The Study On Entropy In Rough Sets And Rough Relational Databases

Using information entropy to measure the uncertainty of rough set prediction is a competing way for predicting a decision variable. Miao and Wang used information entropy to define the corresponding entropy in rough set and proved some basic properties of the entropy. Duntsh and Gediga gave three models for predicting a decision variable, defined by different entropy. In the Chapter 1 of this paper we discuss the formalization of the conditional entropy in rough set theory.It will be proved that certain circumstances, the conditional entropy in rough set theory, given by Duntsch and Gediga, have a connection with roughness of the rough set, which gives a refutation to the claim by Wong, et al. 1985 and rebuild a similar claim. We define anther kind of entropy: the negativeentropy E(P). We can use the negative entropy to represent H'~ (d I Q)Hdet (Q -* d) and Hdet (d I Q) in a concise form.Rough set theory receives much attention recently in data mining because in the rough set theory a new knowledge is formed based on the internal features of the data. To obtain new knowledge from a given data, we need a good measure about the uncertainty of the given data and the uncertainty between the predicting knowledge and the given data. The entropy in information theory, developed by Shannon, seems a competing model for the measurement. Pawlak defined two functions to characterize the imprecision of a rough set X:PR(XY~IRo= KRaR(X) -RxxAlso based on the entropy of information theory, Duntsch and Gediga defined the entropy of rough sets and kinds of the conditional entropy of rough sets for predicting a decision attribute. Beaubouef,et al. defined the entropy for rough sets, rough schema and rough relations in the rough relational database.In the chapter 2 of this paper we mainly discuss the connections between therough entropy PR (X) of rough sets given by Pawlak and the conditionalentropy defined by Duntsch and Gediga. Several properties about the operations on subsets of attributes in an information system and the entropy of the equivalence relations induces by the subsets are discussed. A precise analysis ofIIIentropy gives a mathematical view of deciding which definitions of the entropy is better for measuring the uncertainty of rough sets in the rough set theory. It is still competing to find a good uncertainty measurement of rough sets or rough relations in the rough set theory or the rough relational database.For an information system, entropy is an important accurate numerical description of uncertain information. The uncertainty of information includes indiscemibility, ambiguity and imprecision. Rough set theory captures the notion of indiscernibility or ambiguity instead of a fuzzy type of imprecision. Fuzzy set theory is complementary with the rough set theory. Rough set theory receives much attention recently because rough set theory is based on the internal features of a information system or a relational database.In the rough relational database model, there is an equivalence relation on the domain of attribute j for every attribute j of a relation. A tuple can have multivalues on some attributes, which is restricted in the ordinary relational databases. Beaubouef, et al. defined the corresponding rough relational operators in the rough relational databases as in the ordinary relational databases, and Beaubouef, et al defined the rough entropy E(R) of a rough relation R and gave one example to show that E(R) decreases as R is refined.In the chapter 3 of this paper we discuss the basic properties of the rough entropy of rough relations, and possible connections between the rough entropy and rough relational operators. It will be proved that E(R) is not increasing as the number of tuples in a relation increases; and there is no certain connectionbetween E(R ~ Q) and E(R). E(Q). A precise definition of the entropy ofrough relations in the rough relational databases is given. By using the maximal entropy pri...