1. TEMPORAL ASSOCIATION RULE MINING
Prepared by : Ajit Padukone,
Komal Kapoor

2. Outline Association Rule Mining Applications
Temporal Association Rule Mining
Existing Techniques and their Limitations
Problem Statement
Proposed Approach
Finding Maximal Valid Time Intervals
Finding All Temporally Frequent Itemset
Future Work

3. Motivation Association Rule Mining {onion, potatoes} => {burgers}
{bread, milk} => {butter}
Transaction Data
Frequent itemsets : {onion,potatoes,burgers}, {bread,milk,butter}
Transaction ID
Items 1
bread, milk, butter, cheese, chips 2
onion, capsicum, potatoes, burgers 3
bread, milk, yogurt, butter 4
onion, potatoes, ketchup, burgers 5
soap, shampoo, comb, toothbrush
method for discovering interesting relations between variables in large databases, large scale transaction data supermarkets. The supermarket industry was extremely interested in analyzing the shopping behaviors of customers to find out patterns promotionalpricing or product placements. 

4. Applications Retail Data Analysis Web Usage Mining Intrusion Detection
Bioinformatics

5. Spatial Association Rule Mining
Extract spatial predicates
Find all frequent patterns/predicates/sets
Generate strong rules
E.g. {Contains(Port),crosses(WaterBody)}
Source : Vania Borgony, Enhancing Spatial Association Rule Mining in Geographic Databases, lume.ufrgs.br
Spatial predicates represent materialized spatial relationships between geographic entities such as close, far, contains, within, touches etc.

6. Temporal Association Rule Mining
Chapter 10 of the reference book defines two types of temporal references:
Transaction Time
Valid Time
Time attribute for association rules can also be defined in an analogous way.

7. Existing Technique – Apriori Algorithm
Apriori Algorithm finds the frequent item sets in a set of transaction which satisfy the minimum support threshold.
Support of the item set is defined as the proportion of transactions in the data set which contain the itemset.
Algorithm:
Find all k-itemsets that have transaction support above minimum support (frequent k-itemsets)
Generate candidate k+1-itemsets using large k-itemsets
Prune the candidate k+1-itemsets to obtain frequent k+1-itemsets which have a transaction support above minimum support
If size(frequent k+1-itemsets) > 0, Repeat

8. Apriori Algorithm (contd.)
Universal Set of Items = { A, B, C, D, E, F, G }
Minimum support = 30 % (3 transactions)
Item Set
Count
{ A,B } 4
{ A,C } 2
{ A,E } 1
{ A,F }
{ A,G } 3
{ B,C }
{ B,E }
{ B,F }
{ B,G } 5
{ C,E }
{ C,F }
{ C,G }
{ E,F }
{ E,G }
{ F,G }
Transaction
Items 1
A, B, C 2
B, C, F 3
B, F, G 4
A, C, D, F 5
C, D, E, G 6
A, B, E, G 7
B, C, F, G 8
A, B, G 9
A, B, F, G 10
C, F,G
Item
Count
{ A } 5
{ B } 7
{ C } 6
{ D } 2
{ E }
{ F }
{ G }
Item Set
Count
{ A,B,G } 4
{ B,F,G } 3
{ C,F,G } 2
{ B,C,F }
{ B,C,G } 1
Step 3: 3 – itemsets.
All 3 itemsets with non-frequent 2-item sets as subsets have been pruned. Non-struck out ones are frequent.
Step 1: 1 – itemsets.
Non-struck out ones are frequent.
Table 1: Transaction Database
Step 2: 2 – itemsets.
All 2 itemsets with { D } or { E } as one of the subsets are pruned. Non-struck out ones are frequent.

9. Limitation
The Apriori Algorithm finds the frequent itemsets in the transaction database which satisfy the minimum support threshold for the entire transaction database.
What about those itemsets which are highly frequent over a limited period of time and not over the entire set of transactions?
For e.g. – Turkey-> Pumpkin Pie (Halloween)
The itemsets extracted using the Apriori Algorithm, might not be valid for the entire period over which association rule mining has been performed.

10. Related Work
X. Chen and I. Petrounias, Mining Temporal Features in Association Rules, Proc. Third European Conf. Principles and Practice of Knowledge Discovery in Databases (PKDD '99).
Yingjiu Li, Peng Ning, X. Sean Wang, Sushil Jajodia, Discovering Calendar-based Temporal Association Rules , journal Data & Knowledge Engineering - Special issue: Temporal representation and reasoning archive Volume 44 Issue 2, February 2003.
Kang et. al., Discovering Flow Anomalies: A SWEET Approach, Eighth IEEE International Conference on Data Mining, ICDM

11. Temporal Association Rule Mining
The book also defines
‘Time instants’ or ‘Time Intervals’
‘chronon’ and ‘duration’
e.g.
12th Dec-2009, 11:20:24 {bread, milk, butter, cheese, chips}
12th Dec-2009, 11:27:04 {onion, capsicum, potatoes, burgers}
12th Dec-2009, 12:05:44 {soap, shampoo, comb, toothbrush}
12th Dec-2009, 11th hr {{bread, milk, butter, cheese, chips}, {onion, capsicum, potatoes, burgers}}
12th Dec-2009, 12th hr {{soap, shampoo, comb, toothbrush}}

12. Temporal Association Rule Mining
The book also defines
‘Time instants’ or ‘Time Intervals’
‘chronon’ and ‘duration’
e.g.
12th Dec-2009, 11:20:24 {bread, milk, butter, cheese, chips}
12th Dec-2009, 11:27:04 {onion, capsicum, potatoes, burgers}
12th Dec-2009, 12:05:44 {soap, shampoo, comb, toothbrush}
12th Dec-2009, 11th hr {{bread, milk, butter, cheese, chips}, {onion, capsicum, potatoes, burgers}}
12th Dec-2009, 12th hr {{soap, shampoo, comb, toothbrush}}
Time Unit (chronon)

13. Problem Statement Definitions : Valid Time Intervals
Support of an itemset I over interval (ti,tj) = frequency of I in the interval (ti,tj)/Total number of transaction during the interval (ti,tj)
Valid Time Interval for itemset I: the time interval during which the support of I over the interval is greater than a threshold (lmin_sup)
Maximal Valid Time Interval: A valid interval for an itemset I which not contained in any other valid time interval for I.
Temporally Frequent itemset: A itemset which has atleast one valid time interval associated with it.
Lmin_sup = 0.5
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Valid Time Intervals

14. Problem Statement Definitions : Maximal Valid Time Intervals
Support of an itemset I over interval (ti,tj) = frequency of I in the interval (ti,tj)/Total number of transaction during the interval (ti,tj)
Valid Time Interval for itemset I: the time interval during which the support of I is greater than a threshold (lmin_sup)
Maximal Valid Time Interval: A valid interval for an itemset I which not contained in any other valid time interval for I.
Temporally Frequent itemset: A itemset which has atleast one valid time interval associated with it.
Lmin_sup = 0.5
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Maximal Valid Time Intervals

15. Problem Statement (contd.)
Given: Transaction data D in the format (TU, {T1,T2,…,Tk}) Where TU-> Time Unit Ti-> Transaction Find: All temporally frequent itemsets along with their maximal valid time intervals.

16. Problem Statement (contd.)
So now, along with finding the frequent itemsets we have to find the maximal valid time intervals for each frequent itemset. Complexity of the naive approach for finding maximal valid time intervals for each frequent itemset: O(n2) Where, n= |D|

17. Finding Maximal Valid Time Intervals
Definition :
Valid/Supporting Time Unit for I: Time Unit during which the support of I is greater than lmin_supp.
Non-valid/Non-Supporting Time Unit for I: Time Unit during which the support of I is less than lmin_supp. .
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18. Finding Maximal Valid Time Intervals
Lemma 1: Each valid time interval TUi,TUj should contain atleast 1 valid/supporting time unit for I. Lemma 2: If an interval (TUi,TUj) is not valid for I then the interval (TUi,TUj+1) where TUj+1 is a non-valid time unit cannot be valid. Lemma 3: If an interval (TUi,TUj) is valid for I then the interval (TUi,TUj+1) where TUj+1 is a valid time unit would be valid.
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Using Lemma 3, collapse continuous runs of supporting time units into 1 unit with the average density
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19. Finding Maximal Valid Time Intervals (contd.)
Given: Item set I, Transaction data D <TUi, {T1,T2, …,Tn}>, lmin_sup
Part 1:
Find_maximal_valid_time_intervals(I,D,lmin_sup)
Find STU={TUa1,TUa2,…,TUan} such than TUak is a supporting time unit for I
For i = 1 to n
For j=n to i+1
IF is_valid_time_interval(TUai,TUaj,D,lmin_sup)
break;
End
Lemma 1,3
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20. Finding Maximal Valid Time Intervals (contd.)
Given: Item set I, Transaction data D <TUi, {T1,T2, …,Tn}>, lmin_sup
Part 2:
start = TUai-1+1 , finish=TUaj+1-1
low = start, high = TUaj
While low <= TUai and end < = finish
IF is_valid_time_interval(low,high)
high = high +1
Else
low = low+1
End
Lemma 2

21. Finding Maximal Valid Time Intervals (contd.)
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22. Finding Maximal Valid Time Intervals (contd.)
Further iterations…
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Complexity: O(n’2 + n)

23. Finding All Temporally Frequent Itemset
Given: Transaction data D <TUi,{T1, T2, …,Tn}>, lmin_sup, UI (Universal Itemset)
C->Generate_1-item_candidate_sets(UI,D)
Interval = (1, |D|)
While (|C|>0)
For each candidate set c in C
max_valid_intervals-> find_maximal_valid_time_interval(c,D,lmin_sup)
If |max_valid_intervals|>0
temp_freq_sets.add(<c,max_valid_intervals>)
End
If |temp_freq_sets| > 0
C-> generate_new_candidate_sets(temp_freq_sets , D,lmin_sup)
Else
C-> null

24. Pruning in Candidate Set Generation
Transactions
Item Set T1 T2 T3 T4 T5 T6 T7 T8 T9
L-2
a-b
a-c
C-3
a-b-c

25. Future Work Find cyclic valid time intervals
Identify interesting maximal valid time intervals

26. Questions?