1. On the Discovery of Interesting Patterns in Association Rules
1998년 11월 9일 월요일
DE Lab. 하 미 라

2. Abstract complicated temporal pattern calendar algebra
ex. “first working day of every month”
calendar algebra
calendric association rules

3. Introduction 기존의 association rules cyclic association rules
with no attention paid to segmenting the data over different time intervals
cyclic association rules
simple periodicity
fairy rigid
did not deal with multiple units of time
calendric association rules
complicated pattern  calendar algebra
for “most” but not all the time  mismatch threshold
can handle multiple units of time

4. Problem Definition tj : the jth time unit, j  0
time interval [j • t, (j+1) • t], where t is the unit of time
T[j] : the set of transactions executed in tj
support of an itemset X in T[j]
is the fraction of transactions in T[j] that contain the itemset
confidence of a rule X  Y in T[j]
is the fraction of transactions in T[j] containing X that also contain Y.
An association rule XY holds in time unit tj
if the support of XY in T[j] exceeds supmin and the confidence of XY in T[j] exceeds conmin

5. mis-match threshold : m belong to an association rule XY
calendar C :
set of time intervals {(s1, e1), (s2, e2), … , (sk, ek)}
contain time unit t :
if it contains an interval (sj, ej) such that sj  t  ej
mis-match threshold : m
belong to an association rule XY
if the rule has enough confidence and support for the time units contained in the calendar with at most m mis-matches
belong to an itemset X
if the support of X exceeds supmin in at least w-m time units

6. Example
C = {(31,31), (60,60), (89,89), (121,121), (152,152), (180,180), (213,213), (243,243), (274,274), (305,305), (334,334), (366, 366)}
mismatch threshold : 0
the rule has enough support and confidence on days 31, 60, 89, 121, 152, 180, 213, 243, 274, 305, 334, 366 of year 1996
mismatch threshold : 4
the rule has to hold for at least 12-4 = 8 of the 12 days

7. An association rule can be represented as binary sequence
1 : the time unit in which the rule holds
0 : the time unit in which the rule does not have the minimum support and confidence
ex :
X  Y holds in T[3], T[4], T[8], T[10], T[12]
the calendar {(4,4), (8,8), (12,12)}, which corresponds to the a cycle of length 4, belongs to the rule

8. Calendar Algebra Calendar : a structured collection of intervals
calendar of order 1 : {(s1 , e1), (s2 , e2), … ,(sk , ek)}
calendar of order 2 : a collection of calendars of order 1
and so on...
interval operators (int1 = (s1, e1), int2 = (s2 , e2))
int1 overlaps int2  (s1  s2  e1)  (s2  s1  e2)
int1 during int2  (s1  s2)  (e1  e2)
int1 meets int2  (e1 = s2)
int1  int2  (e1  s2)
int1  int2  (s1  s2)  (e1  e2)

9. Dicing Operations
left : order 1, right : interval  result : order 1
C:R:c’  {c  c’ | c  C  c R c’}/{}
C.R.c’  {c | c  C  c R c’}/{}
left : order 1, right : order 1  result : order 2
C:R:C’  {{c  c’ | c  C  c R c’}/{} | c’  C’}
C.R.C’  {{c | c  C  c R c’}/{} | c’  C’}
ex. WeeksInJan96 = {(-3,4), (5,11), (12,18), (19,25), (26,32)} JanIn1996 = {(1, 31)}
WeeksInJan96 : overlaps : JanIn1996 = {(1,4), (5,11), (12,18), (19,25), (26,31)}
WeeksInJan96.overlaps.JanIn1996 = {(-3,4), (5,11), (12,18), (19,25), (26,32)}

10. Slicing Operations C : calendar, p : integer
(p) / C : replaces each order 1 calendar in C with its pth element
[p] / C : replaces every order 1 calendar in C with a calendar consisting of the pth element
if p is negative, indexing is done from the end of the calendar (ex. (-1) / C)
list of integers
[p1, p2, …, pk] / C : replaces each order 1 calendar in C with a calendar consisting of the element
(p1, p2, …, pk) / C : replaces each order 1 calendar in C with the element

11. ex. WeeksInJan96 = {(-3,4), (5,11), (12,18), (19,25), (26,32)}
flatten operator :
takes an order k calendar
produces an order k-1 calendar
flatten {{(-3,4), (5,11), (12, 18), (19,25), (26,32)}} = {(-3,4), (5,11), (12, 18), (19,25), (26,32)}
flatten {{(1,1)}, {(5,5)}} = {(1,1), (5,5)}

12. The Sequential Algorithm
generate the rules in each time unit with one of the existing methods
and then apply a pattern matching algorithm to discover calendars

13. Pruning, Skipping, Elimination
to reduce the number of itemsets for which support must be caculated
“A calendar that belongs to the rule X  Y also belongs to the itemset XY. ”
skipping
“If time unit tj is not contained in any calendar that belongs to an itemset X, then there is no need to calculate the support for X in time segment T[j].”

14. pruning
Lemma 1 : If a calendar C belongs to an itemset X then it must also belong to all of X’s subsets
for computing the potential calendars of an itemset by merging the calendars of the itemset’s subsets
ex. If we know that the calendar {(4,4), (8,8), (12,12)} is the only calendar that belongs to items A and B,
prunning : the only calendar that can belong to AB is also {(4,4), (8,8), (12,12)}
skipping : we have to calculate the support of AB only in T[4], T[8], and T[12] rather than all the segments

15. elimination
to eliminate certain calendars from further consideration once we have determined then cannot exist.
Lemma 2 : If the support for an itemset X is below the minimum support threshold supmin in m time units contained in a calendar C, where m is the mis-match threshold, then C cannot belong to X.
ex. mis-match threshold : 0, itemset X does not have enough support in T[4]
the calendar {(4,4), (8,8), (12,12)} cannot belong to X

16. The Interleaved Algorithm
First phase :
the calendars belonging to large itemsets are dicovered
using prunning, skipping, elimination
Second phase :
calendric association rules are generated
using the calendars and support of the itemsets without scanning the database
ex.
C = {(4,4), (8,8), (12,12)}
item A :
item B :

17. 2. Time segments are processed sequentially. For each time unit tj :
For each k, k  1 :
1. If k = 1, then all possible calendars are initially assumed to exist for each single itemset. Otherwise(if k > 1), pruning is applied to generate the potential calendars for k-itemsets using the calendars for (k-1)- itemsets. If the list of potential calendars for each k- itemsets is empty, Phase One terminates.
2. Time segments are processed sequentially. For each time unit tj :
2.1 skipping determines, from the set of candidate calendars for k-itemsets, the set of k-itemsets for which support will be calculated in time segment T[j].
2.2 If a k-itemset X chosen in Step 2.1 does not have the minimum support in time segment T[j], then the mis-match count is incremented by 1 for each potential calendar associated with X that contains tj. If this mis-match threshold for a particular calendar, that calendar is eliminated from the list of potential calendars for X.
Fig. 1 : Phase One of the interleaved algorithm