A novel evidence combination method based on stochastic approach for link-structure analysis algorithm and Lance-Williams distance

Basic concepts of D-S evidence theory

Challenges of D-S evidence theory

Due to the diversity and complexity of data, evidence fusion using the D-S evidence theory may lead to situations contrary to the facts, which are mainly divided into the following four categories.

Construction of network of evidence linkages

The evidence association network is a network that can indicate the correlation between evidences. The lower the conflict between two pieces of evidence, the stronger the correlation between evidences and the higher the similarity between the two pieces of evidence. Conversely, the higher the conflict between two pieces of evidence, the weaker the relevance of the evidence and the lower the similarity between evidences. The current methods to measure the degree of conflict between evidences include the amount of evidence agreement and conflict, conflict coefficient, evidence distance, information entropy, and correlation coefficient. Through analysis and comparison, the conflict coefficient and amount of consistency and conflict between evidence cannot accurately measure the degree of conflict between evidence in some cases. Additionally, the evidence distance mainly describes the degree of difference between evidences, which is not equivalent to describing the degree of conflict between evidences. In this paper, we use the correlation coefficient to measure whether there is a conflict between two pieces of evidence. Its specific definition can be found in the study presented in Yong, Li & Zhang (2011), and its calculation formula is described as follows.

The distributions of the random variables X and Y are as follows: ${\displaystyle X=\left(\begin{array}{c}{a}_1\dots a_k\hfill \\ p_1\dots p_k\hfill \end{array}\right),Y=\left(\begin{array}{c}{b}_1\dots b_k\hfill \\ q_1\dots q_k\hfill \end{array}\right).}$

The bias entropy of the random variable X with respect to Y is defined as follows: (5)${\displaystyle H_Y\left(X\right)=\sum _{k=1}^n{q}_k\times e^{-5q_k}}$

The entropy of association between the random variables X and Y is defined as follows: (6)${\displaystyle H\left(X,Y\right)=H_Y\left(X\right)+H_X\left(Y\right)}$

The partial correlation coefficients and correlation coefficients of the random variables are defined as follows:

(7)${\displaystyle r_Y\left(X\right)=\frac{H\left(Y\right)}{H_Y\left(X\right)}}$ (8)${\displaystyle r_X\left(Y\right)=\frac{H\left(X\right)}{H_X\left(Y\right)}}$ (9)${\displaystyle r\left(X,Y\right)=\frac{H\left(X\right)+H\left(Y\right)}{H_Y\left(X\right)+H_X\left(Y\right)}.}$

The formula uses a similar idea to that of entropy, and the proposed concept of entropy-like is to vary the number of 0-1 over a wide range, because e and log are related by an inverse function, so the function e is used instead of log. But the simple e^−p_k can not completely change the number of 0-1 in a wide range, which needs to add a coefficient on top of e^−p_k into the form of e^−5p_k, this coefficient is closer to the log result based on the check calculation.

When the number of correlation coefficients between two pieces of evidence is large, it indicates that the evidence is highly correlated and less conflicting. If the correlation coefficient between two pieces of evidence is small, it indicates that the evidence is less correlated and more conflicting. The calculation of the full conflict paradox example using the correlation coefficient method yields r = 1; this indicates that the two pieces of evidence are consistent, there is no conflict, and the calculation is consistent with the facts.

The evidence involved in the combination is calculated by the above method to calculate the number of correlation links; the evidence with the number of correlation links greater than the average of the number of correlation links of that evidence is connected to build the evidence correlation network. The main idea of the evidence association network is that a high quality evidence is pointed towards by many high quality evidences, and points to many high quality evidences as well.

Authority weights of evidence using SALSA algorithm

The SALSA algorithm is one of the link analysis algorithms which combines the main features of PageRank and HITS algorithms. According to the concept of SALSA algorithm, the evidence association network is transformed into an undirected bipartite graph, and the weights of the evidence are calculated using the formula of authority weights in SALSA algorithm. Then, the weights of the ith evidence E_i are calculated as follows: (10)${\displaystyle w_i=\frac{\left|A_i\right|}{\left|A\right|}\frac{\left|B_{\left(i\right)}\right|}{\left|B_j\right|}}$

where i, j =1 , 2, …, N,  N is the number of evidence, A_i is the number of evidence in the connected graph, A is the number of evidence in the authority subset, $B_{\left(i\right)}$ is the number of incoming chains of evidence nodes, and B_j is the total number of incoming chains in the connected graph.

The number of evidences in the authority subset is $\left|A\right|$. This factor is the same for any evidence node in the authority subset; thus, it does not affect the final ranking of the weights, serves to ensure that the weights are between 0 and 1, and can represent the weights in the form of probabilities. The higher the number of incoming chains $\left|B_{\left(i\right)}\right|$of evidence E_i in the connectivity graph, the higher number of evidence associated with that evidence, and the higher its importance. The greater the ratio $\frac{\left|B_{\left(i\right)}\right|}{\left|B_j\right|}$ between the number of incoming chains $\left|B_{\left(i\right)}\right|$ and the total number of incoming chains $\left|B_i\right|$ contained in the connected graph of evidence E_i, the greater the importance of this evidence in the evidence associated with it and the greater the corresponding weight.

Use of Lance-Williams distance to obtain evidence support

Lance-Williams distance is a quantity that is both dimensionless and unitless. Thus, its numerical magnitude is independent of the chosen unit, overcoming the drawback that Ming’s distance is related to the magnitude of each indicator, while the dimensionless quantity is more suitable for expressing the distance between evidence. The Lance-Williams distance formula is introduced to calculate the distance between individual pieces of evidence, and the distance is transformed into the support of evidence. The specific steps of the method are described as follows.

After defining the identification frame Θ = {A₁, A₂, …, A_n}, Lance-Williams distance between evidence m_i and m_j is as follows: (11)${\displaystyle d\left(m_i,m_j\right)=\frac{1}{N}\sum _{x=1}^N\frac{\left|m_{ix}-m_{jx}\right|}{\left(m_{ix}+m_{jx}\right)}.}$

The distance between individual pieces of evidence can be expressed as a distance matrix: ${\displaystyle D=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill \cdots \hfill & \hfill d_{1n}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill d_{n1}\hfill & \hfill \cdots \hfill & \hfill d_{nn}\hfill \end{array}\right)}$

The similarity between individual pieces of evidence can be expressed in terms of the distance between the evidences: (12)${\displaystyle s_{ij}=s\left(m_i,m_j\right)=1-d\left(m_i,m_j\right).}$

When s_ij is larger, the similarity between the evidence is higher; the smaller the s_ij, the lower the similarity between evidences. The degree of similarity of the evidence can be expressed by a similarity matrix as follows: ${\displaystyle S=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill s_{1n}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill s_{na}\hfill & \hfill \cdots \hfill & \hfill 1\hfill \end{array}\right).}$

The support of the evidence is calculated as SUP_i; if two pieces of evidence are similar, the two pieces of evidence support each other. The higher the degree of similarity, the greater the degree of support. The degree of support of evidence is defined as follows:

(13)${\displaystyle SU{P}_i=\frac{R_i}{\sum _{i=1}^N{R}_i}}$ (14)${\displaystyle R_i=\sqrt{\sum _{j=1,i\ne j}^N{\left(1-d_{ij}\right)}^2}.}$

Correction and integration of evidence

To avoid that a single discount factor is not accurate enough to correct the evidence, this paper adopts a compound discount factor, and combines the SALSA algorithm concept with the support of evidence to jointly construct a new type of correction coefficient to correct the evidence. The correction factor ω_i for the ith evidence is defined as follows: (15)${\displaystyle {\omega }_i=w_i\times SU{P}_i.}$

Normalizing the correction factor is defined as follows: (16)${\displaystyle {\omega }_i=\frac{{\omega }_i}{\sum _{i=1}^N{\omega }_i}.}$

The correction of the underlying probability assignment for each piece of evidence is defined as follows: (17)${\displaystyle \left\{\begin{array}{c}{m}_i^{{}^{\prime }}\left(E_i\right)={\omega }_i\times m_i\left(E_i\right)\hfill \\ m_i^{{}^{\prime }}\left(\Theta \right)=\left(1-{\omega }_i\right)+{\omega }_i\times m_i\left(\Theta \right)\hfill \end{array}\right.}$

The corrected evidence was fused using the D-S combination rule to obtain the final fusion results.

Comparative analysis of synthetic results of single conflict evidence

For the frame of discernment Θ = {A, B, C}, a sensor collects five mutually independent evidences whose corresponding basic probability assignments are as follows: ${\displaystyle \left\{\begin{array}{c}{E}_1:m_1\left(A\right)=0.5,m_1\left(B\right)=0.2,m_1\left(C\right)=0.3\hfill \\ E_2:m_2\left(A\right)=0,m_2\left(B\right)=0.9,m_2\left(C\right)=0.1\hfill \\ E_3:m_3\left(A\right)=0.55,m_3\left(B\right)=0.1,m_3\left(C\right)=0.35\hfill \\ E_4:m_4\left(A\right)=0.55,m_4\left(B\right)=0.1,m_4\left(C\right)=0.35\hfill \\ E_5:m_5\left(A\right)=0.6,m_5\left(B\right)=0.1,m_5\left(C\right)=0.3\hfill \end{array}\right.}$

Data source from literature (Tian, Ye & Wan, 2021).

The evidence association network is constructed according to the method described in section 3.1, as shown in Fig. 2.

We convert the evidence association network into an undirected bipartite graph, as shown in Fig. 3.

The authority weight of the evidence is calculated according to Eq. (10) as follows: ${\displaystyle w={\left[\frac{16}{70}\frac{4}{50}\frac{15}{70}\frac{15}{70}\frac{15}{70}\right]}^T.}$

The support of the evidence can be calculated according to Eqs. (11)–(14) as follows: ${\displaystyle R={\left[0.21820.06960.23810.23810.2360\right]}^T.}$

The correction factor for the evidence is calculated from Eqs. (15)–(16) as follows: ${\displaystyle \omega ={\left[0.23980.02650.24530.24530.2430\right]}^T.}$

From Eq. (17), the discounted basic probability assignment can be calculated as follows: ${\displaystyle \begin{array}{c}{\Theta }^{\prime }=\left\{\left\{A^{\prime }\right\},\left\{B^{\prime }\right\},\left\{C^{\prime }\right\}\right\}\hfill \\ m^{\prime }=\left\{0.53550.14520.3193\right\}.\hfill \end{array}}$

Finally, the amended evidence was fused using the D-S combination rule to obtain the final fusion results as follows: ${\displaystyle \begin{array}{c}\Theta =\left\{\left\{A\right\},\left\{B\right\},\left\{C\right\}\right\}\hfill \\ m=\left\{0.92900.00140.0700\right\}.\hfill \end{array}}$

The following fusion method of this paper is compared with other evidence fusion methods in Table 1.

Observing the above five evidences, it can be seen that the probability that evidences E1, E3, E4 and E5 point to the focal element A is not less than 0.5. However, the probability that evidence E2 points to B is not less than 0.5, which conflicts with the other four evidences.

The analysis of the fusion results obtained by different methods in Table 1 shows that the D-S evidence theory cannot obtain correct fusion results when the evidence is in a highly conflicting or completely conflicting state. Both methods of Deng et al. (2005) and Ali, Dutta & Boruah (2012) can clearly identify the target only when the fourth piece of evidence is added. Additionally, the method of Deng et al. (2005) does not satisfy the exchange and combination laws of evidence fusion and has some limitations. Moreover, the study in Ali, Dutta & Boruah (2012) proposes a new evidence combination method, but it is not apparent for target identification results. The methods in both Tang et al. (2017) and Hu, Zhong & Liu (2016) use information entropy to deal with conflicting evidence and can identify the target, but the identification accuracy is not high when compared with the method proposed in this paper, and only one correction is made. Both Hu, Zhong & Liu (2016) and Tian, Ye & Wan (2021) use iterative methods to correct the evidence, which can accurately identify the target and have high recognition accuracy. However, the higher the accuracy requirement, the more iterations are needed, and the computation and workload are significantly increased; also, the computation is larger when dealing with larger-scale evidence. The comparative analysis shows that the proposed method can identify the target better. When the fourth piece of evidence is added m(A) = 0.8483, the results are more similar to the methods of Deng et al. (2005), Ye & Nie (2015), Tang et al. (2017) and Hu, Zhong & Liu (2016) when the fifth piece of evidence is added. When the fifth piece of evidence is added, the recognition rate of identifying target A is as high as 92.07%, which is higher than the recognition rate of the rest of the synthetic methods, and the accuracy of the method in this paper in identifying the target is higher when the evidence keeps increasing.

Comparison of results of multi-method conflict evidence combination

The following will use several types of typical combination methods to synthesize the four types of typical conflict problems introduced in section 2.2 to compare with the methods in this paper, and the specific combination results are shown in Table 2.

After analyzing Table 2, according to the description in section 2.2, it can be seen that for example 1 the reasonable fusion result is 1 > m(A) = m(B) > 0, the traditional D-S combination rule cannot be fused, the open-framework-based improvement method is inconsistent with this criterion, and the above two methods cannot deal with the full conflict paradox. For example 2, the reasonable fusion result is 1 > m(A) > m(C) > m(B) > 0; according to this criterion, it can be seen that the traditional D-S combination rule, absorption method, weighted average method, and weighted distribution conflict method are obviously inconsistent with this criterion, and the above methods cannot deal with the 0 trust paradox. For example 3, the reasonable fusion result is 1 > m(A) = m(C) = m(D) > m(B) > 0. According to this criterion, only the method proposed in this paper, weighted average method, and conflicting evidence combination method based on the similarity coefficient between evidences can deal with the 1 trust paradox in Table 2; the rest of the methods cannot deal with the 1 trust paradox. For example 4, the reasonable fusion result is 1 > m(B) > m(C∪D) > m(C) > 0 and m(B) < m₂(B) = m₁(B), m(C∪D) < m₁(C∪D), m(C) < m₁(C). According to this criterion the traditional D-S combination rule, absorption method, weighted allocation conflict method, and open frame of discernment combination rule based on evidence distance in Table 2 cannot deal with the paradox of evidence failure. According to the combination results in Table 2, the reasonable situation of each combination method can be determined, and only the conflicting evidence combination method based on the similarity coefficient between evidences and the method in this paper can solve the main problems of D-S evidence theory. In the following section, the advantages and disadvantages of the proposed method and conflicting evidence combination method based on inter-evidence similarity coefficients will be further compared and analyzed through multiple conflicting evidence combinations.

Comparative analysis of results of multi-conflict evidence combination

For the frame of discernment Θ = A, B, C, D a sensor collects seven mutually independent evidences whose corresponding basic probability assignments are as follows: ${\displaystyle \left\{\begin{array}{c}{E}_1:m_1\left(A\right)=0.9,m_1\left(B\right)=0.1,m_1\left(C\right)=0,m_1\left(D\right)=0\hfill \\ E_2:m_2\left(A\right)=0.5,m_2\left(B\right)=0.5,m_2\left(C\right)=0,m_2\left(D\right)=0\hfill \\ E_3:m_3\left(A\right)=0.1,m_3\left(B\right)=0.9,m_3\left(C\right)=0,m_3\left(D\right)=0\hfill \\ E_4:m_4\left(A\right)=0,m_4\left(B\right)=0.6,m_4\left(C\right)=0.4,m_4\left(D\right)=0\hfill \\ E_5:m_5\left(A\right)=0,m_5\left(B\right)=0.9,m_5\left(C\right)=0,m_5\left(D\right)=0.1\hfill \\ E_6:m_6\left(A\right)=0,m_6\left(B\right)=0.5,m_6\left(C\right)=0.5,m_6\left(D\right)=0\hfill \\ E_7:m_7\left(A\right)=0,m_7\left(B\right)=0,m_7\left(C\right)=0.1,m_7\left(D\right)=0.9\hfill \end{array}\right.}$

Data source from literature (Chen & Wang, 2021).

The combination results obtained using the proposed method in this paper and conflicting evidence combination method based on the similarity coefficients between evidences were synthesized separately as shown in Table 3.

For the focal element X, the size of its synthetic basic probability assignment m(X) isinfluenced by two aspects. The first is the overall support of the individual evidence for the focal element, which can be quantified by the concept of basic confidence number, i.e., $\overline{m_i\left(X\right)}=\frac{\sum m_i\left(X\right)}{N}$, where N is the number of evidences. The second is the number of pieces of evidence supporting the focal element X.

Observing the above seven pieces of evidence, we can see that E₂, E₃, E₄, E₅, E₆ all point to element B with a high probability. The basic probability assignment of evidence E₂, E₃ to focal element A is the same as that of evidence E₆, E₇ to focal element C. However, the synthesis result of m₁(A) > m₄(C)is m(A) > m(C) according to common sense. The evidences E₄, E₆, E₇ support 0.4, 0.5, and 0.1 for focal element C, respectively, and evidences E₅, E₇ support 0.1, 0.9, and 0.9 for focal element D, respectively. The average basic credible number of both of them is the same, but the number of evidence supporting focal element C is greater than the number of evidence supporting focal element D. Then, according to common sense, its synthetic result is m(C) > m(D). In summary, its synthetic result is the reasonable standard m(B) > m(A) > m(C) > m(D). The analysis of Table 3 shows that the combination results of the conflicting evidence combination method based on the similarity coefficient between evidences do not meet the reasonable criteria. The method proposed in this paper not only meets the reasonable criteria, but can also identify target B with a higher probability. Therefore, the proposed method is more reasonable.

Comparative analysis of evidence synthesis results in practical applications

To show the effectiveness of the method proposed in this paper in practical applications, an example of missile health status assessment in literature (Li, Liu & Bao, 2022) was used for comparative analysis. Taking a certain type of surface-to-air missile as an example, assume that the test index characterizing its health status is p₁, p₂, p₃, p₄, p₅. In one of the tests in 2020, if it appears that the actual measured value of a test parameter is outside the standard threshold range, the health status of the device is directly determined to be a fault state. Taking one of the missiles as an example, the actual value of the current test, the non-faulty value of the last test, the average value of the historical test, and the standard threshold value for the five test parameters are shown in Table 4.

Assuming that the quality levels corresponding to excellent, very good, good and proposed failure are A, B, C and D, respectively, the BPA is constructed according to the method proposed in the literature (Li, Liu & Bao, 2022). The test parameter p₁, p₂, p₃, p₄, p₅ corresponds to a BPA of m₁, m₂, m₃, m₄, m₅, and the specific values are shown in Table 5.

The method proposed in this paper is combined with other conflict evidence synthesis methods respectively, and the results are shown in Table 6.

Since the value of p₁ for this test is 0. 69, i.e., outside the threshold range, there is a certain degree of conflict in the test message. The data in Table 6 are all four iterations of fusion results, from the fusion results in Table 6 we can see that the missile health state is in a very good state, due to the existence of conflicting evidence, the D-S algorithm can not handle conflicting evidence, so the fusion results can not be obtained, the actual measured value of parameter 1 is about to exceed the standard threshold range, although the overall health state of the missile tends to be good, but if there is an abnormality in a certain test index, in the synthesis, the information should be retained as much as possible to ensure that the synthesis results are true and valid, Then the synthesis results of literature (Kleinberg, 1999)and literature (Gao, Pan & Deng, 2021) have some deviation from the actual situation. The proposed method in this paper has high accuracy of target identification compared to literature (Murphy, 2000), literature (Li, Liu & Bao, 2022), and also retains the information of the existence of anomalies in the test index when fusion.