1. Mining the Most Interesting Rules Roberto J. Bayardo Jr., Rakesh Agrawal Presented by: Mohamed G. Elfeky

2. Introduction Algorithms for mining rules: Constraint-based Heuristic (Predictive rules) Interestingness-metric Several interestingness metrics: confidence, support, laplace, gain, conviction

3. Generic Problem Statement The rule: A  C The input is: (U, D, , C, N) U is a set of conditions for the rule antecedent. D is a data-set.  is a total order on rules. C is a condition for the rule consequent. N is a set of constraints on rules.

4. Optimized Rule Mining Find a set A 1  U such that: A 1 satisfies N,   A 2  U: A 2 satisfies N  A 1 < A 2. Any rule A  C whose A  A 1 is optimal. Generally, this is NP-Hard problem.

5. Partial-Order Optimized Rule Mining Partial order vs. Total order Some rules may be incomparable. Several equivalence classes for optimal rules. Find a set O  P(U) such that:  A  O: A is optimal, For each equivalence class that has a rule that is optimal, exactly one member of this class is within O.

6. Monotonicity f(x) is said to be monotone in x if: x 1 < x 2  f(x 1 )  f(x 2 ) f(x) is said to be anti-monotone in x if: x 1 < x 2  f(x 1 )  f(x 2 )

7. Optimality SC-Optimality PC-Optimality Definition Theoretical Implications Practical Implications

8. SC-Optimality: Definition The partial order  sc For rules r 1 and r 2 : r 1 < sc r 2 if and only if: sup(r 1 )  sup(r 2 )  conf(r 1 ) < conf(r 2 ), or sup(r 1 ) < sup(r 2 )  conf(r 1 )  conf(r 2 ). Also, r 1 = sc r 2 if and only if: sup(r 1 ) = sup(r 2 )  conf(r 1 ) = conf(r 2 ).

9. SC-Optimality: Definition (cont.) The partial order  s  c For rules r 1 and r 2 : r 1 < s  c r 2 if and only if: sup(r 1 )  sup(r 2 )  conf(r 1 ) > conf(r 2 ), or sup(r 1 ) < sup(r 2 )  conf(r 1 )  conf(r 2 ). Also, r 1 = s  c r 2 if and only if: sup(r 1 ) = sup(r 2 )  conf(r 1 ) = conf(r 2 ).

10. SC-Optimality: Definition (cont.) sc-optimal rule s  c-optimal rule non-optimal rule confidence support No optimal rules fall outside the borders

11. SC-Optimality: Theoretical Implications A total order  t is implied by  sc if: r 1  sc r 2  r 1  t r 2 ^ r 1 = sc r 2  r 1 = t r 2 r is optimal for  sc  r is optimal for  t.  t defined by f(r) is implied by  sc if: f(r) is monotone in support, and f(r) is monotone in confidence.

12. SC-Optimality: Theoretical Implications (cont.) Interestingness metrics: laplace(r) = gain(r) = sup(r) (1 –  /conf(r)) conviction(r) =  /(1 – conf(r)) sup(r) + 1 sup(r)/conf(r) + k

13. PC-Optimality: Definition The partial order  pc For rules r 1 and r 2 : r 1 < pc r 2 if and only if: pop(r 1 )  pop(r 2 )  conf(r 1 ) < conf(r 2 ), or pop(r 1 )  pop(r 2 )  conf(r 1 )  conf(r 2 ). Also, r 1 = pc r 2 if and only if: pop(r 1 ) = pop(r 2 )  conf(r 1 ) = conf(r 2 ).

14. PC-Optimality: Definition (cont.) pop(A  C) is the set of records from D that satisfy both A and C. |pop(r)| = sup(r)  |D| Analogously, the definition of  p  c

15. PC-Optimality: Theoretical Implications  sc is implied by  pc and  s  c by  p  c.  pc results in more incomparable rule pairs. pc-optimal rule set will contain more rules than sc-optimal rule set.

16. Optimality: Practical Implications Two algorithms are proposed, one for each type of optimality. Each algorithm produces a set of optimal rules without specifying the interestingness metrics. The produced set is guaranteed to identify the most interesting rules according to several metrics.

17. Optimality: Practical Implications (cont.) These algorithms facilitate interactivity: Examine the optimal rules according to some metric without additional querying or mining. Find the most interesting rule that characterizes any given subset of the population.