Databases Reference
In-Depth Information
Definition 12 (Classification Rule Cover).
Let
R
be the set of frequent
sequential classification rules for
D
. The classification rule cover of
R
is the set
CRC
=
{
r
∈R|
r
:
G
→
c
∧
G
∈G}
,
G
is the set of generator sequences in
.
(1)
D
The next theorem proves that the
CRC
rule set is a sequential classifica-
tion rule cover of
R
. Hence, it is a compact representation of
R
, equivalent to
it for classification purposes.
Theorem 1.
Let
R
be the set of frequent sequential classification rules for
D
.
TherulesetCRC
⊆R
is a sequential classification rule cover of
R
.
Proof.
Consider an arbitrary rule
r
i
∈R
. By Definition 12 and Lemma 2,
there exists at least a rule
r
j
CRC
,
r
j
not necessarily identical to
r
i
,
such that
r
j
is a general rule and
r
i
is a specialization of
r
j
according to
Definition 8. Hence, it follows that the
CRC
rule set satisfies point (i) in
Definition 11. Consider now an arbitrary rule
r
j
∈
∈
CRC
. By removing
r
j
,(at
least)
r
j
itself is no longer represented in
CRC
by Definition 9. Thus,
CRC
is a minimal representation of
R
(point (ii) in Definition 11).
5.3 Compact Classification Rule Set
In this section we present a compact form to encode a classification rule set,
which, differently from the classification rule cover presented in the previ-
ous section, allows the regeneration of the original rule set
. The proposed
representation relies on the notions of both closed and generator sequences.
In the compact form, both general and specialistic rules are explicitly rep-
resented. All the remaining rules are summarized by means of an appropriate
encoding. The compact form consists of a set of elements named
compact
rules
. Each compact rule includes a specialistic rule, a set of general rules,
and encodes a set of rules that are specializations of them.
R
Definition 13 (Compact Rule).
Let M be an arbitrary closed sequence in
D
,and
G
(M) the set of its generator sequences. Let c
∈C
be an arbitrary
class label.
represents all rules
r
:
X → c
i
for D with (i) c
i
=
c and (ii) M ∈CS
(
X
)
, i.e., M belongs to the
sequential closure set of X.
F
:(
G
(
M
)
,M
)
→
c is a compact rule for
D
.
F
By Definition 13, the rule set represented in a compact rule
F
:
(
c
, which is a specialistic
rule since
M
is a closed sequence; (ii) the set of rules
r
:
G
G
(
M
)
,M
)
→
c
includes (i) the rule
r
:
M
→
→
c
that are
general rules since
G
is a generator sequence for
M
(i.e.,
G
∈G
(
M
)); and
(iii) a set of rules
r
:
X
c
that are a specialization of rules in (ii). For rules
in case (iii), the antecedent
X
is a subsequence of
M
(i.e.,
X
→
Ψ
M
)and
it completely includes at least one of the generator sequences in
G
(
M
) (i.e.,
∃
G
∈G
(
M
)
|
G
Ψ
X
).
