Si avvisa che in data 23/11/2017, alle ore 10:00 precise, presso l'Aula
Seminari del III piano del DIpartimento di Matematica del Politecnico di
Milano si svolgerà il seguente seminario:
Titolo: On segmentation with hidden, pairwise and triplet Markov models
Relatore: Juri Lember, University of Tartu, Estonia
Abstract:
The well-known hidden Markov model (HMM) is a two-dimensional stochastic
process (X,Y), where Y is a Markov chain and conditionally on Y, the
X-process consists of independent random variables, the distribution of the
random variable X_t depending on Y_t, only. Over the last decades, HMM's
have become very popular stochastic models with applications to speech
recognition, signal processing, linguistic, computational molecular biology
and so on. Often the Y-process is unobserved (hidden) and the goal of the
inference is to estimate its unobserved realization based on a realization
of X-process. This task is called the segmentation problem and the standard
ways to solve it is to use either maximum likelihood (so-called Viterbi)
path or pointwise maximum likelihood (so-called PMAP) path.
A trivial but important property of HMM is that the process Z=(X,Y) is
itself a Markov process with a product state space. This observation
allows naturally enlarge the class of HMM's to the class of pairwise
Markov models (PMM) as follows: Z=(X,Y) is a PMM if Z has Markov property.
Now it is clear that PMM's are a much larger class of models whose HMM\'s
is just a little subclass. We briefly discuss several PMM's like Markov
switching models and HMM's with dependent noise. It is important to note
that if (X,Y) is a Markov process, then neither X nor Y need to have Markov
property, but conditionally on X, the Y-process is Markov and vice versa.
It turns out that many good properties of HMM's are mainly due to the
Markov property of Z and hence these properties carry on to PMM's as well.
In particular the well-known Viterbi and forward-backward algorithms apply
and so standard segmentation approaches can be applied in the case of
PMM's. Moreover, PMM-models provide a rather flexible and realistic model
for the homology of random sequences. A triplet Markov model (TMM),
introduced by W. Pieczynski, is a three-dimensional Markov process (X,Y,U),
where, as previously, X stands for observations and Y is the hidden state
sequence of interest. But in addition, there is another hidden component
U. Since conditionally on U, the pair (X,Y) is an inhomogeneous PMM, the
U-component models now the change of environment. It turns out that adding
the U-component makes the model really flexible.
We give a general approach to the risk-based segmentation problem that also
applies for PMM's and TMM's, discuss the weaknesses standard approaches
and introduce a way to overcome these problems. We also discuss the
asymptotics of Viterbi segmentation for PMM\'s.
Tutti gli interessati sono invitati a partecipare
Fabio Zucca