In this section, we propose a conflict data management method based on the EBBF and give some examples and related applications to verify its rationality and effectiveness.
5.2. Illustrative Example
Example 1. In the classic example that the FOD is , the two BPAs are given as: In this example, the mass function of the empty set is 0, indicating that the mass function is allocated in the exhaustive FOD. The EBBF value is calculated as follows: It is exactly the same as the value calculated with the base belief function. The BPA of the two groups is modified accordingly, and the results are shown in Table 1. Finally, the results of data fusion are shown in
Figure 2. After consideration and research, the data fusion method we adopted here is the DCR instead of the TBM conjunctive rule. According to the formulas of the two methods, the difference between the two methods can be analyzed as follows: Both the TBM conjunctive rule and the DCR mainly consider the influence of intersection, but the TBM conjunctive rule is not normalized, which makes the sum of belief values less than 1. Obviously, such a result is counter-intuitive, so the TBM conjunctive rule is not suitable for the examples in this paper.
The fusion results show that the DCR rule cannot produce logical results in this example, but both the method in this paper and the base belief function method can produce intuitive results. It can also be seen that under the exhaustive FOD assumption, the method in this paper can be reduced to base belief function.
Example 2. Suppose that the FOD is , and two BPAs are given as: At this point, , showing that the FOD is incomplete, and this example is under the non-exhaustive FOD assumption. The calculation process is shown in Figure 1. Step 1: From the FOD , we can know that the potential of the FOD is: .
Step 2: Bring the empty set mass function value and
N into the formula and calculate the EBBF value as follows:
Step 3: Use the value of the belief function to modify the BPA of each piece of evidence and calculate as follows:
Step 4: Judge whether the empty set mass function is 0 and select a different combination rule.
In this example, the mass function value of the empty set is not 0, so the GCR is used to fuse the BPAs of two sets of evidence, and the following results are obtained.
The result shows a high support for proposition a, which is in line with the actual situation of the evidence. However, at the same time, the results also give b a certain small degree of support, which shows that we have not collected all the evidence; there is still the possibility that b is right, avoiding the data being too absolute.
Example 3. In the FOD , there are two sets of highly conflicting BPAs, which are as follows: Using the proposed method to calculate the belief function value, we can get: Then, is used to modify the BPA value of each piece of evidence, as shown in Table 2. The fusion results obtained by the GCR are shown in
Table 3. The data show that the proposed method gives the same degree of support for
a and
b in two sets of completely conflicting data, which is the correct answer in the intuition. It is proved that the proposed method can still give a good result in accordance with the facts in highly conflicting data.
Example 4. In the FOD , the BPAs are as follows: Using the proposed method to calculate the values of the EBBF, we can get: According to the values of the two EBBFs, the BPA of the two groups was modified accordingly, and the results are shown in
Table 4.
Then, the GCR was used for fusion to obtain the result. At the same time, the BPA from the evidence in the table was fused directly with the GCR, and the results of the two methods were compared, as shown in
Table 5 and
Figure 3.
In this case, the two sets of evidence give high support to a and c, respectively, but the fusion result using only the GCR assigns the highest support to proposition b, which is obviously unreasonable. However, after the BPA was modified with the proposed method in this paper, the fusion results obtained by the GCR were highly supportive of both a and c. Moreover, as the support for c in the second piece of evidence was slightly greater than that in the first piece of evidence, the fusion results also gave a slight advantage to c. This indicates that in the case of highly conflicting data, the right intuitive answer cannot be obtained simply by using the GCR, while reasonable and intuitive results can be obtained by using the method in this paper.
The above examples verify that the proposed method is compatible with the base belief function in the exhaustive FOD and verify the feasibility and effectiveness of the proposed method. The following examples discuss some other properties of the proposed method.
Example 5. In the FOD , the BPAs are as follows: Using the proposed method to calculate the value of the EBBFs, we can get: Then, the BPA of each piece of evidence was modified by using the value of the calculated belief function, and the results are shown in Table 6. Then, the GCR was used for fusion to obtain the result. At the same time, the BPA from the evidence in the table was fused directly with the GCR, and the results of the two methods were compared, as shown in
Table 7 and
Figure 4.
According to
Table 7, when the mass function of each single element subset in the FOD is not 0, the fusion results obtained by using the method in this paper to modify the BPA and by using only the GCR both reflect that
a and
c have equally high confidence. We further calculated the entropy values of the original evidence mass function and the modified mass function. Here, we adopted the extended Deng entropy (EBEOW) [
64], which can calculate the entropy value in the non-exhaustive FOD. According to the calculation, the extended Deng entropy value of the original BPAs is:
The extended Deng entropy of the BPAs modified by the proposed method in this paper is as follows:
Obviously, the entropy corresponding to the fusion results of the proposed method increases significantly, and the increase in entropy indicates that the belief assignment is more dispersed and the uncertainty is increased, which is the consequence of the conflict. In general, when the proposition of a single element subset is not zero and the data are highly conflicting, the method proposed in this paper assigns part of the belief to other multi-subset elements, thus reducing the risk.
Example 6. Suppose that the FOD is and two BPAs are given as: According to the proposed method, the value of the belief function corresponding to evidence and is calculated as follows: Then, the BPA of each piece of evidence was modified by using the value of the calculated belief function, and the results are shown in Table 8. Finally, the fusion results obtained with the GCR and the fusion results with the non-modified BPA obtained by the GCR are shown in
Table 9 and
Figure 5.
It can be seen from the results in
Figure 5 that when all the mass functions of a complete set are nonzero, the fusion results obtained by using the proposed method of modifying the BPA in advance and by using only the GCR both reflect that
a and
b have equally high belief, which is an intuitive result. The extended belief entropies of the original evidence mass function and the modified mass function are further calculated. According to the calculation, the expanded belief entropy of the BPAs is
, and the extended belief entropy of the BPAs modified by the proposed method is
. The entropy value corresponding to the fusion result of this method obviously increases. Therefore, when the mass function of the whole element subset of each evidence is not 0 and the data are highly conflicting, the proposed method allocates part of the belief degree to other sub-propositions, thus reducing the risk.
Example 7. In the FOD , the BPAs are as follows: Using the proposed method to calculate the belief function value, we can get: Then, the GCR was used for fusion to obtain the result. At the same time, the BPA from the evidence in the table was fused directly with the GCR, and the results of the two methods were compared, as shown in
Table 10 and
Figure 6.
When the complete subset mass function of one set of evidence is nonzero, the value of all single element subset mass functions of the other set of evidence is nonzero, and the data are highly conflicting, the fusion results using only the GCR only give a a high confidence degree, while the fusion results using the method in this paper give a and b high support degrees. Obviously, the results of the proposed method are more in line with the objective reality.
5.3. Application to Artificial Data
In order to verify the availability and effectiveness of the conflict data management method proposed in this section, an example in [
65] is used for example analysis, and the calculation results are compared with other methods.
Considering the problem of target recognition, it is assumed that three potential targets are represented as
a,
b, and
c respectively. According to the conflict data management method based on the EBBF proposed in this section, the first step is to model uncertain information evidence. The reports of five sensors are modeled by the BPA, and the results are shown in
Table 11.
Table 11 slightly modifies the data in [
65], causing an expansion from exhaustive FOD to non-exhaustive FOD. Intuitive analysis found that the report from the second sensor is inconsistent and conflicting with the other four sensors’ reports. Moreover, the other four sensors reported that
a was the most likely to be a potential target because its mass function value was the largest and the confidence level was the highest.
Based on the data in
Table 11, the EBBF values of each group of evidence were calculated as follows:
According to the EBBF values, the BPAs were modified, and the modified data of all BPAs in
Table 11 are shown in
Table 12.
Data fusion using the generalized combination rule is as follows:
Then, the GCR was used for fusion to obtain the result. At the same time, the BPAs from the evidence in the table were fused directly with the GCR, and the results of the two methods were compared, as shown in
Table 13 and
Figure 7.
The fusion results show that using the method proposed in this section, it can be concluded that a is the identified target, which is consistent with the intuitive analysis results. However, the result of GCR fusion without the EBBF shows that b is the identified target, while the possibility that a is the target is 0, which is obviously not an intuitive result. Through the above comparison, we can clearly see that the data fusion results verify the effectiveness of the method proposed in this section, which can be used well in conflict data management in practical engineering.