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- Semi-Markov Chains and Hidden Semi-Markov Models toward Applications
- Hidden semi-Markov model
- Markov renewal process

## Semi-Markov Chains and Hidden Semi-Markov Models toward Applications

We review recent advances in the statistical analysis of neuronal spike trains based on Gibbs distributions in a large sense including non stationarity. We evoke some possible applications of Variable Length Markov Chains in this field. Abstract : VLMC allows to model time-series with finite state space and highly-varied dynamics.

In this presentation we consider the situation where the realization of the VLMC are not observed directly but through an observation process. The estimation is done by maximization of a penalized likelihood criteria. The strong consistency of the estimator is proved under general assumptions on the model. The estimator is built using a pruning technique that is combined with an Expectation Maximization based procedure. The semi-Markov chains generalize the Markov chains and renewal chains.

We will present the basic theory of semi-Markov chains with discrete and general state space. Further, estimation of the semi-Markov kernel and the transition function as well as some applications concerning survival and reliability function, earthquake study, DNA analysis will be given.

Finally, some results concerning random evolutions in an asymptotic setting will be presented. Reference : V. Barbu, N. Toward Applications. We consider a non Markovian process with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component.

This class of systems extends both the interacting particle systems, which are Markovian, and the stochastic chains with memory of variable length which have finite state space. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time.

This construction implies uniqueness of the stationary process. G-measures are discrete time stochastic processes generated by conditioning on the past. One-dimensional Gibbs measures correspond to random fields generated by conditioning simultaneously on the past and the future.

The aim of this talk is to review and compare results of both theories. I will define the VLMC models and make an overview of related questions probabilistic properties, connections to other non markov models, random walks. I will also present some applications to text algorithms, data structures and neurobiology.

Interactions between neurons can generate very complex and time-delayed patterns. In fact, neural interactions may reflect a complex anatomical substrate, where chains of activations trigger complex collective and self-organized phenomena. This represents a potential problem in experimental configurations where delayed communications between neurons are taken into consideration.

Standard techniques like correlation analysis are, in many cases, unable to detect such events. Such a problem can be solved by mathematical tools able to model arbitrarily long temporal relationships. A novel framework wherein spike trains with arbitrarily long temporal dependencies are modeled by Markov stochastic models has been proposed. Typically, in regular Markov models, each state depends only on the previous state while higher-order Markov models suffer from high state-space computational complexity.

Such a methodology has showed several interesting potentialities capturing key elements in neural functional dependencies between couples of neurons. Pierre Vallois : Some examples of diffusions with variable memory.

Thierry Dumont : Context tree estimation in variable length hidden Markov models. Antonio G. Markov chains. Spike trains analysis, Gibbs distributions and Variable Length? Context tree estimation in variable length hidden Markov models. Infinite systems of interacting chains with memory of variable length - a stochastic model for biological neural nets. Variable Length Markov Chains as sources: models and applications.

Some examples of diffusions with variable memory. Modeling neural activity by Variable Length Markov Chains.

## Hidden semi-Markov model

This book is concerned with the estimation of discrete-time semi-Markov and hidden semi-Markov processes. Semi-Markov processes are much more general and better adapted to applications than the Markov ones because sojourn times in any state can be arbitrarily distributed, as opposed to the geometrically distributed sojourn time in the Markov case. Another unique feature of the book is the use of discrete time, especially useful in some specific applications where the time scale is intrinsically discrete. The models presented in the book are specifically adapted to reliability studies and DNA analysis. The book is mainly intended for applied probabilists and statisticians interested in semi-Markov chains theory, reliability and DNA analysis, and for theoretical oriented reliability and bioinformatics engineers.

We review recent advances in the statistical analysis of neuronal spike trains based on Gibbs distributions in a large sense including non stationarity. We evoke some possible applications of Variable Length Markov Chains in this field. Abstract : VLMC allows to model time-series with finite state space and highly-varied dynamics. In this presentation we consider the situation where the realization of the VLMC are not observed directly but through an observation process. The estimation is done by maximization of a penalized likelihood criteria. The strong consistency of the estimator is proved under general assumptions on the model.

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Semi-Markov processes are much more general and better adapted to applications than Semi-Markov Chains and Hidden Semi-Markov Models toward Applications PDF · Discrete-Time Renewal Processes. Vlad Barbu, Nikolaos Limnios.

## Markov renewal process

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A hidden semi-Markov model HSMM is a statistical model with the same structure as a hidden Markov model except that the unobservable process is semi-Markov rather than Markov. This means that the probability of there being a change in the hidden state depends on the amount of time that has elapsed since entry into the current state. This is in contrast to hidden Markov models where there is a constant probability of changing state given survival in the state up to that time. The model was first published by Leonard E. Baum and Ted Petrie in

This book is concerned with the estimation of discrete-time semi-Markov and hidden semi-Markov processes. Semi-Markov processes are much more general and better adapted to applications than the Markov ones because sojourn times in any state can be arbitrarily distributed, as opposed to the geometrically distributed sojourn time in the Markov case. Another unique feature of the book is the use of discrete time, especially useful in some specific applications where the time scale is intrinsically discrete. The models presented in the book are specifically adapted to reliability studies and DNA analysis.

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Communications in Statistics-Theory and Methods 33 11 , ,

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