Viterbi_algorithm: Algorithm for Decoding Hidden Markov Models (global)
In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

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

The function decodes a trainded hidden Markov model into a most likely sequence of hidden states. Different to the local_decoding_algorithm, this algorithm determines the sequence of most likely hidden states for all time points simultaneously. See MacDonald & Zucchini (2009, Paragraph 5.3.2) for further details.

1 2	Viterbi_algorithm(x, m, delta, gamma, distribution_class, distribution_theta, discr_logL = FALSE, discr_logL_eps = 0.5)

`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.
`m`	a (finite) number of states in the hidden Markov chain.
`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 (`ncol=nrow=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 by this algorithm: Poisson (`pois`); generalized Poisson (`genpois`); normal (`norm`); geometric (`geom`).
`distribution_theta`	a list object containing the parameter values for the `m` observation distributions 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). Default is `FALSE`.
`discr_logL_eps`	a single numerical value to approximately determine the discrete log-likelihood for a hidden Markov model based on nomal distributions (for `"norm"`). The default value is `0.5`.

Viterbi_algorithm returns a list containing the following two components:

`omega`	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 probabilities (maximum probability to generate the first t members (t=1,...,T) of the given time-series x with the HMM and to stop in state i=1,...,m) calculated by the algorithm. See MacDonald & Zucchini (2009, Paragraph 5.3.2) for further details.
`decoding`	a numerical vector containing the globally most likely sequence of hidden states as decoded by the Viterbi algorithm.

The basic algorithm for a Poisson-HMM can be found in MacDonald & Zucchini (2009, Paragraph A.2.4). 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.

Forney, G.D. (1973). The Viterbi algorithm. Proceeding of the IEE, vol. 61(3), 268–278.

Viterbi, A.J. (1967). Error Bounds for concolutional codes and an asymptotically optimal decoding algorithm. Information Theory, IEEE Transactions on, vol. 13(2), 260–269.

local_decoding_algorithm, HMM_decoding

################################################################
### 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) 


### Train hidden Markov model for m=4

m_trained_HMM <- 
    HMM_training(x = x, 
      min_m = 4, 
      max_m = 4, 
      distribution_class="pois")$trained_HMM_with_selected_m

### Decode the trained HMM using the Viterbi algorithm to get 
### the globally most likely sequence of hidden states for 
### the time-series of observations
global_decoding <- 
    Viterbi_algorithm(x = x, 
      m = m_trained_HMM$m, 
      delta = m_trained_HMM$delta, 
      gamma = m_trained_HMM$gamma, 
      distribution_class = m_trained_HMM$distribution_class, 
      distribution_theta = m_trained_HMM$distribution_theta)

### Most likely sequence of hidden states
print(global_decoding$decoding)
plot(global_decoding$decoding)