Mining Declarative Models using Intervals

1. Mining Declarative Models using Intervals Jan Martijn van der Werf Ronny Mans Wil van der Aalst

2. A service landscape How to combine logs? Merge using time stamps! Are timestamps synchronized in landscape? • Semantics of timestamps? • Time when the event occurred? • Time when it started / completed? • Time when the event is recorded? • Time when the event is stored? • ...

3. Time stamps • Time scale of data? • Dense (time stamps) • Coarse (hour, minute, day) • Reliability of the data? • User entered? • System generated?

4. Events & intervals: “old theory” • Structure of concurrency: • Observe whether an event preceded another event • Observe whether events occurred simultaneously • Implies an order • Interval order! • Position of intervals on the axis!

5. Interval orders b a c d b a a b But only works on level of events! • Define relation > by a > b iff “a occurs wholly after b” • Interval order if: • [ a > b and c > d ] imply [ a > d or c > b ] • Generalization of transitivity • Simultaneousness: ⌐ ( a > b) /\ ⌐ ( b > a)

6. Process mining & intervals • Derive interval for each event • Singleton set (single time stamp) • Accurracy interval ( t ±  ) • Time scale (week, day, hour, minute, ...) • Relate events and intervals to activity • Discover process model

7. Activities & intervals First event until last event Following the life cycle of the activities

8. Activities & intervals • Activities relate to a set of intervals • Many different mappings possible! • Granularity (Density of intervals) • Fine: many small intervals • Coarse: few large intervals • Finest interval function: • Only intervals of single points • Coarsest interval function • Each activity maps to a single interval

9. Process mining & intervals • Derive interval for each event • Singleton set (single time stamp) • Accurracy interval ( t ±  ) • Time scale (week, day, hour, minute, ...) • Relate events and intervals to activity • Many different approaches! • Discover process model

10. Relations on interval sets (1) • Simultaneousness • Weak: there is somewhere some overlap • Dependent: always if A occurs, then B occurs as well • Strong: if A occurs, then B occurs and vice versa

11. Relations on interval sets (2) • Causality • Wholly: all intervals of A before B • Succeeded: each interval of B followed by one of C • Preceeded: each interval of B occurs after one of A

12. Declarative language Succeeds! Preceeds! • Interval relations are highly declarative: • Granularity influences degree of concurrency • Activities occur simultaneously, unless prohibited

13. Declarative language

14. An example

15. Discover declarative model • Derive interval sets • Calculate relations on interval sets • Generate declarative model • Problems: • Simultaneousness relations overlapping • Causality: always finds the transitive closure!

16. Causality & transitive closure Polynomial NP-hard • Transitive reduction: S  S* = R*  R • Minimal edge problem: • Only use “existing” edges for transitive reduction • What are existing arcs in process mining?

17. Next to and betweenness relation b a c a a c b d ? ? • Next to • Weak: there is an interval of A directly followed by A • Strong: all intervals of A are directly followed by B • Betweenness: • interval of B is between two intervals of A • Weak or strong?

18. Conclusions & future work • Approach: • Derive interval for each event • Relate events and intervals to activity • Many possibilities! • Discover process model • Proof of concept implemented in ProM • Apply approach to case studies