forward_backward_algorithm: Calculating Forward and Backward Probabilities and Likelihood
In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

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

The function calculates the logarithmized forward and backward probabilities and the logarithmized likelihood for a discrete time hidden Markov model, as defined in MacDonald & Zucchini (2009, Paragraph 3.1- Paragraph 3.3 and Paragraph 4.1).

forward_backward_algorithm(x, delta, gamma, distribution_class, 
distribution_theta, discr_logL=FALSE, discr_logL_eps=0.5) 

### Default for normal distributions 
### (calculate non-discrete likelihood)

`x`	a vector object containing the time-series of observations that are assumed to be realizations of the (hidden Markov state dependent) observation process of the model.
`delta`	a vector object containing values for the marginal probability distribution of the `m` states of the Markov chain at the time point `t=1`.
`gamma`	a matrix (`nrow=ncol=m`) containing values for the transition matrix of the hidden Markov chain.
`distribution_class`	a single character string object with the abbreviated name of the `m` observation distributions of the Markov dependent observation process. The following distributions are supported: Poisson (`pois`); generalized Poisson (`genpois`); normal (`norm`); geometric (`geom`).
`distribution_theta`	a list object containing the parameter values for the `m` observation distributions of the observation process that are dependent on the hidden Markov state.
`discr_logL`	a logical object. It is `TRUE` if the discrete log-likelihood shall be calculated (for `distribution_class="norm"` ) instead of the general log-likelihood. See MacDonald & Zucchini (2009, Paragraph 1.2.3) for further details. Default is `FALSE`.
`discr_logL_eps`	a single numerical value to approximately determine the discrete log-likelihood for a hidden Markov model based on normal distributions (for `"norm"`). The default value is `0.5`. See MacDonald & Zucchini (2009, Paragraph 1.2.3) for further details.

forward_backward_algorithm returns a list containing the logarithmized forward and backward probabilities and the logarithmized likelihood.

`log_alpha`	a (T,m)-matrix (when T indicates the length/size of the observation time-series and m the number of states of the HMM) containing the logarithmized forward probabilities.
`log_beta`	a (T,m)-matrix (when T indicates the length/size of the observation time-series and m the number of states of the HMM) containing the logarithmized backward probabilities.
`logL`	a single numerical value representing the logarithmized likelihood.
`logL_calculation`	a single character string object which indicates how `logL` has been calculated (see Zucchini (2009) Paragrahp 3.1-3.4, 4.1, A.2.2, A.2.3 for further details).

The basic algorithm for a Poisson-HMM is provided by MacDonald & Zucchini (2009, Paragraph A.2.2). Extension and implementation by Vitali Witowski (2013).

MacDonald, I. L., Zucchini, W. (2009) Hidden Markov Models for Time Series: An Introduction Using R, Boca Raton: Chapman & Hall.

HMM_based_method, HMM_training, Baum_Welch_algorithm, direct_numerical_maximization, initial_parameter_training

################################################################
### Fictitious observations ####################################
################################################################

x <- c(1,16,19,34,22,6,3,5,6,3,4,1,4,3,5,7,9,8,11,11,
  14,16,13,11,11,10,12,19,23,25,24,23,20,21,22,22,18,7,
  5,3,4,3,2,3,4,5,4,2,1,3,4,5,4,5,3,5,6,4,3,6,4,8,9,12,
  9,14,17,15,25,23,25,35,29,36,34,36,29,41,42,39,40,43,
  37,36,20,20,21,22,23,26,27,28,25,28,24,21,25,21,20,21,
  11,18,19,20,21,13,19,18,20,7,18,8,15,17,16,13,10,4,9,
  7,8,10,9,11,9,11,10,12,12,5,13,4,6,6,13,8,9,10,13,13,
  11,10,5,3,3,4,9,6,8,3,5,3,2,2,1,3,5,11,2,3,5,6,9,8,5,
  2,5,3,4,6,4,8,15,12,16,20,18,23,18,19,24,23,24,21,26,
  36,38,37,39,45,42,41,37,38,38,35,37,35,31,32,30,20,39,
  40,33,32,35,34,36,34,32,33,27,28,25,22,17,18,16,10,9,
  5,12,7,8,8,9,19,21,24,20,23,19,17,18,17,22,11,12,3,9,
  10,4,5,13,3,5,6,3,5,4,2,5,1,2,4,4,3,2,1) 


### Assummptions (number of states, probability vector, 
### transition matrix, and distribution parameters)

m <-4

delta <- c(0.25,0.25,0.25,0.25)

gamma <- 0.7 * diag(m) + rep(0.3 / m)

distribution_class <- "pois"

distribution_theta <- list(lambda = c(4,9,17,25))


### Calculating logarithmized forward/backward probabilities 
### and logarithmized likelihood

forward_and_backward_probabilities_and_logL <- 
    forward_backward_algorithm (x = x, 
        delta = delta, 
        gamma = gamma, 
        distribution_class = distribution_class, 
        distribution_theta = distribution_theta)

print(forward_and_backward_probabilities_and_logL)