The Uncertainty Measurement Of General Binary Relation

The information entropy is an effective measure for uncertainty.The Chinese and foreign famous scholars have proposed many methods about the measurement of uncertainty,and have made remarkable achievements,which laid a solid foundation for us accessing to useful knowledge,and improve the efficiency of us to obtain information greatly.Relation,as a fundamental concept in mathematics,represents the connections of a set of elements in the domain.The two definitions of the set and relation constitute the basis of modern mathematics.Additionally,relations have been applied to discrete real-valued data,fuzzy clustering and attribute reduction.Equivalence relations,similarity relations,neighborhood relations are foundations of a sequence of rough set models.We have many good ways to measure the uncertainty of special relations;however the studies of general relations are few.So the studies of this paper are mainly aiming at the uncertainty of general relations.The overall idea of this article is: combining the basic knowledge of information systems and information entropy with the subsequent neighborhood,which puts forward the concepts of the entropy of general binary relation and generalized neighborhood,then makes a detailed discussion on the nature of them,and the work is as follows:1.In order to compute the entropy of general relation,joint entropy,conditional entropy and mutual information,we divide universe mainly through the subsequent neighborhood.Then we give some basic properties and compare with neighborhood entropy.In the end,we prove that the definitions of them are equivalent.2.Through the subsequent neighborhood,we redefine the entropy on the basis of neighborhood entropy,and then put forward the concepts of the entropy,joint entropy,conditional entropy mutual information of generalized neighborhood,and their properties are discussed.In the end,comparing with the entropy of neighborhood entropy,we found it is a kind of entropy that more comprehensive and accurate calculating than neighborhood entropy.