Information Technology Reference
In-Depth Information
Suppose set X
⊆
U, we call R
*
(X)={x:(x
∈
U)
∧
(
m
Definition 11.5
(x)=1)} as
X
lower approximation, R*(X)={x:(x
∈
U)
∧
(
m
(x)>0)} as upper approximation,
X
and BNR(X)=R*(X)-R
*
(X) as boundary.
Apparently, if
)≠
∅
or R
*
(
is a rough concept.
Lower approximation includes all the elements that can be assigned to set
BN
R
(
X
X
)≠R
*
(
X
), then set
X
X
explicitly by using knowledge
, whereas upper approximation includes all the
elements that are possible members of set
R
. The difference between upper and
lower approximation composes the boundary area of the concept.
We can see that a relationship between fuzziness and uncertainty can be
established, that is, fuzziness is defined by uncertainty. In general, not all object
sets can be represented as concepts by given knowledge (the so called rough
concept, inaccurate or similar concepts) for a certain approximation space.
However, a rough concept can be roughly defined by two concepts (upper
approximation and lower approximation) which enables us accurately describe
inaccurate concepts. Moreover, the membership function, upper approximate and
lower approximate can be used to define the inclusion relation and equivalence
relation, etc.
X
11.2 Reduction of Knowledge
This section focus on two basic questions in rough set theory: reduction and
dependency of knowledge. Reduction of knowledge is to investigate the
necessary of every equivalence relation in the approximation space and the
method to delete unnecessary knowledge. Knowledge reduction is of vital
practical importance in information system analysis and data mining.
Dependency among knowledge determines whether knowledge reduction is
needed and knowledge dependency provides important heuristic information for
knowledge reduction.
11.2.1
General Reduction
In the application of rough set theory, reduction and core are two of the most
important basic concepts. Intuitively, the so-called reduction of knowledge is the
essence of knowledge which is enough to define all basic concepts within the
concerned knowledge, whereas the core is its most important component.
