local_decoding_algorithm function

Algorithm for Decoding Hidden Markov Models (local)

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.

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

Arguments

• 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.

Returns

• 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.

Examples

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)

References

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

See Also

Viterbi_algorithm, HMM_decoding

Author(s)

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).