Information Technology Reference
In-Depth Information
Information Systems
Knowledge representation in rough sets is done via information systems, which are a tabular form of an
OBJECT→ATTRIBUTE VALUE relationship. More precisely, an information system,
Γ =<
U V f
, , ,
q
Ω
>
q
q
Ω
, where U is a finite set of objects,
U x x x
=
, and Ω is a finite set of attributes (features). The
attributes in Ω are further classified into disjoint condition attributes
A
and decision attributes
D,
A D
{ , , ,..., }
n
1
2
3
Ω =
. For each q ∈ Ω,
V
q
is a set of attribute values for
q,
each
q
f U
→
is an information func-
tion that assigns particular values from domains of attributes to objects such that
( )
q
f x
∈
for all
q
i
q
x U
∈ ∈Ω
. With respect to a given
q,
the functions partitions the universe into a set of pairwise
disjoints subsets of U:
and
.
∀ ∈
(1)
R
=
{ :
x x U f x q f x q x U
∈ ∧
( , )
=
( , ) }
0
0
Assume a subset of the set of attributes,
P
⊆
A
Two samples,
x
and
y
in
U
, are indiscernible with
respect to
P
if and only if
( , )
= ∀ ∈
The indiscernibility relation for all
P
⊆
A
is written as
IND
(
P
).
U
/
IND
(
P
) is used to denote the partition of
U
given
IND
(
P
) and is calculated as follows:
f x q f y q q P
( , )
.
U IND P
/
( )
= ⊗ ∈
{
q P U IND P q
: /
( )({ })},
(2)
(3)
A B X Y q A Y B X Y
⊗ =
{
∩
:
∀ ∈ ∀ ∈
,
,
∩ ≠
{}}.
Approximation Spaces
A rough set approximates traditional sets using a pair of sets named the lower and upper approxima-
tions of the set. The lower and upper approximations of a set
P
⊆
U,
are defined by equations (4) and
(5), respectively.
(4)
P
Y
=
{ :
X X U IND P X Y
∈
/
( ),
⊆
}
(5)
PY
=
{ :
X X U IND P X Y
∈
/
( ),
∪ ≠
{}}
Assuming
P
and
Q
are equivalence relationships in
U,
the important concept positive region
( )
POS Q
is defined as:
P
=
(6)
POS Q
( )
P
X
P
X Q
∈
A positive region contains all patterns in
U
that can be classified in attribute set
Q
using the infor-
mation in attribute set
P.
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