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Process Mining: Declarative Modeling Approach Using Intervals

This paper explores the semantics and structure of timestamps in a service landscape, emphasizing the relationship between events and intervals. It discusses interval orders, concurrency, and process models derived from timestamps, with a focus on accuracy and granularity. The text delves into the connections between activities and intervals, highlighting different mappings and interval functions. It also covers relations on interval sets, such as simultaneousness and causality, explaining the declarative nature of interval relations and their influence on concurrency. Furthermore, it introduces a declarative language for modeling intervals, including examples and challenges such as dealing with overlapping relations and transitive closure in causality. The text concludes by discussing future research directions and the implementation of the proposed approach in ProM for case studies.

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Process Mining: Declarative Modeling Approach Using Intervals

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  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

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