Journal of Sensors

Volume 2015, Article ID 509385, 9 pages

http://dx.doi.org/10.1155/2015/509385

## Combination of Evidence with Different Weighting Factors: A Novel Probabilistic-Based Dissimilarity Measure Approach

Key Laboratory of Embedded and Network Computing of Hunan Province, College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, China

Received 9 October 2014; Revised 9 February 2015; Accepted 23 February 2015

Academic Editor: Stefania Campopiano

Copyright © 2015 Mengmeng Ma and Jiyao An. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To solve the invalidation problem of Dempster-Shafer 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.

#### 1. Introduction

Multisensor data fusion is a technology that combines information from several sources to form a unified picture [1]. Dempster-Shafer (DS) theory of evidence is one of the most prevalent methods for data fusion and is firstly proposed by Dempster in the 1960s [2] and further developed by Shafer in the 1970s [3]. DS has been widely used in many regions, such as image processing [4, 5], target recognition and tracking [6, 7], fault diagnosis [8], and knowledge discovery [9], to name a few. Unfortunately, in the framework of DS, Dempster’s rule, as an inherent problem, is incapable of managing the high conflicts from various information sources at the step of normalization and will generate counterintuitive results as first highlighted by Zadeh [10].

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 [11–13]. The representative methods are Lefevre’s method [11], Yager’s method [12], and so on. Scholars who put forward this method believe that the cause of high conflict evidence combination failure is due to some defects of the Dempster combination’s rules. The second is to modify the original sources of evidence without changing Dempster’s combination rule [14–19]. The representative methods are Murphy’s method [14], Y. Deng’s method [15], and so on. Our approach is based on the second kind of improvement way, for the improved combination rules cannot meet the commutative law or associative law and the high conflict is not due to Dempster’s rule, and the unreliable evidence is the real cause. The sources of evidence should be discounted according to the reliability. The basic idea of the discounting method is that if one source has great (small) dissimilarity with the others, its reliability should be low (high).

Therefore, the dissimilarity measure between two sources of evidence plays a crucial role in the discounting method. Jousselme et al. [16] proposed a principled distance which regarded the evidence as the vector based on the geometry interpretation, but its computation burden is important. Liu [17] proposed the two-dimensional measure which consists of Shafer’s conflict coefficient and the pignistic probability distance between betting commitments. However, when one factor is large and another is small, the dissimilarity degree cannot be directly assured. Qu et al. [18] proposed a conflict rate to proportion the conflict, but when the two pieces of evidence are equal, the conflict rate will cause a counter-intuitive result. Liu et al. [19] proposed a dissimilarity measure to describe the divergences of two aspects between two pieces of evidence, the difference of beliefs and the difference of hypotheses which two pieces of evidence strongly support. But it is not good enough to capture the difference between BBAs in some cases as it will be seen. Based on the above analysis, the current methods are not adequate to precisely delineate the divergence between two pieces of evidence. This motivates researchers to develop a good and useful measure of dissimilarity.

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 T-conorm rule [20] based on an improved probabilistic transformation. The weighting factors adopted to discount the original sources are automatically determined according to the proposed probabilistic-based dissimilarity measure. The interest of our improved probabilistic transformation, the new dissimilarity measure, and the discounted method to combine conflicting sources of evidence are illustrated through some numerical examples.

The rest of this paper is organized as follows. In Section 2, we briefly review the DS evidence theory. The new method for combining conflict evidence is proposed in Section 3. In Section 4, numerical examples are enumerated to show the performance of the existing alternatives and the proposed method. Section 5 concludes this paper.

#### 2. Theory of Evidence

##### 2.1. Belief Function

The frame of discernment, denoted by , is a finite nonempty set including mutually exclusive and exhaustive elements. denotes the power set composed of all the possible subsets of . A basic belief assignment (BBA) is a function mapping from to and verifies the following conditions:where is the empty set. The subset of with nonzero masses is called the focal elements of . There are also two other definitions in the theory of evidence. They are belief and plausibility functions associated with a BBA and are defined, respectively, as

represents the total amount of probability that is allocated to , while can be interpreted as the amount of support that could be given to . and are the lower and upper limit of the belief level of hypothesis , respectively.

##### 2.2. Dempster’s Combination Rule and the Paradox Problem

Suppose two bodies of evidence and are derived from two information sources; Dempster’s combination rule can be defined aswhere is the conflict coefficient, reflecting the degree of conflict between the 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 [10]. The second is that the conflict coefficient is not very appropriate to really characterize the conflict between BBAs, particularly in case of two equal BBAs as reported in [17].

*Example 1 (Zadeh’s example). *Assume and over are defined as

According to (3) and (4), we get , , and . We can see that and have low support level to hypothesis , but the resulting structure has complete support to . This appears to be counterintuitive.

*Example 2. *Consider two equal and over are defined as

According to (4), we get . This reveals that the two pieces of evidence are of high degree of conflict, but in fact they are equal.

##### 2.3. Pignistic Transformation

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 [22], defined aswhere is the number of elements in subset . transfers the positive mass of belief of each nonspecific element onto the singletons involved in that element according to the cardinal number of the proposition.

#### 3. A Novel Combination Approach of Conflict Evidence

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 well-adopted principle that* the truth lies in the majority opinion*. The sources which are highly conflicting with the majority of other sources will be automatically assigned with a very low weighting factor in order to decrease their bad influence in the fusion process. To determine the weighting factors, the conflict should be well measured first.

The degree of conflict between BBAs has been measured in many works, including conflict coefficient [3], Jousselme’s distance measure [16], pignistic probability distance [17], conflict rate [18], and dissimilarity measure [19]. These measures cannot characterize the conflict comprehensively and accurately. What is more, the distance measures [16, 17, 19] are based on the pignistic transformation [21], but such transformation is only a simple average in mathematics. It considers the role of belief functions while ignoring the effect of the plausibility functions. Therefore, we propose an improved probabilistic transformation to transform BBAs into probabilities to overcome the shortcomings of pignistic transformation. Then based on the improved probabilistic transformation method, a novel dissimilarity degree which integrates the fuzzy nearness and correlation coefficient by Hamacher T-conorm rule is proposed.

##### 3.1. An Improved Probabilistic Transformation

*Definition 3. *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 aswhere is the total value of belief functions, defined as . The uncertain information that can be reallocated can be represented as . can well balance the degree of influence of the belief function and plausibility function. If the value of is big, the certainty information plays a dominant role, so the influence of should be bigger than . On the contrary, if the value of is big, the proportion of uncertain information is larger than the certainty information, so the influence of should be bigger than .

The improved probabilistic transform function satisfies and . It is worthy to note that the presented probabilistic transformation not only includes the special cases described in [22, 23] but also can well transform the BBAs into probabilities in general conditions.(1)If , all the focal elements are single sets, and so the BBAs should remain unchanged. Equation (8) can be simplified to .(2)If , and have the same influence on the allocations of the uncertain information, so (8) degrades into the probabilistic transform function proposed in [24], described as .(3)If , all the focal elements are multiple sets and the allocating of the uncertain information is only based on the , so (8) degrades into the plausibility function-based transform (PFT) proposed by Cobb and Shenoy in [23], described as .

*Example 4. *Let the BBA over the same frame of discernment be as follows:

In Example 4, . We choose Shannon entropy to measure the uncertainty of the probabilities after transformation. The results of probabilities and uncertainties in such general case are listed in Table 1. By analyzing the results, the improved probability transformation can get more effective probability and the smallest information uncertainty compared to the methods proposed in [21–23], since it balances the influence of belief function and plausibility function well with the factor .