Three-way Weighted Complement-entropies Based On Three-layer Granular Structures

Rough set theory is an important theory for the uncertainty information pro-cessing.The information measures have been introduced into rough set theory to provide an effective construction method,especially for uncertainty measurement and attribute reduction.Furthermore,the combination research has theoretical value and applied significance.The de-cision table serves as a basic data background of rough sets analysis,and it has a hierarchical granular structure;however,this structural mechanism of a decision table has not been well regarded when introducing relevant information measures.Thus,this thesis focuses on the three-layer granular structures of a decision table(i.e.,the micro-top,meso-middle and macro-bottom),and it mainly researches the three-way weighted complement-entropies as well as their basic properties.The concrete contents are organized as follows.Firstly,by focusing on the three-way probabilities at the micro-bottom,the three-way complement-entropies are constructed at the meso-middle and their non-monotonicity is proved.The three-way complement-entropies extend the classical complement-entropies,and their non-monotonicity inspires our following research on the weighted complement-entropies and their monotonicity.Secondly,the Bayes' formula at the micro-bottom makes a transformation,and thus the three-way weighted complement-entropies are respectively established at the meso-middle and macro-bottom;moreover,their granular monotonicity and systematicness are achieved.The three-way weighted complement-entropies introduce probability weights into complement-entropies by using the double-quantitable fusion,and they can acquire better uncertainty fea-tures.Thirdly,the classical complement entropy at the micro-top is decomposed,and the complement-entropy,conditional entropy and mutual information are developed at meso-middle.Moreover,the equivalences on the measurement indexes and system relationships of the weighted complement-entropy system and the classical complement-entropy system are proved at both the meso-middle and micro-top.Finally,the three-layer algorithms of three-way weighted complement-entropies are pro-posed,a decision table example is applied to illustrate the three-way weighted complement-entropies and their granular monotonicity,and numerical experiments of three UCI data are implemented to verify the relevant result and algorithm.In summary,based on the three layer structures of a decision table,this thesis construct-s and studies the three-way weighted complement-entropies at different granular layers.The relevant results offer the good three-layer and three-way structures to enrich the granular com-puting and three-way decisions,and thus they provide an in-depth interpretation on uncertainty measures of rough sets.