Boundary Region-based MGRSs And Their Uncertainty Measures

Rough set theory as one of mathematical tools for dealing with uncertain data,characters the maximum probability and the minimum necessity of unknown knowledge by employing the lower and upper approximation operators.So,remaining fundamental properties of the two operators is one of the most important goals in the rough set model's various kinds of generalization.Therefore,this paper tries to give predefined boundary region of the uncertainty,studies single-granulation rough set of a binary relation and covering based on the boundary region.And based on the studing single-granulation rough set by employing boundary region,discusses various kinds of Multi-granulation rough set models based on boundary region and their relevant measures.The main research findings are summarized as follows:Firstly,the general binary relations based boundary operator are proposed,we introduct the lower and upper approximation operators based on the operator,discuss their corresponding properties and axiomatic characterizations.And it is compared with the classcial rough set at the same background,probe them between contacting and difference,the necessary and sufficient conditions of their equipollence are also investigated.As the close links between binary relation and covering,in the same way,introduct boundary operator based on general covering on universe,thus,give the corresponding the lower and upper approximation operators based on covering.It is shown that a covering boundary region based novel rough set and a class of covering rough set proposed by Zakowski are equivalent.We also discuss their corresponding boundary operatos further,the lower and upper approximation operators and their the properties and axiomatic characterizations.Bartol et al characterized a covering based rough set generated by a tolerance relation based rough set.As concerns the generalized rough set based on binary relation and covering,in our works,the relative link between there is gave from boundary region perspective.In other words,the binary relation generalized rough sets based on boundary region are are equivalent to the covering reduction generalized rough sets based on boundary region.Meanwhile,the novel generalized rough sets based on a covering are equivalent to the ones based on a tolerance relation.In addition,with the aid of the novel definitions,we also study relationships between the axiomatic characterizations of the approximation operators based on the binary relations and the ones based on the coverings.As concerns uncertainty measures of the single-granulation rough sets,based on illustration of the existing uncertainty measures,we give the improved axiomatic definition of the knowledge granulation.And a more reasonable knowledge granulation is presented by using the novel boundary operators.It is more effective to characterize the approximation spaces whether a partition or a covering.For a object set,to distinguish the different effect of the same boundaries on various object sets,we also introduce a sets of measure methods in greater detail by defining the inner and outer uncertainty measures.Secondly,based on the approach of boundary region rough set,we study various kinds of Multi-granulation rough set and their relevant properties.Multi-granulation rough set can be divided into two main types: optimistic and pessimistic Multi-granulation rough set.This paper discusses Multi-granulation rough set generated by intersection and union of binary relation single-granulation,thus,further probe optimistic,pessimistic,intersection and union Multi-granulation rough sets and their corresponding properties.It is shown that intersection Multi-granulation rough sets are more optimistic than optimistic ones,and union Multi-granulation rough sets exactly corresponding to pessimistic ones.Therefore,Multi-granulation rough sets can also be divided into three main types: intersection,union(pessimistic)and optimistic Multi-granulation rough sets.Thus,intersection and union Multi-granulation rough sets are corresponding to a binary relation single-granulation rough set on universe,respectively.However,in general,a optimistic Multi-granulation rough set do not corresponding to any binary relation singlegranulation rough set on universe.So the relevant properties of intersection and union Multi-granulation rough sets based on boundary region are similar to the properties of the corresponding binary relation single-granulation rough set.However,the properties of optimistic Multi-granulation rough set based on boundary region are not similar to the properties of any binary relation single-granulation rough set on universe.The common properties of the above mentioned Multi-granulation rough sets is they maintain the basic properties of the classical rough set,example,uncertain object set always between its maximum probability and minimum necessity.Certainly,the Multi-granulation rough sets can be regarded as special fusion strategy of some single-granulation rough sets,we also omparative study the property function between before and after their fusion.In addition,the necessary and sufficient conditions of mutual equivalence of the three Multi-granulation rough sets baed on boundary region and the corresponding original Multi-granulation rough sets are also investigated.Finally,at the basis of the above mentioned conclusions,we review some improtant measure approachs of the classical rough set in the researching of uncertainty.As concerns Multi-granulation rough sets,the essential difference and similarity research among intersection,union(pessimistic)and optimistic Multi-granulation rough sets are concluded based on thorough analysis and comparison.Note that there are some essential difference between union and intersection Multi-granulation rough sets based on equivalence relation.As the intersection of some equivalence relations will remain an equivalence,the union of some equivalence relations will not remain an equivalence relation,at best,which is a reflexive relation.Therefore,intersection Multi-granulation rough set corresponding exactly a classical rough set,whereas union Multi-granulatlion rough set corresponding a generalized rough set based on a reflexive relation.Considering that union Multigranulation rough set is a pessimistic one in nature,thus,in general,in the researching the structural characteristic of the Multi-granulatlion rough sets,we take into account only intersection,union and optimistic Multi-granulation rough sets.To explore their theoretical basis,we give the axiom characterizations of the three Multi-granulatlion rough sets.As concerns the uncertainty measures of Multi-granulation rough sets,which is to itself a kind of fusion mechanism,naturally,there are the corresponding measure structure,information entropy,rough entropy and knowledge granulation etc.By employing some examples,we analyze their advantages and disadvantages in the multi-granulation approximation space.The disadvantages generalized mainly by the partial relation and its limits,The trend is particularly apparent for characterizing the uncertainty of various granulations.To overcome the above mentioned defects,a total ordered relation among various of granulations is introduced in the multi-granulation approximation spaces.It will be better than the original partial relation in expressing uncertainty,which conceals in the approximation space.However,due to missing the monotonicity,the original uncertainty measures do not work well the relevant essential uncertainties even if under the total ordered relation in some exceptional cases.On the foundation of the new total ordered relation,we propose improved information entropy,rough entropy and the axiomatic definition of the fusion knowledge granulation.Consequently,an improved fusion knowledge granulation is defined,and the uncertainty measures,which conceals in the approximation space,produced by the fusion granulation can be expressed effectively.Meanwhile,we give an accurate comparison among the relevant granulation measures of the intersection,union(pessimistic)and optimistic Multi-granulation rough sets via the improved fusion uncertainty measures.