Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 5858272, 10 pages

https://doi.org/10.1155/2018/5858272

## Weighted Evidence Combination Rule Based on Evidence Distance and Uncertainty Measure: An Application in Fault Diagnosis

Correspondence should be addressed to Jun Sang

Received 29 May 2017; Revised 3 September 2017; Accepted 15 November 2017; Published 17 January 2018

Academic Editor: Josefa Mula

Copyright © 2018 Lei Chen 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

Conflict management in Dempster-Shafer theory (D-S theory) is a hot topic in information fusion. In this paper, a novel weighted evidence combination rule based on evidence distance and uncertainty measure is proposed. The proposed approach consists of two steps. First, the weight is determined based on the evidence distance. Then, the weight value obtained in first step is modified by taking advantage of uncertainty. Our proposed method can efficiently handle high conflicting evidences with better performance of convergence. A numerical example and an application based on sensor fusion in fault diagnosis are given to demonstrate the efficiency of our proposed method.

#### 1. Introduction

Information fusion technology (IFT) is utilized to analyze multisource uncertain information comprehensively. Through reserving the common information, IFT can decrease indeterminacy greatly. The Dempster-Shafer theory of evidence [1] (D-S theory, also known as evidence theory or theory of belief functions) is regarded as an efficient model to fuse information in intelligent systems [2]. And Dempster’s combination rule is the most crucial instrument of D-S theory. The theory was firstly proposed by Dempster in 1967 [1] and then developed to its present form by Shafer et al. in 1976 [3]. The Dempster’s combination rule possesses several interesting mathematical properties, such as commutativity and associativity, and it plays a very significant role in evidence theory [4]. Nowadays, the evidence theory is applied widely in many fields, like supplier selection ([5, 6]), target recognition ([7–9]), decision making ([4, 10, 11]), reliability analysis ([12–15]), and so on.

Although D-S theory has a lot of advantages, there also exist some basic problems that still are not completely clarified. One of the most significant issues is that D-S theory will become invalid when using it to fuse highly conflicting evidences, and the counter-intuitive results ([10, 16, 17]) will be generated. To solve such a problem, two major methodologies are popular. One is to preprocess the bodies of evidence (BOEs) ([18–20]), and the other is to modify the combined rule ([21–23]). There are mainly three alternative combination rules belonging to the second type and they are, respectively, Dubois and Prade’s disjunctive combination rule [24], Smets’ unnormalized combination rule [25], and Yager’s combination rule [21]. These three alternatives mentioned above are examined and they all propose a general combination framework. The main work of preprocessing bodies of evidences (BOEs) includes Murphy’s simple average in [19], Yong et al.’s weighted average on the basis of distance of evidence in [26], and Han et al.’s modified weighted average in [27]. In [19], a simple averaging approach of the primitive BOEs is proposed, and in that case all BOEs are seen equally important, which is unreasonable in practice. Yong et al. [26] get a better combination result according to combining the weight average of the masses for times. The approach proposed by Han et al. [27] is a novel weighted evidence combination approach based on the evidence distance and ambiguity measure (AM), which actually modifies Yong et al.’s work [26].

In this paper, a novel weighted evidence combination rule based on evidence distance and uncertainty measure is proposed to address the combination of conflicting evidences. The numerical example and an application in fault diagnosis are given to sufficiently prove the efficiency of our proposed method.

The remaining paper is organized as follows: Section 2 starts with a brief introduction of the Dempster-Shafer theory and evidence distance; the proposed method is presented in Section 3; Sections 4 and 5 give a numerical example and an application in fault diagnosis, respectively, to prove the efficiency of our proposed approach; finally, the conclusion is made in Section 6.

#### 2. Preliminaries

In this section, some preliminaries are briefly introduced below.

##### 2.1. Basics of Evidence Theory

Dempster-Shafer theory of evidence (D-S theory) is used for dealing with uncertainty information as an efficient mathematical model in intelligent systems [1]. In 1967, The definition, D-S theory was proposed by Dempster [1] and then his student Shafer et al. developed this theory [3] in 1976.

([29, 30]) Let be a nonempty finite set and be the set of all subsets of , denoted and .

In Dempster-Shafer theory of evidence [3], a basic probability assignment (BPA) is a mapping: satisfying

If , is called a focal element, and the set consisting of all the focal elements is called one body of evidences (BOEs). When there are more than one independent body of evidences, Dempster’s combination rule, (2) which is a powerful and crucial tool in D-S theory, can be utilized to combine these evidences.where stands for conflict degree, also called normalization constant. What is noted is that if , this combination rule will make no sense. Here, we give a specific example about combination rule and show the corresponding results in Table 1.