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relation is a special case of the weak information generalization relation and
it is disjoint with our data mining generalization relation. This means that
within the framework of our general model we are able to distinguish (as we
should have) the preprocessing generalization from the generalization that
occurs in the data mining proper stage.
Definition 22.
A Weak Generalization Model
is the generalization model
(Definition 1) in which the generalization relation is
reflexive
. We denote the
generalization relation of the weak model by
and call it a
weak generalization
relation
.
Definition 23.
A Strong Generalization Model
is the generalization model
(Definition 1) in which the information generalization relation is not reflexive.
We denote the generalization relation of the strong model by
≺
and call it a
strong generalization relation
.
The relationship between weak and strong generalization relations, gen-
eralization relation (Definition 9) and data mining generalization relation
(Definition 10) and hence between Data Mining Model, Generalization Model,
and Strong and Weak Generalization Models is express by the following
theorem.
Theorem 5.
Let
≺
dm
be the generalization relation as defined by the
Definition 9 and the data mining generalization relation (Definition 10), re-
spectively. The following properties hold.
(i)
and
≺
dm
⊂
(ii)
≺
is a weak information generalization of the Definition 22
(iii)
dm
is a strong information generalization of the Definition 23
The condition (i) is true by definition, condition (ii) follows from
Definition 22 and Theorem 2, and the condition (iii) follows from Definition 23
and Theorem 3.
Given initial information system
I
0
=(
U
0
,A
0
,V
A
0
,f
0
), the object knowl-
edge generalization system (Definition 7)
K
obj
I
0
obj
(
U
0
)
,A,V
A
,g
)
=(
P
is isomorphic with
I
0
i.e.
K
obj
I
0
I
0
by theorem 1 and is also called the
initial
knowledge generalization
system.
Data preprocessing process in the preprocessing stage consists of transfor-
mations the initial knowledge generalization system
K
obj
I
0
I
0
into a certain
K
obj
I
I
0
. Any data mining stage of transformation starts, for uni-
fication purposes with corresponding initial knowledge generalization systems
K
obj
I
I
for
I
⊆
I
obtained by the preprocessing process.
Let
K
be the set of all
knowledge generalization states of
G
M
as defined
prep
in the Definition 8. We define its special subset
K
corresponding to the
preprocessing stage as follows.
