# A Dual Measure of Uncertainty: The Deng Extropy

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Deng Extropy

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 3. The Maximum Deng Extropy

**Theorem**

**1.**

**Proof.**

## 4. Application to Pattern Recognition

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BPA | Basic probability assignment |

PPT | Pignistic probability transformation |

SL | Sepal length in cm |

SW | Sepal width in cm |

PL | Petal length in cm |

PW | Petal width in cm |

Se | Iris Setosa |

Ve | Iris Versicolour |

Vi | Iris Virginica |

## References

- Shannon, C.E. A mathematical theory of communication. Bell Labs. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Lad, F.; Sanfilippo, G.; Agrò, G. Extropy: Complementary dual of entropy. Stat. Sci.
**2015**, 30, 40–58. [Google Scholar] [CrossRef] - Dempster, A.P. Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat.
**1967**, 38, 325–339. [Google Scholar] [CrossRef] - Shafer, G. A Mathematical Theory of Evidence; Princeton University Press: Princeton, NJ, USA, 1976. [Google Scholar]
- Deng, Y. Deng entropy. Chaos Solitons Fractals
**2016**, 91, 549–553. [Google Scholar] [CrossRef] - Fu, C.; Yang, J.B.; Yang, S.L. A group evidential reasoning approach based on expert reliability. Eur. J. Oper. Res.
**2015**, 246, 886–893. [Google Scholar] [CrossRef] - Yang, J.B.; Xu, D.L. Evidential reasoning rule for evidence combination. Artif. Intell.
**2013**, 205, 1–29. [Google Scholar] [CrossRef] - Kabir, G.; Tesfamariam, S.; Francisque, A.; Sadiq, R. Evaluating risk of water mains failure using a Bayesian belief network model. Eur. J. Oper. Res.
**2015**, 240, 220–234. [Google Scholar] [CrossRef] - Liu, H.C.; You, J.X.; Fan, X.J.; Lin, Q.L. Failure mode and effects analysis using D numbers and grey relational projection method. Expert Syst. Appl.
**2014**, 41, 4670–4679. [Google Scholar] [CrossRef] - Han, Y.; Deng, Y. An enhanced fuzzy evidential DEMATEL method with its application to identify critical success factors. Soft Comput.
**2018**, 22, 5073–5090. [Google Scholar] [CrossRef] - Liu, Z.; Pan, Q.; Dezert, J.; Han, J.W.; He, Y. Classifier fusion with contextual reliability evaluation. IEEE Trans. Cybern.
**2018**, 48, 1605–1618. [Google Scholar] [CrossRef] - Smets, P. Data fusion in the transferable belief model. In Proceedings of the Third International Conference on Information Fusion, Paris, France, 10–13 July 2000; Volume 1, pp. PS21–PS33. [Google Scholar]
- Balakrishnan, N.; Buono, F.; Longobardi, M. On weighted extropies. Comm. Stat. Theory Methods. (under review).
- Calì, C.; Longobardi, M.; Ahmadi, J. Some properties of cumulative Tsallis entropy. Physica A
**2017**, 486, 1012–1021. [Google Scholar] [CrossRef] [Green Version] - Calì, C.; Longobardi, M.; Navarro, J. Properties for generalized cumulative past measures of information. Probab. Eng. Inform. Sci.
**2020**, 34, 92–111. [Google Scholar] [CrossRef] - Calì, C.; Longobardi, M.; Psarrakos, G. A family of weighted distributions based on the mean inactivity time and cumulative past entropies. Ricerche Mat.
**2019**, 1–15. [Google Scholar] [CrossRef] - Di Crescenzo, A.; Longobardi, M. On cumulative entropies. J. Stat. Plann. Inference
**2009**, 139, 4072–4087. [Google Scholar] [CrossRef] - Kamari, O.; Buono, F. On extropy of past lifetime distribution. Ricerche Mat.
**2020**, in press. [Google Scholar] [CrossRef] - Longobardi, M. Cumulative measures of information and stochastic orders. Ricerche Mat.
**2014**, 63, 209–223. [Google Scholar] [CrossRef] [Green Version] - Abellan, J. Analyzing properties of Deng entropy in the theory of evidence. Chaos Solitons Fractals
**2017**, 95, 195–199. [Google Scholar] [CrossRef] - Tang, Y.; Fang, X.; Zhou, D.; Lv, X. Weighted Deng entropy and its application in uncertainty measure. In Proceedings of the 20th International Conference on Information Fusion (Fusion), Xi’an, China, 10–13 July 2017; pp. 1–5. [Google Scholar]
- Wang, D.; Gao, J.; Wei, D. A New Belief Entropy Based on Deng Entropy. Entropy
**2019**, 21, 987. [Google Scholar] [CrossRef] [Green Version] - Hohle, U. Entropy with respect to plausibility measures. In Proceedings of the 12th IEEE International Symposium on Multiple-Valued Logic, Paris, France, 25–27 May 1982; pp. 167–169. [Google Scholar]
- Yager, R.R. Entropy and specificity in a mathematical theory of evidence. Int. J. Gen. Syst.
**1983**, 9, 249–260. [Google Scholar] [CrossRef] - Klir, G.J.; Ramer, A. Uncertainty in the Dempster-Shafer theory: A critical re-examination. Int. J. Gen. Syst.
**1990**, 18, 155–166. [Google Scholar] [CrossRef] - Kang, B.; Deng, Y. The Maximum Deng Entropy. IEEE Access
**2019**, 7, 120758–120765. [Google Scholar] [CrossRef] - Dheeru, D.; Karra Taniskidou, E. UCI Machine Learning Repository. 2017. Available online: http://archive.ics.uci.edu/ml (accessed on 20 May 2020).
- Cui, H.; Liu, Q.; Zhang, J.; Kang, B. An Improved Deng Entropy and Its Application in Pattern Recognition. IEEE Access
**2019**, 7, 18284–18292. [Google Scholar] [CrossRef] - Kang, B.Y.; Li, Y.; Deng, Y.; Zhang, Y.J.; Deng, X.Y. Determination of basic probability assignment based on interval numbers and its application. Acta Electron. Sin.
**2012**, 40, 1092–1096. [Google Scholar] - Tran, L.; Duckstein, L. Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy Sets Syst.
**2002**, 130, 331–341. [Google Scholar] [CrossRef]

**Figure 1.**$E{X}_{d}\left(m\right)$ in function of n with basic probability assignment (BPA) defined in Example 4.

A | Deng Extropy | Deng Entropy |
---|---|---|

$\left\{1\right\}$ | 28.104 | 2.6623 |

$\{1,2\}$ | 27.904 | 3.9303 |

$\{1,2,3\}$ | 27.704 | 4.9082 |

$\{1,\dots ,4\}$ | 27.504 | 5.7878 |

$\{1,\dots ,5\}$ | 27.304 | 6.6256 |

$\{1,\dots ,6\}$ | 27.104 | 7.4441 |

$\{1,\dots ,7\}$ | 26.903 | 8.2532 |

$\{1,\dots ,8\}$ | 26.702 | 9.0578 |

$\{1,\dots ,9\}$ | 26.500 | 9.8600 |

$\{1,\dots ,10\}$ | 26.295 | 10.661 |

$\{1,\dots ,11\}$ | 26.086 | 11.462 |

$\{1,\dots ,12\}$ | 25.866 | 12.262 |

$\{1,\dots ,13\}$ | 25.621 | 13.062 |

$\{1,\dots ,14\}$ | 25.304 | 13.862 |

Item | SL | SW | PL | PW |
---|---|---|---|---|

$Se$ | [4.4,5.8] | [2.3,4.4] | [1.0,1.9] | [0.1,0.6] |

$Ve$ | [4.9,7.0] | [2.0,3.4] | [3.0,5.1] | [1.0,1.7] |

$Vi$ | [4.9,7.9] | [2.2,3.8] | [4.5,6.9] | [1.4,2.5] |

$Se,Ve$ | [4.9,5.8] | [2.3,3.4] | – | – |

$Se,Vi$ | [4.9,5.8] | [2.3,3.8] | – | – |

$Ve,Vi$ | [4.9,7.0] | [2.2,3.4] | [4.5,5.1] | [1.4,1.7] |

$Se,Ve,Vi$ | [4.9,5.8] | [2.3,3.4] | – | – |

Item | SL | SW | PL | PW | Combined BPA |
---|---|---|---|---|---|

$m\left(Se\right)$ | 0.1098 | 0.1018 | 0.0625 | 0.1004 | 0.0059 |

$m\left(Ve\right)$ | 0.1703 | 0.1303 | 0.1839 | 0.2399 | 0.4664 |

$m\left(Vi\right)$ | 0.1257 | 0.1385 | 0.1819 | 0.3017 | 0.4656 |

$m(Se,Ve)$ | 0.1413 | 0.1663 | 0.0000 | 0.0000 | 0.0000 |

$m(Se,Vi)$ | 0.1413 | 0.1441 | 0.0000 | 0.0000 | 0.0000 |

$m(Ve,Vi)$ | 0.1703 | 0.1527 | 0.5719 | 0.3580 | 0.0620 |

$m(Se,Ve,Vi)$ | 0.1413 | 0.1663 | 0.0000 | 0.0000 | 0.0000 |

Deng extropy | 5.2548 | 5.2806 | 5.1636 | 4.9477 |

Item | SL | SW | PL | PW | Combined BPA |
---|---|---|---|---|---|

$m\left(Se\right)$ | 0.0808 | 0.0730 | 0.0504 | 0.1004 | 0.0224 |

$m\left(Ve\right)$ | 0.1252 | 0.0934 | 0.1482 | 0.2399 | 0.4406 |

$m\left(Vi\right)$ | 0.0925 | 0.0993 | 0.1465 | 0.3017 | 0.4451 |

$m(Se,Ve)$ | 0.1039 | 0.1192 | 0.0000 | 0.0000 | 0.0000 |

$m(Se,Vi)$ | 0.1039 | 0.1033 | 0.0000 | 0.0000 | 0.0000 |

$m(Ve,Vi)$ | 0.1252 | 0.1095 | 0.4608 | 0.3580 | 0.0919 |

$m(Se,Ve,Vi)$ | 0.3684 | 0.4023 | 0.1942 | 0.0000 | 0.0000 |

Item | Setosa | Versicolor | Virginica | Global |
---|---|---|---|---|

Kang’s method | 100% | 96% | 84% | 93.33% |

Method based on Deng extropy | 100% | 96% | 86% | 94% |

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Buono, F.; Longobardi, M.
A Dual Measure of Uncertainty: The Deng Extropy. *Entropy* **2020**, *22*, 582.
https://doi.org/10.3390/e22050582

**AMA Style**

Buono F, Longobardi M.
A Dual Measure of Uncertainty: The Deng Extropy. *Entropy*. 2020; 22(5):582.
https://doi.org/10.3390/e22050582

**Chicago/Turabian Style**

Buono, Francesco, and Maria Longobardi.
2020. "A Dual Measure of Uncertainty: The Deng Extropy" *Entropy* 22, no. 5: 582.
https://doi.org/10.3390/e22050582