Mu T, Zhang X, Mo Z. Double-Granule Conditional-Entropies Based on Three-Level Granular Structures.
ENTROPY (BASEL, SWITZERLAND) 2019;
21:e21070657. [PMID:
33267371 PMCID:
PMC7515153 DOI:
10.3390/e21070657]
[Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/30/2019] [Revised: 06/26/2019] [Accepted: 06/27/2019] [Indexed: 06/12/2023]
Abstract
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.
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