in this paper, which is based on the belief function and the plausibility function. The proposed function can be calculated with the difference between the belief function and the plausibility function, we call it multiscale probability function and denote it as a function . In the multiscale probability function, a factor based on the Tsallis entropy tsallis1988possible is used to make the multiscale probabilities diversified. When the value of equals to 0, the proposed multiscale probability transformation can be degenerated as the pignistic probability transformation.
The rest of this paper is organized as follows. Section 2 introduces some basic Preliminaries about the Dempster-Shafer theory and the pignistic probability transformation. In section 3 the multiscale probability transformation is presented. Section 4 uses an example to illustrate the effectiveness of the multiscale probability transformation. Conclusion is given in Section 5.
2.1 Dempster-Shafer theory of evidence
Dempster-Shafer theory of evidence dempster1967upper ; shafer1976mathematical , also called Dempster-Shafer theory or evidence theory, is used to deal with uncertain information. As an effective theory of evidential 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.
2.2 Pignistic probability transformation
In the transferable belief model (TBM) smets1994transferable , pignistic probabilities are used for decision making. The definition of the pignistic probability transformation is shown as follows.
Let be a BPA on the frame of discernment . Its associated pignistic probability function is defined as:
is the cardinality of subset A. The process of pignistic probability transformation(PPT) is that basic probability assignment transferred to probability distribution. Therefore, the pignistic betting distance can be easily obtained by PPT.
3 Multiscale probability transformation of basic probability assignment
In the transferable belief model (TBM) smets1994transferable , pignistic probabilities are used for decision making. The transferable belief model is presented to represent quantified beliefs based on belief functions. The main idea of the pignistic probability transformation is to transform the multi-elements subsets into singleton subsets by an average method. Though the pignistic probability transformation is widely used, it is not reasonable in the Example 1.
Example 1. Suppose there is a frame of discernment of a, b, c, the BPA is given as follows.
, , , .
In the pignistic probability transformation, for , the result will be . Actually it is not reasonable, means the sensor can not judge the target belongs to which classes, it represents a meaning of “unknown ”. In other word, only according to , nothing can be obtained except “unknown ”. In this situation, average is used in the pignistic probability transformation, which is one of the methods to solve the problem. Compared with the average, weighted average is more reasonable in many situations. In this paper, the weighted average is represented by the difference between the belief function and the plausibility function, whose definition is shown as follows.
Let be a BPA on the frame of discernment . The difference function is defined as:
The weight is defined as:
Based on the weighted average idea, a factor , which is proposed in the Tsallis entropy tsallis1988possible , is used to highlight the weights. Thus, the definition of multiscale probability function is shown as follows.
Let be a BPA on the frame of discernment . Its associated multiscale probability function on is defined as:
where is the cardinality of subset A. is a factor based on the Tsallis entropy to amend the proportion of the interval. The transformation between and is called the multiscale probability transformation.
Actually, the part of the Eq. 10 denotes the weight of element based on normalization, which is replaced the averaged in the pignistic probability function.
Theorem 3.1: Let be a BPA on the frame of discernment . Its associated multiscale probability on is degenerated as the pignistic probability when equals to 0.
Proof: When equals to 0, equals to 1, the multiscale probability function will be calculated as follows:
Then, it can obtain:
Theorem 3.2: Let be a BPA on the frame of discernment . If the belief function equals to the plausibility function, its associated multiscale probability on is degenerated as the pignistic probability .
Proof: Given a BPA on the frame of discernment , for each , when the belief function equals to the plausibility function, namely , the bel is a probability distribution P smets1994transferable , then MulP is equal to BetP.
For example, let be a frame of discernment and , if it satisfies with , , , the BPA on the frame must be satisfied with . In this situation, the multiscale probability will be degenerated as the pignistic probability.
Corollary: If bel is a probability distribution P, then MulP is equal to P.
Theorem 3.3: Let be a BPA on the frame of discernment . If the differences between the belief function and the plausibility function is the same, the multiscale probability transformation can be degenerated as the pignistic probability transformation.
Proof: The same as the proof of theorem 3.1.
An illustrative example is given to show the calculation of the multiscale probability transformation step by step.
Example 2. Let be a frame of discernment with 3 elements. We use , , to denote element 1, element 2, and element 3 in the frame. One body of BPA is given as follows:
Step 2 Calculate the difference between the belief function and the plausibility function:
Step 3 Calculate the weight of each element in . Assumed that the value of equals to 1.
Step 4 The value of the multiscale probability function can be obtained based on above steps.
4 Case study
In this section, an illustrative example is given to show the effection of the multiscale probability function when the value of changes.
Example 3. Let be a frame of discernment with 3 elements, namely .
Given one body of BPAs:
Based on the pignistic probability transformation, the results of the pignistic probability function is shown as follows:
According to the proposed function in this paper, the results of the multiscale probability function can be obtained through the follow steps.
Firstly, the values of belief function and the plausibility function can be obtained as follows:
Then, the differences between the belief functions and the plausibility functions can be calculated:
Based on the definition of the multiscale probability transformation, the values of can be obtained. There are 20 cases where the values of starting from Case 1 with and ending with Case 11 when as shown in Table 1. The values of for these 20 cases is detailed in Table 1 and graphically illustrated in Fig. 1.
According to the Table 1 and the Fig. 1, on one hand, when the value of increased, the probability of the element which has larger weight is increased, and the probability of the element which has smaller weight is decreased. For example, the element starting with probability 0.5500, and ending with probability 0.8250. The element starting with probability 0.2500, and ending with probability 0.1061.
On the other hand, according to the Table 1, the option ranking of the the values of can be obtained. It is starting with , and ending with . It is mainly because is impact of the values of . This principle makes the multiscale probability function has the ability to highlight the proportion of each element in the frame of discernment.
Note that when the value of equals to 0, the values of pignistic probability is the same as the values of multiscale probability , which is proposed in this paper. In other word, the multiscale probability function is a generalization of the pignistic probability function.
In the transferable belief model(TBM), pignistic probabilities are used for decision making. In this paper, a multiscale probability transformation of basic probability assignment based on the belief function and the plausibility function, which is a generalization of the pignistic probability transformation is proposed. In the multiscale probability function, a factor is proposed to make the multiscale probability function has the ability to highlight the proportion of each element in the frame of discernment. When the value of equals to 0, the multiscale probability transformation can be degenerated as the pignistic probability transformation. An illustrative case is provided to demonstrate the effectiveness of the multiscale probability transformation.
The work is partially supported by 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).
Conflict of interests
The authors declare that there is no conflict of interests regarding the publication of this article.
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Yong Deng received his Ph.D. degree from Shanghai Jiao Tong University, China, 2003. He is a full professor in School of Computer and Information Science of Southwest University, Chongqing, China and a visiting professor in School of Engineering of Vanderbilt University, TN, USA. Now he is a member of the editorial board for The Scientific World Journal. So far, he has published more than 100 peer-reviewed journal. His research interests include uncertain information modeling, risk and reliability analysis, information fusion and optimization under uncertain environment.