边界区域定义的粗糙集与不确定性度量研究

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"这篇研究论文探讨了在近似空间中基于边界区域的粗糙集理论及其不确定性度量。文章由Zhouming Ma和Jusheng Mi共同撰写，发表于Information Sciences期刊，探讨了如何通过预定义的边界区域来定义新的近似操作符，并与原始概念进行了比较，研究了它们等价的必要条件和充分条件。关键词包括边界区域、公理化、不确定性度量和粗糙集。" 在粗糙集理论中，近似操作符是核心概念，它们通常通过邻域系统来定义。这篇论文引入了一种新的方法，即通过二元关系预定义的边界区域来定义这些操作符。这一创新性的定义旨在提供更深入的理解和更精细的分析工具，特别是在处理数据的不确定性和复杂性时。边界区域是粗糙集理论中的关键元素，它用于区分对象的确定性和不确定性。在传统的粗糙集中，边界区域通常是通过上下近似的差异来确定的，而本文提出的方法则直接将边界作为基础来构建近似操作符。这可能使我们能够更精确地捕捉数据集中的模糊性和不完整性。论文接下来的部分对比了新定义的近似操作符与传统的粗糙集概念，探索了它们之间的等价性。等价性条件的研究对于确保新模型的理论一致性至关重要，同时也有助于验证其在实际应用中的效用。作者们可能讨论了各种情况下的转换规则，以及如何在新旧框架之间建立桥梁。不确定性度量是粗糙集理论中另一个重要的研究领域，它帮助量化数据的不确定性。通过新定义的边界区域，论文可能提出了新的不确定性度量方法，这些方法可能更加直观或者更能适应复杂的决策问题。这样的度量对于理解和处理实际数据集中的不确定性和不精确性非常有用。这篇研究论文为粗糙集理论提供了新的视角，尤其是通过对边界区域的深入利用，以及对近似操作符和不确定性度量的重新定义。这些发展对于提升粗糙集理论在数据分析、决策支持系统、人工智能和模式识别等领域的应用具有潜在价值。

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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