| forwardback {HiddenMarkov} | R Documentation |
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)
x |
is a vector of length n containing the observed process. |
Pi |
is the m times m 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). |
fortran |
logical, if TRUE (default) use the Fortran code, else use the R code. |
Denote the n times m matrices containing the forward and backward probabilities as A and B, respectively. Then the (i,j)th elements are
α_{ij} = Pr{ X_1 = x_1, cdots, X_i = x_i, C_i = j }
and
β_{ij} = Pr{ X_{i+1} = x_{i+1}, cdots, X_n = x_n ,|, C_i = j } ,.
Further, the diagonal elements of the product matrix A B^prime 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 probabilities, logalpha and logbeta, respectively, and the log-likelihood (LL).
The functions backward and forward return a matrix containing the forward and backward probabilities, logalpha and logbeta, respectively.
David Harte, 2005. The algorithm has been taken from Zucchini (2005).
MacDonald, I.L. & Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-valued Time Series. Chapman and Hall/CRC, Boca Raton.
Zucchini, W. (2005). Hidden Markov Models Short Course, 3–4 April 2005. Macquarie University, Sydney.
# 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)