A new method of conflicting evidence management based on non-extensive entropy and Lance distance in uncertain scenarios

Abstract

Handing uncertain information is one of the research focuses currently. For the sake of great ability of handing uncertain information, Dempster-Shafer evidence theory (D-S theory) has been widely used in various fields of uncertain information processing. However, when highly contradictory evidence appears, the results of the classical Dempster combination rules (DCR) can be counterintuitive. Aiming at this defect, by considering the relationship between the evidence and its own characteristics, the proposed method is a new method of conflicting evidence management based on non-extensive entropy and Lance distance in uncertain scenarios. Firstly, the Lance distance function is used to measure the degree of discrepancy and conflict between evidences, and the credibility of evidence is expressed by matrix. Introducing non-extensive entropy to measure the amount of information about evidence and express the uncertainty of evidence. Secondly, the discount coefficient of the final fusion evidence is measured by considering the credibility and uncertainty of the evidence, and the original evidence is modified by the discount coefficient. Then, the final result is obtained by evidence fusion with DCR. Finally, two numerical examples are provided to illustrate the efficiency of the proposed method, and the utility of our work is demonstrated through an application of the active lane change to avoid obstacles to the autonomous driving of new energy vehicles. The proposed method has a better identification accuracy, reaching 0.9811.

Keywords

Dempster-Shafer evidence theory conflicting evidences information fusion Lance distance non-extensive entropy

1 Introduction

Uncertainty is an integral part of the big data and data science environment [1], and how to measure and deal with uncertain information and data conflicts has attracted considerable attention [2]. As a method of uncertainty reasoning, Dempster [3] proposed the concept of evidence theory, Shafer [4, 5] introduced trust functions on the concept proposed by Dempster, thus supplementing and expanding evidence theory, So this theory is also called Dempster-Shaffer theory (D-S theory). Due to the need of prior data in the evidence theory more intuitive than the theory of probability reasoning, more readily available, plus the Dempster combination rules can synthesize data from different experts or data sources, which makes the evidence theory in the information fusion [6], Multi-Sensor fusion [7, 8], target recognition [9, 10], uncertainty reasoning [11], multiattribute decision analysis [12 –14], risk analysis and assessment [15], It has been widely used in fault diagnosis [16] and other fields [17 –20].

The same evidence with different sourcestends to get different basic probability assignments (BPA). Dempster proposed to integrate these BPAs with orthogonal sums, which is the core of evidence theory. Dempster’s combination rule (DCR) [21] has good interchangeability and relevance. Nevertheless, improper handling of classical DCR in conflict allocation sometimes makes the results counterintuitive or even incorrect [22]. To solve this problem, scholars proposed to modify the Dempster’s combination rule to solve the conflict allocation problem. In Yager’s method [23], the conflict is assigned to the complete collection, which solved this problem to a certain extent. However, this method does not have the commutativity and associativity that DCR has, therefore it is difficult to popularize in practice. Murphy [24] proposed to replace the mean of all mass function values with all the evidence, and then obtain the final results with DCR to improve the above method. Inspired by the Murphy approach, Deng et al [25]. proposed to obtain the corresponding weight of evidence by calculate the distance of evidence, then use the weighted average method to process the evidence, finally fused the evidence N - 1 times with DCR. A cost-aware, fault-tolerant and reliable strategy, called CaFtR is proposed by Xiao [26] to improving the performance of the fuzzy complex event processing-based decision-making systems. Moreover, by bridging complex evidence theory (CET) and quantum mechanics, Xiao [27] propose a new complex evidential quantum dynamical (CEQD) model to predict interference effects on human decision-making behaviors. the proposed CEQD method provides a new perspective to study and explain the interference effects involved in human decision-making behaviors.

Scholars have already achieved remarkable results in the measurement of information uncertainty, for exemple: Shannon entropy [28], Deng entropy [29 –32], generalized information entropy [33], divergence [34] Non-extensive entropy [35], etc. The Non-extensive entropy has been widely used in fractal systems [36]. On the basis of Shannon’s entropy, Non-extensive entropy introduces non-generic parameters, thus constructing a new entropy function. Gao et al. [37]. extended the Non-extensive entropy to D-S theory, effectively expressing the uncertainty caused by multivariate sets.

With the increasing application of D-S theory, many methods for measuring the distance between evidence have emerged, including Jousselme distance [38], Minkowski distance [39], and Mahalanobis distance [40], etc. The Jousselme distance function is affected by the dispersion degree of evidence BPA and defective in measuring the evidence conflict [41]. When calculating the Minkowski distance function, the attribute dimensions must be the same, which makes the evidence lose consistency; the Mahalanobis distance function is realized by calculating the covariance of the matrix, excessive calculation, not suitable for processing large-scale data, which is difficult to apply to the actual scenes. The Lance distance was proposed by Lance and Williams [42], which was measured in terms of ratios, independent of the unit of each variable. For data with large attribute value deviation, the Lance distance formula is superior to other distance formulas, which is a good distance measurement method in data analysis.

Aiming at the shortcomings of the current methods, such as low recognition accuracy, high computational complexity and counterintuitive results, this paper proposes a new method of conflicting evidence management based on non-extensive entropy and Lance distance in uncertain scenarios. The main innovation of this paper are the proposed method uses the weighted average rule to combine the evidence, and the weight of evidence is influenced by the difference between evidence distribution and the amount of information of evidence. Lance distance is used to measure the difference between evidence distribution, Non-extensive entropy is used to measure the information, and numerical example is given to verify the effectiveness of the proposed method. In addition, the method is applied to the case of active lane change to avoid obstacles to the autonomous driving of new energy vehicles, which shows that the method has a better prospect of practical application.

The structure of this article is as follows. Section 2 briefly combs and reviews classical theories. Section 3 mainly introduces the conflict evidence combination method that combines Lance distance and non-extensive entropy proposed in this paper. In section 4, two example sare used to prove the effectiveness and superiority of this method. In section 5, the proposed evidence combination method is applied to the case of active lane change to avoid obstacles in the autonomous driving of new energy vehicles. Finally, section 6 gives a conclusion.

2 Preliminaries

2.1 Dempster-Shafer evidence theory

2.1.1 Frame of discernment

Let H be a frame of discernment (FOD) composed of mutually exclusive sets, denoted as: [43]: $H = {h_{1}, h_{2}, \dots, h_{i}, \dots, h_{M}}$ (1) The set of powers in H constitutes a set of propositions of 2^H, which contains all the possible assumptions of H, and each is mutually exclusive [44]: $\begin{matrix} 2^{H} = & {\emptyset, {h_{1}}, \dots, {h_{M}}, {h_{1}, h_{2}}, \dots, \\ {h_{1}, h_{2}, \dots, h_{i}}, \dots, H} \end{matrix}$ (2) in which ∅ denotes an empty set.

2.1.2 Basic probability assignment(BPA)

A mass function called a BPA in a frame of discernment H is a mapping m from 2^H to [0, 1], denoted as: [29] m : 2^H → [0, 1] satisfying the conditions m (∅) =0, and ∑_A∈2^Hm (A) =1. and m (A) is a mass function, which represents the degree of support and trust for evidence A. If m (A) >0, A is called a focalel ement.

D-S theory divides the evidence interval into supporting interval, uncertainty interval, and rejecting interval, according to the relationship between Bel (A) and Pl (A), which are shown in Fig. 1.

Fig.1

Relationship diagram of Bel (A) and Pl (A).

2.1.3 Belief function and plausibility function

From the BPA of assumed A calculates the available belief function and plausibility functions constitute the supporting interval, which indicates the uncertainty of a certain assumption. Let A be a proposition in the frame of discernment H.

A belief function is defined by: $Bel (A) = \sum_{B \subseteq A} m (B) A \in H, B \neq \emptyset$ (3) A plausibility function is defined by: $Pl (A) = 1 - Bel (\bar{A}) = \sum_{B \cap A \neq \emptyset} m (B) A \in H, B \neq \emptyset$ (4) It can be seen that Bel (A) and Pl (A) are the lower and upper limit functions, respectively.

According to the relationship between Bel (A) and Pl (A), D-S theory divides the evidence interval into three intervals, namely support, rejection, and uncertainty intervals, which are shown in Fig. 1.

2.1.4 Dempster’s combination rule

The BPA assignment of the two evidence on the frame of discernment H is assumed as m₁,m₂, respectively. The Dempster’s combination rule denoted as m = m₁ ⊕ m₂, is defined by $m (A) = {\begin{matrix} \frac{1}{1 - K} \sum_{B \cap C = A} m_{1} (B) m_{2} (C), & A \neq \emptyset \\ 0, & A = \emptyset \end{matrix}$ (5) with $K = \sum_{B \cap C = \emptyset} m_{1} (B) m_{2} (C)$ (6) Among them, B and C are also elements in 2^H, where $K$ represents the conflict coefficient between the two evidences. It is worth noting that the Dempster combination rule requires $K < 1$ .

2.2 The weighted average combination of evidence

The evidence weighted fusion method proposed by Murphy [24] is as follows: calculate the average value of the mass function values of all the evidence, replace it with the mass function values of all the evidence, and then use Dempster’s combination rule for N - 1 times to get the final result(N represents the total amount of evidence used for the fusion). Deng et al. [25] made an improvement on this basis, and obtained the corresponding weight of the evidence by calculating the distance of the evidence, processed the evidence by using the weighted average method, and finally conducted N - 1 times fusion by using Dempster’s combination rule. In this paper, the final fusion weight was determined by calculating the credibility and uncertainty of evidence, then use Dempster’s combination rules to perform N - 1 times fusions.

Assuming there are N mass functions, according to [45] the weighted mass function can be obtained: $m_{Avg} (A) = \sum_{i = 1}^{N} w_{i} m_{i} (A)$ (7) $\sum_{i = 1}^{n} w_{i} (A) = 1$ (8) where A ∈ 2^H.

Definition 1. The calculated mass function value m_Avg (A) is used to modify the mass function of the original evidence source. Then, applying the Dempster’s combination rule, the mass function value is combined N - 1 times (when there are N original evidences in the system), the obtained evidence fusion result m_Fus is: $m_{Fus} = {(({({(m_{Avg} \oplus m_{Avg})}_{1} \oplus \dots)}_{i}) \oplus m_{Avg})}_{N - 1}$ (9)

2.3 Lance distance

Lance distance was proposed by Lance and Williams [42], which is independent of the unit of each variable, and it is measured in the form of ratio, which is less sensitive to bias data because the ratio is less affected by the extreme value.Lance distance formula is better than other distance formula for data with large deviation of attribute value, so it is a common method to measure distance in data analysis [46, 47].

Definition 2. The FOD is defined as: H ={ h₁, h₂, …, h_M }, has N evidences:E₁, E₂, …, E_N and the corresponding mass functions are: m₁, m₂, …, m_N respectively. m_i ={ m_i (h₁) , m_i (h₂) , …, m_i (h_N) } Let two evidences be E_i, E_j. The lance distance between evidences E_i and E_j is defined as: [42] $d (E_{i}, E_{j}) = \frac{1}{M} \sum_{k = 1}^{M} \frac{| m_{i} (A_{k}) - m_{j} (A_{k}) |}{[m_{i} (A_{k}) + m_{j} (A_{k})]}$ (10) where i, j = 1, 2, …, N, k = 1, 2, …, M.

Definition 3. The Lance distance matrix D represents the distance between any two evidences, which is defined as follows: $D = [\begin{matrix} 0 & d_{12} & \dots & d_{1 j} & \dots & d_{1 N} \\ d_{21} & 0 & \dots & d_{2 j} & \dots & d_{2 N} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ d_{i 1} & d_{i 2} & \dots & d_{ij} & \dots & d_{iN} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ d_{N 1} & d_{N 2} & \dots & d_{Nj} & \dots & 0 \end{matrix}]$ (11)Definition 4. The degree of similarity between them can be measured by calculating the evidence distance, and 1 - d_ij can measure the similarity between evidences,defined as s (E_i, E_j) with: $\begin{matrix} s (E_{i}, E_{j}) = s_{ij} = 1 - d_{ij} = 1 - d (E_{i}, E_{j}) \\ 0 \leq s_{ij} \leq 1 \end{matrix}$ (12) The similarity matrix S is defined as: $S = [\begin{matrix} 1 & s_{12} & \dots & s_{1 j} & \dots & s_{1 N} \\ s_{21} & 1 & \dots & s_{2 j} & \dots & s_{2 N} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ s_{i 1} & s_{i 2} & \dots & s_{ij} & \dots & s_{iN} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ s_{N 1} & s_{N 2} & \dots & s_{Nj} & \dots & 1 \end{matrix}]$ (13)Definition 5. The element s_ij in the similarity matrix S reflects the degree of consistency between E_i and E_j of evidence. The larger s_ij value is, the higher the degree of consistency is, otherwise the lower. The uncertainty of evidence will increase with the increase of uncertain information. Considering that the weights of different evidences are different in the identification framework, the credibility Rel_i of evidence E_i is defined as: ${Rel}_{i} = \frac{\sqrt{\sum_{j = 1, j \neq i}^{N} s_{ij}^{2}}}{\sum_{i = 1}^{N} \sqrt{\sum_{j = 1, j \neq i}^{N} s_{ij}^{2}}}$ (14)

2.4 Non-extensive entropy

Non-extensive entropy has now been widely used in fractal systems [36]. The non-extensive entropy is an extension of Shannon entropy, introducing an undetermined coefficient q, which represents the non-extensiveness of entropy. The definition of non-extensive entropy is as follows:

Definition 6. Assuming P ={ p₁, p₂, …, p_n } is a probability distribution, the non-extensive entropy of P is as follows [35]: $Eq (p) = \frac{k}{q - 1} (1 - \sum_{i = 1}^{n} p_{i}^{2})$ (15)

The selection of the non-extensive coefficient q is of the utmost importance for the calculation of non-extensive entropy. q improves the pertinence and flexibility of information measurement.

Definition 7. Given a FOD H ={ h₁, h₂, …, h_M }. The non-extensive entropy in D-S theory is defined as follows [37]: $\begin{matrix} E_{TDS} (m) \\ = \sum_{A \in 2^{H}} \frac{(2^{| A |} - 1) \times m (A) \times (1 - {(\frac{m (A)}{2^{| A |} - 1})}^{| A | - 1})}{| A | - 1} \end{matrix}$ (16) When |A|→1, the non-extensive entropy is equivalent to the Shannon entropy. Thus, the expansion of non-extensive entropy as a Shannon entropy is equally applicable for systems with a wide range of features and is also used to analyze stochastic complex signals. $\begin{matrix} lim_{| A | \to 1} \frac{(2^{| A |} - 1) \times m (A) \times (1 - {(\frac{m (A)}{2^{| A |} - 1})}^{| A | - 1})}{| A | - 1} \\ = - m (A) {log}_{2} m (A) \end{matrix}$ (17)

The greater the non-extensive entropy of the calculated A, the greater the uncertainty, the smaller the weight corresponding to the evidence. Otherwise, the smaller the non-extensive entropy, the greater the weight.

3 The proposed method

This section covers the work of the proposed method obtains the final evidence and the proposed method algorithmic form, respectively.

According to the algorithm generated by the proposed method, the flow chart of the uncertain information fusion system is shown in Fig. 2.

Fig.2

The flow chart for uncertain information fusion system.

Step 1.Calculate the distance between each evidence

Given two mass functions m₁ and m₂. The Lance distance can be calculated by Equation 9. According to Equation 8, the distance between any two evidences can be obtained in the form of distance matrix D.

Step 2. Calculate each evidence’s credibility

The similarity measure between m₁ and m₂ can be calculated by Equation 11. Assuming there are N mass functions, a N × N similarity matrix S can be constructed according to Equation 12.

Step 3. Calculate each evidences amount of information

Amount of information is denoted as AI_i it reflects the degree of uncertainty of information m_i. In order to avoid the obstruction to the calculation of information fusion method caused by the exponential term, the amount of information should be normalized. The non-extensive entropy E_TDS (m) of m_i is expressed by the following equation: $A I_{i} = \frac{e^{E_{TDS} (m_{i})}}{\sum_{j = 1}^{x} e^{E_{TDS} (m_{j})}}$ (18) Step 4. Generate the average weighted evidence

The final weight of the evidence is determined by the following equation and normalized. $w_{i} = \frac{{Rel}_{i} \times A I_{i}}{\sum_{j = 1}^{x} {Rel}_{i} \times A I_{i}}$ (19) Then, the weighted average of evidence is calculated using the Equation 7.

Step 5. Obtain final evidence

Assuming that there are N mass functions, the final evidence is obtained by fusing the evidence using Equation 9. The final evidence will be the basis for decision-making.

3.6. Algorithmic

Suppose that a group of original evidence sources collected are E ={ E₁, E₂ …… E_N },The corresponding mass function is m ={ m₁, m₂ …… m_N }. After N evidences are collected, fusion results are generated through algorithm calculation to provide support for decision making. As shown in Algorithm 1:

Table 12

Algorithm 1 A new method of conflicting evidence
management based on non-extensive entropy in
uncertain scenarios
Input: Mass functions for a set of original evidence
sources m ={ m₁, ⋯ , m_i, …, m_N }
Output: Final evidence m_Fus
1: Step 1
2: fori = 1 ; i ≤ Ndo
3: forj = 1 ; j ≤ Ndo
4: The Lance distance d_ij according to E_i,E_j
calculated by using Equation 10;
5: end for
6: end for
7: The lance distance matrix D is constructed by using
Equation 11;
8: Step 2
9: fori = 1 ; i ≤ Ndo
10: The degree of similarity s_ij obtained by using
Equation 12;
11: end for
12: Step 3
13: fori = 1 ; i ≤ Ndo
14: The reliability degree Rel_i calculated by using
Equation 14;
15: end for
16: Step 4
17: fori = 1 ; i ≤ Ndo
18: The exponential form of non-extensive entropy
E_TDS (m) calculated by using Equation 16;
19: end for
20: Step 5
21: fori = 1 ; i ≤ Ndo
22: Each evidences amount of information AI_m
calculated by using Equation 18;
23: end for
24: Step 6
25: fori = 1 ; i ≤ Ndo
26: The average weighted evidence m_Avg calculated by
using Equation 7 and Equation 19;
27: end for
28: Step 7
29: fori = 1 ; i ≤ N - 1 do
30: The final fusion evidence m_Fus obtained by
using Equation 9;
31: end for

4 Numerical example

Two classic numerical examples are recalled in this section to illustrate the effectiveness and feasibility of the proposed method.

Example 1. Suppose the FOD is $H = {A, B, C}$ . Three evidences are m₁, m₂ and m₃, as shown in Table 1.

Table 1
BPAs in Example 1

m ₁ m ₂ m ₃

$A$ 0.99 0 0.01

$B$ 0.01 0.01 0.01

$ℂ$ 0 0.99 0.01

	m ₁	m ₂	m ₃
$A$	0.99	0	0.01
$B$	0.01	0.01	0.01
$ℂ$	0	0.99	0.01

When highly conflicting evidence emerged, the fused BPAs calculated by the classic DCR are: $m (A) = 0$ , $m (B) = 1$ , $m (ℂ) = 0$ . The results indicate that proposition $B$ is the most likely to occur and proposition $A$ and proposition $ℂ$ will never occur, which is obviously counter-intuitive and unreasonable. The classic DCR proved to be inescapably flawed in this example. The proposed method can obtain more reasonable fusion results, the specific steps are as follows:

First, calculate the credibility of each evidences. $d (E_{A}, E_{B}) = \frac{1}{3} \sum_{i = 1}^{3} \frac{| m_{1} (E_{A}) - m_{2} (E_{A}) |}{[m_{1} (E_{A}) + m_{2} (E_{A})]} = 0.6667$ $d (E_{A}, E_{ℂ}) = \frac{1}{3} \sum_{i = 1}^{3} \frac{| m_{1} (E_{A}) - m_{3} (E_{A}) |}{[m_{1} (E_{A}) + m_{3} (E_{A})]} = 0.6667$ $d (E_{B}, E_{ℂ}) = \frac{1}{3} \sum_{i = 1}^{3} \frac{| m_{2} (E_{B}) - m_{3} (E_{B}) |}{[m_{2} (E_{B}) + m_{3} (E_{B})]} = 0.6667$ $s (E_{A}, E_{B}) = s_{12} = 1 - d (E_{A}, E_{B}) = 0.3333$ ${Rel}_{1} = \frac{\sqrt{\sum_{j = 1, j \neq i}^{3} s_{12}^{2}}}{\sum_{i = 1}^{3} \sqrt{\sum_{j = 1, j \neq i}^{3} s_{12}^{2}}} = 0.3322$ ${Rel}_{2} = \frac{\sqrt{\sum_{j = 1, j \neq i}^{3} s_{23}^{2}}}{\sum_{i = 1}^{3} \sqrt{\sum_{j = 1, j \neq i}^{3} s_{23}^{2}}} = 0.3322$ ${Rel}_{3} = \frac{\sqrt{\sum_{j = 1, j \neq i}^{3} s_{13}^{2}}}{\sum_{i = 1}^{3} \sqrt{\sum_{j = 1, j \neq i}^{3} s_{13}^{2}}} = 0.3355$ Then the information content of the three mass functions can be obtained. $\begin{matrix} E_{TDS} (m_{1}) \\ = \sum_{A \in 2^{H}} \frac{(2^{| A |} - 1) \times m (A) \times (1 - {(\frac{m (A)}{2^{| A |} - 1})}^{| A | - 1})}{| A | - 1} \\ = 0.0808 \end{matrix}$ $\begin{matrix} E_{TDS} (m_{2}) \\ = \sum_{B \in 2^{H}} \frac{(2^{| B |} - 1) \times m (B) \times (1 - {(\frac{m (B)}{2^{| B |} - 1})}^{| B | - 1})}{| B | - 1} \\ = 0.1993 \end{matrix}$ $\begin{matrix} E_{TDS} (m_{3}) \\ = \sum_{ℂ \in 2^{H}} \frac{(2^{| ℂ |} - 1) \times m (ℂ) \times (1 - {(\frac{m (ℂ)}{2^{| ℂ |} - 1})}^{| ℂ | - 1})}{| ℂ | - 1} \\ = 0.0808 \end{matrix}$ $A I_{1} = \frac{e^{E_{TDS} (m_{1})}}{\sum_{j = 1}^{3} e^{E_{TDS} (m_{j})}} = 0.3199$ $A I_{2} = \frac{e^{E_{TDS} (m_{2})}}{\sum_{j = 1}^{3} e^{E_{TDS} (m_{j})}} = 0.3602$ $A I_{3} = \frac{e^{E_{TDS} (m_{3})}}{\sum_{j = 1}^{3} e^{E_{TDS} (m_{j})}} = 0.3199$ Finally, the final weight is obtained and the weighted average mass function is calculated. $w_{1} = \frac{{Rel}_{1} \times A I_{1}}{\sum_{j = 1}^{3} {Rel}_{1} \times A I_{1}} = 0.3189$ $w_{2} = \frac{{Rel}_{2} \times A I_{2}}{\sum_{j = 1}^{3} {Rel}_{2} \times A I_{2}} = 0.3590$ $w_{3} = \frac{{Rel}_{3} \times A I_{3}}{\sum_{j = 1}^{3} {Rel}_{3} \times A I_{3}} = 0.3221$ $m_{Avg} (A) = \sum_{i = 1}^{3} w_{A} m_{1} (A) = 0.3189$ $m_{Avg} (B) = \sum_{i = 1}^{3} w_{B} m_{2} (B) = 0.0108$ $m_{Avg} (ℂ) = \sum_{i = 1}^{3} w_{ℂ} m_{3} (ℂ) = 0.3221$

The mass function of the final form can be calculated by Equation 9. $m_{Fus} (A) = 0.4926$ , $m_{Fus} (B) = 0.0001$ , $m_{Fus} (ℂ) = 0.5073$ . The results show the effectiveness of the proposed method in classical examples. Obviously, the result is more reasonable. Next, the superiority over the proposed method will be further verified through Example 2.

Example 2. Hypothesis the FOD is $H = {A, B, C}$ . Five evidences are m₁, m₂ … m₅, they are shown in Table 2.

Table 2

BPAs in Example 2

	m ₁	m ₂	m ₃	m ₄	m ₅
$A$	0.61	0.8	0	0.7	0.65
$B$	0	0.1	0.7	0.05	0.15
$C$	0.39	0.1	0.3	0.25	0.2

Table 3

The credibility of evidence

Rel ₁	Rel ₂	Rel ₃	Rel ₄	Rel ₅
0.1681	0.2098	0.1667	0.2242	0.2311

Table 4

The non-extensive entropy of each evidence

E_TDS (m₁)	E_TDS (m₂)	E_TDS (m₃)	E_TDS (m₄)	E_TDS (m₅)
0.9648	0.6644	0.8813	1.0763	1.2789

Table 5

The uncertainty of evidence

AI ₁	AI ₂	AI ₃	AI ₄	AI ₅
0.1943	0.1439	0.1787	0.2172	0.2660

Table 6

The final weight of each evidence

w ₁	w ₂	w ₃	w ₄	w ₅
0.1611	0.1488	0.1469	0.2401	0.3031

Table 7

The average weighted evidence

$m_{Avg} (A)$	$m_{Avg} (B)$	$m_{Avg} (C)$
0.5824	0.1752	0.2424

Table 8

The final mass function

$m_{Fus} (A)$	$m_{Fus} (B)$	$m_{Fus} (C)$
0.9853	0.0024	0.0123

Step 1. Calculate the credibility of Evidence.

Calculate the distance between each evidence by Equation 10: The lance distance matrixD is: $D = [\begin{matrix} 0 & 0.5755 & 0.7101 & 0.4292 & 0.4513 \\ 0.5755 & 0 & 0.7500 & 0.2762 & 0.2123 \\ 0.7101 & 0.7500 & 0 & 0.6525 & 0.6157 \\ 0.4292 & 0.2762 & 0.6525 & 0 & 0.2160 \\ 0.4513 & 0.2123 & 0.6157 & 0.2160 & 0 \end{matrix}]$ The similarity matrix S is: $S = [\begin{matrix} 1 & 0.4245 & 0.2899 & 0.5708 & 0.5487 \\ 0.4245 & 1 & 0.2500 & 0.7238 & 0.7877 \\ 0.2899 & 0.2500 & 1 & 0.3475 & 0.3843 \\ 0.5708 & 0.7238 & 0.3475 & 1 & 0.7840 \\ 0.5487 & 0.7877 & 0.3843 & 0.7840 & 1 \end{matrix}]$ Calculate the credibility by Equation 14:

Step 2. Calculate Uncertainty of Evidence.

First, each evidence’s non-extensive entropy can be calculated by Equation 16:

Then, calculate uncertainty of evidence by Equation 18:

Step 3. Generate the average weighted evidence

First, According to Equation 18, The final weight of each piece of evidence can be calculated as follows:

Next, The average weighted evidence can be calculated by Equation 7:

Step 4. Obtain the final evidence.

The mass function of the final form can be calculated by Equation 9:

According to the analysis of the results in Table 9: The Dempster’s method is counter-intuitive for the calculation results of conflicting evidence, and no effective results can be obtained. The other three methods can achieve effective identification of $m (A)$ . Deng and Zhangs method are very similar to the method proposed in this article, but the proposed method requires less calculation and is more efficient. Yuan’s method has poorer recognition accuracy than the other three methods. After comparison, the proposed method has higher recognition accuracy, which proves the effectiveness of this method in dealing with conflicting evidence. Therefore, the proposed method has stronger practicability.

Table 9

Fusion results with different method in Example 2

	$m (A)$	$m (B)$	$m (C)$
Dempster’s method [49]	0	0	1
Deng et al.’s method [25]	0.9820	0.0039	0.0141
Zhang et al.’s method [48]	0.9820	0.0033	0.0147
Yuan et al.’s method [55]	0.9815	0.0012	0.0173
The proposed method	0.9853	0.0024	0.0123

5 Application

This section illustrates the effectiveness and feasibility of the proposed method of a case of a new energy vehicle actively avoiding obstacles to autonomous driving.

Vhicle initiative lane change research belongs to the category of the autopilot, and is the main principle of relying on the advanced on-board sensors for the road ahead, the target vehicle and the vehicle state information, with the initiative lane change path planning algorithm to analyze status information processing, timely change driving path, in order to avoid or reduce the security of the problems of traffic accidents.

In the automatic driving condition recognition system, five sensors are assumed to detect the target and make a soft decision on its category, The categories of objects to be identified are known as “left-sided obstruction”, “right-sided obstruction”, or “both sides obstruction”. The frame of discernment (FOD) is H ={ A (Left) , B (Right) , C (Both) }, The mass functions formed by the evidence collected by the five sensors are shown in Table 10.

Table 10
Mass functions collected by the five sensors

m ₁ m ₂ m ₃ m ₄ m ₅

A 0.75 0.5 0.68 0.75 0

B 0.1 0.2 0.32 0 0.3

C 0.15 0.3 0 0.25 0.7

	m ₁	m ₂	m ₃	m ₄	m ₅
A	0.75	0.5	0.68	0.75	0
B	0.1	0.2	0.32	0	0.3
C	0.15	0.3	0	0.25	0.7

Observing the five pieces of evidence shows that four pieces of evidence support the target A(Left) with a greater probability, while the evidence m₂ supports the target B(Right) with a greater probability, which obviously has a conflict of evidence. The results of fusion using the proposed method are compared with other methods in Table 11. The recognition results of the three recognition targets in each method are shown in Fig. 3, 4, 5.

Table 11

Fusion results with different method

Fusion method	Fusion Results
	m₁ ⊕ m₂	m₁ ⊕ m₂ ⊕ m₃	m₁ ⊕ m₂ ⊕ m₃ ⊕ m₄	m₁ ⊕ m₂ ⊕ m₃ ⊕ m₄ ⊕ m₅
Dempster’s method [49]	m(A)=0.7668	m(A)=0.9455	m(A)=1	m(A)=NaN
	m(B)=0.0776	m(B)=0.0545	m(B)=0	m(B)=NaN
	m(C)=0.1621	m(C)=0	m(C)=0	m(C)=NaN
Yager’s method [23]	m(A)=0.2950	m(A)=0.2034	m(A)=0.1565	m(A)=0
	m(B)=0.0244	m(B)=0.0105	m(B)=0	m(B)=0
	m(C)=0.0600	m(C)=0	m(C)=0	m(C)=0
Sun et al. [50]	m(A)=0.4655	m(A)=0.4412	m(A)=0.4457	m(A)=0.3594
	m(B)=0.0778	m(B)=0.1142	m(B)=0.0766	m(B)=0.0864
	m(C)=0.1523	m(C)=0.0814	m(C)=0.0989	m(C)=0.1660
Li et al. [51]	m(A)=0.6345	m(A)=0.6921	m(A)=0.6762	m(A)=0.6032
	m(B)=0.1342	m(B)=0.1889	m(B)=0.1112	m(B)=0.1169
	m(C)=0.2313	m(C)=0.1190	m(C)=0.2126	m(C)=0.2799
Li and Gou [52]	m(A)=0.6345	m(A)=0.6856	m(A)=0.6669	m(A)=0.5763
	m(B)=0.1342	m(B)=0.1866	m(B)=0.1563	m(B)=0.1548
	m(C)=0.2313	m(C)=0.1278	m(C)=0.1768	m(C)=0.2689
Jiang et al. [44]	m(A)=0.8645	m(A)=0.9739	m(A)=0.9942	m(A)=0.9578
	m(B)=0.0275	m(B)=0.0164	m(B)=0.0017	m(B)=0.0016
	m(C)=0.1080	m(C)=0.0097	m(C)=0.0041	m(C)=0.0406
Zhang et al. [48]	m(A)=0.7401	m(A)=0.9352	m(A)=0.9876	m(A)=0.9633
	m(B)=0.0769	m(B)=0.0562	m(B)=0.0062	m(B)=0.0054
	m(C)=0.1830	m(C)=0.0086	m(C)=0.0062	m(C)=0.0313
Tang et al. [53]	m(A)=0.7562	m(A)=0.9453	m(A)=0.9778	m(A)=0.9477
	m(B)=0.0697	m(B)=0.0496	m(B)=0.0074	m(B)=0.0047
	m(C)=0.1741	m(C)=0.0051	m(C)=0.0148	m(C)=0.0476
Lin et al. [54]	m(A)=0.7521	m(A)=0.9390	m(A)=0.9854	m(A)=0.9658
	m(B)=0.0649	m(B)=0.0447	m(B)=0.0052	m(B)=0.0040
	m(C)=0.1830	m(C)=0.0163	m(C)=0.0094	m(C)=0.0302
The proposed method	m(A)=0.7765	m(A)=0.8976	m(A)=0.9556	m(A)=0.9811
	m(B)=0.0729	m(B)=0.0258	m(B)=0.0084	m(B)=0.0027
	m(C)=0.1506	m(C)=0.0767	m(C)=0.0360	m(C)=0.0163

Fig.3

Line graph of the effect of each method to identify target A.

Fig.4

Line graph of the effect of each method to identify target B.

Fig.5

Line graph of the effect of each method to identify target C.

Through the comparison of the fusion results of different algorithms and the proposed method in Table 11 and Fig. 3, 4, 5, the classical D-S theory method can not recognize the correct target because of the conflict of evidence. Yager’s method can’t get the fusion result when the fourth evidence appears. Sun’s method leads to high uncertainty in the identification results, Less significantly in the case of multi-evidence fusion, It is difficult to make an effective judgment. Li and guo’s method of modifying the model leads to a relatively conservative fusion result. Both Lin and Tang used information entropy to process conflicting evidence, with remarkable results, but both were combined only once. Jiang and Zhangs method can handle the evidence with a high degree of conflict, but the results are reduced after the fifth evidence is merged. The proposed method measures the association between the evidences through a variety of measurement parameters, and makes full use of the reliability of the original evidence, and iteratively revises the fusion on the basis of the original evidence. The recognition effect is better than other methods. As shown in the Fig. 6 stability at multi-evidence fusion is significantly better than other approaches. Prove that the proposed method has great application prospect.

Fig.6

Fusion results of the proposed method.

6 Conclusion

The classic DCR is an effective combination method of D-S theory in dealing with information under uncertain conditions and environments. But when the evidence collected from different sensors is highly conflicting, the conclusion drawn by the classic DCR is often counter-intuitive. A conflict evidence combination method based on non-extensive entropy and Lance distance is proposed. To solve this problem. The proposed method uses the weighted average rule to combine the evidence, and the weight of evidence is influenced by the difference of evidence distribution and the amount of information of evidence.Lance distance is used to measure the difference of evidence distribution, and the amount of information is measured by non-extensive entropy, hence the evidence can be fully discriminated. The numerical examples and applications show that the proposed method can make accurate decisions without considering the unreliability of sensors and the conflict of evidence. The proposed method can obtain more reasonable, correct and intuitive results in the case of more evidence and more complex conflict relations.

Furthermore, The proposed method still has limitations of dealing with the index explosion, and other problems in D-S theory. In the following research, we will focus on reducing the computational complexity and seeking a faster combination method. In order to reduce the recognition system with the increase of the number of sensors brought by the huge computing burden, and strive to achieve on-line and on-time decision-making system. It is also our future research direction to further study whether there is a more effective conflict-evidence identification method and how to combine other factors to increase the reliability of weighting when determining the weight.

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A new method of conflicting evidence management based on non-extensive entropy and Lance distance in uncertain scenarios

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Dempster-Shafer evidence theory

2.1.1 Frame of discernment

3.6. Algorithmic

Table 1 BPAs in Example 1 m 1 m 2 m 3 A 0.99 0 0.01 B 0.01 0.01 0.01 ℂ 0 0.99 0.01

Table 10 Mass functions collected by the five sensors m 1 m 2 m 3 m 4 m 5 A 0.75 0.5 0.68 0.75 0 B 0.1 0.2 0.32 0 0.3 C 0.15 0.3 0 0.25 0.7

References

Table 1
BPAs in Example 1

m ₁ m ₂ m ₃

$A$ 0.99 0 0.01

$B$ 0.01 0.01 0.01

$ℂ$ 0 0.99 0.01

Table 10
Mass functions collected by the five sensors

m ₁ m ₂ m ₃ m ₄ m ₅

A 0.75 0.5 0.68 0.75 0

B 0.1 0.2 0.32 0 0.3

C 0.15 0.3 0 0.25 0.7