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.
Estep(x, Pi, delta, distn, pm, pn = NULL)is a vector of length \(n\) containing the observed process.
is the current estimate of the \(m \times m\) transition probability matrix of the hidden Markov chain.
is a list object containing the current (Markov dependent) parameter estimates associated with the distribution of the observed process (see dthmm).
is a list object containing the observation dependent parameter values associated with the distribution of the observed process (see dthmm).
is the current estimate of the marginal probability distribution of the \(m\) hidden states.
A list object is returned with the following components.
an \(n \times m\) matrix containing estimates of the conditional expectations. See “Details”.
an \(n \times m \times m\) array containing estimates of the conditional expectations. See “Details”.
the current value of the log-likelihood.
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 \(i = 1, \cdots, n\).
Cited references are listed on the HiddenMarkov manual page.