Discovering Calendarbased Temporal Association Rules

1
Discovering Calendar-based Temporal Association
Rules
TIME 01, 8th International Symposium on Temporal
Representation and Reasoning

• SHOU Yu Tao
• May. 21st, 2003

2
Outline of the Presentation

• Background
• Temporal Association Rule Mining w.r.t Precise
  Match
• Temporal Association Rule Mining w.r.t Fuzzy
  Match
• Experiments
• Conclusions
• References
• Q A

3
Background

• Temporal Association RuleAssociation rules
  along with their temporal intervals
• E.g. turkey? pumpkin pie is a temporal
  association rule along with the temporal interval
  within the week before thanksgiving.
• Why interested in temporal association rule
  mining?
• We may discover different association rules
  regarding different time intervals. Some
  association rules may hold during some intervals
  but not during others.
• ? this may lead to useful information.

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Calendar Schema

• Relational schema R (fnDn, fn-1Dn-1, .
  f1D1) together with a valid constraint
• fi a calendar unit name like year, month, etc.
• Di a finite subset of the positive integers.
• A constraint valid a boolean function on DnD1
  specifying which combinations of the values in
  DnD1 are valid.

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Calendar Schema

• E.g.a calendar schema (year1995,1996,..2002,
  month1,2,..,12, day1,2,..,31) with the
  constraint valid that evaluates to True
  only if the combination gives a valid date.
• e.g., is not valid
• Simply stated, a calendar schema is determined by
  a hierarchy of calendar concept
• e.g. (year, month, day)

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Calendar Pattern

• Defines a set of time intervals based on the
  calendar schema
• e.g is a calendar pattern based on
  the calendar schema
  corresponding to the time intervals consisting of
  all the 16th days of all months in year 2000
• Time intervals or periodic cycles can be easily
  described by calendar patterns with appropriate
  calendar schemas.
• E.g. the periodic cycle every seven days can
  be expressed by a calendar pattern , where
  1
  day) depending on which day the cycle starts

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Problem Formulation

• Given a calendar schema R, a set T of timestamped
  transactions and a match ratio (optional), we
  want to discover all interesting association
  rules w.r.t.
• Precise Match
• Fuzzy Match
• Assumption we are not interested in the
  association rules that only hold during basic
  time intervals. Indeed, such rules do not reveal
  much interesting information in terms of time.
• E.g. if the calendar schema is (year, month,
  day), we are not going to find the association
  rules hold during each single day.
• -- Basic time interval a calendar pattern with
  no wild-card symbol

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Problem Formulation

• Temporal Association Rule w.r.t. Precise Match
• Given a calendar schema R and a set T of
  timestamped transactions, a temporal association
  rule (r,e) hold if and only if the association
  rule r holds for each basic time interval t
  covered by star calendar pattern e.
• -- Star calendar pattern a calendar pattern with
  at least one wild-card symbol
• E.g., given the calendar schema (year, month,
  Thursday), we may have a temporal association
  rule (turkey?pumpkin pie, ) that holds
  w.r.t precise match. The rule means that the
  association rule (turkey?pumpkin pie) holds on
  all Thanksgiving days, which is the 4th Thursday
  in November of every year.

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Problem Formulation

• Temporal Association Rule w.r.t. Fuzzy Match
• Given a calendar schema R, a set T of timestamped
  transactions and a match ratio m, a temporal
  association rule (r,e) hold if and only if the
  association rule r holds for at least 100m of
  basic time interval t covered by star calendar
  pattern e.
• E.g., given the calendar schema (year, month,
  Thursday) and match ratio m0.8, we may have a
  temporal association rule (turkey?pumpkin pie,
  ) that holds w.r.t fuzzy match. This
  means that the association rule (turkey?pumpkin
  pie) holds on at least 80 of Thanksgiving days.

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Temporal Association Rule Mining

• Two sub-problems
• Finding all large itemsets for all the star
  calendar patterns on the given calendar schema
  (based on Apriori AS94) crux of the discovery
  of temporal association rules.
• Generating temporal association rules using the
  large itemsets and their calendar patterns the
  same as traditional association rule generation
  approach AS94.

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Outline of the Algorithm (for both precise and
fuzzy match)
critical step! Because fewer candidate large
itemsets, less time for phase II needed.
The same as traditional approach
The same here
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Phase III for Precise Match

• After the basic time interval e0 is processed in
  pass k, the large k-itemsets for all the calendar
  patterns e that covers e0 are updated as follows,
• If Lk(e) is updated for the first time
  (i.g.,Lk(e) NULL), let Lk(e)Lk(e0)
• Else Lk(e) Lk(e) Lk(e0)
• E.g.
• given calendar patterns (1995, , 1) and (,2,)
  and L2(1995, , 1) AB, DE and L2(,2,)
  AB, BC, DE. suppose after processing basic time
  interval (1995,2,1), we get L2 AC, BC, DE
• ? L2(1995, , 1) DE
• ? L2(,2,) BC, DE
• So after all the basic time intervals are
  processed, the set of large k-itemsets for each
  calendar pattern could be discovered.

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Phase III for Fuzzy Match

• Associate a counter c_update with each candidate
  for each star calendar pattern.
• Counters are initially set to 1
• When Lk(e0) is used to update Lk(e) in phase III,
  the counters of the itemsets in Lk(e) that are
  also in Lk(e0) are increment by 1
• Suppose there are totally N basic time intervals
  covered by e and this is the nth update to Lk(e),
  an itemset cannot be large for e if its counter
  c_update does not satisfy c_update (N-n) mN

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Phase III for Fuzzy Match

• Example
• Calendar schema R (week, day)
• fuzzy match ratio m 0.8
• Consider calendar pattern , suppose there
  are only 5 basic time intervals covered. (N5)
• This is the 3rd time that L2() is updated
  (n3)
• So we only keep the itemsets with c_update mN
  (N-n) 2

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Candidate Generation (Phase I)

• Direct-Apriori A naïve approach to generate
  candidate itemsets is to treat each basic time
  interval individually and directly apply
  Aprioris candidate generation approach.
• For both precise match and fuzzy match

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Candidate Generation for Precise MatchTemporal
AprioriGen

• Since we are not interested in the large itemsets
  for basic time intervals, if a Ck(e0) cannot be
  large for any of the star calendar patterns that
  cover the basic time interval e0, simply ignore
  it.
• So, we can generate the candidate Ck (k1) as
  follows

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Candidate Generation for Precise MatchTemporal
AprioriGen

• Example
• Consider the calendar schema R
  (week1,..,5, day1,..,7). Suppose we already
  have L2()AB,AC,AD,AE,BC,BD,CD,CE
  L2() AB,AC,AD,BC,BD,CE
  L2()AB,AC,AD,BD,CD.
• By using temporal aprioriGen C3()ABC,ABD
  C3()ABD,ACD
• C3()C3() U C3()ABC,ABD,ACD
• B y using Direct-Apriori,
• C3() ABC,ABD,ACD,ACE,BCD

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Candidate Generation for Precise
MatchHorizontal Pruning

• If an itemset l in Ck(e0) does not appear in any
  of the tentative Lk(e1), where e1 is a 1-star
  pattern that covers e0, then l cannot be large
  for any star pattern e that covers e0.
  Therefore, we drop l from Ck(e0)

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Candidate Generation for Precise
MatchHorizontal Pruning

• Example
• suppose when the
  basic time interval is being processed, we
  already have
• L3()ABD
• L3()ABD,ACD.
• we get C3() ABC,ABD,ACD after using
  temporal aprioriGen, we can further prune it by
• C3()C3() (L3() U L3())
• ABD, ACD

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Candidate Generation for Fuzzy Match Temporal
AprioriGen
Temporal AprioriGen for precise match cannot be
directly applied to solve the fuzzy match
problem, because an itemset may be large for a
star calendar e even if it is not large for any
1-star pattern covered by e.
For example Consider a schema R (week, day)
and fuzzy ratio m 0.8. We can see and
is large and is not large
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Candidate Generation for Fuzzy Match Temporal
AprioriGen

• Change the temporal aprioriGen to apply to fuzzy
  match as follows,
• Change blue underline part to Lk-1(e) when memory
  is the critical resource.

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Candidate Generation for Fuzzy Match Temporal
AprioriGen

• Example
• Suppose we already have
• L2() AB,AC,AD,AE,BD,CD,CE
• L2() AB,AC,AD,BC,BD,CE
• L2() AB,AC,AD,BD,CD
• L2() AB,AD,BD,CD,AC,AE
• LT L2() L2() AB,AC,AD,BD,CE
• C3() aprioriGen(LT) ABD
• Similarly, we can get C3()ABD,ACD and
  C3()ABD,ACE
• ?C3() C3() U C3() U C3()
  ABD,ACD,ACE

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Candidate Generation for Fuzzy MatchHorizontal
Pruning

• The pruning idea is to discard the candidate
  itemsets that cannot be large for calendar
  pattern e even if they are large for basic time
  interval e0.

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Candidate Generation for Fuzzy MatchHorizontal
Pruning

• Example
• Suppose we already have
• C3() ABD,ACD,ACE
• L3(), L3(), L3() have been
  updated once.
• C3()ABD,ABE C3()ABD,ACDC3()ABD
• then C3() can be pruned as
• C3() C3() (C3()UC3()UC3(,))
• ABD,ACD

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Experiments

• Real Data set
• Data file consists of homepage request records,
  each of which contains attribute values
  describing the request and the person who sent
  the request.
• Data file records are from Jan 30 to Mar 31,2000
• Calendar schema used R week, day, timeofday,
  where timeofday contains
  (0am-8am), daytime (8am-4pm), evening (4pm-12pm)
• Data set contains 777,480 transactions, 23.4
  items per transaction on average.

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Experiments

• Real data set result

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Experiments

• Synthetic Data Set
• Extend the data generator propose in AS94 to
  incorporate temporal features.

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Experiments

• Synthetic Data Set Result

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Conclusions

• Develops a new representation mechanism for
  temporal association rules on the basis of
  calendars and identify two classes of interesting
  temporal association rules w.r.t. precise match
  and fuzzy match.
• The representation requires less prior knowledge
  and resulting time intervals are easier to
  understand
• Extend the algorithm Apriori and develop two
  optimization techniques to discover both classes
  of temporal association rules
• Experiments show that the optimization techniques
  are effective

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Possible Future Works

• It requires for a calendar schema (fn,fn-1,,f1),
  each calendar unit of fi is uniquely contained in
  a unit of fi1, where 0
• E.g., (year, month, week) is NOT allowed because
  a week may not be contained in a unique month
• Consider temporal patterns in other data mining
  problems such as clustering, etc.

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References

• Y. Li, P. Ning, X. S. Wang, and S. Jajodia.
  Discovering calendar-based temporal association
  rules. In the Eighth International Symposium on
  Temporal Representation and Reasoning (TIME 01)
• AS94 R. Agrawal and R. Srikant. Fast algorithms
  for mining association rules in large databases.
  VLDB 94
• S. Ramaswamy, S. Mahajan and A. Silberschatz. On
  the discovery of interesting patterns in
  association rules. VLDB98

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Questions and Answers

• Any Questions?

?