Mathematical Problems in Engineering

Volume 2017, Article ID 7635972, 15 pages

https://doi.org/10.1155/2017/7635972

## A Method to Identify the Incomplete Framework of Discernment in Evidence Theory

School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Wen Jiang; nc.ude.upwn@newgnaij

Received 30 July 2017; Revised 16 October 2017; Accepted 5 November 2017; Published 10 December 2017

Academic Editor: Anna M. Gil-Lafuente

Copyright © 2017 Wen Jiang et al. 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

Dempster-Shafer evidence theory is a very useful tool in dealing with the decision-making of uncertain information. However, the classical evidence theory is no longer applicable when the frame of discernment (FOD) is incomplete. Moreover, incomplete FOD is an important reason for the emergence of conflict. So it is necessary to identify whether the FOD of the system is complete or not. In this paper, a method is proposed to identify the incomplete FOD under framework of the generalized evidence theory dealing with incomplete information. Within the proposed method, pieces of evidence are generated from the attributes of each sample firstly; then three criteria are used to identify weather the FOD is incomplete according to these evidence. The main parameters of the criteria are the amount of being a focal element in generated evidence, the mass of in the weighted average evidence from generated evidence, and the mass of in the combination of generated evidence. Some experiments are used to demonstrate the effectiveness of the proposed method.

#### 1. Introduction

Dempster-Shafer evidence theory (D-S evidence theory) [1, 2] is widely used in many fields such as decision-making [3–6], evidential reasoning [7, 8], uncertainty measure [9, 10], and others [11–15] because of its advantages in handling uncertainty information. This theory is also widely used in practical applications, such as fault diagnosis [16, 17], knowledge acquisition [18], risk and reliability analysis [19, 20], and failure mode [21–23]. While the counterintuitive results can be obtained when the given evidences highly conflict with each other, hundreds of methods have been proposed to address this issue [24]. In D-S evidence theory, conflict management is still an important issue. In general, there are two main reasons that may lead to conflict: one is the incomplete frame of discernment (FOD) and the other is that the sensors are disturbed. In order to better implement the combination of conflicting evidence, it is necessary and important to identify whether the FOD is complete or not.

In previous studies, Lefevre et al. [25] proposed a unified belief function combination method to manage the conflict, mainly considering the issue of conflict redistribution. Haenni’s view is to get the pretreatment evidence and then use Dempster combination rule to manage the conflict [26]. Murphy [27] presented a method to combine belief functions named averaging to balance multiple evidence. But it does not offer convergence toward certainty. Based on this, an improved method is presented in [28]. While these studies ignored the fact that the incomplete FOD is also an important cause that may lead to the conflict, for that, Smets and Kennes [29] proposed the TBM model under the open world assumption. Recently, a generalized evidence theory was presented in [30], addressing the combination of conflicting evidence in open world. It greatly expands the application of evidence theory like fuzzy [31–33] and game theory [34] and jumps out of the original model in dealing with conflicts. But the research aforementioned also did not tell under which condition the system is in incomplete FOD. According to previous studies, the research on identifying the incomplete FOD is still an open issue and has not been given deserved attention.

In this paper, a method is proposed to identify the incomplete FOD under framework of the generalized evidence theory. Generalized evidence theory [30] is a novel theory which can express and deal with uncertain information in an incomplete FOD. In consideration of empty set can express the information of incomplete FOD; three parameters of are used in the proposed method. They are the mass of in the weight average evidence from generated pieces of evidence, the amount of being a focal element in generated pieces of evidence, and the mass of in the combination of generated pieces of evidence. Within the proposed method, pieces of evidence can be generated from the attributes of each sample firstly; then three criteria are used to identify weather the FOD is incomplete according to these evidence.

In this paper, a method to identify the incomplete FOD, which takes into consideration the information in both the evidence and the samples, is proposed. The method uses the correlation coefficient [35] which has a better performance compared with other coefficients to express the similarity of evidence. From three aspects the proposed method collects the information about the FOD of the system, respectively, the mass of in one piece of evidence, the mass distribution of in weighted average evidence, and the mass distribution of in combination result. Some experiments are used to demonstrate the effectiveness of the proposed method. The experiments show that, for a collected sample, if the criteria can be satisfied, the system is seen incomplete. If it is not satisfied, it is seen as a complete FOD.

The rest of this paper is organised as follows. In Section 2, the preliminaries about D-S evidence and generalized evidence theory are briefly introduced. Section 3 presents the proposed method with three criteria. In Section 4, some experiments are shown to demonstrate the effectiveness of our method. An application about the motor rotor fault diagnosis is shown in Section 5. Finally, a brief conclusion is made in Section 6.

#### 2. Preliminaries

##### 2.1. Dempster-Shafer (D-S) Evidence Theory

D-S evidence theory is introduced by Dempster [1] and then developed by Shafer [2]. Owing to its outstanding performance in uncertainty model and process, this theory is widely applied to decision-making, optimization and reliability, and risk analysis.

*Definition 1. *Let be a finite nonempty set of mutually exclusive hypotheses, indicated by where set is called a frame of discernment. The power set of , , is indicated as follows:

*Definition 2. *A mass function is a mapping from to , formally noted by which satisfies the following condition: When , is called a focal element of the mass function.

*Definition 3. *Evidence combination in D-S evidence theory is noted as . Assume that there are two BPAs indicated by and ; the evidence combination of the two BPAs with Dempster’s combination rule [1] is formulated as follows:where reflects the conflict between the two BPAs and .

When given pieces of evidence, the evidence fusion with Dempster’s combination rule can be shown in (8). It attributes to the merit of the commutativity and associativity of the combination rule

Recently, Jiang proposed a correlation coefficient [35] to measure the degree of evidence.

*Definition 4. *For a discernment frame with elements, suppose the mass of two pieces of evidence denoted by . A correlation coefficient is defined as follows: where is degree of correlation denoted as where ; is the focal elements of mass, respectively; and is the cardinality of subset.

##### 2.2. Generalized Evidence Theory

Generalized evidence theory (GET) [30] eliminates the constraint close world and builds framework on an open world. To some extent, GET is the extension of the D-S theory and can express and deal with more uncertain information in the open world, comparing with D-S theory.

*Definition 5. *Let be a frame of discernment (FOD) in an open world. Its power set is composed of propositions. For , the mass function is a mapping : that satisfies then is the generalized basic probability assignment (GBPA) of the FOD .

The difference between GBPA and classical BPA is the restriction of (4), which means the empty set should also be regarded as a focal element and represents the union of the focal element out of the given FOD. And if , the GBPA degenerates to a classical BPA.

The same as GET, TBM model also assigns mass to empty set to represent unknown information. But the difference is the generation process of the mass of empty set. TBM model is simply to remove the normalization process of the Dempster’s combination rule and assigns the value of conflict coefficient to empty set. When generating evidence, the mass of empty set is still 0, while in GET, when generating evidence, mass can be assigned to empty set. This means there is no the restriction of (4).

*Definition 6. *Given two GBPAs ( and ), and are regarded as conflict with each other and the mass distribution of should be assigned to conflict. The generalized combination rule (GCR) is defined as follows: with Equation (13) defines the generalized conflict coefficient, and when means the framework of discernment is complete, the generalized conflict coefficient degenerates to a classical conflict coefficient.

While Jiang and Zhan pointed that there are two shortcomings of GCR in [30], one is the way to obtain that is unreasonable and lacks specific physical meaning. The other is that the way to obtain generalized conflict coefficient in (16) is not consistent with the GBPA. So the modified generalized combination rule (mGCR) in GET was proposed in [30].

*Definition 7. *In mGCR, and are considered as a support for . The orthogonal sum of and should also be normalized like other focal elements. Given two GBPAs ( and ), the mGCR is defined as follows: with

Also the distance between two bodies of evidence based on GET is proposed the same as in D-S evidence theory.

*Definition 8. *Let and be the two GBPAs on the framework of discernment ; the distance between and can be defined as follows: where is an dimension matrix and its element is expressed as follows:and its computing method is where , .

Equation (19) can be used in the situation when the frame of discernment is complete, and the result is similar to the distance using the definition in [36].

#### 3. The Proposed Method

Generally speaking, the empty set indicates that no elements are included. In the classical evidence theory, there is no mass assignment to . While in GET [30], it is considered to indicate the elements that are not in the framework, which presents the information that is out of the FOD. Based on this idea, a method mainly employing the mass of is proposed under the GET framework to identify the incomplete FOD.

An incomplete FOD means that there are targets or classes or anything else that are not included in the current FOD. Let us consider a classification problem. Assume it is known that all known samples belong to classes which constitute a FOD . And it is assumed that each sample has attributes. Now a new sample is obtained. How can we identify the completeness of FOD according to ? In this paper, for the sample , at first GBPAs which allow the empty set to have mass (i.e., ), denoted as , will be generated from the attributes. Then, three criteria are used as follows.

##### 3.1. Criterion 1

*Criterion 1. *Let be a variable taking value in , . If in GBPA ; then ; else, . Sample supports that FOD is incomplete if . Otherwise, FOD is said to be complete.

This criterion illustrates that if the amount of the initial generated GBPA of whose value exceeds 0.5 is more than half of the quantity of evidence, it is considered in an incomplete FOD. There is a physical meaning of the criterion. Firstly, the parameter of a generated GBPA represents the evidence’s confidence to support the incomplete FOD. That is because is treated as a focal element which expresses the elements that are not in the FOD. That is to say if the distribution of is larger, there is a larger support that the FOD is incomplete. Then assuming 0.5 is a threshold, if exceeds the threshold, this evidence is judged supporting an incomplete FOD. Therefore, the amount of evidence which satisfies , equivalent to , is used to identify the FOD. If the criteria can be satisfied, which says more than half of the evidence support is out of the FOD, the FOD is incomplete.

##### 3.2. Criterion 2

*Criterion 2. *Let be the weighted average of . FOD is incomplete if . Otherwise, FOD is said to be complete.

The criterion indicates if the mass of in weighted average evidence is more than 0.5, it is considered in an incomplete FOD. In this criterion, a weighted average of , namely, , is calculated by considering that the evidence generated from different attributes should have different importance. represents the total support of the incomplete FOD, taking the correlativity and difference of the generated evidence into account. That is to say the value of also is an information to identify the FOD. And if the value of is larger, there is a larger support that the FOD is incomplete. Assuming 0.5 is a threshold, if , it is judged as supporting an incomplete FOD.

As shown above, this criterion is based on the weighted average evidence . In this paper, Deng’s approach given in [28] is used to obtain . The process is given as below.

*Step 1. *For each pair of generated GBPA and , the similarity between and is denoted as . Deng proposed to calculate based on the distance of evidence. While Jiang discussed in [35] that the correlation coefficient she proposed has a better performance compared with distance of evidence, so in this paper the correlation coefficient is used to measure the similarity measure .

*Definition 9. *Let be a frame of discernment (FOD) in an open world, containing mutually exclusive and exhaustive hypotheses. The similarity measure is expressed as where where ; is the focal elements of mass, respectively; and is the cardinality of subset, especially, .

*Step 2. *For the generated GBPA, we can calculate the similarity measure between and (). So a similarity measure matrix (SMM) can be constructed to give the insight into the agreement between the pieces of evidence:

*Step 3. *After obtaining the similarity measure matrix SMM, the support degree of each evidence () is defined by Then, the credibility degree of evidence (i.e., ) is obtained For each piece of evidence its credibility degree is seen as its weight.

*Step 4. *Finally, the modified weighted average evidence is given as

Once is obtained, can be known as well. According to this criterion, if , it can be judged that the FOD is incomplete.

##### 3.3. Criterion 3

*Criterion 3. *Let be the combination of by using the modified generalized rule (mGCR). FOD is incomplete if . Otherwise, FOD is said to be complete.

This criterion shows that if the mass of in the combination of generated pieces of evidence with mGCR is more than 0.5, it is considered in an incomplete FOD. That is because the parameter represents the pieces of evidence’s total confidence of the incomplete FOD, and the combination rule takes all generated pieces of evidence into account to get a final result to identify which set the sample belongs to. If the assignment of is large, which means the assignment of other set will be inversely small, it is supporting the set and the incomplete FOD. Assuming 0.5 is a threshold, if , this system is judged supporting the incomplete FOD.

##### 3.4. A Numerical Example for the Three Criteria

The proposed method can be used to identify the incomplete FOD, as long as we obtain the three parameters , , . In this subsection, an illustrative example is given to show the identification result according to the three criteria.

*Example 10. *Assume there is a FOD . A new sample with attributes is obtained. It is assumed that four GBPAs are generated from this sample; these GPBAs are shown as follows: Now, we can use the proposed criteria to identify the completeness of FOD in terms of , , , .

At first, according to Criterion 1, FOD is incomplete, since , , which means .

Then, from Criterion 2, FOD is also identified as incomplete; the process is shown as follows. *Steps **1 and 2*. The similarity measure matrix (SMM) can be calculated with (22) and (24) firstly:*Step **3*. Then the weights of four pieces of evidence , , , are calculated according to (25) and (26): *Step **4*. Finally, the weighted average evidence of the four pieces of evidence is Therefore, according to Criterion 2, , which shows that the FOD is still incomplete.

At last, let us use Criterion 3 to identify the completeness of . By using mGCR to combine the four pieces of evidence, the result is where . According to Criterion 3, the FOD is incomplete. To express intuitively, these results are all shown in Table 1.