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?• ...

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)

b a

cd

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

• 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?

ba

c

aa

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