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Double-Granule Conditional-Entropies Based on Three-Level Granular Structures

1,2, 1,2,* and 1,2
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China
Institute of Intelligent Information and Quantum Information, Sichuan Normal University, Chengdu 610066, China
Author to whom correspondence should be addressed.
Entropy 2019, 21(7), 657;
Received: 30 April 2019 / Revised: 26 June 2019 / Accepted: 27 June 2019 / Published: 3 July 2019
(This article belongs to the Special Issue Information-Theoretical Methods in Data Mining)
PDF [1480 KB, uploaded 9 July 2019]


Rough set theory is an important approach for data mining, and it refers to Shannon’s information measures for uncertainty measurements. The existing local conditional-entropies have both the second-order feature and application limitation. By improvements of hierarchical granulation, this paper establishes double-granule conditional-entropies based on three-level granular structures (i.e., micro-bottom, meso-middle, macro-top ), and then investigates the relevant properties. In terms of the decision table and its decision classification, double-granule conditional-entropies are proposed at micro-bottom by the dual condition-granule system. By virtue of successive granular summation integrations, they hierarchically evolve to meso-middle and macro-top, to respectively have part and complete condition-granulations. Then, the new measures acquire their number distribution, calculation algorithm, three bounds, and granulation non-monotonicity at three corresponding levels. Finally, the hierarchical constructions and achieved properties are effectively verified by decision table examples and data set experiments. Double-granule conditional-entropies carry the second-order characteristic and hierarchical granulation to deepen both the classical entropy system and local conditional-entropies, and thus they become novel uncertainty measures for information processing and knowledge reasoning. View Full-Text
Keywords: rough set theory; information theory; conditional entropy; uncertainty; granular computing; three-level granular structures rough set theory; information theory; conditional entropy; uncertainty; granular computing; three-level granular structures

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Mu, T.; Zhang, X.; Mo, Z. Double-Granule Conditional-Entropies Based on Three-Level Granular Structures. Entropy 2019, 21, 657.

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