Information Technology Reference
In-Depth Information
11.5.1
Variable Precision Rough Set Model
In original rough set model, universe U is assumed to be known, and the resulting
conclusions are only tenable for the objects in U. However, in real applications,
the assumptions are very difficult to be satisfied. To solve the problem, some
new methods are needed to be developed, so that conclusions can be made from
fewer objects and used to more applications. Besides, the conclusions are hoped
to be tenable only for known objects, and for the whole universe, the conclusions
should be viewed as uncertain or fuzzy. Based on the idea, W.Ziarko proposed an
extended rough set model called as variable precision rough set model, which
provides a classification strategy whose error rate is less a given value. Under the
given error rate, new definitions of positive region, boundary region and negative
region are presented, and related properties are discussed. Furthermore, J.D.
Katzberg and W. Ziarko developed a VPRS model with asymmetry boundaries.
The work generalized VPRS models and extended the application fields of VPRS.
In the following, we will introduce the main idea of VPRS in brief.
In general, the fact that set
X
is included by
Y
does not give a degree
measuring how much
X
belongs to
Y
. Considering this, VPRS define the degree:
C(
)>0
C(
X, Y
)=0 if
card
(
x
)=0
X, Y
)=1-
card
(
X
∩
Y
)/
card
(
X
), if
card
(
x
C
(
X, Y
) is the rate that set
X
is falsely classified into set
Y
, that is, there are
C
)×100% objects that are classified with mistake. Given a false classifying
rate β(0≤β<0.5), according to above definition, we have
(
X, Y
X
⊆
β
Y
, if and only if
C
(
X,
Y
)≤β.
Based on these, suppose
U
is a universe,
R
is an equivalence relation on
U
,
and
U/R
=
A
={
X
1
, X
2
, ···, X
k
}. Thus, the β-lower approximate set of definable set
X
is defined as
⊆
β
R
β
X
=∪
X
i
(
X
X
,
i
=1, 2, ···,
k
)
i
or
R
β
X
=∪
X
i
(
C
(
X
i
, X
)≤β,
i
=1, 2, ···,
k
).
Here,
R
β
X
is called as the β-positive region of
X
. The β-lower approximate set of
R
β
X
=∪
X
i
(
C
(
X
i
, X
)<1-β,
i
=1, 2, ···,
k
).
Thus, the β-boundary region is
X
is defined as
BNR
β
X
=∪
X
i
(β<
C
(
X
i
, X
)<1-β).
β-negative region is
NEGR
β
X
=∪
X
i
(
C
(
X
i
, X
)≥1-β).
Furthermore, we can define
β-dependency degrees,
β-reducts and many
concepts related with traditional rough set models.
