R: E-Step of EM Algorithm for DTHMM

Estep {HiddenMarkov} | R Documentation

## 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*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

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

and `v[i,j,k]` is

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

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

### Value

A `list` object is returned with the following components.

`u` | an n*m matrix containing estimates of the conditional expectations. See “Details”.
`v` | an n*m*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`

[Package HiddenMarkov version 1.8-8 Index]