We want to obtain a probabilistic graph of
transitions between states (clusters) with the length-
of-stay in each state (temporal state representations).
It is also interesting to study cluster succession of
length k (for example, the 3 last states of resident’s
clusters, a k timed series). Probabilistic automata are
used in various areas in pattern recognition or in
fields to which pattern recognition is linked.
Different concept learning algorithms have been
developed for different types of concepts. The
learning of deterministic finite automata (DFA), also
called regular inference is based on acceptance of
regular languages which allow to model the
behaviour of systems. The aim consists in
constructing a DFA from information about the set
of words it accepts. There are many algorithms for
regular inference (Angluin, 1987); (Garcia and
Vidal, 1990a); (Rivest and Sphapire, 1993); (Balczar
et al., 1997); (Parekh et al., 1998); (Parekh and
Honavar, 2001); (Bugalho and Oliviera, 2005)...
A finite automaton with transition probabilities
represents a distribution over the set of all strings
defined over a finite alphabet. The articles presented
by (Rico-Juan et al., 2000) and (Vidal et al., 2005)
present a survey and a study of the relations and
properties of probabilistic finite-automata and tree.
(Dupont et al., 2005) clarify the links between
probabilistic automata and hidden Markov models.
In a first part of this work, the authors present:
the probabilities distributions generated by
these models,
the necessary and sufficient conditions for an
automaton to define a probabilistic language.
The authors show that one the one hand,
probabilistic deterministic finite automata (PDFA)
form a proper subclass of probabilistic non-
deterministic automata (PNFA) and the other hand,
PNFA and hidden Markov models are equivalent.
We assume that our problem could be modelled
as a state transition graph (probabilistic deterministic
finite automaton). Consequently, the pattern
recognition of sequences and the corresponding
probabilities could be inductively learned via an
inference algorithm. The k-TSSI (k-Testable
Languages in the Strict Sense Inference) algorithm
(Garcia et al., 1990a, 1990b) could be useful,
convenient and suitable for two reasons: the
simplicity of implementation and the possibility to
take into account memory effects (timed macro-
states). The inductive inference of the class of k –
testable languages in the strict sense (k-TLSS) has
been studied and adapted to local languages, N-
grams and tree languages. A k-TLSS is essentially
defined by a finite set of substrings of length k that
are permitted to appear in the strings of then
language. Given a size k of memory, the objective is
to find an automaton for the language. This subclass
of language called k-testable language has the
property that the next character is only dependent on
the previous k-1 characters. In our case, it is
interesting to be able to identify the substrings
(memory) of length k. But, our goal is to infer a
timed model and an automaton inferred by the k-
TSSI algorithm does not take into account the timed-
state. The interesting question is how to infer timed
automata and very few works exist in the domain
(Alur et al., 1990, 1991); (Alur and Dill, 1994);
(Grinchtein et al., 2005); (Verwer et al., 2007,
2011). Timed automata correspond to finite state
models where explicit notion of time is taken into
account and is represented by timed events. Time
can be modelled in different ways, e.g. discrete or
continuous. The more recent works (Verwer et al.,
2007, 2011) propose an algorithm for learning
simple timed automata, known as real-time automata
where the transitions of real-time automata can have
a temporal constraint on the time of occurrence of
the current symbol relative to the previous symbol.
The problem is also that it is difficult to take into
account a set of substrings of length k (k>1) and the
algorithm is not generalized to probabilistic timed-
automata. In this section we propose a model in
order to take into account the concept of time in the
automaton inferred by the k-TSSI algorithm (i.e. the
duration of time a resident spends in a particular
cluster). In the next section, we present the
implementation of the model.
5 DEVELOPMENT OF
PATIENTS’ PROFILES:
MODEL IMPLEMENTATION
The method consists in:
1. Learning a deterministic finite automata (DFA)
using k-TSSI algorithm.
2. Transforming this DFA into a probabilistic
DFA.
3. Converting this probabilistic DFA in a Markov
chain model.
5.1 Preliminaries
The aim of grammatical inference is to learn models
of languages from examples of sentences of these
languages. Sentences can be any structured
composition of primitive elements or symbols,
ContributionofProbabilisticGrammarInferencewithk-TestableLanguageforKnowledgeModeling-Applicationon
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