Estep {HiddenMarkov}

E-Step of EM Algorithm for DTHMM

Description

Performs the expectation step of the EM algorithm for a dthmm process. This function is called by the BaumWelch function. The Baum-Welch algorithm referred to in the HMM literature is a version of the EM algorithm.

Usage

Estep(x, Pi, delta, distn, pm, pn = NULL)

Arguments

x

is a vector of length n containing the observed process.

Pi

is the current estimate of the m \times m transition probability matrix of the hidden Markov chain.

distn

is a character string with the distribution name, e.g. "norm" or "pois". If the distribution is specified as "wxyz" then a probability (or density) function called "dwxyz" should be available, in the standard R format (e.g. dnorm or dpois).

pm

is a list object containing the current (Markov dependent) parameter estimates associated with the distribution of the observed process (see dthmm).

pn

is a list object containing the observation dependent parameter values associated with the distribution of the observed process (see dthmm).

delta

is the current estimate of the marginal probability distribution of the m hidden states.

Details

Let u_{ij} be one if C_i=j and zero otherwise. Further, let v_{ijk} be one if C_{i-1}=j and C_i=k, and zero otherwise. Let X^{(n)} contain the complete observed process. Then, given the current model parameter estimates, the returned value u[i,j] is

\widehat{u}_{ij} = \mbox{E}[u_{ij} \, | \, X^{(n)}] = \Pr\{C_i=j \, | \, X^{(n)} = x^{(n)} \} \,,

and v[i,j,k] is

\widehat{v}_{ijk} = \mbox{E}[v_{ijk} \, | \, X^{(n)}] = \Pr\{C_{i-1}=j, C_i=k \, | \, X^{(n)} = x^{(n)} \}\,,

where j,k = 1, \cdots, m and i = 1, \cdots, n.

Value

A list object is returned with the following components.

u

an n \times m matrix containing estimates of the conditional expectations. See “Details”.

v

an n \times m \times m array containing estimates of the conditional expectations. See “Details”.

LL

the current value of the log-likelihood.

Author(s)

The algorithm has been taken from Zucchini (2005).

References

Cited references are listed on the HiddenMarkov manual page.

See Also

BaumWelch, Mstep