2016 International Conference on Computational Modeling, Simulation and Applied Mathematics (CMSAM 2016) ISBN: 978-1-60595-385-4
A Variable Precision Fuzzy Rough Set Approach to a
Fuzzy-Rough Decision Table
Li-rong JIAN
1and Ming-yang LI
21College of Economics and Management, Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China
2
College of Forestry, Nanjing Forestry University Nanjing 210037, China
Keywords: Fuzzy-rough decision tables, Probabilistic rules, Variable precision rough set, Fuzzy set.
Abstract. In a real world, non-empty boundaries between classes may be both rough and fuzzy. In order to make decision in fuzzy approximation space, a fuzzy VPRS (variable precision rough set) approach is proposed based on substitution of the indiscernibility relation by a fuzzy indiscernibility relation in the rough approximation of decision classes, which can obtain probabilistic rules from fuzzy decision tables. Some set theoretic properties of the proposed approach are discussed.
Introduction
The rough set theory and fuzzy set theory are extensions of classical set theory, and they are related but distinct and complementary theories. The rough set theory[1,2] is mainly focused on crisp information granulation, while its basic concept is indiscernibility, for example, an indiscernibility between different objects deduced by different attribute values of described objects in the information system; whereas the fuzzy set theory is regarded as a mathematical tool for imitating the fuzziness in the human classification mechanism, which mainly deals with fuzzy information granulation. Because of its simplicity and similarity with the human mind, its concept is always used to express quantity data expressed by language and membership functions in the intelligent system. In fuzzy sets, the attributes of elements may be between yes and no. For example, a beautiful scenery, we cannot simply classify the beautiful scenery into a category between yes and no. For the set of beautiful scenery, there does not exist good and definite border. The fuzzy sets cannot be described with any precise mathematical formula, but it is included in the physical and psychological process of human's way of thinking, because the physiology of human reasoning is never used any precise mathematical formula during the physical process of reasoning, and fuzzy sets is important in the pattern classification. Essentially, these two theories both study the problems of information granularity. The rough set theory[3,4] studies rough non-overlapping type and the rough concept; while the fuzzy set theory studies the fuzziness between overlapping sets, and these naturally lead to investigating the possibility of the "hybrid" between the rough sets and the fuzzy set. The hybrid of rough set and fuzzy set can be divided into three kinds of approximations that are the approximations of fuzzy sets in a crisp space, the approximations of crisp sets in fuzzy approximate space, and the approximations of fuzzy sets in fuzzy approximate space[8,9]. The paper mainly discusses the second case. In order to simulate the situation of this type, Dubios introduced the concept of fuzzy rough sets (Dubois and Prade, 1990)[6], which is an extension of rough set approximation deduced from a crisp set in fuzzy approximate space.
Variable Precision Fuzzy Rough Set Model
When the knowledge in knowledge base is fuzzy while the approximate concept is clear, fuzzy variable precision rough set can be applied to solve the problems of probabilistic decision.
Fuzzy Equivalence Relation
Definition 1. Suppose FS=( , )U R is a fuzzy approximation space, U is a non-empty universe, R is the
fuzzy relation on U, and the membership function of R is denoted by µR, if R satisfies all of the
following three properties:
(1) Reflexivity: µR( , ) 1x x = , ∀ ∈x U;
(2) Symmetry: µR( , )x y =µR( , )y x , ∀ ∈x y, ∈U;
(3)Transitivity:∀ ∈x y z, , ∈U, ∀ ∈λ [0,1], if ( , )y z ≥λ, µR( , )x y ≥λ, then µR( , )x z ≥λ.
Then, R is called the fuzzy equivalence relation on
U
.Variable Precision Fuzzy Rough Set Model
Definition 2. Suppose FS=( , )U R is a fuzzy approximation space, U is non-empty universe, and R is the fuzzy equivalence relation of U, while Rλ is the
λ
- cut set of R and X⊆U. Fuzzy set FX based on fuzzy equivalence relation is defined as follows:| ( ) | {( , ( )) : , ( ) }
| ( ) |
X X
X F F
R x X
F x x x U x
R x
λ λ
µ µ ∩
= ∈ =
(1) where ( )
X
F x
µ is the membership function of fuzzy set
X
F and denotes the degree that element x belongs to fuzzy set FX. R xλ( ) is fuzzy equivalence class containing element x. R xλ( ) is the sum of the
membership degree of elements contained in fuzzy equivalence class R xλ( ).
For every λ-cut set in fuzzy equivalence relation R, variable precision rough set model can be applied, thus β -upper approximation and β-lower approximation of variable precision rough set can
be derived from λ horizontal fuzzy equivalence relation for a specified subset X⊆U . Every approximation is still a fuzzy set among them.
Definition 3. FS=( , , , )U A V f is a fuzzy decision table, Uis non-empty universe, and A C= ∪D,
P⊆C is the fuzzy equivalence relation on U, while Ris the fuzzy equivalence relation on U, Rλ is
the λ-cut set of R, x U∈ , X⊆U, 0.5<β≤1.Then β-upper approximation and β-lower approximation of
X are defined respectively by:
{
}
(X) sup ( )
X
P F
Rβ =∪ x∈U：µ x ≥β
(2)
{
}
X sup ( ) 1
X
P F
R（ ）β =∪ x U∈ ：µ x ≥ −β
(3) when two different subsets have non-empty fold in the universe U, the lower approximation of X can
be interpreted as the union of the equivalence classes that membership function ( )
X
F x
µ
is not below the supremum of β; while the upper approximation of X can be interpreted as the union of the equivalence classes membership function µFX( )x is above the infimum of 1−β. Obviously when R isequivalence relation, it is degenerated into variable precision rough set model. Due to β> −1 β, the key
property RP X RP X
β β
⊆
（ ） （ ） still holds.
Proposition 1. For any subset X⊆U in fuzzy approximation space FS= U R（ ， ）, then 0 ( ) 1
X
F x
µ
≤ ≤
Proof:φ ⊆R xλ( )∩X⊆R xλ( ), thus we can infer
| ( ) |
0 1
| ( ) |
R x X
R x λ
λ
∩
≤ ≤ , that is to say, 0 ( ) 1
X
F x
µ
≤ ≤ .
Proposition 2. For any two subsets X⊆U and Y⊆U in fuzzy approximation space FS=（ ， ）U R , Ris
the fuzzy equivalence relation onU, Rλ is the
λ
-cut set of, then(1) FX Y∪ ⊇FX∪FY
(2) If X⊆Y or Y⊆X, then FX∪Y =FX∪FY
(3) FX∩Y⊆FX∩FY
(4) If X ⊆Y or Y⊆X, then FX Y∩ ⊆FX∩FY
Proof: (1) for ∀ ∈x U,
| ( ) ( ) | | ( ( ) ) ( ( ) ) | ( )
| ( ) | | ( ) |
X Y
F
R x X Y R x X R x Y
x
R x R x
λ λ λ
λ λ
µ ∪ = ∩ ∪ = ∩ ∪ ∩
max{| ( ) |,| ( ) |} | ( ) | ( ) |
max ,
| ( ) | | ( ) | | ( ) |
R x X R x Y R x X R x Y
R x R x R x
λ λ λ λ
λ λ λ
∩ ∩ ∩ ∩ ≥ =
{
}
max ,X Y X Y
F F F F
µ µ µ ∪
= =
Therefore FX∪Y ⊇FX∪FY
(2) The proof of (2) can be derived from the proof of (1). (3) (x)
Y X F∩ µ = | ) ( | | ) ( ) ( | x R Y X x R λ
λ ∩ ∩ =
| ) ( | | ) ) ( ( ) ) ( ( | x R Y x R X x R λ λ λ ∩ ∩ ∩
| ) ( | |} ) ( | |, ) ( min{| x R Y x R X x R λ λ
λ ∩ ∩
≤ =min ∩ ∩ | ) ( | | ) ( , | ) ( | | ) ( | x R Y x R x R X x R λ λ λ λ =min{ X F µ , Y F
µ }=
µ
FX∩FYTherefore FX∩Y ⊆FX∩FY
(4) The proof of (4) can be derived from the proof of (3).
According to the definition of variable precision fuzzy roughset, given a subset X ⊆U and a fuzzy equivalence relationRonU, for confident threshold value 0.5<β≤1,variable X can be classified into the following four classes:
(1) If RPβ(X)≠φ, RP(X)≠U
β , then
Xis partly definable.
(2) If RβP(X)≠φ, RβP(X)=U, then X is internally definable. (3) If RPβ(X)=φ, RP(X)≠U
β
, then X is externally definable. (4) If RPβ(X)=φ, RP(X)=U
β
, then X is completely not definable.
Such classification is actually a fuzzy extension of classifications in the variable precision rough set.
Probabilistic Decision Rules Acquisition of Fuzzy Rough Decision Table
The sets of all fuzzy condition elements in the universe are called condition classes in FS, denoted by ( 1, 2,..., )
i
X
F i= k ; the sets of all decision elements in the universe are called decision classes in FS, denoted by Yj(j=1, 2,..., )k , and FXi∩Yj=φ
, then, the rule r: CONC(FXi)→DECD(Yj)
β is called probabilistic
rules of ( ,C D) with confident degreeβ, denoted by
{ }
rij . The syntax of the rules is as follows:If f(x,q1)=rq1∧f(x,q2)=rq2∧...∧f(x,qp)=rqp , then x Y∈ j with β where {q1,q2,...qp}⊆C ,
} ,..., 2 , 1 { , ... } ,..., ,
(rq1 rq2 rqp ∈Vq1×Vq2× Vqp j∈ m presents a certain class of given decision attribute.
A Case
Table 1. A fuzzy decision table.
U SP Conditional attributes DP Decision attributes BP
n1 0.9/N 0.9/N N
n2 0.1/N+0.75/H 0.4/N H
n3 0.85/N 0.3/N+0.4/H N
n4 1/L 1/L L
n5 1/H 0.16/N+0.6/H H
n6 0.4/N 1/H H
n7 0.5/L+0.1/N 0.4/N L
The equivalence class generated from decision attribute BP is as follows:
} , , { /D XNXHXL
U =
Where XN=
{
n , n1 3}
, XH ={
n , n , n2 5 6}
, XL={
n , n4 7}
.According to fuzzy conditional attribute SP and DP based on that a
λ
- cut set can be applied on fuzzy equivalence relationR in fuzzy conditional attribute, fuzzy equivalence class is as follows:.{
}
{
}
{
}
{
}
{
}
{
1 2 3 7 2 5 4 7 3 5 6 1 2 3 7}
/ ( n ,n ,n ,n ,0.1),( n ,n ,0.75),( n ,n ,0.5),( n ,n ,n ,0.4),( n ,n ,n ,n ,0.3)U D= We can obtain the
membership function sup
{
: ( )}
X
F
x∈U µ x of every decision class with fuzzy equivalence class. For
instance, we can get the following for decision class XN:
} | ) ( | | )) ( ( | sup{ x R X x R λ
λ ∩ =sup }
| } , , , { | | } , { } , , , { | { 7 3 2 1 3 1 7 3 2 1 n n n n n n n n n
n ∩ =sup
} 4 . 0 3 . 0 4 . 0 9 . 0 3 . 0 9 . 0 , 1 . 0 85 . 0 1 . 0 9 . 0 85 . 0 9 . 0 { + + + + + + + + =0.897
Similarly, sup
{
: ( )}
X
F
x U∈ µ x of other decision class can be acquired. Suppose confidence level
β=0.85, we can get β-lower approximation, β-upper approximation, linear fuzzy degree, binary time fuzzy exponent and fuzzy entropy of subsets XN, XH, XL shown in table 2.
Table 2. Approximation of decision class.
Decision class β- lower approximation β- upper approximation XN {({ n1, n2, n3, n7},0.1)} {({ n1, n2, n3, n7},0.1)}
XH {({ n2, n5},0.75)} {({ n2, n5},0.75), ( {n3, n5, n6,},0.4)}
XL {(n4, n7),0.5} {(n4, n7),0.5}
From table 2, we can know that the uncertainty of the knowledge inXN is maximal, and the
uncertainty of the knowledge inXHis medium while that is minimal inXL. The minimal probabilistic decision rules of β-lower approximation can be generated from the table 2 and shown in table 3.
Table 3. Minimal probabilistic decision rules.
Rules Degree of confidence
SP=N∧DP=N 89 →.7% BP=N 89.7%
SP=H100 →% BP=H 100%
SP=L100 →% BP=L 100%
Summary
[image:4.612.99.508.527.594.2]set model to solving such decision problems. As a result of the inner fuzziness of the decision-maker’s thinking and the existence of noisy data, sometimes it is impossible to acquire decision rules but likely to acquire strong probabilistic decision rules. In order to deduce the probabilistic decision rules when the knowledge in knowledge base or approximated concept is clear, we have studied hybrid model between variable precision rough set and fuzzy set. For fuzzy rough decision table, we convert fuzzy equivalence relation into equivalence relation through λ-cut set, and we construct variable precision fuzzy rough set model based on this, and integrate variable precision rough set and fuzzy set, then develop the concept of variable precision rough set, further extend the function of variable precision rough set method and analyze some properties among them.
Acknowledgement
This research was financially supported by the National Nature Science Foundation of China (71573124, 71173104); the Key Foundation of University Philosophic and Social Sciences of Jiangsu Province (No. 2012ZDIXM030).
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