Databases Reference
In-Depth Information
With respect to the definition of specialistic rule proposed in [11, 19, 30],
our definition is more restrictive. In particular, both rules are required to have
the same confidence, support and class label, similarly to [7] in the context of
associative classification.
Based on Definition 8, we now introduce the concept of general rule. This
is the rule with the shortest antecedent, among all rules having same class
label, support and confidence.
Definition 9 (General Rule).
Let
R
be the set of frequent sequential clas-
sification rules for
D
,andr
i
∈R
an arbitrary rule. r
i
is a general rule in
R
iff
r
j
∈R
, such that r
i
is a specialization of r
j
.
In the example dataset,
BA → c
2
is a contiguous general rule with respect
to the rules
DBA
c
2
. The next lemma formalizes the
concept of general rule by means of the concept of generator sequence.
→
c
2
and
ADBA
→
Lemma 2 (General Rule).
Let
R
be the set of frequent sequential classifi-
cation rules for
D
,andr
∈R
, r
:
X
→
c, an arbitrary rule. r is a general
rule in
R
iff X is a generator sequence in
D
.
Proof.
We first prove the su
cient condition. Let
r
i
:
X
→
c
be an arbitrary
rule in
,where
X
is a generator sequence. By Definition 7, if
X
is a generator
sequence then
R
Ψ
X
it is
sup
Φ
(
Y
)
>sup
Φ
(
X
). Thus,
r
i
is a general rule according to Definition 9. We now prove the necessary
condition. Let
r
i
:
X
∀
r
j
:
Y
→
c
in
R
with
Y
. For the sake of
contradiction, let
X
not be a generator sequence. It follows that
→
c
be an arbitrary general rule in
R
∃
r
j
:
Y
→
c
in
R
, with
Y
Ψ
X
and
sup
Φ
(
X
)=
sup
Φ
(
Y
). Hence, from Property 2,
{
, and thus
sup
Φ
(
r
i
)=
sup
Φ
(
r
j
). It follows that
r
i
is not a general rule according to
Definition 9, a contradiction.
(
SID,S,c
)
∈D|
Y
Φ
S
}
=
{
(
SID,S,c
)
∈D|
X
Φ
S
}
, we can identify some par-
ticular rules which are not specializations of any other rules in
By applying iteratively Definition 8 in set
R
. These are
the rules with the longest antecedent, among all rules having same class label,
support and confidence. We name these rules
specialistic rules
.
R
Definition 10 (Specialistic Rule).
Let
R
be an arbitrary set of frequent
sequential classification rules for
D
,andr
i
∈R
an arbitrary rule. r
i
is a
specialistic rule in
R
iff
r
j
∈R
such that r
j
is a specialization of r
i
.
c
2
is a contiguous specialistic rule in the example
dataset, with support 33
.
33% and confidence 50%. The contiguous rules
ACBA
For example,
B
→
→
c
2
and
ADCBA
→
c
2
which include it have support equal to
33
.
33% and confidence 100%.
The next lemma formalizes the concept of specialistic rule by means of the
concept of closed sequence.
