Information Technology Reference
In-Depth Information
It is obvious that, when β=0, VPRS models are transformed to traditional
rough set models, so classical rough sets are special cases of VPRS. Moreover,
VPRS is a directly extension of rough set, so it inherits the properties and
advantages of rough sets. Consequently, it extends the application fields of rough
set theory.
11.5.2
Similarity Based Model
In the case that there are lost attribute values in data set (the cases are very
popular in databases), indiscernibility relations or equivalence relations cannot
deal with these cases. To extend the ability of rough set, a lot of researchers use
similarity relations but not indiscernibility relations as the foundation of rough
set theory.
After replacing indiscernibility relations with similarity relations in rough set
theory, the generated similar classes are not a partition of a set, because they are
overlapped. Similar with equivalence classes, we can define a set SIM
b
(
) that
includes all elements that are similar with
x
in attribute set B. Note, the elements
in SIM
b
(
x
) may belong to different decision classes, so the definition of similar
decision classes is needed. Similar decision classes are the decision classes
corresponding to similar sets.
Because the elements in a similar set may not belong to identical decision
class, thus the relative-absorption term set is defined. Subset
x
is called as
relative-absorption term set, if for each
x
∈
U
, there exists
y
∈
Y
that is similar with
x
Y
⊆
U
. Obviously, relative-absorption term set
can be used to reduce data. With the help of relative-absorption term set, positive
region can be easily defined. Positive region is the union of similar sets included
by decision classes. Dependency degrees and reducts can be defined similarly
with classical sets.
In practice, similarity models have better performances than traditional rough
set models. In the case that there is lost data in databases, a simple similar
relation can be defined as follows (symbol “? ” represents that the value is
unknown or unconcerned.):
τ
C
(
and has the same decision value with
x
x
,
y
)={
x
∈U,
y
∈U|∀a∈C, a(
x
)=a(
y
) or a(
x
)=? or a(
y
)=?}
