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Article

The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets

College of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
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Symmetry 2019, 11(3), 386; https://doi.org/10.3390/sym11030386
Received: 14 February 2019 / Revised: 10 March 2019 / Accepted: 12 March 2019 / Published: 16 March 2019
The processing of uncertain information is increasingly becoming a hot topic in the artificial intelligence field, and the information measures of uncertainty information processing are also becoming of importance. In the process of decision-making, decision-makers make decisions mostly according to information measures such as similarity, distance, entropy, and cross-entropy in order to choose the best one. However, we found that many researchers apply cross-entropy to multi-attribute decision-making according to the minimum principle, which is in accordance with the principle of distance measures. Thus, among all the choices, we finally chose the one with the smallest cross-entropy (distance) from the ideal one. However, the relation between cross-entropy and distance measures in fuzzy sets or neutrosophic sets has not yet been verified. In this paper, we mainly consider the relation between the discrimination measure of fuzzy sets and distance measures, where we found that the fuzzy discrimination satisfied all the conditions of distance measure; that is to say, the fuzzy discrimination was found to be consistent with distance measures. We also found that the cross-entropy, which improved when it was based on the fuzzy discrimination, satisfied all the conditions of distance measure, and we finally proved that cross-entropy, including fuzzy cross-entropy and neutrosophic cross-entropy, was also a distance measure. View Full-Text
Keywords: discrimination; cross-entropy; distance measure; consistency discrimination; cross-entropy; distance measure; consistency
MDPI and ACS Style

Wang, Y.; Yang, H.; Qin, K. The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets. Symmetry 2019, 11, 386. https://doi.org/10.3390/sym11030386

AMA Style

Wang Y, Yang H, Qin K. The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets. Symmetry. 2019; 11(3):386. https://doi.org/10.3390/sym11030386

Chicago/Turabian Style

Wang, Yameng, Han Yang, and Keyun Qin. 2019. "The Consistency between Cross-Entropy and Distance Measures in Fuzzy Sets" Symmetry 11, no. 3: 386. https://doi.org/10.3390/sym11030386

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