Mining Quantitative Association Rules in Large Relational Tables

1. Mining Quantitative Association Rules in Large Relational Tables ACM SIGMOD Conference 1996 Authors: R. Srikant, and R. Agrawal Presented by: Biyu Liang March 29, 2006

2. Outline • Review of Association Analysis • Introducing Quantitative AR Problem • Partitioning Quantitative Attributes • Identifying the Interesting Rules • Extending the Apriori Algorithm • Conclusions

3. Association Rule • Item sets X and Y • Rule X => Y • Support = Pr(XUY) • Confidence= Pr(Y|X)= Pr(XUY)/Pr(X) • Find rules that have MinSup and MinConf

4. Boolean Association Rules • Attribute has a value of “1” if the transaction contains the corresponding item; “0” otherwise.

5. Outline • Review of Association Analysis • Introducing Quantitative AR Problem • Partitioning Quantitative Attributes • Identifying the Interesting Rules • Extending the Apriori Algorithm • Conclusions

6. Quantitative Association Rules • <Age: 30..39> and <Married: Yes> => <NumCars: 2> • Support = 40%, Conf = 100%

7. Mapping to Boolean Association Rules Problem • Using <attribute: value> as new attribute, which has only boolean values

8. Problems with Direct Mapping • MinSup: If number of intervals is large, the support of a single interval can be lower • MinConf: Information lost during partition values into intervals. Confidence can be lower as number of intervals is smaller

9. The Tradeoff • Increase the number of intervals (to reduce information lost) while combining adjacent ones (to increase support) • ExecTime blows up as items per record increases • ManyRules: Number of rules also blows up. Many of them will not be interesting

10. The Proposed Approach • Partition quantitative attribute values and combining adjacent partitions as necessary • Partial Completeness Measure for deciding the partitions • Interest Measure (pruning) to address the “ManyRules” problem • Extend the Apriori Algorithm

11. 5 Steps of the Proposed Approach • Determine the number of partitions for each quantitative attribute • Map values/ranges to consecutive integer values such that the order is preserved • Find the support of each value of the attributes, and combine when support is less than MaxSup. Find frequent itemsets, whose support is larger than MinSup • Use frequent set to generate association rules • Pruning out uninteresting rules

12. 5 Steps of the Proposed Approach • Determine the number of partitions for each quantitative attribute • Map values/ranges to consecutive integer values such that the order is preserved • Find the support of each value of the attributes, and combine when support is less than MaxSup. Find frequent itemsets, whose support is larger than MinSup • Use frequent set to generate association rules • Pruning out uninteresting rules

13. Outline • Review of Association Analysis • Introducing Quantitative AR Problem • Partitioning Quantitative Attributes • Identifying the Interesting Rules • Extending the Apriori Algorithm • Conclusions

14. Partial Completeness • R : rules obtained before partition • R’: rules obtained after partition • Partial Completeness measures the maximum distance between a rule in R and its closest generalization in R’ • is a generalization of itemset X: if • The distance is defined by the ratio of support

15. K-Complete • C : the set of frequent itemsets • For any K ≥ 1, P is K-complete w.r.t C if: • P C • For any itemset X (or its subset) in C, there exists a generalization whose support is no more than K times that of X (or its subset) • The smaller K is, the less the information lost

16. Theoretical Results • Lemma 1: If P is K-complete set w.r.t C, then any rule R obtained from C has a generalization R’ from P, such that conf(R’) is bounded by [conf(R)/K, K*conf(R)] • For given partial completeness level K, equi-depth partitioning satisfies the completeness level with minimum number of intervals: 2n/[m(K-1)],and MaxSup for each interval is m(K-1)/(2n)

17. Outline • Review of Association Analysis • Introducing Quantitative AR Problem • Partitioning Quantitative Attributes • Identifying the Interesting Rules • Extending the Apriori Algorithm • Conclusions

18. Example of Uninteresting Rule • Suppose a quarter of people in age group 20..30 are in the age group 20..25 • <Age: 20..30> => <Cars: 1..2>, with 8% sup, 70% conf • <Age: 20..25> => <Cars: 1..2>, with 2% sup, 70% conf • The second rule doesn’t give any additional information, and is less general than the first rule

19. Expected Values Based on Generalization • Itemset Z = {<z1, l1, u1>, …, <z1, l1, u1>} • The expected support of Z based on the support of its generalization is defined as

20. Expected Values Based on Generalization • The expected confidence of the rule X => Y based on the confidence of its generalization is defined as

21. Interest Measure • Itemset X is R-interesting w.r.t its generalization if • The support of X is no less than R times the expected supports based on , and • For any specialization X' of , X – X' is R-interesting w.r.t • Rule X => Y is R-interesting w.r.t its generalization if the support or confidence is R times that of , and the itemset is R-interesting w.r.t

22. Outline • Review of Association Analysis • Introducing Quantitative AR Problem • Partitioning Quantitative Attributes • Identifying the Interesting Rules • Extending the Apriori Algorithm • Conclusions

23. Candidate Generation • Given the set Lk-1 of all frequent (k-1)-itemset, generate the set of Lk • The process has three parts: • Join Phase • Subset Prune Phase • Interest Prune Phase

24. Join Phase • Lk-1 joined with itself • Join condition: k-2 items are the same, the remaining ones have different attribute • Example, L2: • {<Married:Yes> <Age:20..24>} • {<Married:Yes> <Age:20..29>} • {<Married:Yes> <Cars:0..1>} • {<Age:20..29> <Cars:0..1> } • Result of self-join, C3: • {<Married:Yes> <Age:20..24><Cars:0..1>} • {<Married:Yes> <Age:20..29><Cars:0..1>}

25. Subset Prune Phase • Make sure any (k-1)-subset is in Lk-1 • Example, L2: • {<Married:Yes> <Age:20..24>} • {<Married:Yes> <Age:20..29>} • {<Married:Yes> <Cars:0..1>} • {<Age:20..29> <Cars:0..1> } • Result of self-join, C3: • {<Married:Yes> <Age:20..24><Cars:0..1>} • {<Married:Yes> <Age:20..29><Cars:0..1>} • Delete the first itemset in C3 since <Age:20..24><Cars:0..1> is not in L2

26. Interest Prune Phase • Given user-specified interest level R • Delete any itemset that contains a item with support greater than 1/R • Lemma 5 guarantees that such itemsets cannot be R-interesting w.r.t to their generalizations

27. Outline • Review of Association Analysis • Introducing Quantitative AR Problem • Partitioning Quantitative Attributes • Identifying the Interesting Rules • Extending the Apriori Algorithm • Conclusions

28. Conclusions • This paper introduced the problem of mining quantitative association rules in large relational tables • It dealt with quantitative attributes by fine-partitioning the values and combining adjacent partitions as necessary • Partial completeness quantifies the info lost, and help decide the partitions • Interest measure to identify interesting rules

29. Thanks! Question?

30. Final Exam Questions • What is Partial Completeness? (p.14-15) • Determine a number of intervals, where there 3 quantitaive attributes, .70 min support and a 1.5 partial completeness level? (p.16) • If Intervals are too large, rules may not have MinConf, and if they are too samll, rules may not have MinSupp, how Do you go about solving this catch 22 problem? (p.8-9)