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

2. 2 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 Presentation Outline

3. 3 Review: Association Rules Mining Itemset I={i 1, …, i k } Find all the rules X  Y with 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%) Transaction -id Items bought 10A, B, C 20A, C 30A, D 40B, E, F

4. 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. 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 Items Bought CoffeeNo Coffee Sum (row) Tea20525 No Tea70575 Sum (col.) 9010100

6. 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. 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. 8 Computing Correlation Rules: Chi-squared Test for Independence For an itemset I={i 1,…,i k }, construct a k-dimensional contingency table R= {i 1,i 1 ’} x … x {i k,i k ’} We need to test whether each cell r= r 1,…,r k 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. 9 Example: Computing the Chi-squared Statistic CoffeeNo Coffee Sum (row) Tea20525 No Tea70575 Sum (col.) 9010100 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 Since this value is greater than the cut-off value (2.71 at 90% significance level), we reject the independence assumption  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

10. 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. CoffeeNo Coffee Tea0.892 No Tea1.030.66 Measures of Interest  Define measures of interest for each cell I(r) = O(r) / E(r)  Thus, [No Coffee,Tea] contributes the most to the correlation, indicating that buying tea might inhibit buying coffee Cell Counts CoffeeNo Coffee Tea205 No Tea705 = 70/67.5

11. 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. 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. 13 A level-wise algorithm for finding correlation rules

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

15. 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. 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. 17 Causality Hot-Dogs Hamburgers 33% Association Rule: Hot-Dogs  BBQ Sauce [33%, 50%] Causality Rule: Hamburgers  BBQ Sauce

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

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

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

21. 21 First Rule of Causality 1)Suppose we have three pair wise dependent variables: 2)And two variables become independent when conditioned on the third one Independent

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

23. 23 dependent independent dependent Second Rule of Causality dependent 1)Suppose we have three variables with these relationships 2)And the two independent variables become dependent when conditioned on the third variable

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

25. 25 Finding Causality 1)Construct a graph where each variable is a vertex 2)Perform a Chi-squared test to 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 a causality rule can be applied CC C C C C C C U C

26. 26 Weaknesses of the Algorithm Causality rules do not cover all possible causality relationships The X 2 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. 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. 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. 29 Experiments (Text Data) 416 distinct frequent words 86320 pairs of words, 10% are correlated CorrelationCausality Rules Nelson, Mandelaupi, not reuter area, provinceIraqi, Iraq area, secretary, warunited, states area, secretary, theyprime, minister

30. 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. 31 Association Rules with Constraints Correlation can be seen as a monotone constraint Algorithm obtained by modifying algorithms for mining constrained association rules At least one item is meat

32. 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. 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. 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. 35 Open Problems How to discover hidden variables in causality How to resolve bi-directional causality for disambiguation e.g: prime  minister minister  prime How do we find causal patterns for more than 3 variables

36. 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. 37 From Association Rules To Causality Questions