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

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?... 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?...

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

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!

Interval orders 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) ba c d b a b a But only works on level of events!

Process mining & intervals 1.Derive interval for each event Singleton set (single time stamp) Accurracy interval ( t ±  ) Time scale (week, day, hour, minute,...) 2.Relate events and intervals to activity 3.Discover process model

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

Activities & intervals First event until last event Following the event life cycle Based on event bursts:...

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

Process mining & intervals 1.Derive interval for each event Singleton set (single time stamp) Accurracy interval ( t ±  ) Time scale (week, day, hour, minute,...) 2.Relate events and intervals to activity Many different approaches! 3.Discover process model

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

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

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

Declarative language

An example

Discover declarative model 1.Derive interval sets 2.Calculate relations on interval sets 3.Generate declarative model −Problems: −Simultaneousness relations overlapping −Causality: always finds the transitive closure!

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

Next to and betweenness relation 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? b a c a a c b d ??

Conclusions & future work Approach: 1.Derive interval for each event 2.Relate events and intervals to activity −Many possibilities! 3.Discover process model Proof of concept implemented in ProM Apply approach to case studies