边界区域粗糙集与近似空间不确定性度量的新定义

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本文主要探讨了近似空间中基于边界区域的粗糙集和不确定性度量。在粗糙集理论中，抽象操作符起着关键作用，它们通常通过邻域系统来定义。作者周明马和朱胜米在2016年的《信息科学》期刊上发表了一篇研究论文，提出了一个新颖的预定义边界区域概念来重新定义粗糙集的抽象操作符。在传统的粗糙集理论中，抽象操作符的定义依赖于数据对象之间的相似性或邻近性，而这篇论文则尝试打破这种常规，引入了一个固定的、预先确定的边界区域，这个区域是基于二元关系来构建的。这种新的边界区域提供了对数据结构的另一种理解方式，它可能更适用于某些特定场景，比如在处理不确定性和模糊性较高的数据集时，边界区域可以更好地捕捉到对象间的边界效应。论文首先定义了这种新的边界区域概念，并对比了它与传统粗糙集理论中的抽象操作符定义，分析了它们之间的异同。接着，作者深入探讨了新定义下的抽象操作符的必要性和充分条件，即探讨了这些新型操作符是否能保持原有的性质，如稳定性、有效性等，以及它们是否具有等价性，即是否能够达到相同的结论。进一步的研究可能涉及到边界区域如何影响粗糙集的精化算子和精简算子，以及这种改变如何影响粗糙集的决策表表示和知识约简过程。不确定性度量在粗糙集中扮演着量化信息不确定性的角色，因此论文也可能会讨论新的边界区域概念下不确定性度量的计算方法和其在决策分析中的应用。这篇论文对于粗糙集理论的扩展和改进具有重要意义，它挑战了传统方法，并可能为处理复杂数据集提供一种新的思考角度和工具。通过理论分析和实证研究，作者揭示了边界区域在近似空间中的潜在价值，这将有助于推进粗糙集理论在实际应用中的优化和拓展。

242 Z. Ma, J. Mi / Information Sciences 370–371 (2016) 239–255

The above mentioned three types of approximation operators are equivalent to each other if R is an equivalence relation.

We also have shown that these different types of operators are different if R is a non-equivalence relation. And these types

of approximation operators are different pathways to extend the classical rough set. In other words, when R is not an

equivalence relation, they will generate the corresponding binary relations based generalized rough sets. A basic unit is a

set which can express the minimal ensemble. From an intuitive point of view, the basic units are the minimal units among

all of the precise knowledge. We also think the basic units are rudiments of information units.

Proposition 3. For any x ∈ U , R

(x ) = { y ∈ U| xRy } is a basic unit if and only if R is an equivalence relation on U.

Proof. “⇐ ”It is obvious from Deﬁnition 1 .

“⇒ ”For any x ∈ U , if R

(x ) = { y ∈ U| xRy } is a basic unit, we need to prove that R is an equivalence relation on U .

∀ x ∈ U , if R

( x ) is a basic unit, then R

( x )  = ∅ from Deﬁnition 1 . Suppose y ∈ R

( x ), ( x , y ) ∈ R , we have { x , y } ⊆R

( x ), namely

x ∈ R

( x ), ( x , x ) ∈ R . Thus R is a reﬂexive relation on U .

Suppose ( x , y ) ∈ R , namely y ∈ R

( x ), since R is a reﬂexive relation on U , y ∈ R

( y ), y ∈ R

( x ) ∩ R

( y )  = ∅ . Moreover, R

( x )

and R

( y ) are basic units on U , we have R

(x ) = R

(y ) , x ∈ R

(x ) = R

(y ) holds from Proposition 1 , namely ( y , x ) ∈ R holds.

Thus R is a symmetric relation on U .

Suppose ( x , y ) ∈ R and ( y , z ) ∈ R , we have y ∈ R

( x ) and y ∈ R

( y ) from the above conclusion. Moreover R

( x ) and R

( y )

are basic units on U , so R

(x ) = R

(y ) . Therefore, z ∈ R

(y ) = R

(x ) , namely ( x , z ) ∈ R holds. Thus R is a transitive relation

on U .

Finally, we conclude that R is an equivalence relation on U . 

Proposition

3 also shows, when R is a non-equivalence relation on U , there exists x ∈ U , R

( x ) is not a basic unit. In this

sense, it is more reasonable to deﬁne the approximation operators by means of the basic unit than successor neighborhood

of x . In general, the precision of the latter deﬁnition is lower than one of the former’s.

Example 2. Let U = { a, b, c, d, e, f, g} . R = { (a, b) , (c, d) , (d, e ) , (e, f ) , ( f, g) } is a binary relation on U . Then U

{{ a, b} , { c, d, e, f, g}} . R

(a ) = { b} , R

(b) = ∅ , R

(d ) = { e } , R

(e ) = { f } , R

( f ) = { g} , R

(g) = ∅ . And let X =

{ a, b, c, d } , thus,

(X ) = { a, b, c, g} , R

(X ) = { a, c} , P

(X ) = 2 is no sense;

(X ) = { b, d} , R

(X ) = { b, d} , P

(X ) = 1 is no sense because of X = { a, b, c, d};

(X ) = { a, b, c, d} , R

(X ) = { a, b, c, d, e } , P

(X ) = 0 . 8 .

Example 2 illustrates that the 0-type and 1-type approximation operators already lose the sense of “low” and “up”,

namely the lower approximation of X do not have to be contained in set X , and X do not have to be contained in the upper

approximation of X . The 2-type approximation operators remain the sense, but it is more rough to describe information of

U from the point of view of classiﬁcation. So, we hope to ﬁnd a more reasonable deﬁnition.

The essence of the classical rough sets lies in approximation. The difference of the upper and lower approximations is

called the boundary. Now, we introduce the following deﬁnition.

Deﬁnition 2. ∀ X ⊆U , we call

(X ) = { x ∈ X |∃ y ∈∼ X, xRy ∨ yRx } ,

(X ) = { x ∈∼ X |∃ y ∈ X, xRy ∨ yRx }

the lower and upper boundaries of set X based on binary relation R , respectively. For any X , it is apparent that BR

( X )  = U

and BR

( X )  = U ; BR

(X ) = ∅ if and only if BR

(X ) = ∅ . And BR (X ) = BR

(X ) ∪ BR

(X ) is called the boundary of set X based

on binary relation R , which is called the boundary for short.

Obviously, the boundary

BR (X ) = { x ∈ U|∃ y ∈ U, xRy ∨ yRx, (x ∈ X ∧ y ∈∼ X ) ∨ (x ∈∼ X ∧ y ∈ X ) } .

Proposition 4. For any X , Y ⊆U , the boundary BR ( X ) satisﬁes the following properties:

(1) BR (∅ ) = ∅ ;

(2) BR (U) = ∅ ;

(3) BR (X ) = BR (∼ X ) ;

(4) BR ( X ∩ Y ) ⊆BR ( X ) ∪ BR ( Y );

(5) BR ( X ∪ Y ) ⊆BR ( X ) ∪ BR ( Y );

(6) ∀ x ∈ X , R

( x ) ∪ R

( x ) ⊆X ⇒ BR (X) = ∅ ;

(7) if X is a precise set, then BR (X ) = ∅ ;

(8) if X is a basic unit, then BR (X ) = ∅ ;

(9) BR (X ) = X ⇔ X = ∅ .

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