Uncertainty Measures of Roughness of
Knowledge and Rough Sets in Ordered
Information Systems
Wei-Hua Xu1,Hong-zhi Yang2,and Wen-Xiu Zhang3
1Institute for Information and System Sciences,
Xi’an Jiaotong University,Xi’an,Shaan’xi710049,P.R.China
2He’nan Pingyuan University,Xinxiang453003,P.R.China
Institute for Information and System Sciences,
Xi’an Jiaotong University,Xi’an,Shaan’xi710049,P.R.China
3Faculty of Science,Institute for Information and System Sciences, Xi’an Jiaotong University,Xi’an,Shaan’xi710049,P.R.China
wxzhang@mail.xjtu.edu
Abstract.Rough t theory has been considered as a uful tool to deal
with inexact,uncertain,or vague knowledge.However,in real-world,
most of information systems are bad on dominance relations,called
ordered information systems,in stead of the classical equivalence for
various factors.So,it is necessary tofind a new measure to knowledge
and rough t in ordered information systems.In this paper,we address
uncertainty measures of roughness of knowledge and rough ts by intro-男人鼻子
ducing rough entropy in ordered information systems.We prove that the
rough entropy of knowledge and rough t decreas monotonously as the
granularity of information becomesfiner,and obtain some conclusions,
which is every helpful in future rearch works of ordered information
systems.
Keywords:Rough t,Information systems,Dominance relation,Rough
entropy,Rough degree.
1Introduction
The rough t theory,propod by Pawlak in the early1980s[1],is an extension of the classical t theory for modeling uncertainty or imprecision information. The rearch has recently roud great interest in the theoretical and application fronts,such as machine learning,pattern recognition,data analysis,and so on [2-6].
In Pawlak’s original rough t theory,partition or equivalence(indiscernibility relation)is a important and primitive concept.However,partition or equivalence relation,as the indiscernibility relation in Pawlak’s original rough t theory,is still restrictive for many applications.To address this issue,veral interesting
D.-S.Huang,L.Heutte,and M.Loog(Eds.):ICIC2007,LNAI4682,pp.759–769,2007.
c Springer-Verlag Berlin Heidelberg2007
760W.-H.Xu,H.-z.Yang,and W.-X.Zhang
and meaningful extensions to equivalence relation have been propod in the past,such as tolerance relations [17],similarity relations [16],others [18-20].Particularly,Greco,Matarazzo,and Slowinski[7-11]propod an extension rough ts theory,called the dominance-bad rough ts approach(DRSA)to take into account the ordering properties of attributes.This innovation is mainly bad on substitution of the indiscernibility relation by a dominance relation.In DRSA condition attributes and class are preference ordered.And many studies have been made in DRSA[12-15].
On the other hand,the concept of entropy,originally defined by Shannon in 1948for communication theory,gives a measure of uncertainty about the struc-ture of a system.It has been uful concept for characterizing information content in a great diversity of models and applications.Attempts have been made to u Shannon’s entropy to measure uncertainty in rough t theory [21-24].Moreover,information entropy is introduced into incomplete information systems,and a kind of new rough entropy is defined to describe the incomplete information systems and roughness of roug
h t.While,most of information systems are bad on dominance ,ordered information systems.Hence,consid-eration of the uncertain measure about entropy in ordered information systems is needed.This paper discusd the problem mainly.
In this paper,we address uncertainty measures of roughness of knowledge and rough ts by introducing rough entropy in ordered information systems.We prove that the rough entropy of knowledge and rough t decreas monotonously as the granularity of information becomes finer,and obtain some conclusions,which is every helpful in future rearch works of ordered information systems.2Rough Sets and Ordered Information Systems
The following recalls necessary concepts and preliminaries required in the quel of our work.Detailed description of the theory can be found in [4,15].
The notion of information system (sometimes called data tables,attribute-value systems,knowledge reprentation systems etc.)provides a convenient tool for the reprentation of objects in terms of their attribute values.
An information system is an ordered quadruple I =(U,A,F ),where U ={x 1,x 2,···,x n }is a non-empty finite t of objects called the univer,and A ={a 1,a 2,···,a p }is a non-empty finite t of attributes,s
uch that there exists a map f l :U →V a l for any a l ∈A ,where V a l is called the domain of the attribute a l ,and denoted F ={f l |a l ∈A }.In an information systems,if the domain of a attribute is ordered according to a decreasing or increasing preference,then the attribute is a criterion.
Definition 2.1.An information system is called an ordered information sys-tem(OIS)if all condition attributes are criterions.
Assumed that the domain of a criterion a ∈A is complete pre-ordered by an outranking relation a ,then x a y means that x is at least as good as y with respect to criterion a .And we can say that x dominates y .In the following,
Uncertainty Measures of Roughness of Knowledge and Rough Sets 761without any loss of generality,we consider criterions having a numerical domain,that is,V a ⊆R (R denotes the t of real numbers).We define x y by f (x,a )≥f (y,a )according to increasing preference,where a ∈A and x,y ∈U .For a subt of attributes B ⊆A ,x B y means that x a y for any a ∈B ,and that is to say x dominates y with respect to all attributes in B .Furthermore,we denote x B y by xR ≥B y .In general,we denote a ordered information systems by I =(U,A,F ).Thus the following definition can be obtained.
Definition 2.2.Let I =(U,A,F )be an ordered information,for B ⊆A ,denote R B
={(x,y )∈U ×U |f l (x )≥f l (y ),∀a l ∈B };R B are called dominance relations of ordered information system I .
Let denote
[x i ] B ={x j ∈U |(x j ,x i )∈R B }
={x j ∈U |f l (x j )≥f l (x i ),∀a l ∈B };
U/R B ={[x i ] B |x i ∈U },
where i ∈{1,2,···,|U |},then [x i ] B will be called a dominance class or the gran-
ularity of information,and U/R B be called a classification of U about attribute
t B .
The following properties of a dominance relation are trivial by the above definition.
Proposition 2.1.Let R A
be a dominance relation.(1)R A is reflexive,transitive,but not symmetric,so it is not a equivalence relation.(2)If B ⊆A ,then R A ⊆R B .(3)If B ⊆A ,then [x i ] A ⊆[x i ] B (4)If x j ∈[x i ] A ,then [x j ] A ⊆[x i ] A and [x i ] A =∪{[x j ] A |x j ∈[x i ] A }.(5)[x j ] A =[x i ] A ifff (x i ,a )=f (x j ,a )(∀a ∈A ).
(6)|[x i ] B |≥1for any x i ∈U .(7)U/R B constitute a covering of U ,i.e.,for every x ∈U we have that [x ] B =φand x ∈U [x ] B =U .
where |·|denote cardinality of the t.
For any subt X of U ,and A of I define
R A (X )={x ∈U |[x ] A ⊆X };R A (X )={x ∈U |[x ] A ∩X =φ},
R A (X )and R A (x )are said to be the lower and upper approximation of X with respect to a dominance relation R A .And the approximations have also some properties which are similar to tho of Pawlak approximation spaces.
762W.-H.Xu,H.-z.Yang,and W.-X.Zhang
Proposition2.2.Let I =(U,A,F)be an ordered information systems and X,Y⊆U,then its lower and upper approximations satisfy the following prop-erties.
(1)R
A (X)⊆X⊆R
A
(X).
(2)R
A (X∪Y)=R
A
(X)∪R
A
(Y);
R
A (X∩Y)=R
A
(X)∩R
A
(Y).
节奏的英文
(3)R
A (X)∪R
A
(Y)⊆R
A
青兰(X∪Y);
R
A (X∩Y)⊆R
A
(X)∩R
A
(Y).
(4)R
X (∼X)=∼R
A
(X);R
A
(∼X)=∼R
A
(X).
(5)R
A (U)=U;R
A
(φ)=φ.
(6)R
A (X)⊆R
A
(R
A
(X));R
A
(R
鹿图片大全A
(X))⊆R
A
(X).
(7)If X⊆Y,then R
A (X)⊆R
A
(Y)and R
A
(X)⊆R
A
(Y).
where∼X is the complement of X.
Definition2.3.For a ordered information system I =(U,A,F)and B,C⊆A.
(1)If[x]
B =[x]
C
才气无双for any x∈U,then we call that classification U/R
B
is
equal to R/R
C ,denoted by U/R
B
=U/R
C
.
(2)If[x]
B
⊆[x] C for any x∈U,then we call that classification U/R B is
finer than R/R
C ,denoted by U/R
B
⊆U/R C.
(3)If[x]
B
⊆[x] C for any x∈U and[x] B=[x] C for some x∈U,then we
call that classification U/R
B is properlyfiner then R/R
C
,denoted by U/R
B
⊂
U/R
C烤三文鱼的做法
.
For a ordered information system I =(U,A,F)and B⊆A,it is obtained
that U/R
A ⊆U/R
B directly by Proposition2.1(3)and above definition.And
an ordered information systems I =(U,A,F)be regarded as knowledge repre-
ntation system U/R
A or knowledge A,as is same to classical rough t bad
on equivalence relation.
Example2.1.Given an ordered information system in Table1.
Table1An ordered information system
U×A a1a2a3
x1121
x2322
x3112
x4213
x5332
x6323
If denote B={a1,a2},from the table we have
[x1]
A
={x1,x2,x5,x6};
Uncertainty Measures of Roughness of Knowledge and Rough Sets763
[x2]
A
={x2,x5,x6};
[x3]
A
={x2,x3,x4,x5,x6};
[x4]
A
={x4,x6};
[x5]
A
={x5};
[x6]
A
={x6};
and
[x1]
B
={x1,x2,x5,x6};
[x2]
B
={x2,x5,x6};
[x3]
B
={x1,x2,x3,x4,x5,x6};
[x4]
B
={x2,x4,x5,x6};
[x5]
B
={x5};
[x6]
B
={x5,x6}.
Thus,it is obviously that U/R
A
⊆U/R ,classification U/R A isfiner
than classification U/R
B .
For simple description,in the following information systems are bad on dominance relations ,ordered information systems.
3Rough Entropy of Knowledge in Ordered Information Systems
In classical rough t theory,knowledge be regarded as partition of t of objects to an information system.However,it is known that equality relations is replaced by dominance relations in an ordered information system.Thus,knowledge be regarded as classification of t of objects to an ordered information system by ction2.
In this ction,we will introduce rough entropy of knowledge and establish relationships between roughness of knowledge and rough entropy in ordered in-formation systems.
Firstly,let give the concept of rough entropy of knowledge in ordered infor-mation systems.
Definition3.1.Let I =(U,A,F)be an ordered information systems and B⊆A.The rough entropy of knowledge B is defined as follows:
E(B)=|U|
i=1
|[x i] B|
|U|
·log2|[x i] B|,
where|·|is the cardinality of ts.
764W.-H.Xu,H.-z.Yang,and W.-X.Zhang
Example3.1.In Example2.1,the rough entropy of knowledge A={a1,a2,a3} can be calculated by above definition,which is
同性恋文
E(A)=4
6
·log24+3
6
·log23+5
6
·log25+ 2
6
·log22+1
6
·log21+1
6
·log21
=2
3
·2+1
2
·log23+5
6
·log25+1
3
=4.39409
Proposition3.1.Let I =(U,A,F)be an ordered information systems and B⊆A.The following hold.
(1)E(B)can obtain its maximum|U|·log2|U|,iffU/R B=U.
(2)E(B)can obtain its minimum0,iffU/R
B
={{x1},{x2},···,{x|U|}}. Proof.It is straightforward by Definition3.1.
From Proposition3.1,it can be concluded that information quantity provided by knowledge B is zero when its rough entropy reaches maximum,and its cannot distinguish any two objects in U,when the classification of ordered informa-tion systems is no meaning.When the rough entropy of knowledge B obtains its minimum,the information quantity is the most and every objects can be discriminated by B in the ordered information systems.
Theorem 3.1.Let I =(U,A,F)be an ordered information systems and
B1,B2⊆A.If U/R B
1⊂U/R B
2
,then E(B1)<E(B2).
Proof.Becau of U/R
B1⊂U/R B
2
,we have that[x i]
B1
⊆[x i] B
2
for every
x i∈U.Thus there exists some x j∈U such that|[x j] B
1|<|[x j] B
2
|.Hence,by
the Proposition2.1and Definition3.1we can obtain
|U|
i=1|[x i] B
1
|·log2|[x i] B
1
|<
|U|
i=1
|[x i] B
2
|·log2|[x i] B
2
|,
<,
E(B1)<E(B2).瘦的成语
From Theorem3.1,we canfind that rough entropy of knowledge decread monotonously as the granularity of information became smaller throughfiner classifications of objects t U.
Corollary 3.1.Let I =(U,A,F)be an ordered information systems and B1,B2⊆A.If B2⊆B1,then E(B1)≤E(B2).
Theorem 3.2.Let I =(U,A,F)be an ordered information systems and
B1,B2⊆A.If U/R B
1=U/R
B2
,then E(B1)=E(B2).
Proof.Since U/R
B1=U/R
B2
,we have that[x i]
B1
=[x i]
B2
for every x i∈U.
Thus,it is obtain E(B1)=E(B2)directly.
