Hidden Markov models : Différence entre versions
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−  Markov chains are a useful tool for analyzing categorical longitudinal data. However, sometimes the Markov process cannot be directly observed, though some output, dependent on the  +  [http://en.wikipedia.org/wiki/Markov_chain Markov chains] are a useful tool for analyzing categorical longitudinal data. However, sometimes the [https://en.wikipedia.org/wiki/Markov_process Markov process] cannot be directly observed, though some output, dependent on the 
(hidden) state, is visible. More precisely, we assume that the distribution of this observable output depends on the underlying hidden state. Such models are called hidden Markov models (HMMs).  (hidden) state, is visible. More precisely, we assume that the distribution of this observable output depends on the underlying hidden state. Such models are called hidden Markov models (HMMs).  
HMMs can be applied in many contexts and have turned out to be particularly pertinent in several biological contexts. For example, they are useful when characterizing diseases for which the existence of several discrete stages of illness is a realistic assumption, e.g., epilepsy and migraines.  HMMs can be applied in many contexts and have turned out to be particularly pertinent in several biological contexts. For example, they are useful when characterizing diseases for which the existence of several discrete stages of illness is a realistic assumption, e.g., epilepsy and migraines.  
−  Here, we will consider a parametric framework with [http://en.wikipedia.org/wiki/Markov_chain Markov chains]  +  Here, we will consider a parametric framework with [http://en.wikipedia.org/wiki/Markov_chain Markov chains] in a discrete and finite state space $\mathbf{K} = \{1,\ldots,K\}$. 
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−  HMMs have been developed to describe how a given system moves from one state to another over time, in situations where the successive visited states are unknown and a set of observations is the only available information to describe the dynamics of the system. HMMs can be seen as a variant of mixture models that allow for possible memory in the sequence of hidden states. An HMM is thus defined as a pair of processes $(z_j,y_j, j=1,2,\ldots)$, where the latent sequence $(z_j)$ is a Markov chain and where the distribution of the observation $y_j$ at time $t_j$ depends on the state $z_j$.  +  HMMs have been developed to describe how a given system moves from one state to another over time, in situations where the successive visited states are unknown and a set of observations is the only available information to describe the dynamics of the system. HMMs can be seen as a variant of mixture models that allow for possible memory in the sequence of hidden states. An HMM is thus defined as a pair of processes $(z_j,y_j, j=1,2,\ldots)$, where the latent sequence $(z_j)$ is a [http://en.wikipedia.org/wiki/Markov_chain Markov chain] and where the distribution of the observation $y_j$ at time $t_j$ depends on the state $z_j$. 
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reference=(1) }}  reference=(1) }}  
−  For each individual $i$, $z_i$ is a Markov chain whose probability distribution is defined by  +  For each individual $i$, $z_i$ is a [http://en.wikipedia.org/wiki/Markov_chain Markov chain] whose probability distribution is defined by 
Version actuelle en date du 21 juin 2013 à 10:13
Sommaire 
[modifier] Introduction
Markov chains are a useful tool for analyzing categorical longitudinal data. However, sometimes the Markov process cannot be directly observed, though some output, dependent on the (hidden) state, is visible. More precisely, we assume that the distribution of this observable output depends on the underlying hidden state. Such models are called hidden Markov models (HMMs). HMMs can be applied in many contexts and have turned out to be particularly pertinent in several biological contexts. For example, they are useful when characterizing diseases for which the existence of several discrete stages of illness is a realistic assumption, e.g., epilepsy and migraines.
Here, we will consider a parametric framework with Markov chains in a discrete and finite state space $\mathbf{K} = \{1,\ldots,K\}$.
[modifier]
HMMs have been developed to describe how a given system moves from one state to another over time, in situations where the successive visited states are unknown and a set of observations is the only available information to describe the dynamics of the system. HMMs can be seen as a variant of mixture models that allow for possible memory in the sequence of hidden states. An HMM is thus defined as a pair of processes $(z_j,y_j, j=1,2,\ldots)$, where the latent sequence $(z_j)$ is a Markov chain and where the distribution of the observation $y_j$ at time $t_j$ depends on the state $z_j$.
Dynamics of a hidden Markov model

In a population approach, HMMs from several individuals can be described simultaneously by considering mixed HMMs.
Let $y_i=\left(y_{i,1},\ldots,y_{i,n_i}\right)$ and $z_i= \left(z_{i,1}, \ldots,z_{i,n_i}\right)$ denote respectively the sequences of observations and hidden states for individual $i$.
We suppose that the joint distribution of $(z_i,y_i)$ is a parametric distribution that depends on a vector of parameters $\psi_i$ and can be decomposed as
\(
\pcyzipsii(z_i,y_i  \psi_i) = \pczipsii(z_i \psi_i) \, \pcyizpsii(y_i  z_i,\psi_i) .
\)

(1) 
For each individual $i$, $z_i$ is a Markov chain whose probability distribution is defined by
 the distribution $ \pi_{i,1} = (\pi_{i,1}^{k},\ k=1,2,\ldots,K)$ of the first state $z_{i,1}$:
 the sequence of transition matrices $(Q_{i,j} \ ; \, j=2,3,\ldots)$, where for each $j$, $Q_{i,j} = (q_{i,j}^{\ell,k} \ ; \, 1\leq \ell,k \leq K)$ is a matrix of size $K \times K$ such that $q_{i,j}^{\ell,k} = \prob{z_{i,j} = k  z_{i,j1}=\ell , \psi_i}$.
Transitions of a Markov chain with 3 states

The conditional distribution $\qcyizpsii$ depends on the model for the observations: for each state, observation $y_{ij}$ has a certain distribution. Let us see some examples:
[modifier] Examples
1. In a continuous data model, one possibility is that the residual error model is a hidden Markov model that can randomly switch between $K$ possible residual error models.
2. In a Poisson model for count data, the Poisson parameter might randomly switch between $K$ intensities. Such models have been used for describing the evolution of seizures in epileptic patients:
[modifier] Distributions of observations
Assuming that the $N$ individuals are independent, the joint pdf is given by:
\(
\pcypsi(y_1,\ldots,y_N  \psi_1,\ldots,\psi_N ) = \prod_{i=1}^{N}\pcyipsii(y_i  \psi_i).
\)

(2) 
Then, computing the conditional distribution of the observations $\qcyipsii$ for any individual $i$ requires integration of the joint conditional distribution $\qcyzipsii$ over the states:
Though this looks complicated, it turns out that forward recursion of the BaumWelch algorithm provides a quick way to numerically compute it.
[modifier] Bibliography
Albert, P. S.  A two state Markov mixture model for a time series of epileptic seizure counts
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 Statistics and Computing 19:381393,2009
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 Journal of the Royal Statistical Society  Series A. 171, Part 3:739753,2008