Download - Mining Declarative Models using Intervals
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Mining Declarative Models using Intervals
Jan Martijn van der WerfRonny MansWil van der Aalst
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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?• ...
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Time stamps
• Time scale of data?• Dense (time stamps)• Coarse (hour, minute, day)
• Reliability of the data?• User entered?• System generated?
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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!
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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!
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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 activity3. Discover process model
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Activities & intervals
• First event until last event
• Following the life cycle of the activities
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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
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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
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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
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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
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Declarative language
• Interval relations are highly declarative:• Granularity influences degree of concurrency
• Activities occur simultaneously, unless prohibited
Succeeds!
Preceeds!
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Declarative language
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An example
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Discover declarative model
1. Derive interval sets2. Calculate relations on interval sets3. Generate declarative model
− Problems: − Simultaneousness relations overlapping− Causality: always finds the transitive closure!
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• 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
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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?
bac
aa
c
b
d? ?
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Conclusions & future work
• Approach:1. Derive interval for each event2. Relate events and intervals to activity
− Many possibilities!3. Discover process model
• Proof of concept implemented in ProM• Apply approach to case studies