  Discovering patterns in a sequence is an important aspect of data mining. One
popular choice of such patterns are episodes, patterns in sequential data
describing events that often occur in the vicinity of each other. Episodes also
enforce in which order the events are allowed to occur.
  In this work we introduce a technique for discovering closed episodes.
Adopting existing approaches for discovering traditional patterns, such as
closed itemsets, to episodes is not straightforward. First of all, we cannot
define a unique closure based on frequency because an episode may have several
closed superepisodes. Moreover, to define a closedness concept for episodes we
need a subset relationship between episodes, which is not trivial to define.
  We approach these problems by introducing strict episodes. We argue that this
class is general enough, and at the same time we are able to define a natural
subset relationship within it and use it efficiently. In order to mine closed
episodes we define an auxiliary closure operator. We show that this closure
satisfies the needed properties so that we can use the existing framework for
mining closed patterns. Discovering the true closed episodes can be done as a
post-processing step. We combine these observations into an efficient mining
algorithm and demonstrate empirically its performance in practice.
