local_decoding_algorithm: Algorithm for Decoding Hidden Markov Models (local)
In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

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

The function decodes a hidden Markov model into a most likely sequence of hidden states. Different to the Viterbi_algorithm, this algorithm determines the most likely hidden state for each time point seperately.

1 2	local_decoding_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`.

local_decoding_algorithm returns a list containing the following two components:

`state_probabilities`	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 (conditional probability of a state i=1,...,m at a time point t=1,...,T given all observations x) calculated by the algorithm. See MacDonald & Zucchini (2009, Paragraph 5.3.1) for further details.
`decoding`	a numerical vector containing the locally most likely sequence of hidden states as decoded by the local_decoding_algorithm.

The basic algorithm for a Poisson-HMM can be found in MacDonald & Zucchini (2009, Paragraph A.2.6). 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.

Viterbi_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 local-decoding algorithm 
### to get the locally most likely sequence of hidden states 
### for the time-series of observations
local_decoding <- 
    local_decoding_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(local_decoding$decoding)
plot(local_decoding$decoding)