  Process mining is concerned with deriving formal models capable of
reproducing the behaviour of a given organisational process by analysing
observed executions collected in an event log. The elements of an event log are
finite sequences (i.e., traces or words) of actions. Many effective algorithms
have been introduced which issue a control flow model (commonly in Petri net
form) aimed at reproducing, as precisely as possible, the language of the
considered event log. However, given that identical executions can be observed
several times, traces of an event log are associated with a frequency and,
hence, an event log inherently yields also a stochastic language. By exploiting
the trace frequencies contained in the event log, the stochastic extension of
process mining, therefore, consists in deriving stochastic (Petri nets) models
capable of reproducing the likelihood of the observed executions. In this
paper, we introduce a novel stochastic process mining approach. Starting from a
"standard" Petri net model mined through classical mining algorithms, we employ
optimization to identify optimal weights for the transitions of the mined net
so that the stochastic language issued by the stochastic interpretation of the
mined net closely resembles that of the event log. The optimization is either
based on the maximum likelihood principle or on the earth moving distance.
Experiments on some popular real system logs show an improved accuracy w.r.t.
to alternative approaches.
