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Open AccessArticle

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

by 1,2, 1,2,* and 1,2
1
School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China
2
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; https://doi.org/10.3390/e21070657
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)
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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