Decision-theoretic Rough Sets Theory And Method For Multiple Costs

Professor Pawlak is the first one who proposed the classical rough set theory in Poland.The theory is constructed on the basis of strict indiscernibility relation, and it is based on the assumption of complete information systems, in which the attribute values are discrete and they have high integrity and accuracy. However, because of the limitation of the measurement error and data acquisition in the real world, the information system is not complete, so the incomplete information system is introduced into rough set. But, a lot of rough set methods in incomplete information systems do not consider the data costs which are more important in the practical engineering application. It is worth mentioning that the research on the cost sensitive learning is very important in machine learning and data mining area. So the research of rough set method based on cost sensitive will be helpful to rough set theory.In this paper, the main idear researches on incomplete information system, uses the rough set models as the mathematical tool and their properties to deeply investigated the cost sensitivity of rough set in incomplete information system. There are two primary innovations as follows:(1) The decision-theoretic rough set is proposed by Yao, in which a pair of the thresholds is derived by the wrong classification cost and delay decision costs matrix for the construction of probabilistic rough set in complete information system. It is worth noting that the decision rough set model of Yao is constructed in the complete information system.However, because of the mission of data, the limitation of the data, and the subjective understanding of the data, the information system that the researchers get is often incomplete in the application of real world. Therefore, the classical decision rough set theory in solving such problems has shown the shortage of one side. For the unknown attribute values, the tolerance relation is applied to decision-theoretic rough set, and the decision-theoretic rough set based on tolerance relation is constructed. For the unknown attribute values are missing or absent, the maximal consistent blocks relation is applied to the decision-theoretic rough set, and the decision-theoretic rough set model based on the maximal consistent blocks is constructed.(2) At the same time, Yao's decision-theoretic rough set model only uses a cost matrix to describe the costs, which have some limitations in dealing with practical problems. It indicates that Yao's decision-theoretic rough set did not consider costs inherent diversity and variability. To solve the above problems, this paper uses multiple cost matrixes to construct a new decision-theoretic rough set method. Then the optimistic and pessimistic multi-costdecision-theoretic rough set models are proposed based on it in incomplete information systems, respectively. Moreover, the paper explores the relationships between the two new decision-theoretic rough sets and the single cost matrix based decision-theoretic rough set.Furthermore, we describe the formulas of the whole decision costs of optimistic and pessimistic multi-cost decision-theoretic rough set models, which based on the definitions of the optimism and pessimism. Finally, Experimental results show that the optimistic multi-cost decision-theoretic rough set model can generate the lowest decision cost, and the decision costs will keep a steady value at last.