A belief logarithmic similarity measure based on Dempster-Shafer theory and its application in multi-source data fusion

Abstract

Dempster-Shafer theory (DST) has attracted widespread attention in many domains owing to its powerful advantages in managing uncertain and imprecise information. Nevertheless, counterintuitive results may be generated once Dempster’s rule faces highly conflicting pieces of evidence. In order to handle this flaw, a new belief logarithmic similarity measure ( $BLSM$ ) based on DST is proposed in this paper. Moreover, we further present an enhanced belief logarithmic similarity measure ( $EBLSM$ ) to consider the internal discrepancy of subsets. In parallel, we prove that $EBLSM$ satisfies several desirable properties, like bounded, symmetry and non-degeneracy. Finally, a new multi-source data fusion method based on $EBLSM$ is well devised. Through its best performance in two application cases, specifically those pertaining to fault diagnosis and target recognition respectively, the rationality and effectiveness of the proposed method is sufficiently displayed.

Keywords

Dempster-Shafer theory basic belief assignment logarithmic similarity measure belief entropy data fusion

1 Introduction

Multi-source data fusion (MSDF) can effectively integrate information from diverse sources, which is conducive to improving the credibility and efficiency of decision-making [1 –3]. MSDF, in recent years, has attracted extensive concern in various domains, such as pattern recognition [4, 5], fault diagnosis [6 –8], risk analysis [9, 10] and so on [11 –14]. Nevertheless, the information gathered from various sources may often be imperfect (i.e., uncertain and imprecise), which is usually attributed to the interference of environmental conditions [15, 16]. How to handle such uncertain and imprecise information is still a challenge. Fortunately, over the past few years, a variety of theories have been devoted to dealing with this problem, including Bayesian theory [17], fuzzy set theory [18], rough set theory [19], Dempster-Shafer theory [20, 21], Z-numbers [22], random permutation set [23], intuitionistic fuzzy sets [24] and complex evidence theory [25 –27].

Dempster-Shafer theory (DST), also known as evidence theory or belief functions theory, was initially put forward by Dempster [20] and then developed by Shafer [21]. As an effective mathematical tool for modeling uncertainty and imprecision, DST has several unique advantages. Specifically, DST characterizes imprecision and uncertainty by assigning probabilities to basic belief assignment (BBA) [28]. Additionally, Dempster’s rule of combination follows the associative and commutative laws, making its implementation simple [29]. Despite the benefits mentioned above, DST is limited by dealing with highly conflicting pieces of evidence, as counterintuitive results tend to be generated [30].

As such, a huge number of methods have been put forward to address this issue, which can be mainly divided into two types [31]. One is to modify Dempster’s rule of combination, the other is to preprocess the pieces of evidence. Some researches that devote to the first type include Smets’s unnormalized combination rule [32], Dubois and Prade’s disjunctive combination rule [33], and Yager’s combination rule [34]. However, these rules violate the excellent properties of Dempster’s rule, like associative and commutative laws.

Hence, preprocessing the pieces of evidence becomes the choice of many researchers. To date, various methods have been proposed for this purpose, each with their own strengths and weaknesses. One such method is Murphy’s approach, which takes a simple arithmetic average of the pieces of evidence [35]. While this may be effective in some cases, other researchers, like Deng et al., have opted for a more nuanced approach. They have adopted a weighted average based on the Jousselme distance, which is a measure of the evidence’s similarity and conflict [36]. However, as research progresses, it has become clear that even the Jousselme distance fails in certain situations. Jiang has highlighted this issue and proposed a new correlation coefficient to measure the discrepancies between BBAs [37]. This new approach provides greater accuracy in handling conflicts between different pieces of evidence. Recently, Xiao has presented a novel method that uses a belief divergence measure of evidence, taking into account the information volume of evidence in terms of belief entropy [38]. Unfortunately, this approach neglects the internal discrepancy of subsets and treats multiple-element subsets the same as singleton-element subsets, which also occurs in [39 –41]. In a word, managing conflicts in DST is such an essential and challenging task that has remained an open issue so far. While various approaches have been proposed, there is still much room for improvement in MSDF.

In this paper, a new belief logarithmic similarity measure ( $BLSM$ ) based on DST is first proposed. To take the internal discrepancy of subsets into consideration, the enhanced belief logarithmic similarity measure ( $EBLSM$ ) is proposed subsequently. Then, a new MSDF method is designed based on $EBLSM$ . In general, it comprehensively considers each piece of evidence’s credibility and information volume weight. To further demonstrate the rationality and robustness of the proposed method, several numerical examples and two applications in fault diagnosis and target recognition are illustrated.

The main contributions are summarized below:

(1) A new $EBLSM$ based on DST is proposed to measure the similarity between the pieces of evidence.

(2) The proposed $EBLSM$ takes the internal discrepancy of subsets into account so that it is able to distinguish the multiple-element subset from the singleton-element subset.

(3) The proposed $EBLSM$ has some ideal mathematical properties since it is bounded and has symmetry and non-degeneracy. More noteworthy is that all of these properties have been proved in detail in this paper.

(4) Based on $EBLSM$ , a new MSDF method is well devised. Additionally, two application cases for fault diagnosis and target recognition are provided to prove the effectiveness and robustness of the proposed method.

The organization of the rest of this paper is as follows. In section 2, DST and belief entropy are briefly reviewed. A new $BLSM$ based on DST and $EBLSM$ are proposed in section 3 with some numerical examples for demonstration of the rationality and effectiveness. After that, a new MSDF method is proposed in section 4. Then the application based on the proposed method is illustrated in section 5. Finally, section 6 makes a conclusion.

2 Preliminaries

2.1 Dempster–Shafer theory

Dempster-Shafer theory (DST), as a generalization of conventional Bayesian theory, is a useful mathematical tool for modeling and solving uncertain and imprecise information, which has been widely used in many fields [42 –46]. Here we will briefly review the basic concepts of DST.

Definition 1. [Framework of Discernment] Assume Θ is a set that contains N exhaustive and mutually exclusive elements. Θ is called the frame of discernment (FOD), and it can be defined as: $Θ = {ϑ_{1}, ϑ_{2}, . . ., ϑ_{N}}$ (1)

The power set of Θ, defined as 2^Θ, which are expressed as follows: $2^{Θ} = {\emptyset, {ϑ_{1}}, . . ., {ϑ_{N}}, {ϑ_{1}, ϑ_{2}}, . . ., Θ}$ (2)

Definition 2. [Basic Belief Assignment] A basic belief assignment (BBA), also called mass function, is defined as a mapping m of the power set 2^Θ to [0, 1], which meets: $\sum_{ϑ_{i} \subseteq 2^{Θ}} m (ϑ_{i}) = 1 and m (\emptyset) = 0$ (3) where m (ϑ_i) denotes the degree of support to ϑ_i.

Definition 3. [Dempster’s Rule] Let m₁ and m₂ are two independent BBAs. Dempster’s rule of combination is defined as: $m (ϑ_{i}) = {\begin{matrix} \frac{1}{1 - K} \sum_{ϑ_{j} \cap ϑ_{k} = ϑ_{i}} m_{1} (ϑ_{j}) \cdot m_{2} (ϑ_{k}) & ϑ_{i} \neq \emptyset \\ 0 & ϑ_{i} = \emptyset ∥ \end{matrix}$ (4) with $K = \sum_{ϑ_{j} \cap ϑ_{k} = \emptyset} m_{1} (ϑ_{j}) \cdot m_{2} (ϑ_{k})$ (5) where $K$ represents the degree of conflict between m₁ and m₂, and $K$ must be satisfied that $0 \leq K < 1$ . When K = 0, no conflict occurs between m₁ and m₂. When K = 1, m₁ and m₂ are completely conflict and Dempster’s rule is not applicable in this case.

2.2 Belief entropy

Belief entropy was first proposed by Deng in [47]. As a generation of Shannon entropy, it has superiority in measuring the uncertainty of information.

Definition 4. [Belief Entropy] Let ϑ_i be a hypothesis of the mass function m, |ϑ_i| is the cardinality of the set ϑ_i. Belief entropy E_d of set ϑ_i is defined as: $E_{d} = - \sum_{ϑ_{i} \subseteq Θ} m (ϑ_{i}) log \frac{m (ϑ_{i})}{2^{| ϑ_{i} |} - 1}$ (6) In the case of a singleton-element subset, belief entropy degenerates to Shannon entropy in terms of Eq.(7). $\begin{matrix} E_{d} & = - \sum_{ϑ_{i} \subseteq Θ} m (ϑ_{i}) log \frac{m (ϑ_{i})}{2^{| ϑ_{i} |} - 1} \\ = - \sum_{ϑ_{i} \subseteq Θ} m (ϑ_{i}) log m (ϑ_{i}) \end{matrix}$ (7) It should be noted that the greater the uncertainty of the piece of evidence, the greater the belief entropy.

3 The proposed similarity measure

In this section, two new belief logarithmic similarity measures will be proposed, respectively. Additionally, some numerical examples will be applied to compare the two similarity measures.

3.1 A new belief logarithmic similarity measure

Various existing similarity measures are based on the strategy that calculates some distance between the pieces of evidence. Inspired by these strategies, a new belief logarithmic similarity measure ( $BLSM$ ) is defined as follows.

Definition 5. [Belief Logarithmic Similarity Measure] $BLSM (m_{1}, m_{2}) = {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| m_{1} (ϑ_{i}) - m_{2} (ϑ_{i}) |}{2})$ (8) The similarity between m₁ and m₂ can be basically measured with the proposed $BLSM$ . However, obviously, it is defective as it only takes the external discrepancy of BBAs like many other measures but ignores the internal discrepancy of BBAs, which means the multiple-element subsets will be treated completely the same as the singleton-element subsets. Consequently, an enhanced version that considers the internal discrepancy will be proposed in subsection 3.2.

3.2 An enhanced belief logarithmic similarity measure

With regard to the drawbacks of $BLSM$ , an enhanced version of $BLSM$ , called $EBLSM$ , is put forward below.

Definition 6. [Enhenced Belief Logarithmic Similarity Measure] $EBLSM (m_{1}, m_{2}) = {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2})$ (9) Apparently, the multiple-element subset can be distinguished from the singleton-element subset with $EBLSM$ as it takes the cardinality of the subset into account. What is more, $EBLSM$ has several ideal mathematical properties. The introduction and proof are shown as follows.

Property 1. The proposed $EBLSM$ satisfies the following properties: 1. Bounded: $0 \leq EBLSM (m_{1}, m_{2}) \leq 1$ .

2. Symmetry: $EBLSM (m_{1}, m_{2})$ = $EBLSM (m_{2}, m_{1})$ .

3. Non-degeneracy: $EBLSM (m_{1}, m_{2}) = 1$ iff m₁ = m₂.

Proof. 1. For two arbitrary BBAs m₁ and m₂ in Θ, Let us consider $X = 2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2}$ (10)

Namely, $X$ reaches the maximum value 2 as m₁ = m₂ while it reaches the minimum value 1 as the two BBAs are completely contradictory. Since $X \in [1, 2]$ , we thus obtain: $\begin{matrix} 0 \leq {log}_{2} (X) \leq 1 \Leftrightarrow \\ 0 \leq {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2}) \leq 1 \end{matrix}$ (11) Clearly, we obtain $0 \leq EBLSM (m_{1}, m_{2}) \leq 1$ . □ Proof. 2. For two arbitrary BBAs m₁ and m₂ in Θ, we have: $EBLSM (m_{1}, m_{2}) = {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2})$ (12) and $EBLSM (m_{2}, m_{1}) = {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2})$ (13) Since $| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} | = | \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |$ , it is easy to obtain $EBLSM (m_{1}, m_{2})$ = $EBLSM (m_{2}, m_{1})$ . □ Proof. 3. For two same BBAs m₁ and m₂ in Θ, i.e. m₁ = m₂. Thus, we have: $\begin{matrix} EBLSM (m_{1}, m_{2}) & = {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2}) \\ = {log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2}) \\ = 1 \end{matrix}$ (14) Conversely, assume that $EBLSM (m_{1}, m_{2}) = 1$ , we thus have: ${log}_{2} (2 - \sum_{ϑ_{i} \subseteq Θ} \frac{| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |}{2}) = 1$ (15) According to Eq. (15), we can conclude ∑_{ϑ_i⊆Θ} $| \frac{m_{1} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} - \frac{m_{2} (ϑ_{i})}{2^{| ϑ_{i} |} - 1} |$ = 0, which also means that m₁ = m₂.

□

3.3 Numerical examples

In this section, several numerical examples will be applied to illustrate the performance of $EBLSM$ and further prove its mathematical properties. And some numerical examples will be used to compare $EBLSM$ with $BLSM$ in order to show the rationality and strengths of $EBLSM$ .

Example 1. Suppose that m₁ and m₂ are two BBAs in Θ = {ϑ₁, ϑ₂, ϑ₃}. $\begin{matrix} m_{1} : m_{1} ({ϑ_{1}}) = α, m_{1} ({ϑ_{2}}) = β \\ m_{1} ({ϑ_{3}}) = 1 - α - β \\ m_{2} : m_{2} ({ϑ_{1}}) = 0.4, m_{2} ({ϑ_{2}}) = 0.6 \end{matrix}$ Where α, β ∈ [0, 1] and α + β ∈ [0, 1]. As seen in Fig 1, when m₁ = m₂, that is, α = 0.4 and β = 0.6, $EBLSM$ just reaches 1 which is the maximum value. Meanwhile, it can be observed that the minimum value is 0. These phenomena are in good agreement with the previously mentioned properties that $EBLSM$ is bounded and has non-degeneracy.

Fig. 1

The results of $EBLSM$ varying with α and β in Example 1.

Example 2. Suppose that m₁ and m₂ are two BBAs in Θ = {ϑ₁, ϑ₂}. $\begin{matrix} m_{1} : m_{1} ({ϑ_{1}}) = α, m_{1} ({ϑ_{2}}) = β \\ m_{1} ({ϑ_{1}, ϑ_{2}}) = 1 - α - β \\ m_{2} : m_{2} ({ϑ_{1}}) = β, m_{2} ({ϑ_{2}}) = α \\ m_{2} ({ϑ_{1}, ϑ_{2}}) = 1 - α - β \end{matrix}$ Where α, β ∈ [0, 1] and α + β ∈ [0, 1]. It can be seen from Fig 2, with the change of α and β, the values of the two BBAs m₁ and m₂ change symmetrically, while the values of $EBLSM$ are equal, which is reasonable because it is consistent with the symmetry of $EBLSM$ .

Fig. 2

The results of $EBLSM$ varying with α and β in Example 2.

Example 3. Suppose that m₁ and m₂ are two BBAs defined on FOD, where Θ_T indicates the set of variables shown in Table 1 and variable α is between 0 and 1.

Table 1

Variable set

t	Θ_T
1	{ϑ₁}
2	{ϑ₁, ϑ₂}
3	{ϑ₁, ϑ₂, ϑ₃}
4	{ϑ₁, ϑ₂, ϑ₃, ϑ₄}
5	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅}
6	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆}
7	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇}
8	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈}
9	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉}
10	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀}
11	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁}
12	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂}
13	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃}
14	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄}
15	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄, ϑ₁₅}
16	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄, ϑ₁₅, ϑ₁₆}
17	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄, ϑ₁₅, ϑ₁₆, ϑ₁₇}
18	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄, ϑ₁₅, ϑ₁₆, ϑ₁₇, ϑ₁₈}
19	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄, ϑ₁₅, ϑ₁₆, ϑ₁₇, ϑ₁₈, ϑ₁₉}
20	{ϑ₁, ϑ₂, ϑ₃, ϑ₄, ϑ₅, ϑ₆, ϑ₇, ϑ₈, ϑ₉, ϑ₁₀, ϑ₁₁, ϑ₁₂, ϑ₁₃, ϑ₁₄, ϑ₁₅, ϑ₁₆, ϑ₁₇, ϑ₁₈, ϑ₁₉, ϑ₂₀}

$\begin{matrix} m_{1} : m_{1} ({ϑ_{2}}) = 0.95, m_{1} (Θ_{T}) = 0.05 \\ m_{2} : m_{2} ({ϑ_{2}}) = α, m_{2} (Θ_{T}) = 1 - α \end{matrix}$ Where α ∈ [0, 1] and t ∈ {1, 2, . . . , 20}. The results of $EBLSM$ varying with α and t is given in Fig 3. When t = 1, which means the Θ_T is a singleton set, there is no intersection between {ϑ₂} and Θ_T. Consequently, the value of $EBLSM$ becomes smaller. As the element in Θ_T increases from 2 to 20, the values of $EBLSM$ ascend as shown in Fig 3(b). Since the influence of the number of elements is exponential, the greater the number of elements, the greater the uncertainty and the smaller the change of $EBLSM$ value. With the regard of α, when α = 0.95, which means m₁ = m₂, the $EBLSM$ get the maximum value no matter how the t changes. Therefore, this example is reasonable and indicates $EBLSM$ is effective under the variation of the support of proposition and the number of elements.

Fig. 3

The results of $EBLSM$ varying with α and t in Example 3.

Example 4. Suppose that m₁ and m₂ are two BBAs defined on FOD Θ = {ϑ₁, ϑ₂, . . . , ϑ₂₀}, where Θ_T indicates the set of variables shown in Table 1 and variable α is between 0 and 1. $\begin{matrix} m_{1} : m_{1} ({ϑ_{2}, ϑ_{3}, ϑ_{4}, ϑ_{5}}) = α, m_{1} (Θ_{T}) = 1 - α \\ m_{2} : m_{2} ({ϑ_{2}, ϑ_{3}, ϑ_{4}, ϑ_{5}}) = 0.9, m_{2} (Θ_{T}) = 0.1 \end{matrix}$

Where α ∈ [0, 1] and t ∈ {1, 2, . . . , 20}. As seen in the Fig 4, it is similar to Example 3.3 as a whole. However, since the subset contains multiple elements, the value of $EBLSM$ will be greater. This example illustrates the effectiveness of $EBLSM$ when the subsets all contain multiple elements.

Fig. 4

The results of $EBLSM$ varying with α and t in Example 4.

Example 5. Suppose that m₁ and m₂ are two BBAs defined on FOD, where Θ_T is a variable set including t elements shown in Table 1. $\begin{matrix} m_{1} : m_{1} ({ϑ_{2}}) = 0.05, m_{1} (Θ_{T}) = 0.95 \\ m_{2} : m_{2} ({ϑ_{2}}) = 0.95, m_{2} (Θ_{T}) = 0.05 \end{matrix}$

Where t ∈ {1, 2, . . . , 20}. As seen in Fig 5, when t = 1, the Θ_T becomes a singleton set that has no intersection with {ϑ₂}. In the light of Example 3.3, it is easy to figure out the result of $EBLSM$ in Fig 5 is reasonable. In contrast, the value of $BLSM$ keeps invariant no matter how the t changes. That is because $BLSM$ treat the multiple-element subset as a singleton-element subset, which ignores the effect of the multiple-element subset. The same phenomenon also manifests in the other three existing measure being compared. To be specific, we have made a comparison of the existing measures in [6 , 40] which correspond to SM, BJS, and DSM in Fig 5 respectively. A notable observation is that regardless of how t changes, their values for measuring the similarity or dissimilarity between evidence remain unchanged, indicating their inferiority compared to $EBLSM$ .

Fig. 5

The comparison results of $EBLSM$ and conventional measures varying with t in Example 5.

4 A new

EBLSM

based MSDF method

In this section, a new MSDF method based on $EBLSM$ is well devised. As it takes the influence of multiple-element subsets into consideration, it can unfold a more granular map of similarities between two pieces of evidence. Moreover, it also considers the information volume of evidence, which makes it handle the uncertainty reasonably. Next, the procedure of the proposed method will be introduced in detail, and the flowchart is shown in Fig 6.

Fig. 6

Flowchart of the proposed MSDF method.

Step 1: Calculate the ${EBLSM}_{ij} (i \neq j)$ between m_i (i = 1, 2, . . . , n) and m_j (j = 1, 2, . . . , n) in the light of Eq.(9). Therefore, the similarities measured matrix $SMM = ({EBLSM}_{ij})_{n \times n}$ is constructed as follow: $SMM = [\begin{matrix} 1 & \dots & {EBLSM}_{1 i} & \dots & {EBLSM}_{1 n} \\ ⋮ & ⋱ & \dots & ⋮ \\ {EBLSM}_{i 1} & \dots & 1 & \dots & {EBLSM}_{in} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {EBLSM}_{n 1} & \dots & {EBLSM}_{ni} & \dots & 1 \end{matrix}]$ (16)

Step 2: The support degree of m_i is calculated by $Sup (m_{i}) = \sum_{j = 1, j \neq i}^{n} \frac{{EBLSM}_{ij}}{n - 1}$ (17)

Step 3: The credibility weight ${\tilde{Crd}}_{i}$ of m_i is calculated by ${\tilde{Crd}}_{i} = \frac{Sup (m_{i})}{\sum_{k = 1}^{n} Sup (m_{k})}$ (18)

Step 4: By utilizing belief entropy, the information volume of m_i is defined as below: ${IV}_{i} = e^{E_{d_{i}}}$ (19) The greater the belief entropy of m_i, the more information it harbors, and hence, the larger the information volume it represents. A considerable information volume for m_i signifies its pivotal role in the ultimate synthesis of results. Therefore, the belief entropy E_di of m_i (i = 1, 2, . . . , n) is calculated by leveraging Eq.(6). ${IV}_{i} = e^{E_{d_{i}}} = e^{- \sum_{ϑ_{i} \subseteq Θ} m (ϑ_{i}) log \frac{m (ϑ_{i})}{2^{| ϑ_{i} |} - 1}}$ (20) where N denotes the number of elements in FOD.

Step 5: The information volume weight $\tilde{{IV}_{i}}$ is then obtained as below: ${\tilde{IV}}_{i} = \frac{{IV}_{i}}{\sum_{k = 1}^{n} {IV}_{k}}$ (21)Step 6: The weight of each piece of evidence gets adjusted based on the information volume ${\tilde{IV}}_{i}$ and credibility ${\tilde{Crd}}_{i}$ , which is denoted as ${\tilde{ACrd}}_{i}$ : ${\tilde{ACrd}}_{i} = \frac{{\tilde{Crd}}_{i} \times {\tilde{IV}}_{i}}{\sum_{i = 1}^{n} {\tilde{Crd}}_{i} \times {\tilde{IV}}_{i}}$ (22)

Step 7: The weighted average evidence $\tilde{m}$ is obtained in the light of the final weight ${\tilde{ACrd}}_{i}$ of m_i as follow: $\tilde{m} = \sum_{i = 1}^{n} \tilde{{ACrd}_{i}} \times m_{i}$ (23) Namely, by means of weighted averaging, $\tilde{m}$ merges n distinct BBAs to yield one single BBA.

Step 8: Obtain the combination result through fusing $\tilde{m}$ with Dempster’s rule Eq.(4) by n - 1 times if there are n pieces of evidence. The whole process is also given in pseudo-code as seen in Algorithm 1.

Algorithm 1

Multi-source data fusion method based on $EBLSM$

Input: The n pieces of BBAs in Θ which contains N elements

Output: Recognition result τ

1: for i = 1 → n do

2: for j = 1 → n do

3: Calculate the ${EBLSM}_{ij}$ between m_i and m_j in terms of Eq.(9).

4: end for

5: end for

6: The similarities measured matrix is obtained according to Eq. (16)

7: Measure the support degree Sup (m_i) of m_i by Eq.(17)

8: Obtain the credibility weight ${\tilde{Crd}}_{i}$ of m_i by Eq. (18)

9: for i = 1 → n do

10: for j = 1 → N do

11: Calculate the information volume of m_i in the light of Eq.(20)

12: end for

13: end for

14: Obtain the information volume weight by Eq.(21)

15: Calculate the adjusted weight ${\tilde{ACrd}}_{i}$ in terms of Eq.(22)

16: Obtain the weighted average evidence $\tilde{m}$ by Eq.(23)

17: for i = 1 → n - 1 do

18: Combine the weighted average evidence $\tilde{m}$ in terms of Dempster’s rule of combination

19: end for

20: return τ

5 Application

This section will illustrate two application cases involving fault diagnosis and target recognition to compare the proposed method with related methods.

5.1 Statement of application I

Suppose a multi-sensor-based fault diagnosis problem is associated with the data collected by five different types of sensors S₁, S₂, S₃, S₄ and S₅. The data are modeled as the BBAs given in Table 2, where three kinds of faults ϑ₁, ϑ₂, ϑ₃ make up of the FOD Θ = {ϑ₁, ϑ₂, ϑ₃}. And the data from [7] are used to make comparisons with related methods.

Table 2
BBAs modeled from sensors

{ϑ₁} {ϑ₂} {ϑ₃} Θ

m ₁ 0.7 0.1 0 0.2

m ₂ 0.7 0 0 0.3

m ₃ 0.65 0.15 0 0.2

m ₄ 0.75 0 0.05 0.2

m ₅ 0 0.2 0.8 0

	{ϑ₁}	{ϑ₂}	{ϑ₃}	Θ
m ₁	0.7	0.1	0	0.2
m ₂	0.7	0	0	0.3
m ₃	0.65	0.15	0	0.2
m ₄	0.75	0	0.05	0.2
m ₅	0	0.2	0.8	0

5.1.1 Implementation based on the proposed method

Step 1: Calculate the similarities measured matrix SMM is constructed as follow: $SMM = [\begin{matrix} 1 & 0.9582 & 0.9635 & 0.9260 & 0.2458 \\ 0.9582 & 1 & 0.9206 & 0.9582 & 0.1745 \\ 0.9635 & 0.9206 & 1 & 0.8875 & 0.3053 \\ 0.9260 & 0.9582 & 0.8875 & 1 & 0.1836 \\ 0.2458 & 0.1745 & 0.3053 & 0.1836 & 1 \end{matrix}]$

Step 2: Calculate the support degree Sup (m_i) of m_i.

Step 3: Generate the credibility weight ${\tilde{Crd}}_{i}$ of m_i.

Step 4: Measure the information volume IV_i of m_i.

Step 5: Obtain the information volume weight ${\tilde{IV}}_{i}$ of m_i.

Step 6: Adjust weight with ${\tilde{Crd}}_{i}$ and ${\tilde{IV}}_{i}$ of m_i.

Step 7: Obtain the weighted average evidence: $\begin{matrix} \tilde{m} ({ϑ_{1}}) = 0.6772 \tilde{m} ({ϑ_{2}}) = 0.0722 \\ \tilde{m} ({ϑ_{3}}) = 0.0317 \tilde{m} (Θ) = 0.2189 \end{matrix}$

Step 8: Combine the weighted average evidence in terms of Dempster’s rule four times, and the result is given in Table 4. And all of the intermediate results are presented in Table 3

Table 3
All of the intermediate results of application I

m ₁ m ₂ m ₃ m ₄ m ₅

Sup 0.7734 0.7529 0.7692 0.7388 0.2273

$\tilde{Crd}$ 0.2371 0.2308 0.2358 0.2265 0.0697

IV 5.5748 5.56041 6.2989 4.7267 2.0584

$\tilde{IV}$ 0.2298 0.2310 0.2596 0.1948 0.0848

$\tilde{ACrd}$ 0.2487 0.2434 0.2795 0.2014 0.0270

	m ₁	m ₂	m ₃	m ₄	m ₅
Sup	0.7734	0.7529	0.7692	0.7388	0.2273
$\tilde{Crd}$	0.2371	0.2308	0.2358	0.2265	0.0697
IV	5.5748	5.56041	6.2989	4.7267	2.0584
$\tilde{IV}$	0.2298	0.2310	0.2596	0.1948	0.0848
$\tilde{ACrd}$	0.2487	0.2434	0.2795	0.2014	0.0270

5.1.2 Discussion

All the results of the above numerical examples are presented in Table 4. From Table 4, it can be seen that even though the first four pieces of evidence support ϑ₁, Dempster’s rule of combination eventually generates counterintuitive results under the influence of the last one that supports ϑ₃. Therefore, it can not effectively deal with the data fusion problem when the pieces of evidence are highly conflicting. Whereas the other methods all support ϑ₁, which is in sync with logic. Additionally, the proposed method is more efficient in handling highly conflicting evidence with the greatest belief (0.9956), as shown in Table 4. That is because the proposed method not only considers the internal and external discrepancies among the subsets of the piece of evidence but also adopts the belief entropy to measure the uncertainty of the piece of evidence, which is helpful in obtaining more reliable weighted average evidence and more ideal result.

Table 4
Combination results of application I

Methods m _1,2 m _1,2,3 m _1,2,3,4 m _1,2,3,4,5 Fault type

Dempster’s rule [20] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9598 m ({ϑ₁}) =0.9906 m ({ϑ₁}) =0.0000 ϑ₃

m ({ϑ₂}) =0.0323 m ({ϑ₂}) =0.0249 m ({ϑ₂}) =0.0053 m ({ϑ₂}) =0.3443

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0008 m ({ϑ₃}) =0.6557

m (Θ) =0.0645 m (Θ) =0.0153 m (Θ) =0.0033 m (Θ) =0.0000

Murphy’s method [35] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9598 m ({ϑ₁}) =0.9899 m ({ϑ₁}) =0.9715 ϑ₁

m ({ϑ₂}) =0.0296 m ({ϑ₂}) =0.0241 m ({ϑ₂}) =0.0058 m ({ϑ₂}) =0.0055

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0008 m ({ϑ₃}) =0.0222

m (Θ) =0.0672 m (Θ) =0.0161 m (Θ) =0.0035 m (Θ) =0.0008

Deng et.al.’s method [36] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9597 m ({ϑ₁}) =0.9899 m ({ϑ₁}) =0.9933 ϑ₁

m ({ϑ₂}) =0.0296 m ({ϑ₂}) =0.0243 m ({ϑ₂}) =0.0058 m ({ϑ₂}) =0.0030

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0008 m ({ϑ₃}) =0.0029

m (Θ) =0.0672 m (Θ) =0.0160 m (Θ) =0.0035 m (Θ) =0.0008

Lin et.al.’s method [6] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9597 m ({ϑ₁}) =0.9899 m ({ϑ₁}) =0.9894 ϑ₁

m ({ϑ₂}) =0.0296 m ({ϑ₂}) =0.0244 m ({ϑ₂}) =0.0058 m ({ϑ₂}) =0.0037

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0008 m ({ϑ₃}) =0.0061

m (Θ) =0.0672 m (Θ) =0.0159 m (Θ) =0.0035 m (Θ) =0.0008

Jiang’s method [37] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9598 m ({ϑ₁}) =0.9899 m ({ϑ₁}) =0.9952 ϑ₁

m ({ϑ₂}) =0.0296 m ({ϑ₂}) =0.0242 m ({ϑ₂}) =0.0058 m ({ϑ₂}) =0.0025

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0008 m ({ϑ₃}) =0.0015

m (Θ) =0.0672 m (Θ) =0.0161 m (Θ) =0.0035 m (Θ) =0.0008

Xiao’s method [38] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9575 m ({ϑ₁}) =0.9889 m ({ϑ₁}) =0.9955 ϑ₁

m ({ϑ₂}) =0.0295 m ({ϑ₂}) =0.0285 m ({ϑ₂}) =0.0070 m ({ϑ₂}) =0.0028

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0006 m ({ϑ₃}) =0.0008

m (Θ) =0.0673 m (Θ) =0.0140 m (Θ) =0.0035 m (Θ) =0.0008

Gao and Xiao’s method [4] m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9577 m ({ϑ₁}) =0.9890 m ({ϑ₁}) =0.9955 ϑ₁

m ({ϑ₂}) =0.0295 m ({ϑ₂}) =0.0282 m ({ϑ₂}) =0.0068 m ({ϑ₂}) =0.0029

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0006 m ({ϑ₃}) =0.0008

m (Θ) =0.0673 m (Θ) =0.0141 m (Θ) =0.0036 m (Θ) =0.0009

The proposed method m ({ϑ₁}) =0.9032 m ({ϑ₁}) =0.9590 m ({ϑ₁}) =0.9892 m ({ϑ₁}) =0.9956 ϑ₁

m ({ϑ₂}) =0.0295 m ({ϑ₂}) =0.0251 m ({ϑ₂}) =0.0066 m ({ϑ₂}) =0.0027

m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0000 m ({ϑ₃}) =0.0007 m ({ϑ₃}) =0.0008

m (Θ) =0.0673 m (Θ) =0.0159 m (Θ) =0.0036 m (Θ) =0.0009

Methods	m _1,2	m _1,2,3	m _1,2,3,4	m _1,2,3,4,5	Fault type
Dempster’s rule [20]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9598	m ({ϑ₁}) =0.9906	m ({ϑ₁}) =0.0000	ϑ₃
	m ({ϑ₂}) =0.0323	m ({ϑ₂}) =0.0249	m ({ϑ₂}) =0.0053	m ({ϑ₂}) =0.3443
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0008	m ({ϑ₃}) =0.6557
	m (Θ) =0.0645	m (Θ) =0.0153	m (Θ) =0.0033	m (Θ) =0.0000
Murphy’s method [35]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9598	m ({ϑ₁}) =0.9899	m ({ϑ₁}) =0.9715	ϑ₁
	m ({ϑ₂}) =0.0296	m ({ϑ₂}) =0.0241	m ({ϑ₂}) =0.0058	m ({ϑ₂}) =0.0055
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0008	m ({ϑ₃}) =0.0222
	m (Θ) =0.0672	m (Θ) =0.0161	m (Θ) =0.0035	m (Θ) =0.0008
Deng et.al.’s method [36]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9597	m ({ϑ₁}) =0.9899	m ({ϑ₁}) =0.9933	ϑ₁
	m ({ϑ₂}) =0.0296	m ({ϑ₂}) =0.0243	m ({ϑ₂}) =0.0058	m ({ϑ₂}) =0.0030
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0008	m ({ϑ₃}) =0.0029
	m (Θ) =0.0672	m (Θ) =0.0160	m (Θ) =0.0035	m (Θ) =0.0008
Lin et.al.’s method [6]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9597	m ({ϑ₁}) =0.9899	m ({ϑ₁}) =0.9894	ϑ₁
	m ({ϑ₂}) =0.0296	m ({ϑ₂}) =0.0244	m ({ϑ₂}) =0.0058	m ({ϑ₂}) =0.0037
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0008	m ({ϑ₃}) =0.0061
	m (Θ) =0.0672	m (Θ) =0.0159	m (Θ) =0.0035	m (Θ) =0.0008
Jiang’s method [37]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9598	m ({ϑ₁}) =0.9899	m ({ϑ₁}) =0.9952	ϑ₁
	m ({ϑ₂}) =0.0296	m ({ϑ₂}) =0.0242	m ({ϑ₂}) =0.0058	m ({ϑ₂}) =0.0025
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0008	m ({ϑ₃}) =0.0015
	m (Θ) =0.0672	m (Θ) =0.0161	m (Θ) =0.0035	m (Θ) =0.0008
Xiao’s method [38]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9575	m ({ϑ₁}) =0.9889	m ({ϑ₁}) =0.9955	ϑ₁
	m ({ϑ₂}) =0.0295	m ({ϑ₂}) =0.0285	m ({ϑ₂}) =0.0070	m ({ϑ₂}) =0.0028
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0006	m ({ϑ₃}) =0.0008
	m (Θ) =0.0673	m (Θ) =0.0140	m (Θ) =0.0035	m (Θ) =0.0008
Gao and Xiao’s method [4]	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9577	m ({ϑ₁}) =0.9890	m ({ϑ₁}) =0.9955	ϑ₁
	m ({ϑ₂}) =0.0295	m ({ϑ₂}) =0.0282	m ({ϑ₂}) =0.0068	m ({ϑ₂}) =0.0029
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0006	m ({ϑ₃}) =0.0008
	m (Θ) =0.0673	m (Θ) =0.0141	m (Θ) =0.0036	m (Θ) =0.0009
The proposed method	m ({ϑ₁}) =0.9032	m ({ϑ₁}) =0.9590	m ({ϑ₁}) =0.9892	m ({ϑ₁}) =0.9956	ϑ₁
	m ({ϑ₂}) =0.0295	m ({ϑ₂}) =0.0251	m ({ϑ₂}) =0.0066	m ({ϑ₂}) =0.0027
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0007	m ({ϑ₃}) =0.0008
	m (Θ) =0.0673	m (Θ) =0.0159	m (Θ) =0.0036	m (Θ) =0.0009

5.2 Statement of application II

Suppose there are three types of aircraft, denoted as ϑ₁, ϑ₂ and ϑ₃, which make up the FOD. The set of radars given by R = {R₁, R₂, R₃, R₄, R₅} are used to gather the data. In addition, the collected data from five different types of radar are modeled as BBAs shown in Table 5, where the five BBAs m₁, m₂, m₃, m₄, m₅ are reported from the five radars R₁, R₂, R₃, R₄, R₅ respectively. And the data from [3] are used to make a comparison with related methods.

Table 5
BBAs based on five radars

{ϑ₁} {ϑ₂} {ϑ₃} Θ

m ₁ 0.4 0.6 0 0

m ₂ 0 0.7 0.3 0

m ₃ 0.85 0 0 0.15

m ₄ 0.4 0.6 0 0

m ₅ 0.75 0 0 0.25

	{ϑ₁}	{ϑ₂}	{ϑ₃}	Θ
m ₁	0.4	0.6	0	0
m ₂	0	0.7	0.3	0
m ₃	0.85	0	0	0.15
m ₄	0.4	0.6	0	0
m ₅	0.75	0	0	0.25

5.2.1 Implementation based on the proposed method

Step 1: Calculate the similarities measured matrix SMM is constructed as follow: $SMM = [\begin{matrix} 1 & 0.6781 & 0.5502 & 1 & 0.5918 \\ 0.6781 & 1 & 0.0899 & 0.6781 & 0.1468 \\ 0.5502 & 0.0899 & 1 & 0.5502 & 0.9582 \\ 1 & 0.6781 & 0.5502 & 1 & 0.5918 \\ 0.5918 & 0.1468 & 0.9582 & 0.5918 & 1 \end{matrix}]$

Step 2: Calculate the support degree Sup (m_i) of m_i.

Step 3: Generate the credibility weight ${\tilde{Crd}}_{i}$ of m_i.

Step 4: Measure the information volume IV_i of m_i.

Step 5: Obtain the information volume weight ${\tilde{IV}}_{i}$ of m_i. Step 6: Adjust weight with ${\tilde{Crd}}_{i}$ and ${\tilde{IV}}_{i}$ of m_i.

Step 7: Obtain the weighted average evidence: $\begin{matrix} \tilde{m} ({ϑ_{1}}) = 0.5368 \tilde{m} ({ϑ_{2}}) = 0.3308 \\ \tilde{m} ({ϑ_{3}}) = 0.0328 \tilde{m} (Θ) = 0.0996 \end{matrix}$

Step 8: Combine the weighted average evidence in terms of Dempster’s rule four times, and the result is given in Table 7 and Fig 7(a). Especially the mass of belief for ϑ₁ of different methods is shown in Fig 7(b). And all of the intermediate results are presented in Table 6.

Table 6
All of the intermediate results of application II

m ₁ m ₂ m ₃ m ₄ m ₅

Sup 0.7050 0.3982 0.5371 0.7050 0.5722

$\tilde{Crd}$ 0.2416 0.1365 0.1841 0.2416 0.1961

IV 2.6405 2.4140 2.8037 2.6405 4.5409

$\tilde{IV}$ 0.1756 0.1605 0.1864 0.1756 0.3019

$\tilde{ACrd}$ 0.2118 0.1094 0.1714 0.2118 0.2956

	m ₁	m ₂	m ₃	m ₄	m ₅
Sup	0.7050	0.3982	0.5371	0.7050	0.5722
$\tilde{Crd}$	0.2416	0.1365	0.1841	0.2416	0.1961
IV	2.6405	2.4140	2.8037	2.6405	4.5409
$\tilde{IV}$	0.1756	0.1605	0.1864	0.1756	0.3019
$\tilde{ACrd}$	0.2118	0.1094	0.1714	0.2118	0.2956

Fig. 7

Results of application II.

Table 7

Combination results of application II

Methods	m _1,2	m _1,2,3	m _1,2,3,4	m _1,2,3,4,5	Target
Dempster’s rule [20]	m ({ϑ₁}) =0.0000	m ({ϑ₁}) =0.0000	m ({ϑ₁}) =0.0000	m ({ϑ₁}) =0.0000	ϑ₂
	m ({ϑ₂}) =1.0000	m ({ϑ₂}) =1.0000	m ({ϑ₂}) =1.0000	m ({ϑ₂}) =1.0000
	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000	m ({ϑ₃}) =0.0000
	m (Θ) =0.0000	m (Θ) =0.0000	m (Θ) =0.0000	m (Θ) =0.0000
Murphy’s method [35]	m ({ϑ₁}) =0.0825	m ({ϑ₁}) =0.4663	m ({ϑ₁}) =0.3723	m ({ϑ₁}) =0.7273	ϑ₁
	m ({ϑ₂}) =0.8711	m ({ϑ₂}) =0.5182	m ({ϑ₂}) =0.6263	m ({ϑ₂}) =0.2720
	m ({ϑ₃}) =0.0464	m ({ϑ₃}) =0.0149	m ({ϑ₃}) =0.0014	m ({ϑ₃}) =0.0007
	m (Θ) =0.0000	m (Θ) =0.0006	m (Θ) =0.0000	m (Θ) =0.0000
Deng et.al.’s method [36]	m ({ϑ₁}) =0.0825	m ({ϑ₁}) =0.3501	m ({ϑ₁}) =0.2440	m ({ϑ₁}) =0.7261	ϑ₁
	m ({ϑ₂}) =0.8711	m ({ϑ₂}) =0.6389	m ({ϑ₂}) =0.7554	m ({ϑ₂}) =0.2736
	m ({ϑ₃}) =0.0464	m ({ϑ₃}) =0.0108	m ({ϑ₃}) =0.0005	m ({ϑ₃}) =0.0003
	m (Θ) =0.0000	m (Θ) =0.0003	m (Θ) =0.0000	m (Θ) =0.0000
Lin et.al.’s method [6]	m ({ϑ₁}) =0.0825	m ({ϑ₁}) =0.3991	m ({ϑ₁}) =0.2805	m ({ϑ₁}) =0.6901	ϑ₁
	m ({ϑ₂}) =0.8711	m ({ϑ₂}) =0.5905	m ({ϑ₂}) =0.7190	m ({ϑ₂}) =0.3096
	m ({ϑ₃}) =0.0464	m ({ϑ₃}) =0.0101	m ({ϑ₃}) =0.0006	m ({ϑ₃}) =0.0003
	m (Θ) =0.0000	m (Θ) =0.0003	m (Θ) =0.0000	m (Θ) =0.0000
Jiang’s method [37]	m ({ϑ₁}) =0.0825	m ({ϑ₁}) =0.3405	m ({ϑ₁}) =0.2499	m ({ϑ₁}) =0.7328	ϑ₁
	m ({ϑ₂}) =0.8711	m ({ϑ₂}) =0.6512	m ({ϑ₂}) =0.7496	m ({ϑ₂}) =0.2669
	m ({ϑ₃}) =0.0464	m ({ϑ₃}) =0.0081	m ({ϑ₃}) =0.0005	m ({ϑ₃}) =0.0002
	m (Θ) =0.0000	m (Θ) =0.0002	m (Θ) =0.0000	m (Θ) =0.0000
Xiao’s method [38]	m ({ϑ₁}) =0.0903	m ({ϑ₁}) =0.4404	m ({ϑ₁}) =0.2887	m ({ϑ₁}) =0.8393	ϑ₁
	m ({ϑ₂}) =0.8673	m ({ϑ₂}) =0.5526	m ({ϑ₂}) =0.7111	m ({ϑ₂}) =0.1605
	m ({ϑ₃}) =0.0424	m ({ϑ₃}) =0.0067	m ({ϑ₃}) =0.0002	m ({ϑ₃}) =0.0002
	m (Θ) =0.0000	m (Θ) =0.0003	m (Θ) =0.0000	m (Θ) =0.0001
Gao and Xiao’s method [4]	m ({ϑ₁}) =0.0903	m ({ϑ₁}) =0.4509	m ({ϑ₁}) =0.2991	m ({ϑ₁}) =0.8504	ϑ₁
	m ({ϑ₂}) =0.8673	m ({ϑ₂}) =0.5421	m ({ϑ₂}) =0.7007	m ({ϑ₂}) =0.1494
	m ({ϑ₃}) =0.0424	m ({ϑ₃}) =0.0066	m ({ϑ₃}) =0.0002	m ({ϑ₃}) =0.0002
	m (Θ) =0.0000	m (Θ) =0.0004	m (Θ) =0.0000	m (Θ) =0.0001
The proposed method	m ({ϑ₁}) =0.0903	m ({ϑ₁}) =0.4023	m ({ϑ₁}) =0.2946	m ({ϑ₁}) =0.8759	ϑ₁
	m ({ϑ₂}) =0.8673	m ({ϑ₂}) =0.5901	m ({ϑ₂}) =0.7050	m ({ϑ₂}) =0.1238
	m ({ϑ₃}) =0.0424	m ({ϑ₃}) =0.0074	m ({ϑ₃}) =0.0004	m ({ϑ₃}) =0.0003
	m (Θ) =0.0000	m (Θ) =0.0003	m (Θ) =0.0000	m (Θ) =0.0001

5.2.2 Discussion

As seen in Table 5 and Table 7, m₂ supports ϑ₂, which makes the Dempster’s rule of combination support ϑ₂ while the rest of methods support ϑ₁. In addition, as for ϑ₁, the support degree of Murphy’s method, Deng et.al.’s method, Lin et.al.’s method and Jiang’s method are 0.7273, 0.7261, 0.6901 and 0.7328 respectively, which are lower than those take the information volume of evidence into consideration. And all of them ignore the effect of multiple-element subsets and treat them completely the same as singleton-element subsets. Furthermore, the support degree of Xiao’s method, Gao and Xiao’s method and the proposed method are 0.8393, 0.8504 and 0.8759, respectively, which means the proposed method outperforms the other methods. That is because the proposed method considers the number of elements of the subsets and distinguishes the multiple-element subset from the singleton-element subset. It is worth noting that the proposed method does not consider the intersection and union information between subsets. However, the correlation coefficient proposed by Jiang utilizes the well-known Jaccard coefficient to reflect the intersection and union information among subsets. The result should be further improved if the proposed method also uses this information.

6 Conclusion

In this paper, a new $BLSM$ based on DST is proposed to deal with the highly conflicting pieces of evidence in MSDF. For the sake of utilizing the internal discrepancy of the subsets to facilitate solving the conflict problem, an enhanced version, $EBLSM$ , that distinguishes the multiple-element subset from the singleton-element subset is subsequently proposed. On top of $EBLSM$ , a new MSDF method based on $EBLSM$ is well devised. The proposed method is applied in several numerical examples to validate its rationality and effectiveness. And the results indicate that the proposed method can effectively resolve conflict issues and achieve better results. In order to verify the proposed method further, the proposed method is applied in two cases, including fault diagnosis and target recognition. Compared with the conventional methods, the proposed approach exhibits a clear superiority, attaining its optimal performance in both application cases. All of these results reveal that the proposed method is a promising MSDF method. In the future, we intend to delve deeper into the application of the proposed method across a broader realm of domains.

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A belief logarithmic similarity measure based on Dempster-Shafer theory and its application in multi-source data fusion

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Dempster–Shafer theory

3.1 A new belief logarithmic similarity measure

5.1 Statement of application I

Table 2 BBAs modeled from sensors {ϑ1} {ϑ2} {ϑ3} Θ m 1 0.7 0.1 0 0.2 m 2 0.7 0 0 0.3 m 3 0.65 0.15 0 0.2 m 4 0.75 0 0.05 0.2 m 5 0 0.2 0.8 0

Table 3 All of the intermediate results of application I m 1 m 2 m 3 m 4 m 5 Sup 0.7734 0.7529 0.7692 0.7388 0.2273 Crd ˜ 0.2371 0.2308 0.2358 0.2265 0.0697 IV 5.5748 5.56041 6.2989 4.7267 2.0584 IV ˜ 0.2298 0.2310 0.2596 0.1948 0.0848 ACrd ˜ 0.2487 0.2434 0.2795 0.2014 0.0270

Table 5 BBAs based on five radars {ϑ1} {ϑ2} {ϑ3} Θ m 1 0.4 0.6 0 0 m 2 0 0.7 0.3 0 m 3 0.85 0 0 0.15 m 4 0.4 0.6 0 0 m 5 0.75 0 0 0.25

Table 6 All of the intermediate results of application II m 1 m 2 m 3 m 4 m 5 Sup 0.7050 0.3982 0.5371 0.7050 0.5722 Crd ˜ 0.2416 0.1365 0.1841 0.2416 0.1961 IV 2.6405 2.4140 2.8037 2.6405 4.5409 IV ˜ 0.1756 0.1605 0.1864 0.1756 0.3019 ACrd ˜ 0.2118 0.1094 0.1714 0.2118 0.2956

6 Conclusion

References

Table 2
BBAs modeled from sensors

{ϑ₁} {ϑ₂} {ϑ₃} Θ

m ₁ 0.7 0.1 0 0.2

m ₂ 0.7 0 0 0.3

m ₃ 0.65 0.15 0 0.2

m ₄ 0.75 0 0.05 0.2

m ₅ 0 0.2 0.8 0

Table 5
BBAs based on five radars

{ϑ₁} {ϑ₂} {ϑ₃} Θ

m ₁ 0.4 0.6 0 0

m ₂ 0 0.7 0.3 0

m ₃ 0.85 0 0 0.15

m ₄ 0.4 0.6 0 0

m ₅ 0.75 0 0 0.25