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為頻繁單變量不確定樣式產生摘要
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pair of consecutive bound values in each attribute (Liu, 2012).
Table 4 The Base Intervals
Attribute
Base Interval
A
1
{
BI
1
, [13, 15]} {
BI
2
, [15, 30]} {
BI
3
, [30, 31]}
A
2
{
BI
4
, [52, 54]} {
BI
5
, [54, 78]}
Example 1
. In Table 1, the bound values of
A
1
are 13, 15, 30, and 31; and they form
three base intervals: [13, 15], [15, 30], and [30, 31]. The bound values of
A
2
are 52, 54, and
78; and they form three base intervals: [52, 54], and [54, 78]. Table 4 presents the base
intervals of both attributes, where a serial number is given to each base interval.
Definition 3
. Suppose an attribute
A
in a transaction
T
is associated with a quantitative
interval
I
A
and a probability density function
P
A
. The
existential probability
of an interval
I
AS
ϵ
I
A
is the possibility that the values in
I
AS
appear in
T
. This is defined as the integral of the
density over
I
AS
, denoted by
ExProb(I
AS
, T)
. (Liu, 2012)
Definition 4
. A quantitative interval
I
of attribute
A
i
with a range from
m
to
n
is
represented by
A
i
:[
m
,
n
]. A
U2 pattern
comprises one or more non-repeated attributes, each
of which is associated with an interval; and a U2 pattern comprised of interval
I
1
,
I
2
, …,
I
j
is
denoted as [
I
1
,
I
2
, …,
I
j
]. A U2 pattern is
frequent
if its expected support (see Definition 5)
exceeds the minimum support specified by the user. Thus, a
frequent U2 pattern
represents
the intervals where the actual values locate with high probability. (Liu, 2012)
Definition 5
. For a given U2 pattern
Pat
, the
expected support
, i.e., the expected
number of transactions that contain
Pat
in the database, is denoted as
ExSupport(Pat)
. Let
T
Pat
be the set of transactions containing
Pat
, and let an interval
x
of a transaction
TRA
i
ϵ
T
Pat
correspond to one of the intervals in
Pat
. Thus,
(1)
where |
T
Pat
| denotes the number of transactions in
T
Pat
. (Liu, 2012)
Example 2
. In Table 1, let the probability density function of each interval be a uniform
distribution. The existential probability of U2 pattern [
BI
1
], which represents [
A
1
:[13, 15]], in
T
2
is 0.111 ((15 – 13)/(31 – 13)). The expected support of U2 pattern [
BI
1
] is also 0.111
because
T
1
and
T
2
do not contain [
BI
1
]; the expected support of U2 pattern [
BI
1
,
BI
2
,
BI
5
],