Extended two-dimensional belief function based on divergence measurement

Abstract

The two-dimensional belief function (TDBF = (m_A, m_B)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, m_B includes support degree, non-support degree and reliability unmeasured degree of m_A. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

Keywords

Two-dimensional belief function;divergence Dempster-Shafer evidence theory evidence conflict

1 Introduction

Dempster-Shafer evidence theory, also known as D-S evidence theory. The D-S evidence theory of evidence was first proposed by Dempester [1] in 1967, and then further promoted by his student Shafer [2] in 1976. As an extension of probability theory, D-S evidence theory can better describe and deal with uncertain information, so it is widely used. Bappy [3] et al. used D-S evidence theory to combine AHP and HER to evaluate supply chain sustainability; Han [4] et al. used D-S evidence theory for fuzzy reasoning, extending the method of random modeling; Liu [5] et al. used extended D-S evidence theory for multi-attribute data fusion; Fei [6] et al. combined D-S evidence theory with the Elimination and choice translating reality (ELECTRE) method to solve the decision-making problem of supplier selection; others [7 –9]. Although D-S evidence theory has many advantages, there are still some problems need to be solved.

Conflict evidence fusion is a research hotspot of D-S evidence theory, and there have been many research results. One type of method is to modify the Dempester’s fusion rule. Yager [10] proposed a method of average support for propositions, and Deng’s method considered evolutionary combination rule [11]. Another method is to modify the idea of the source of evidence. The method D-S under this idea include the arithmetic average method of Murphy [12], and the weighted average method proposed by Deng et al. [13, 14]. Some researchers focus on the D-S evidence theory framework itself [15], such as the D number concept proposed by Deng et al. [16]. The D number theory breaks the limitations of mutual exclusion in the identification framework of traditional evidence theory and requires complete conditions [17 –20]. It provides a good reference for the fusion of conflict evidence.

The proposal of Z-number [21] improves the reliability of information and has been studied and improved by many scholars. Kang [22] et al. proposed a method based on OWA weights and maximum entropy to determine Z-number; Qiao [23] et al. constructed a comprehensive weighted cross entropy to compare two discrete Z-numbers; Li [24] et al. proposed a new method to measure the uncertainty of discrete Z-number based on Shannon entropy; Deng [25] et al. discussed the method of measuring the uncertainty of Z-number by considering the influence of fuzzy measure and fuzzy set cardinality, and many more.

The two-dimensional belief function (TDBF) [26] is composed of a pair of ordered basic probability distribution functions (BPA), which is more flexible and more rich than traditional discount coefficients to express the evaluation value of experts, and can also solve part of the conflict of evidence. However, only collecting expert assessments is single and one-sided.

The measurement of the divergence of belief functions is also a research hotspot. S. Kullback proposed Kullback-Leibler divergence [27] in 1997 to measure the difference between two probabilities. Based on Kullback-Leibler divergence, Deng et al. [28, 29] proposed a new method for measuring the divergence of belief functions, considering the influence of multiple subsets. In addition, Deng [30] also proposed a new method of divergence measurement based on Deng entropy. Xiao [31] proposed a new divergence measure called the confidence degree of the Johnson-Shannon divergence.

This paper considers the differences between the belief functions themselves, and incorporates the divergence measure of the belief function into the TDBF, fully considering the subjectivity and objectivity of uncertain information, so that TDBF can better describe and deal with uncertainty. Numerical examples show that the extended TDBF performs well in target recognition, conflict management and special situations.

The rest of this article is arranged as follows. In Section 2, the preliminary knowledge is briefly introduced, including D-S evidence theory, TDBF, divergence of belief function and other theories involved in this article. The proposed model is described in detail in Section 3. In Section 4, numerical examples are given to prove the usefulness of the proposed model. Finally, this article is summarized in Section 5.

2 Preliminaries

2.1 D-S evidence theory

Dempster-Shafer evidence theory, also known as D-S evidence theory. Dempster-Shafer evidence theory is used to describe and process uncertain, incomplete, and inaccurate information.

Definition 1 [8]. The frame of discernment Θ is a set of mutually exclusive and collectively exhaustive events. We use $\overset{Θ}{2}$ to represent the power set of Θ, defined as:

$\begin{matrix} Θ = {\underset{1}{H}, \underset{2}{H}, . . ., \underset{N}{H}} \overset{Θ}{2} = {ϕ, {H_{1}}, . . ., {HN}, \\ {H_{1}, H_{2}}, . . ., {H_{1}, H_{2}, . . . H_{i}}, . . ., Θ} \end{matrix}$ (1)

Definition 2 [8]. Let Θ be a recognition frame, and the mass function m is defined as $m : \overset{Θ}{2} \to [0, 1]$ (2)

It meets the following conditions: $m (ϕ) = 0$ (3) $\sum_{A \in \overset{Θ}{2}} m (A) = 1$ (4) where ϕ is an empty set and A is a subset of Θ. m (A) describes how much evidence supports A.

Definition 3 [8]. The discount coefficient α in evidence theory is usually used to deal with evidence conflicts, and α is between 0 and 1. And $\overset{α}{m}$ means the evidence is considered by probability α, defined as: $\overset{α}{m} (A) = α \times m (A), A \in Θ$ (5) $\overset{α}{m} (Θ) = (1 - α) + α \times m (Θ)$ (6)

Definition 4 [8]. Assume that m₁ and m₂ are two mass functions. The fusion rule of Demspter are defined as: $m = m_{1} \oplus m_{2}$ (7) $m (A) = \frac{\sum_{B \cap C = A} \underset{1}{m} (B) \underset{2}{m} (C)}{1 - K}$ (8) $K = \sum_{B \cap C = ϕ} m_{1} (B) m_{2} (C)$ (9)

Where ϕ is an empty set. B and C are focal elements. K is called the conflict coefficient and is used to indicate the degree of conflict between the two evidences and K < 1.

2.2 Two-dimensional belief function (TDBF)

TDBF, T = (m_A, m_B), consisting of two BPAs. Both $\underset{A}{m}$ and $\underset{B}{m}$ are classic BPAs, an $\underset{B}{m}$ is also a measure of the reliability (determinism) of $\underset{A}{m}$ [26].

The frame of discernment of $\underset{B}{m}$ set Θ = {Y, N}, where “Y” means “support” and “N” means “not supported”. Θ = {Y, N} is set by two monomers {Y} and {N}, a common Θ and an empty set ϕ. Then m ({Y}) indicates how reliable $\underset{A}{m}$ is, m ({N}) indicates how unreliable $\underset{A}{m}$ is, and m(Θ) indicates that the reliability of $\underset{A}{m}$ is not measured. According to the definition of TDBF, if a set of evidence is close to the actual value, its m ({Y}) will be high and m ({N}) will be low. Conversely, if a set of evidence is far from the actual value, its m ({Y}) will be low and m ({N}) will be high. Compared with the single-valued discount coefficient, this model is more reasonable, flexible and comprehensive.

Given the TDBF, combinations rule is defined as follows [26]: ${\begin{matrix} m_{Z} (A) = m_{A} ({x_{i}}) \times m_{B} (Y) + m_{B} (N) \times (1 - m_{A} ({x_{i}})) \forall x_{i} \subset \overset{Θ}{2} \\ m_{Z} (A) = m_{A} (A_{i}) \times m_{B} (Y) \forall A_{i} \subset \overset{Θ}{2} \\ m_{Z} (Θ) = m_{A} (Θ) \times m_{B} (Y) + m_{B} (Θ) \end{matrix}$ (10) where $\underset{i}{A}$ is multisubset of Θ, and $\underset{i}{x}$ is single subset of Θ, i = 1, 2, 3, ... , n.

2.3 Divergence of belief function

Based on Kullback-Leibler divergence, Deng et al. [29] considered the influence of multiple subsets in the belief function and proposed a new method for measuring divergence.

Assume that m₁ and m₂ are two mass functions, the divergence between m₁ and m₂ is defined as follows [29]:

$D i v (m_{1}, m_{2}) = \sum_{i} \frac{1}{2^{| F_{i} |} - 1} m_{1} (F_{i}) l o g (\frac{m_{1} (F_{i})}{m_{2} (F_{i})}) (i = 1, 2, ..., n)$ (11)

It can be seen from the above formula that the belief of each focus element F_i is divided by a term (2^{F
_i} - 1) representing the number of potential states in F_i. Compared with the Kullback-Leibler divergence in probability theory, the elements of D-S evidence theory can be composed of multiple subsets. Deng et al. considered the influence of multiple subsets, so the divergence measure of beliefs is allocated to multiple subsets.

$\begin{matrix} D (m_{1}, m_{2}) = D (m_{2}, m_{1}) \\ = \frac{Div (m_{1}, m_{2}) + Div (m_{2}, m_{1})}{2} \end{matrix}$ (12)

Equation (12) is a symmetric divergence proposed in consideration of symmetry [29]. Where F_i is the focal element of m, |F_i| is the cardinality of F_i.

3 Extended two-dimensional belief function

Based on the divergence measurement of the belief function, we propose an extended two-dimensional belief function (ETDBF). The proposed method is as follows:

Step 1. Calculate the divergence between belief functions based on formulas (11) (12) to obtain the divergence matrix DMM. $DMM = (\begin{matrix} 0 & \underset{12}{D} & . . . & \underset{1 n}{D} \\ \underset{21}{D} & \underset{2 n}{D} \\ ⋮ & ⋮ \\ \underset{m_{1}}{D} & \underset{m_{2}}{D} & \dots & 0 \end{matrix})$ (13)

Where D_ij represents the divergence measurement between m_i and m_j.

Step 2. Calculate the support degree Y₁ of m_i based on the divergence matrix. $\underset{1}{Y} = \frac{1}{\frac{\sum_{j = i}^{n} D_{ij}}{n - 1}}$ (14)

The larger the value of Y₁, the higher the support for m_i from other evidences, and the smaller the conflict with other evidences.

Step 3. Calculate the unsupported degree N₁ of m_i based on the divergence matrix. $D_{i} = \frac{\sum_{j = 1}^{n} D_{ij}}{n - 1}, N_{1} = \frac{D_{i}}{\sum_{i = 1}^{m} D_{i}}$ (15)

Where D_i is divergence of m_i based on the divergence matrix.

Step 4. Calculate the reliability of the unmeasured degree Θ₁ of m_i based on the divergence matrix. $Θ_{1} = 1 - Y_{1} - N_{1}$ (16)

Step 5. Fusion of evaluation value of expert team and reliability of evidence based on belief function divergence measurement.

T = (m_A, m_B), where the three monomers in m_B are respectively Y, N and Θ, and the expert ’s evaluation value is respectively Y₂, N₂ and Θ₂. $Y = (1 - α) Y_{1} + α Y_{2}$ $N = (1 - α) N_{1} + α N_{2}$ $Θ = (1 - α) Θ_{1} + α Θ_{2}$ (17)

Where α is the binding coefficient, and α = 0.5 is used for calculations unless otherwise stated in this paper.

Step 6. Using Equation (10) to combine m_A and m_B of T = (m_A, m_B) and standardize.

Step 7. If there are n pieces of evidence, we can use the classical Dempster’s rule to combine the new masses n –1 times.

4 Numerical examples

Examples 1 and 2 in this paper use several examples from Deng’s [26] article to prove the rationality and effectiveness of the proposed model.

Example 1. Target Recognition. Three targets are known to exist in a target recognition system, x₁, x₂, x₃, Tables 1 and 2 are the TDBF of the evidence given by the sensor. Then determine the target based on the evidence given.

Table 1
m_A of T = (m_A, m_B) in Example 1

Source

m _A,1 m_A,1 ({x₁}) =0.5 m_A,1 ({x₁, x₂}) =0.2 m_A,1 ({Θ}) =0.3

m _A,2 m_A,2 ({x₁}) =0.2 m_A,2 ({x₃}) =0.2 m_A,2 ({x₁, x₃}) =0.5

m _A,3 m_A,3 ({x₁}) =0.6 m_A,3 ({x₂}) =0.1 m_A,3 ({x₁, x₃}) =0.3

Source
m _A,1	m_A,1 ({x₁}) =0.5	m_A,1 ({x₁, x₂}) =0.2	m_A,1 ({Θ}) =0.3
m _A,2	m_A,2 ({x₁}) =0.2	m_A,2 ({x₃}) =0.2	m_A,2 ({x₁, x₃}) =0.5
m _A,3	m_A,3 ({x₁}) =0.6	m_A,3 ({x₂}) =0.1	m_A,3 ({x₁, x₃}) =0.3

Table 2

m_B of T = (m_A, m_B) in Example 1

Source
m _B,1	m_B,1 ({Y}) =0.8	m_B,1 ({N}) =0.1	m_B,1 ({Θ}) =0.1
m _B,2	m_B,2 ({Y}) =0.5	m_B,2 ({N}) =0.2	m_B,2 ({Θ}) =0.3
m _B,3	m_B,3 ({Y}) =0.7	m_B,3 ({N}) =0.2	m_B,3 ({Θ}) =0.1

Calculate the divergence matrix DMM based on formulas (11) and (12). $DMM = (\begin{matrix} 0 & 7.3960 & 3.7585 \\ 7.3960 & 0 & 5.2300 \\ 3.7585 & 5.2300 & 0 \end{matrix})$

Calculate the support degree, non-support degree and reliability unmeasured degree based on the divergence of belief function based on formula (14) (15) and (16). The results are shown in Table 3.

Table 3

Y₁, N₁, Θ₁ in Example 1

Source
m _B,1	m_B,1 ({Y}) =0.1793	m_B,1 ({N}) =0.3404	m_B,1 ({Θ}) =0.4803
m _B,2	m_B,2 ({Y}) =0.1584	m_B,2 ({N}) =0.3853	m_B,2 ({Θ}) =0.4563
m _B,3	m_B,3 ({Y}) =0.2225	m_B,3 ({N}) =0.2743	m_B,3 ({Θ}) =0.5032

The support degree, non-support degree and reliability unmeasured degree based on divergence measurement are combined with the evaluation value of experts using formula (17). The results are shown in Table 4.

Table 4

Y, N, Θ in Example 1

Source
m _B,1	m_B,1 ({Y}) =0.48965	m_B,1 ({N}) =0.2202	m_B,1 ({Θ}) =0.29015
m _B,2	m_B,2 ({Y}) =0.3292	m_B,2 ({N}) =0.29265	m_B,2 ({Θ}) =0.37815
m _B,3	m_B,3 ({Y}) =0.46125	m_B,3 ({N}) =0.23715	m_B,3 ({Θ}) =0.3016

Based on formula (10), the given TDBF is combined and standardized. The results are shown in Tables 5 and 6.

Table 5

The data is combined with the rules of TDBF

Source
m ₁	$\begin{matrix} m ({x_{1}}) = 0.3549, m ({x_{2}}) = 0.2202, m ({x_{3}}) = 0.2202, \\ m ({x_{1}, x_{2}}) = 0.0979, m ({Θ}) = 0.4370 \end{matrix}$
m ₂	$\begin{matrix} m ({x_{1}}) = 0.3000, m ({x_{2}}) = 0.29265, m ({x_{3}}) = 0.3036, \\ m ({x_{1}, x_{3}}) = 0.1646, m ({Θ}) = 0.37815 \end{matrix}$
m ₃	$\begin{matrix} m ({x_{1}}) = 0.3716, m ({x_{2}}) = 0.2596, m ({x_{3}}) = 0.23715, \\ m ({x_{1}, x_{3}}) = 0.1384, m ({Θ}) = 0.3016 \end{matrix}$

Table 6

Table 5 standardized data

Source
m ₁	$\begin{matrix} m ({x_{1}}) = 0.2668, m ({x_{2}}) = 0.1655, m ({x_{3}}) = 0.1655, \\ m ({x_{1}, x_{2}}) = 0.0736, m ({Θ}) = 0.3285 \end{matrix}$
m ₂	$\begin{matrix} m ({x_{1}}) = 0.2085, m ({x_{2}}) = 0.2034, m ({x_{3}}) = 0.2110, \\ m ({x_{1}, x_{3}}) = 0.1144, m ({Θ}) = 0.2628 \end{matrix}$
m ₃	$\begin{matrix} m ({x_{1}}) = 0.2840, m ({x_{2}}) = 0.1984, m ({x_{3}}) = 0.1813, \\ m ({x_{1}, x_{3}}) = 0.1058, m ({Θ}) = 0.2305 \end{matrix}$

By combining these three mass functions twice using the classic Dempster combining rule, the following results can be obtained. $\begin{matrix} m : m ({x_{1}}) = 0.4259, m ({x_{2}}) = 0.1984, \\ m ({x_{3}}) = 0.2579, m ({x_{1}, x_{2}}) = 0.0097, \\ m ({x_{1}, x_{3}}) = 0.0473, m ({Θ}) = 0.0532 \end{matrix}$

If only the evaluation value of experts is considered using traditional TDBF, the results are as follows: $\begin{matrix} m : m ({x_{1}}) = 0.6643, m ({x_{2}}) = 0.1067, \\ m ({x_{3}}) = 0.1547, m ({x_{1}, x_{2}}) = 0.0056, \\ m ({x_{1}, x_{3}}) = 0.0566, m ({Θ}) = 0.0121 \end{matrix}$

It can be clearly seen that the results obtained by the two methods are the same, the target is x₁. Evidence 2 has a low degree of support for the target x₁, because Evidence 2 has a large divergence from other evidences, so it has a low degree of support and has no effect on the results. The ETDBF can effectively solve the target recognition problem.

Example 2. Evidence conflict management. Table 7 and Table 8 are the three pieces of evidence given by the sensor. It can be seen that there is a clear conflict between evidence 1 and evidence 2. If you use the traditional discount method to process the data and then use Dempster’s combination rules, it will produce counterintuitive results. So we use the ETDBF to process the evidence given. The calculation results are as follows:

Table 7

m_A of T = (m_A, m_B) in Example 2

Source
m _A,1	m_A,1 ({x₁}) =0.9	m_A,1 ({x₂}) =0.1	m_A,1 ({x₃}) =0
m _A,2	m_A,2 ({x₁}) =0	m_A,2 ({x₂}) =0.9	m_A,2 ({x₃}) =0.1
m _A,3	m_A,3 ({x₁}) =0.6	m_A,3 ({x₂}) =0.2	m_A,3 ({x₁, x₃}) =0.2

Table 8

m_B of T = (m_A, m_B) in Example 2

Source
m _B,1	m_B,1 ({Y}) =0.8	m_B,1 ({N}) =0.1	m_B,1 ({Θ}) =0.1
m _B,2	m_B,2 ({Y}) =0.4	m_B,2 ({N}) =0.2	m_B,2 ({Θ}) =0.4
m _B,3	m_B,3 ({Y}) =0.7	m_B,3 ({N}) =0.2	m_B,3 ({Θ}) =0.1

Calculate the divergence matrix DMM based on formulas (11) and (12). $DMM = (\begin{matrix} 0 & 13.1585 & 0.9462 \\ 13.1585 & 0 & 10.4822 \\ 0.9462 & 10.4822 & 0 \end{matrix})$

By combining these three mass functions twice using the classic Dempster’s combining rule, the results are as follows: $\begin{matrix} m : m ({x_{1}}) = 0.4801, m ({x_{2}}) = 0.2126, \\ m ({x_{3}}) = 0.2351, m ({x_{1}, x_{3}}) = 0.0147, \\ m ({Θ}) = 0.0580 \end{matrix}$

The ETDBF takes into consideration the evidence support degree, non-support degree and the unmeasured degree of reliability based on the divergence measurement and the evaluation value directly from the experts, which is more objective. It can be seen from the Table 11 that the results obtained using the ETDBF are consistent with the calculation results of Murphy ’s average combination rule, Deng et al ’s weighted average combination rule, and TDBF ’s combination. It can be explained that the method we proposed can partially solve the conflict of evidence.

Table 9

Y, N, Θ in Example 2

Source
m _B,1	m_B,1 ({Y}) =0.4709	m_B,1 ({N}) =0.1934	m_B,1 ({Θ}) =0.3357
m _B,2	m_B,2 ({Y}) =0.2423	m_B,2 ({N}) =0.3404	m_B,2 ({Θ}) =0.4173
m _B,3	m_B,3 ({Y}) =0.4375	m_B,3 ({N}) =0.2162	m_B,3 ({Θ}) =0.3463

Table 10

The data is combined with the rules of TDBF and standardization

Source
m ₁	$\begin{matrix} m ({x_{1}}) = 0.3714, m ({x_{2}}) = 0.1853, m ({x_{3}}) = 0.1621, \\ m ({Θ}) = 0.2813 \end{matrix}$
m ₂	$\begin{matrix} m ({x_{1}}) = 0.2540, m ({x_{2}}) = 0.1881, m ({x_{3}}) = 0.2466, \\ m ({Θ}) = 0.3113 \end{matrix}$
m ₃	$\begin{matrix} m ({x_{1}}) = 0.2771, m ({x_{2}}) = 0.2068, m ({x_{3}}) = 0.1717, \\ m ({x_{1}, x_{3}}) = 0.0695, m ({Θ}) = 0.2750 \end{matrix}$

Table 11

The results of combination by different rules

Combination rule
Dempster	m ({x₁}) =0, m ({x₂}) =1, m ({x₃}) =0, m ({x₁, x₃}) =0
Murphy ’s average	$\begin{matrix} m ({x_{1}}) = 0.7143, m ({x_{2}}) = 0.2657, m ({x_{3}}) = 0.0011, \\ m ({x_{1}, x_{3}}) = 0.0189 \end{matrix}$
Deng et al ’s weighted average	$\begin{matrix} m ({x_{1}}) = 0.9066, m ({x_{2}}) = 0.0895, m ({x_{3}}) = 0.0018, \\ m ({x_{1}, x_{3}}) = 0.0021 \end{matrix}$
TDBF	$\begin{matrix} m ({x_{1}}) = 0.7270, m ({x_{2}}) = 0.1625, m ({x_{3}}) = 0.0911, \\ m ({x_{1}, x_{3}}) = 0.0113, m ({Θ}) = 0.0081 \end{matrix}$
extended TDBF	$\begin{matrix} m ({x_{1}}) = 0.4801, m ({x_{2}}) = 0.2126, m ({x_{3}}) = 0.2351, \\ m ({x_{1}, x_{3}}) = 0.0147, m ({Θ}) = 0.0580 \end{matrix}$

Example 3. Knowing the identification framework Θ = (x₁, x₂, x₃), Tables 12 and 13 are the TDBF of the evidence given by the sensor. Identify the target based on the given data.

Table 12

m_A of T = (m_A, m_B) in Example 3

Source
m _A,1	m_A,1 ({x₁, x₂}) =0.7054	m_A,1 ({Θ}) =0.2946
m _A,2	m_A,2 ({x₂}) =0.2037	m_A,2 ({Θ}) =0.7963
m _A,3	m_A,3 ({x₁}) =0.4419	m_A,3 ({x₂, x₃}) =0.2628	m_A,3 ({Θ}) =0.2953

Table 13

m_B of T = (m_A, m_B) in Example 3

Source
m _B,1	m_B,1 ({Y}) =0.6	m_B,1 ({N}) =0.2	m_B,1 ({Θ}) =0.2
m _B,2	m_B,2 ({Y}) =0.4	m_B,2 ({N}) =0.3	m_B,2 ({Θ}) =0.3
m _B,3	m_B,3 ({Y}) =0.5	m_B,3 ({N}) =0.2	m_B,3 ({Θ}) =0.3

Using traditional TDBF for data fusion can get the following results: $\begin{matrix} m : m ({x_{1}}) = 0.3199, m ({x_{2}}) = 0.3186, \\ m ({x_{3}}) = 0.1846, m ({x_{1}, x_{2}}) = 0.0778 \\ m ({x_{2}, x_{3}}) = 0.0203, m ({Θ}) = 0.0693 \end{matrix}$

It can be seen from the results that the support levels of x₁ and x₂ are very close, and it is difficult to determine the final result or even wrong judgment.

The calculation using the ETDBF is as follows: $\begin{matrix} m : m ({x_{1}}) = 0.2843, m ({x_{2}}) = 0.3157, \\ m ({x_{3}}) = 0.2387, m ({x_{1}, x_{2}}) = 0.0481 \\ m ({x_{2}, x_{3}}) = 0.0146, m ({Θ}) = 0.0908 \end{matrix}$

When α = 1 / 3using the ETDBF fusion data, the results are as follows: $\begin{matrix} m : m ({x_{1}}) = 0.2804, m ({x_{2}}) = 0.3134, \\ m ({x_{3}}) = 0.2638, m ({x_{1}, x_{2}}) = 0.0365 \\ m ({x_{2}, x_{3}}) = 0.0116, m ({Θ}) = 0.0943 \end{matrix}$

After using the ETDBF, we can see from the results that the support of the three targets is relatively close, but the target can still be identified as x₂. Change the combination coefficient, the result does not change with it, the target is still x₂. When the target cannot be determined using TDBF, the ETDBF based on divergence measurement proposed in this paper can better solve such problems.

5 Conclusions

The divergence of the belief function can measure the difference between the two belief functions and has a strong objectivity. TDBF can better solve the problems in uncertain environment due to its flexibility and richness in describing and handling uncertain information. But only considering the evaluation value from experts is very subjective and one-sided. In this paper, the divergence of the belief function is used to measure the support degree, non-support degree and reliability unmeasured degree, and the combination coefficient is combined with the evaluation value of experts. Numerical examples prove that the proposed ETDBF can effectively deal with uncertain information, and can also partially solve the problem of evidence conflict, and even better solve the problem that TDBF cannot handle.

Since Deng proposed TDBF, no one has studied its practical application. In future research, TDBF and ETDBF can be applied to actual problems in uncertain environments, such as supplier selection, FMEA, fault diagnosis, and risk assessment. Take advantage of their greater value in describing and processing uncertain information.

Footnotes

Acknowledgments

This work was supported in part by the Fund for Shanxi “1331 Project” Key Innovative Research Team 2017, and in part by “The Discipline Group Construction Plan for Serving Industries Innovation”, Shanxi, China: The Discipline Group Program of Intelligent Logistics Management for Serving Industries Innovation 2018.

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Extended two-dimensional belief function based on divergence measurement

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 D-S evidence theory

Table 1 m A of T = (m A , m B ) in Example 1 Source m A,1 mA,1 ({x1}) =0.5 mA,1 ({x1, x2}) =0.2 mA,1 ({Θ}) =0.3 m A,2 mA,2 ({x1}) =0.2 mA,2 ({x3}) =0.2 mA,2 ({x1, x3}) =0.5 m A,3 mA,3 ({x1}) =0.6 mA,3 ({x2}) =0.1 mA,3 ({x1, x3}) =0.3

Footnotes

Acknowledgments

References

Table 1
m_A of T = (m_A, m_B) in Example 1

Source

m _A,1 m_A,1 ({x₁}) =0.5 m_A,1 ({x₁, x₂}) =0.2 m_A,1 ({Θ}) =0.3

m _A,2 m_A,2 ({x₁}) =0.2 m_A,2 ({x₃}) =0.2 m_A,2 ({x₁, x₃}) =0.5

m _A,3 m_A,3 ({x₁}) =0.6 m_A,3 ({x₂}) =0.1 m_A,3 ({x₁, x₃}) =0.3