Structure-Based Attribute Reduction: A Rough Set Approach - Feature Selection for Data and Pattern Recognition - page 147

BN A ( X i ) = BN C ( X i ) for all i ∈ V d , or equivalently,

(DB )

X i X i BN A ( ) = BN C ( ) for all i ∈ V d .

Yang et al. [ 52 ] independently proposed four kinds of reducts in DRSM with
unknown attribute values, which are application of distribution reducts of Mi
et al. [ 33 ]. Those reducts preserve lower/upper approximations of upward/
downward unions. Hence, they correspond to L ≥ -, L ≤ -, U ≥ -, and U ≤ -reducts of
ours. However, Yang et al. did not consider boundaries and combinations of differ-
ent types of reducts.
From ( 7.23 ), we know that (DL ≥ ) and (DU ≤ ) are equivalent. Similarly, (DL ≤ )
and (DU ≥ ) are also equivalent. Therefore, (DL ) is equivalent to (DU ). Moreover,
since condition (DL ) implies conditions (DL ≥ ) and (DL ≤ ), any L -reduct satisfies
(DL ≥ ) and also (DL ≤ ). Similarly, since condition (DU ) implies conditions (DU ≥ )
and (DU ≤ ), any U -reduct satisfies (DU ≥ ) and also (DU ≤ ). Therefore, we have the
following theorem.

Theorem 7 ([ 25 , 52 ]) Let A be a subset of C . The following statements hold.

AisaU ≥ -reduct if and only if A is an L ≤ -reduct. •

AisaU ≤ -reduct if and only if A is an L ≥ -reduct.

•

AisaU -reduct if and only if A is an L -reduct.

•

AisaB -reduct if and only if A is an L -reduct.

•

If A is an L -reduct then A satisfies ( DL ≥ ), ( DL ≤ ), ( DU ≥ ), and ( DU ≤ ).

•

As the result of the discussion, we obtain 3 different types of reducts based on
the structure induced from rough set operations on unions. They are represented by
L ≥ -reduct, L ≤ -reduct and L -reduct.
Now, we are ready to define other types of structure-based reducts, considering
approximations of decision classes. The first kind of reducts, called L-reduct, pre-
serves the lower approximations of decision classes, the second kind of reducts,
called U-reduct, preserves the upper approximations of decision classes, the third
kind of reduct, called B-reduct, preserves the boundary regions of decision classes,
and the fourth kind of reduct, called P-reduct, preserves the positive region. They are
parallel to L-, U-, B-, P-reducts discussion in the classical RSM.

Definition 17 ([ 31 ]) We define four types of reducts as follows.

• An L-reduct in DRSM is a minimal condition attribute subset A ↆ C satisfying

the following condition:

LA A ( X i ) = LA C ( X i ) for all i ∈ V d . (DL)

• A U-reduct in DRSM is a minimal condition attribute subset A ↆ C satisfying the

following condition:

UA A ( X i ) = UA C ( X i ) for all i ∈ V d . (DU)