3. rough set extensions in the rough set literature, several extensions have been developed that...
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3. Rough set extensions
In the rough set literature, several extensions have been de-veloped that attempt to handle better the uncertainty present in real world data.
Variable precision rough set is generalized model of rough sets, allowing a controlled degree of misclassification by relax-ing the subset operator.
Fuzzy rough sets and tolerance rough sets handle real-valued data by replacing the traditional equivalence classes of crisp rough set theory with alternatives that are better suited to dealing with this type of data.
3.1 Variable precision rough sets Variable precision rough sets (VPRS) attempts to improve upon rough
set theory by relaxing the subset operator. It was proposed to ana-lyze and identify data patterns which represent statistical trends rather than functional.
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The main idea of VPRS is to allow objects to be classified with an er-ror smaller than a certain predefined level.
This approach is arguably easiest to be understood within the framework of classification. Let X, Y U, the relative classification er-ror is defined by
.1,X
YXYXc
Observe that c(X, Y)=0 if and only if XY. A degree of inclusion can be achieved by allowing a certain level of error, , in classification:
.5.00,),( YXciffYX
Using instead of , the -upper and -lower approximations of a set X can be defined as
.1),]([|/][
][|/][
XxcRUxXR
XxRUxXR
RR
RR
Note that for =0.XRXR
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The positive, negative and boundary regions in the original rough set theory can now be extended to
)17(.)(
)16()(
)15()(
,
,
,
XRXRXBND
XRUXNEG
XRXPOS
R
R
R
In the example dataset in Table 1, (15) can be used to calculate the -positive region for R={b, c}, X={e} and =0.4.
U/R={{2}, {0, 4}, {3}, {1,6,7}, {5}}U/X={{0}, {1, 3, 6}, {2, 4, 5, 7}}
For each set AU/R and BU/X, the value of c(A,B) must be less than if the equivalence class A is to be included in the -positive region. Considering A={2} gives
c({2}, {0})=1> c({2}, {1, 3, 6})=1>
c({2}, {2, 4, 5, 7})=0<
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So object 2 is added to the -positive region as it is a -subset of {2, 4, 5, 7}
Taking A={1, 6, 7}, a more interesting case is encountered:c({1, 6, 7}, {0})=1>
c({1, 6, 7}, {1, 3, 6})=0.3333<c({1, 6, 7}, {2, 4, 5, 7})=0.6667>
Here the objects 1, 6 and 7 included in the -positive region as the set {1, 6, 7} is a -subset of {1, 3, 6}.
Calculating the subsets in this way leads to the following -positive region:
Compare this with the positive region generated previously: {2, 3, 5}. Objects 1, 6 and 7 are now included due to the relaxation of the subset operator.
Consider a decision table (U, CD), where C is the set of conditional attributes and D is the set of decision attributes. The -positive re-gion of an equivalence relation Q on U may be determined by
}.7,6,5,3,2,1{)(, XPOSR
XRQPOS QUXR /, )(
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where R is also an equivalence relation on U. This can then be used to calculate dependencies and thus determine -reducts. This depen-dency function become
.)(
)(,
, U
QPOSQ
R
R
3.2 Tolerance rough sets Another way of attempting to handle imprecision is to introduce a
measure of similarity of attribute values and define the lower and upper approximations based on these similarity measures.
3.2.1 Similarity measures In this approach, suitable similarity relations must be defined for each at-
tribute, although the same definition can be used for all attributes if applica-ble. A standard measure for this purpose is
where a is the attribute under consideration, and amax and amin denote the maximum and minimum values respectively for this attribute.
)18()()(
1),(minmax aa
yaxaYXSIM a
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When considering more than one attribute, the defined similarities must be combined to provide a measure of the overall similarity of objects. For a subset of attributes, P, this can be achieved in many ways; two commonly adopted approaches are
)20(),(
),(
)19(),(),(
,
,
P
yxSIMiffSIMyx
yxSIMiffSIMyx
Paa
P
PaaP
where is a global similarity threshold. From this, the so-called tolerance classes that are generated by a given simi-
larity relation for an object x are defined as
)21(.),(|)( ,, PP SIMyxUyxSIM
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3.3.2 Approximations and dependency Lower and upper approximations are then defined in a similar way to tradi-
tional rough set theory:
The tuple is called a tolerance rough set.
Positive region and dependency functions then become
)23(.)(|
)22()(|
,
,
XxSIMxXP
XxSIMxXP
P
P
XPXP ,
)25(.)(
)(
)24()(
,,
/,
U
QPOSQ
XPQPOS
PP
QUXP
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3.3 Fuzzy-rough sets
3.3.1 Main approaches In the same way that crisp equivalence classes are central to rough sets, fuzzy
equivalence classes are central to the fuzzy-rough set approach. The concept of crisp equivalence classes can be extended by the inclusion of a
fuzzy similarity relation S on the universe, which determines the extent to which two elements are similar in S.
The usual properties of reflexivity (S(X, Y)=1), symmetry (S(x, y)=S(y, x)) and transitivity (S(x, z)S(x, y)S(y, z), where is a t-norm) hold.
Using such a fuzzy similarity relation S, the fuzzy equivalence class [x]S for the objects close to x can be defined:
[Def. 3.1] Suppose FS=(U, A, V, f) is a knowledge representation system, and I is defined as an equivalence relation on U, while FX is a fuzzy set on U, PA, FXU, then a pair of the lower approximation and the upper approxi-mation of the knowledge representation system FS about FX is described, re-spectively, as
)26().,()(][ yxy Sx S
)(FXaprP
)(FXapr P
)(:)(sup)(
)(:)(inf)(
xxIxFXapr
xxIxFXapr
FXP
FXP
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Where FX(x) is the membership degree that x belongs to FX, and the upper approximation and the lower approximation of the fuzzy set are still fuzzy sets. Could be understood as the membership degree that the object definitely belongs to the fuzzy set FX, while could be understood as the membership degree that the object possibly belongs to the fuzzy set.
[Ex. 5.1] Suppose U={n1, n2, n3, n4, n5, n6, n7, n8} is a group con-sisted of eight students studied. They are divided into three equiva-lent classes, denoted by U/I={N, M}, U/I={{n1, n6}, {n2, n7, n8}, {n3, n4, n5}}. Suppose the fuzzy set FX expresses the fuzzy concept “height,” and its membership function is FX(X)={n1/0.6, n2/0.4, n3/0.4, n4/0.7, n5/0.6, n6/0.8, n7/1, n8/0.9}, calculates the upper ap-proximation, the lower approximation, and the classification accu-racy of the rough fuzzy set FX.
[Solution]
)(FXaprP
)(FXapr P
5337.07.6
6.3
118.07.07.07.018.0
4.04.06.04.04.04.04.06.0
)(
)()(
1,
1,8.0
,7.0
,7.0
,7.0
,1,8.0
)(
4.0,4.0
,6.0
,4.0
,4.0
,4.0
,4.0
,6.0
)(
87654321
87654321
FXapr
FXaprFX
nnnnnnnnFXapr
nnnnnnnnFXapr
P
PP
P
P
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[Def. 5.2] FS=(U, R) is a fuzzy approximation space, Where U is a nonempty universe, and R is an approximate relation on U, xU, yU, PA, FXU, the input class F1, F2, …, Fn is a fuzzy cluster originated from a fuzzy equivalence relation, and n is the number of the clus-ter. Every Fi is a fuzzy equivalent class. Thus, a pair of the upper ap-proximation and the lower approximation of the fuzzy rough set FX on U is
)(),(minsup)(
)(),(1maxinf)(
xxFXR
xxFXR
FXFP
FXFP
i
i
Obviously, when the equivalence relation is clear, a fuzzy rough set will degrade into a rough fuzzy set. When all fuzzy equivalence rela-tions are clear, it will further degrade into a classical rough set.