forwardback: Forward and Backward Probabilities of DTHMM
In HiddenMarkov: Hidden Markov Models

Description Usage Arguments Details Value Author(s) References See Also Examples

These functions calculate the forward and backward probabilities for a dthmm process, as defined in MacDonald & Zucchini (1997, Page 60).

backward(x, Pi, distn, pm, pn = NULL)
forward(x, Pi, delta, distn, pm, pn = NULL)
forwardback(x, Pi, delta, distn, pm, pn = NULL, fortran = TRUE)
forwardback.dthmm(Pi, delta, prob, fortran = TRUE, fwd.only = FALSE)

`x`	is a vector of length n containing the observed process.
`Pi`	is the mm* transition probability matrix of the hidden Markov chain.
`delta`	is the marginal probability distribution of the m hidden states.
`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`).
`prob`	an nm* matrix containing the observation probabilities or densities (rows) by Markov state (columns).
`fortran`	logical, if `TRUE` (default) use the Fortran code, else use the R code.
`fwd.only`	logical, if `FALSE` (default) calculate both forward and backward probabilities; else calculate and return only forward probabilities and log-likelihood.

Denote the n*m matrices containing the forward and backward probabilities as A and B, respectively. Then the (i,j)th elements are

alpha_{ij} = Pr{ X_1 = x_1, ..., X_i = x_i, C_i = j }

and

beta_{ij} = Pr{ X_{i+1} = x_{i+1}, ..., X_n = x_n | C_i = j } .

Further, the diagonal elements of the product matrix AB' are all the same, taking the value of the log-likelihood.

The function forwardback returns a list with two matrices containing the forward and backward (log) probabilities, logalpha and logbeta, respectively, and the log-likelihood (LL).

The functions backward and forward return a matrix containing the forward and backward (log) probabilities, logalpha and logbeta, respectively.

The algorithm has been taken from Zucchini (2005).

Cited references are listed on the HiddenMarkov manual page.

logLik

#    Set Parameter Values

Pi <- matrix(c(1/2, 1/2,   0,   0,   0,
               1/3, 1/3, 1/3,   0,   0,
                 0, 1/3, 1/3, 1/3,   0,
                 0,   0, 1/3, 1/3, 1/3,
                 0,   0,   0, 1/2, 1/2),
             byrow=TRUE, nrow=5)

p <- c(1, 4, 2, 5, 3)
delta <- c(0, 1, 0, 0, 0)

#------   Poisson HMM   ------

x <- dthmm(NULL, Pi, delta, "pois", list(lambda=p), discrete=TRUE)

x <- simulate(x, nsim=10)

y <- forwardback(x$x, Pi, delta, "pois", list(lambda=p))

# below should be same as LL for all time points
print(log(diag(exp(y$logalpha) %*% t(exp(y$logbeta)))))
print(y$LL)

#------   Gaussian HMM   ------

x <- dthmm(NULL, Pi, delta, "norm", list(mean=p, sd=p/3))

x <- simulate(x, nsim=10)

y <- forwardback(x$x, Pi, delta, "norm", list(mean=p, sd=p/3))

# below should be same as LL for all time points
print(log(diag(exp(y$logalpha) %*% t(exp(y$logbeta)))))
print(y$LL)