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臺大管理論叢
第
27
卷第
2S
期
39
which represents [
A
1
:[13, 30],
A
2
:[54, 78]], is 0.944.
Definition 6.
An
across-attribute extension
of a U2 pattern
Pat
is a U2 pattern that
extends
Pat
by inserting one or more intervals belonging to attributes that do not appear in
Pat
. A
within-attribute extension
of
Pat
is a U2 pattern that extends
Pat
by extending one or
more intervals belonging to attributes that are already in
Pat
. Extending an interval
I
of an
attribute
A
means adding base intervals of
A
to
I
that were not originally in
I
. (Liu and Wang,
2013)
Example 3
. Suppose
Pat
is only comprised of
BI
1
in Fig. 3. Pattern [
BI
1
,
BI
4
] is an
across-attribute extension of
Pat
, while [
BI
1
,
BI
2
] is a within-attribute extension of
Pat
. In
this study, we do not consider U2 patterns that cover disjunctive intervals. Therefore, [
BI
1
,
BI
3
] is not a within-attribute extension of
Pat
. The rationale for not considering such patterns
is that an attribute of real-world data usually occupies a continuous interval.
3.2 Distance Measure
In this section, we discuss the distance measure used in the study. Because we adopt the
clustering technique to derive the representative FU2Ps, the distance (or dissimilarity)
between two FU2Ps, i.e.,
P
A
and
P
B
, is required to be defined. The distance measure
comprises two components: the expected support part (
D
ExSup
) and the appearance part (
D
App
).
The expected support part measures the equality of the two FU2Ps' expected supports, which
is defined as follows:
(2)
In equation (2), min(
．
,
．
) returns the minimum of the two arguments, and max(
．
,
．
)
returns the maximum of the two arguments. If the expected supports of
P
A
and
P
B
are the
same,
D
ExSup
is equal to 0; in contrast, if the difference between the expected supports of
P
A
and
P
B
becomes larger,
D
ExSup
also becomes larger. The range of
D
ExSup
is between 0 and 1. We
propose
D
ExSup
because it is beneficial for checking if the FU2Ps in a cluster have similar
expected supports.
The second component of the distance measure considers the appearances of the FU2Ps.
We compare the intervals of attributes in
P
A
and
P
B
to compute
D
App
. In detail, let the number
of possible attributes be
M
,
D
j
be the partial distance between
P
A
and
P
B
on attribute
j
.
D
App
is
defined as follows: