Databases Reference
In-Depth Information
Definition 3.
Given two information systems I
0
=(
U
0
,A
0
,V
A
0
,f
0
)
and I
=
(
U,A,V
A
,f
)
, we say that I
is a subsystem of
I
0
and denote it as
I
⊆
I
0
if and only if the following conditions are satisfied
(i)
|
U
|
=
|
U
0
|
(ii) A
V
0
(iii) The information functions f and f
0
are such that
⊆
A
0
, V
A
∼
v
a
∈
V
0
(
f
0
(
x,a
)=
v
a
∩
v
a
∼
∀
x
∈
U
∀
a
∈
A
(
f
(
x,a
)=
v
a
⇔∃
v
a
))
In the data analysis, preprocessing and data mining we start the process
with the input data. We assume here that they are represented in a format of
information system table. We hence define the lowest level of information gen-
eralization as the relational table. The meaning of the intermediate and final
results are considered to be of a higher level of generalization. We represent
those levels of generalization by a sets of objects of the given (data mining)
universe
U
, as in [1, 7].
This approach follows the granular view of the data mining and is formal-
ized within a notion of knowledge generalization system, defined as follows.
Definition 4.
A knowledge generalization system
based on the information
system I
=(
U,A,V
A
,f
)
is a system
K
I
=(
P
(
U
)
,A,E,V
A
,V
E
,g
)
where
E is a finite set of
knowledge attributes
(k-attributes) such that A
∩
E
=
∅
.
V
E
is a finite set of
values of k- attributes
.
g is a partial function called
knowledge information function
(k-function)
g
:
P
(
U
)
×
(
A
∪
E
)
−→
(
V
A
∪
V
E
)
such that
(i) g
(
x∈U
{
|
x
}×
A
)=
f
(ii)
∀
S∈P
(
U
)
∀
a∈A
((
S,a
)
∈
dom
(
g
)
⇒
g
(
S,a
)
∈
V
A
)
(iii)
∀
S∈P
(
U
)
∀
e∈E
((
S,e
)
∈
dom
(
g
)
⇒
g
(
S,e
)
∈
V
E
)
Any set
S
∈P
(
U
) i.e.
S
⊆
U
is often called a granule or a group of objects.
Definition 5.
The set
Gr
K
=
{
S
∈P
(
U
):
∃
b
∈
(
E
∪
A
)((
S,b
)
∈
dom
(
g
))
}
is called
a granule universe
of K
I
.
Observe that
g
is a total function on
Gr
K
.
