HMM_decoding: Algorithm for Decoding Hidden Markov Models (local or global)

In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models

Algorithm for Decoding Hidden Markov Models (local or global)

Description

The function decodes a hidden Markov model into a most likely sequence of hidden states. Furthermore this function provides estimated observation values along the most likely sequence of hidden states. See Details for more information.

Usage

HMM_decoding(
  x,
  m,
  delta,
  gamma,
  distribution_class,
  distribution_theta,
  decoding_method = "global",
  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	integer; (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.
decoding_method	a string object to choose the applied decoding-method to decode the HMM given the time-series of observations x. Possible values are "global" (for the use of the Viterbi_algorithm) and "local" (for the use of the local_decoding_algorithm). Default value is "global".
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.

Details

More precisely, the function works as follows:

Step 1: In a first step, the algorithm decodes a HMM into the most likely sequence of hidden states, given a time-series of observations. The user can choose between a global and a local approch.
If decoding_method="global" is applied, the function calls Viterbi_algorithm to determine the sequence of most likely hidden states for all time points simultaneously.
If decoding_method="local" is applied, the function calls local_decoding_algorithm to determine the most likely hidden state for each time point seperately.

Step 2: In a second step, this function links each observation to the mean of the distribution, that corresponds to the decoded state at this point in time.

Value

HMM_decoding returns a list containing the following two components:

decoding_method

a string object indicating the applied decoding method.

decoding

a numerical vector containing the most likely sequence of hidden states as decoded by the Viterbi_algorithm (if "global" was applied) or by the local_decoding_algorithm (if "local" was applied).

decoding_distr_means

a numerical vector of estimated oberservation values along the most likely seuquence of hidden states (see decoding and Step 2).

Author(s)

Vitali Witowski (2013).

References

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

See Also

local_decoding_algorithm, Viterbi_algorithm

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) 
  
 # Set graphical parameters 
 old.par <- par(no.readonly = TRUE) 
 par(mfrow = c(1,1)) 

# i) Train hidden Markov model ----- 
#    for different number of states m=2,...,6 and select the optimal model

 m_trained_HMM <- 
    HMM_training(x = x, 
             min_m = 2, 
             max_m = 6, 
     distribution_class = "pois")$trained_HMM_with_selected_m

# ii) Global decoding -----
# Decode the trained HMM using the Viterbi algorithm to get 
# the estimated sequence of hidden physical activity levels
global_decoding <- 
    HMM_decoding(
        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,
        decoding_method = "global")
        
# Globally most likely sequence of hidden states, 
# i.e. in this case sequence of activity levels

global_decoding$decoding
plot(global_decoding$decoding)

# Plot the observed impulse counts and the most likely 
# sequence (green) according to the Viterbi algorithm that 
# generated these observations

plot(x)
lines(global_decoding$decoding_distr_means, col = "green")

# iii) Local decoding 
# Decode the trained HMM using the local decoding algorithm 
# to get the estimated sequence of hidden physical activity levels

local_decoding <- 
   HMM_decoding(
      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,
      decoding_method = "local")
      
 # Locally most likely sequence of hidden states, 
 # i.e. in this case sequence of activity levels
 # local_decoding$decoding
 plot(local_decoding$decoding)
 
 # Plot the observed impulse counts and the most likely 
 # sequence (green) according to the local decoding algorithm 
 # that generated these observations
 plot(x)
 lines(local_decoding$decoding_distr_means, col = "red")
 
 # iv) Comparison of global and local decoding -----
 # Comparison of global decoding (green), local decoding (red) 
 # and the connection to the closest mean (blue)
 print(global_decoding$decoding)  
 print(local_decoding$decoding)
 
 # Plot comparison 
 par(mfrow = c(2,2))
 plot(global_decoding$decoding[seq(230,260)], col = "green", 
      ylab = "global decoding", 
      main = "(zooming)")
 plot(x[seq(230,260)], ylab = "global decoding", 
 main = "(zooming x[seq(230,260)])")
 lines(global_decoding$decoding_distr_means[seq(230,260)], col = "green")
 plot(local_decoding$decoding[seq(230,260)], col = "red", 
      ylab = "local decoding", main = "(zooming)")
 plot(x[seq(230,260)], ylab = "local decoding", 
 main = "(zooming x[seq(230,260)])")
 lines(local_decoding$decoding_distr_means[seq(230,260)], col = "red")
 
 par(old.par)
 

HMMpa documentation built on April 3, 2025, 10:29 p.m.