Databases Reference
In-Depth Information
Definition 6.
We call the system K
=(
Gr
K
,E,V
E
,g
)
a
granule knowledge
generalization
system.
The condition (i) of Definition 4 says that when
E
=
∅
the k-function
g
is
total on the set
{{x}
:
x ∈ U}×A
and
∀
x
∈
U
∀
a
∈
A
(
g
(
{
x
}
,a
)=
f
(
x,a
))
.
Definition 7.
The set
obj
(
U
)=
P
{{
x
}
:
x
∈
U
}
is called an object universe. The knowledge generalization system
K
obj
=(
obj
(
U
)
,A,
obj
(
U
)
,A,V
A
,g
)
P
∅
,V
A
,
∅
,g
)=(
P
is called an
object knowledge generalization
system.
Theorem 1.
For any information system I
=(
U,A,V
A
,f
)
, the object knowl-
edge generalization system K
obj
I
=(
P
obj
(
U
)
,A,V
A
,g
)
is isomorphic with I.
We denote it by
I K
ob
I
.
obj
(
U
)
,F
(
x
)=
establishes (by condition
(i) of Definition 4) the required isomorphism of
K
obj
I
The function
F
:
U
−→ P
{
x
}
and
I
.
2.2 Universe and Knowledge Generalization States
Any Data Mining process starts with a certain initial set of data. The model
of such a process depends on representation of this data and we represent it
in a form information system table.
We assume hence that the data mining process we model starts with an
initial information system
I
0
=(
U
0
,A
0
,V
A
0
,f
0
)
and we adopt the
universe U
0
as the universe of the model
, i.e.
G
M
=(
U
0
,
K
,
G
,
)
.
Data Mining process consists of transformations the initial
I
0
into an ini-
tial knowledge generalizations systems
K
0
that in turn is being transformed
into some knowledge generalizations systems
K
I
, all of them based on some
subsystems
I
of the input system
I
0
,whatwedenoteby
I
I
0
. The formal
definition of the notion of subsystem
I
of the input system
I
0
is presented
in [13]. These transformations of the initial input data (system
I
0
) in practice
are defined by different Data Preprocessing and Data Mining algorithms, and
in our model by appropriate generalization operators. We hence adopt the
following definition.
⊆
