From Association Rules To Causality

1. From Association Rules To Causality Presenters: Amol Shukla, University of Waterloo Claude-Guy Quimper, University of Waterloo

2. From Association Rules To Causality Presentation Outline • Limitations of Association Rules and the Support-Confidence Framework • Generalizing Association Rules to Correlations • Scalable Techniques for Mining Causal Structures • Applications of Correlation and Causality • Summary

3. Review: Association Rules Mining • Itemset I={i1, …, ik} • Find all the rules XYwith min confidence and support • support, s, probability that a transaction contains XY • confidence, c,conditional probability that a transaction having X also contains Y, i.e., P(Y|X) • Let min_support = 50%, min_conf = 50%. • Two example association rules are: • A  C (50%, 66.7%) • C  A (50%, 100%)

4. Limitations of Association Rules using Support-Confidence Framework • Negative implications or dependencies are ignored • Consider the adjoining database. • X and Y: positively related, • X and Z: negatively related • support and confidence of X=>Z dominates • Only the presence of items is taken into account

5. Limitations of Association Rules using Support-Confidence Framework • Another market basket data example Buys Tea => Buys Coffee (support=20%,confidence=80%) • Is this rule really valid? • Pr(Buys Coffee)=90% • Pr(Buys Coffee|Buys Tea)=80% • Negative correlation between buying tea and buying coffee is ignored

6. From Association Rules To Causality • Limitations of Association Rules and the Support-Confidence Framework • Generalizing Association Rules to Correlations • Scalable Techniques for Mining Causal Structures • Applications of Correlation and Causality • Summary

7. What is Correlation? • P(A): Probability that event A occurs P(A’): Probability that event A does not occur P(AB): Probability that events A and B occur together. • Events A and B are said to be independent if P(AB) = P(A) x P(B) Otherwise A and B are dependent • Events A and B are said to be correlated if any of AB, A’B , AB’, A’B’ are dependent • A correlation rule is a set of items that are correlated

8. Computing Correlation Rules: Chi-squared Test for Independence • For an itemset I={i1,…,ik}, construct a k-dimensional contingency table R= {i1,i1’} x … x {ik,ik’} • We need to test whether each cell r= r1,…,rk in this table is dependent • Let O(r) denote the observed value of cell r in this table, and E(r) be its expected value. • The chi-squared statistic is the computed as: • If 2= 0, the cells are independent. If 2 > cut-off value,reject the independence assumption

9. Example: Computing the Chi-squared Statistic E(Coffee,Tea)= (90 x 25)/100 = 22.5 E(No Coffee,Tea) = (10 x 25)/100 = 2.5 E(Coffee,No Tea)= (90 x 75)/100 = 67.5 E(No Coffee,No Tea)=(10 x 75)/100=7.5 2 = (20-22.5)2/22.5 + (5-2.5)2/2.5+ (70-67.5)2/67.5 + (5-7.5)2/7.5 = 0.28 + 2.5 + 0.09 + 0.83 = 3.7 Since this value is greater than the cut-off value (2.71 at 90% significance level), we reject the independence assumption

10. Determining the Cause of Correlation • I(r)>1 indicates positive dependence and I(r)<1 indicates negative dependence • The farther I(r) is from 1, the more a cell contributes to the 2 value, and the correlation. • Define measures of interest for each cell I(r) = O(r) / E(r) Cell Counts • Thus, [No Coffee,Tea] contributes the most to the correlation, indicating that buying tea might inhibit buying coffee Measures of Interest = 70/67.5

11. Properties of Correlation • If a set of items is correlated, all its supersets are also correlated. Thus, correlation is upward-closed • We can focus on minimal correlated itemsets to reduce our search space • Support is downward-closed. A set has minimum support only if all its subsets have minimum support • We can combine correlation with support for an effective pruning strategy

12. Combining Correlation with Support • Support-confidence framework looks at only the top-left cell in the contingency table. To incorporate negative dependence, we must consider all the cells in the table • Combine correlation with support by defining “CT-support” • Let s be a user specified min-support threshold. Let p be a user-specified cut-off percentage value • An itemset I is CT-supported if at least p% of the cells in its contingency table have support not less than s • An itemset is significant if it is CT-supported and minimally correlated

13. A level-wise algorithm for finding correlation rules

14. Steps performed by the algorithm at level k Start Is the Itemset CT-supported? No Construct Contingency Table for next itemset at the level Add to the set NOTSIG Yes Done processing all itemsets at level k No Is 2 greater than cut-off value? Generate itemset(s) of size k+1 such that all of its subsets are in NOTSIG Mark the itemset as ‘significant’ Yes

15. Limitations of Correlation • Correlation might not be valid for ‘sparse’ itemsets. At least 80% of the cells in the contingency table must have expected value greater than 5. • Finding correlation rules is computationally more expensive than finding association rules. • Only indicates that the existence of a relationship. Does not specify the nature of the relationship, i.e., the cause and effect phenomenon is ignored. • Identifying causality is important for decision-making.

16. From Association Rules to Causality • Limitations of Association Rules and the Support-Confidence Framework • Generalizing Association Rules to Correlations • Scalable Techniques for Mining Causal Structures • Applications of Correlation and Causality • Summary

17. Hamburgers Hot-Dogs Causality 33% 33% 33% Association Rule: Hot-Dogs  BBQ Sauce [33%, 50%] Causality Rule: Hamburgers  BBQ Sauce

18. Bayesian Networks • What is the best topology of a Bayesian network that describes the observed data? • Problem: Very expensive to compute

19. Simplifying Causal Relationships • Knowing the existence of a causal relationship is as good as knowing the relationship

20. Causality vs Correlation • Two correlated variables can have either: • A causal relationship • A common ancestor

21. Independent Independent First Rule of Causality 1) Suppose we have threepair wise dependentvariables: 2) And two variables become independent when conditionedon the third one

22. First Rule of Causality Then we have one of these following configurations

23. independent dependent dependent dependent dependent Second Rule of Causality • Suppose we havethree variables withthese relationships 2) And the two independent variables become dependentwhen conditioned on the third variable

24. Second Rule of Causality • Then the two independent variables cause the third variable.

25. U C C C C C C C C C Finding Causality 1) Construct a graph whereeach variable is a vertex 2) Perform a Chi-squared testto determine correlation 3) Add an edge labeled “C”for each correlated test 4) Add an edge labeled “U”for each uncorrelated test 5) For each triplet, check if acausality rule can be applied

26. Weaknesses of the Algorithm • Causality rules do not cover all possible causality relationships • The X2 test with confidence set to 95% is expected to fail 5 times every 100 tests • Some variables might not be reported correlated or uncorrelated

27. From Association Rules to Causality • Limitations of Association Rules and the Support-Confidence Framework • Generalizing Association Rules to Correlations • Scalable Techniques for Mining Causal Structures • Applications of Correlation and Causality • Summary

28. Experiments (Census) • Correlation rules • Not a native English speaker  Not born in the U.S • Served in the military  Male • Married  more than 40 years old • Causality Rules • Male  Moved Last 5 years, Support-Job • Native-Amer.  $20-$40K  House Holder • Asian, Laborer  < $20K

29. Experiments (Text Data) • 416 distinct frequent words • 86320 pairs of words, 10% are correlated Correlation Causality Rules Nelson, Mandela upi, not reuter area, province Iraqi, Iraq area, secretary, war united, states area, secretary, they prime, minister

30. Beyond Correlation and Causality • Correlation and causality seem to be stronger mathematical model than confidence and support • It is possible to apply these concepts where confidence and support were previously applied

31. At least one item is meat Association Rules with Constraints • Correlation can be seen as a monotone constraint • Algorithm obtained by modifying algorithms for mining constrained association rules

32. From Association Rules to Causality • Limitations of Association Rules and the Support-Confidence Framework • Generalizing Association Rules to Correlations • Scalable Techniques for Mining Causal Structures • Applications of Correlation and Causality • Summary

33. Conclusion (Good news) • Correlation and causality are stronger mathematical models to retrieve interesting association rules • Allow to detect negative implications • Causality explains why there is a correlation

34. Conclusion (Bad news) • Difficult to precisely detect correlation (especially in sparse data cubes) • Not all causality relationships can be found • Are the results really better than with support and confidence?

35. Open Problems • How to discover hidden variables in causality • How to resolve bi-directional causality for disambiguatione.g: prime  minister minister prime • How do we find causal patterns for more than 3 variables

36. References Papers • “Beyond Market Baskets: Generalizing Association Rules to Correlations” - Brin, Motwani, Silverstein; SIGMOD 97 • “Scalable Techniques for Mining Causal Structures” - Silverstein, Brin, Motwani, Ullman; VLDB 98 • “Efficient Mining of Constrained Correlated Sets” - Grahne, Lakshmanan, Wang; ICDE 2000 • “A Simple Constraint-Based Algorithm for Efficiently Mining Observational Databases for Causal Relationships” - Cooper; Data Mining and Knowledge Discovery, vol 1, 1997 Textbook • “Causality: models, reasoning, and inference” - Judea Pearl; Cambridge University Press, 2000

37. From Association Rules To Causality Questions