Double-granule Conditional-entropies Based On Three-level Granular Structures

Rough set theory is an important approach for data mining,and it has referred to Shannon's information measures for uncertainty measurement.As far as conditional-entropies are concerned,they are extensively applied in rough set theory from multiple pointcuts,while uncertainty measurement and reduction construction still serve as two basic issues.The existing local conditional-entropies have both the second-order feature and application limitation,and they lack the condition granulation to restrict their uncertainty measurement function.By improvements of hierarchical granulation,this paper establishes double-granule conditionalentropies based on three-level granular structures(i.e.,Micro-Bottom,Meso-Middle,MacroTop),and then investigates relevant properties.The concrete contents are organized as follows.Firstly,in terms of the decision table and its decision classification,double-granule conditional-entropies are proposed at Micro-Bottom as a basis of hierarchical development 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.Secondly,the new measures acquire their number distribution,calculation algorithm,three bounds and granulation non-monotonicity at three corresponding levels.There,this paper obtains an upper bound and a lower bound by increasing or decreasing the weight coefficient of the double-granule conditional-entropies,and gets another upper bound by using the properties of the convex function.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.