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SCIENCE CHINA Information Sciences, Volume 63 , Issue 11 : 210201(2020) https://doi.org/10.1007/s11432-020-3006-9

Uncertainty measure in evidence theory

Yong DENG 1,2,*
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  • ReceivedMay 27, 2020
  • AcceptedJul 1, 2020
  • PublishedOct 20, 2020
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Abstract


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61973332).


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    Figure 1

    (Color online) The question of the team's championship.

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    Figure 2

    (Color online) Comparison results of Case $1$.

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    Figure 3

    (Color online) Comparison results of Case $2$.

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    Figure 4

    (Color online) Comparison results of Case $3$. (a) Definition 1; (b) definition 2; (c) definition 3; (d) definition 4; (e) definition 5; (f) definition 6; (g) definition 7; (h) definition 8; (i) definition 9; (j) definition 10; (k) definition 11; (l) definition 12; (m) definition 13; (n) definition 14; (o) definition 15; (p) definition 16; (q) Deng entropy.

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    Figure 5

    (Color online) The statistical distribution (%) of the application fields of Deng entropy.

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    Figure 6

    (Color online) Development statistics of the applications of Deng entropy.

  • Table 1  

    Table 1A summary of the properties of entropies in evidence theory
    Proposer Probability consistency Set consistency Maximum entropy Additivity Subadditivity
    H${\rm~\ddot{o}}$hle[32] Yes No NoYesNo
    Smets[33] No No NoYesNo
    Yager[34] Yes No NoYesNo
    Dubois and Prade[35] No Yes YesYesYes
    Lamata and Moral[36] Yes Yes NoYesNo
    Klir and Ramer[37] Yes Yes NoYesNo
    Klir and Parviz[38] Yes Yes NoYesNo
    Pal et al.[39,40] Yes Yes NoYesNo
    George and Pal[41] Yes Yes NoYesNo
    Jousselme et al.[42] Yes Yes NoYesNo
    Jirou${\rm~\check{s}}$ek and Shenoy[43] Yes Yes YesYesNo
    Pan et al.[44] Yes Yes YesYesNo
  • Table 2  

    Table 2The belief distribution of maximum Deng entropy [76]
    Frame of discrement$|X|=1$$|X|=2$$|X|=3$ $|X|=4$
    $\Theta~=~\{~A~\}$ 1 – – –
    $\Theta~=~\{~A,B~\}$ $(\frac{1}{5},\frac{1}{5})$ $\frac{3}{5}$ ––
    $\Theta~=~\{~A,B,C~\}$ $(\frac{1}{19},\frac{1}{19},\frac{1}{19})$$(\frac{3}{19},\frac{3}{19},\frac{3}{19})$ $\frac{7}{19}$–
    $\Theta~=~\{~A,B,C,D~\}$ $(\frac{1}{65},\frac{1}{65},\frac{1}{65},\frac{1}{65})$$(\frac{3}{65},\frac{3}{65},\frac{3}{65},\frac{3}{65},\frac{3}{65},\frac{3}{65})$$(\frac{7}{65},\frac{7}{65},\frac{7}{65},\frac{7}{65})$ $\frac{15}{65}$
  • Table 3  

    Table 3The Pseudo-Pascal triangle of the maximum Deng entropy [76]
    $2^{|m|-1}$$2^{|1|-1}$ $2^{|2|-1}$ $2^{|3|-1}$ $2^{|4|-1}$ $2^{|5|-1}$ $2^{|6|-1}$ $2^{|7|-1}$ $2^{|8|-1}$ $2^{|9|-1}$ $2^{|10|-1}$ $2^{|11|-1}$ $2^{|12|-1}$
    $N$ = 1 1 –––––––––––
    $N$ = 2 2 1 ––––––––––
    $N$ = 3 3 3 1 –––––––––
    $N$ = 4 4 6 4 1 ––––––––
    $N$ = 5 5 10 10 5 1 –––––––
    $N$ = 6 6 15 20 15 6 1 ––––––
    $N$ = 7 7 21 35 35 21 7 1 –––––
    $N$ = 8 8 28 56 70 56 28 81 ––––
    $N$ = 9 9 36 84 126 126 84 3691 –––
    $N$ = 10 10 45 120 210 252 210 12045101––
    $N$ = 11 11 55 165 330 462 462 33016555111 –
    $N$ = 12 12 66 220 495 792 924 79249522066 121
  • Table 4  

    Table 4Distribution of the applications of Deng entropy
    Application subsector Frequency Percentage (%) Reference
    Multi-sensor information fusion 16 36.17[28,77,78,81-94]
    Fault diagnosis5 10.64 [95-99]
    Pattern recognition 5 10.64 [100-104]
    Failure mode and effects analysis4 8.51 [105-108]
    Risk assessment 3 6.38 [109-111]
    Multi-criteria decision-making 3 6.38 [26,112,113]
    Decision-making2 4.26 [114,115]
    Emergency management 2 4.26 [116,117]
    Quantum decision 2 4.26 [118,119]
    Organizational management 1 2.13 10.1007/978-3-030-14815-735
    IoT applications 1 2.13 [121]
    Language model 1 2.13 [122]
    Medical diagnosis 1 2.13 [98]