Paradox Elimination in Dempster–Shafer Combination Rule with Novel Entropy Function: Application in Decision-Level Multi-Sensor Fusion
Abstract
:1. Introduction
- A single sensor may provide faulty, erroneous results, and there is no way to modify that other than by changing the sensor. A multi-sensor system provides results with diverse accuracy. With the help of a proper fusion algorithm, faulty sensors can be easily detected.
- A multi-sensor system receives information with wide variety and characteristics. Thus, it helps to create a more robust system with less interference.
2. Dempster–Shafer Evidence-Based Combination Rule
2.1. Frame of Discernment (FOD)
2.2. Basic Probability Assignment (BPA)/Mass Function
2.3. Dempster–Shafer Rule of Combination
2.4. Belief and Plausibility Function
3. Paradoxes (Source of Conflicts) in DS Combination Rule
3.1. Completely Conflicting Paradox:
3.2. “One Ballot Veto” Paradox
3.3. “Total Trust” Paradox
4. Eliminating the Paradoxes of DS Combination Rule
4.1. Modification of DS Combination Rule
4.2. Revision of Original Evidence before Combination
4.3. Hybrid Technique Combining Both Modification of DS Rule and Original Evidence
5. Entropy in Information Theory under DS Framework
Properties of Proposed Entropy Function
6. Proposed Steps to Eliminate Paradoxes
- Step 1: Build a multi-sensor information matrix. Assume, for a multi-sensor system, there are N evidences (sensors) in the frame (objects to be detected).
- Step 2: Measure the relative distance between evidences. Several distance function can be used to measure the relative distance. They all have their own advantages and disadvantages regarding runtime and accuracy. We have used Jousselme’s distance [28] function. Jousselme’s distance function uses cardinality in measuring distance which is an important metric when multiple elements are present in one BPA under DS framework. The effect of different distance functions (Euclidean, Jousselme, Minkowsky, Manhttan, Jffreys, and Camberra distance function) on simulation time and information fusion can be found in the literature [29]. Assuming that there are two mass functions indicated by and on the discriminant frame , the Jousselme distance between and is defined as follows:
- Step 3: Calculate sum of evidence distance for each sensor.
- Step 4: Calculate global average of evidence distance.
- Step 5: Calculate belief entropy for each sensor by using Equation (11), and normalize.
- Step 6: The evidence set is divided into two parts: the credible evidence and the incredible evidence. From Equations (14) and (15):
- Step 7: Modify the original evidences.
- Step 8: Combine modified evidence for () times (for this example, 4 times) with DS combination rule by using Equations (5) and (6). How to apply the fusion rule is important. For this example, if evidences and are fused with modified evidence, then . Now, to get , if values are fused with values using Equations (5) and (6), that would be wrong. To get , values should be fused with the original modified evidence from step 7. It is also evident that, for single elements, if that element has higher value after step 7, it will have highest value after fusing () times. The higher the value after step 7, the higher the value after fusion.
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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A | B | C | A,B | A,C | B,C | A,B,C | |
---|---|---|---|---|---|---|---|
Bel(.) | 0.48 | 0.24 | 0.08 | 0.72 | 0.56 | 0.32 | 1.0 |
Pl(.) | 0.68 | 0.44 | 0.28 | 0.92 | 0.76 | 0.52 | 1.0 |
a | b | c | a,b | a,c | b,c | a,b,c | |
---|---|---|---|---|---|---|---|
Bel(.) | 1/7 | 1/7 | 1/7 | 3/7 | 3/7 | 3/7 | 1.0 |
Pl(.) | 4/7 | 4/7 | 4/7 | 6/7 | 6/7 | 6/7 | 1.0 |
m(A) | m(B) | m(C) | m(A,C) | |
---|---|---|---|---|
Bel = 0.42, Pl = 0.41 | Bel = 0.29, Pl = 0.29 | Bel = 0.3, Pl = 0.3 | Bel = 0, Pl = 0 | |
Bel = 0, Pl = 0 | Bel = 0.9, Pl = 0.9 | Bel = 0.1, Pl = 0.1 | Bel = 0, Pl = 0 | |
Bel = 0.93, Pl = 0.93 | Bel = 0.07, Pl = 0.07 | Bel = 0, Pl = 0.35 | Bel = 0.93, Pl = 0.93 | |
Bel = 0.9, Pl = 0.9 | Bel = 0.1, Pl = 0.1 | Bel = 0, Pl = 0.35 | Bel = 0.9, Pl = 0.9 | |
Bel = 0.9, Pl = 0.9 | Bel = 0.1, Pl = 0.1 | Bel = 0, Pl = 0.3 | Bel = 0.9, Pl = 0.9 |
Combination Rule | ||||
---|---|---|---|---|
Dempster [8] | m(A) = 0, m(B) = 0.8969, m(C) = 0.1031 | m(A) = 0, m(B) = 0.8969, m(C) = 0.1031 | m(A) = 0, m(B) = 0.8969, m(C) = 0.1031 | m(A) = 0, m(B) = 0.8969, m(C) = 0.1031 |
Murphy [21] | m(A) = 0.0964, m(B) = 0.8119, m(C) = 0.0917, m(AC) = 0 | m(A) = 0.4619, m(B) = 0.4497, m(C) = 0.0794, m(AC) = 0.0090 | m(A) = 0.8362, m(B) = 0.1147, m(C) = 0.0410, m(AC) = 0.0081 | m(A) = 0.9620, m(B) = 0.0210, m(C) = 0.0138, m(AC) = 0.0032 |
Deng [30] | m(A) = 0.0964, m(B) = 0.8119, m(C) = 0.0917, m(AC) = 0 | m(A) = 0.4974, m(B) = 0.4054, m(C) = 0.0888, m(AC) = 0.0084 | m(A) = 0.9089, m(B) = 0.0444, m(C) = 0.0379, m(AC) = 0.0089 | m(A) = 0.9820, m(B) = 0.0039, m(C) = 0.0107, m(AC) = 0.0034 |
Han [31] | m(A) = 0.0964, m(B) = 0.8119, m(C) = 0.0917, m(AC) = 0 | m(A) = 0.5188, m(B) = 0.3802, m(C) = 0.0926, m(AC) = 0.0084 | m(A) = 0.9246, m(B) = 0.0300, m(C) = 0.0362, m(AC) = 0.0092 | m(A) = 0.9844, m(B) = 0.0023, m(C) = 0.0099, m(AC) = 0.0034 |
Wang [32] recalculated | m(A) = 0.0964, m(B) = 0.8119, m(C) = 0.0917, m(AC) = 0 | m(A) = 0.6495, m(B) = 0.2367, m(C) = 0.1065, m(AC) = 0.0079 | m(A) = 0.9577, m(B) = 0.0129, m(C) = 0.0200, m(AC) = 0.0094 | m(A) = 0.9867, m(B) = 0.0008, m(C) = 0.0087, m(AC) = 0.0035 |
Jiang [20] | m(A) = 0.0964, m(B) = 0.8119, m(C) = 0.0917, m(AC) = 0 | m(A) = 0.7614, m(B) = 0.1295, m(C) = 0.0961, m(AC) = 0.0130 | m(A) = 0.9379, m(B) = 0.0173, m(C) = 0.0361, m(AC) = 0.0087 | m(A) = 0.9837, m(B) = 0.0021, m(C) = 0.0110, m(AC) = 0.0032 |
Proposed | m(A) = 0.00573, m(B) = 0.96906, m(C) = 0.02522, m(AC) = 0 | m(A) = 0.7207, m(B) = 0.1541, m(C) = 0.1178, m(AC) = 0.007 | m(A) = 0.9638, m(B) = 0.0019, m(C) = 0.0224, m(AC) = 0.0117 | m(A) = 0.9877, m(B) = 0.0002, m(C) = 0.0087, m(AC) = 0.0034 |
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Khan, M.N.; Anwar, S. Paradox Elimination in Dempster–Shafer Combination Rule with Novel Entropy Function: Application in Decision-Level Multi-Sensor Fusion. Sensors 2019, 19, 4810. https://doi.org/10.3390/s19214810
Khan MN, Anwar S. Paradox Elimination in Dempster–Shafer Combination Rule with Novel Entropy Function: Application in Decision-Level Multi-Sensor Fusion. Sensors. 2019; 19(21):4810. https://doi.org/10.3390/s19214810
Chicago/Turabian StyleKhan, Md Nazmuzzaman, and Sohel Anwar. 2019. "Paradox Elimination in Dempster–Shafer Combination Rule with Novel Entropy Function: Application in Decision-Level Multi-Sensor Fusion" Sensors 19, no. 21: 4810. https://doi.org/10.3390/s19214810
APA StyleKhan, M. N., & Anwar, S. (2019). Paradox Elimination in Dempster–Shafer Combination Rule with Novel Entropy Function: Application in Decision-Level Multi-Sensor Fusion. Sensors, 19(21), 4810. https://doi.org/10.3390/s19214810