HMMpa (version 1.0.2)

HMM_training: Training of Hidden Markov Models

Description

Function to estimate the model specific parameters (delta, gamma, distribution_theta) for a hidden Markov model, given a time-series and a user-defined distribution class. Can also be used for model selection (selecting the optimal number of states m). See Details for more information.

Usage

HMM_training(
  x,
  distribution_class,
  min_m = 2,
  max_m = 6,
  n = 100,
  training_method = "EM",
  discr_logL = FALSE,
  discr_logL_eps = 0.5,
  Mstep_numerical = FALSE,
  dynamical_selection = TRUE,
  BW_max_iter = 50,
  BW_limit_accuracy = 0.001,
  BW_print = TRUE,
  DNM_max_iter = 50,
  DNM_limit_accuracy = 0.001,
  DNM_print = 2
)

Value

HMM_training returns a list containing the following components:

trained_HMM_with_selected_m

a list object containing the key data of the optimal trained HMM (HMM with selected m) -- summarized output of the Baum_Welch_algorithm or
direct_numerical_maximization algorithm, respectively.

list_of_all_initial_parameters

a list object containing the plausible starting values for all HMMs (one for each state m).

list_of_all_trained_HMMs

a list object containing all trained m-state-HMMs. See Baum_Welch_algorithm or direct_numerical_maximization for training_method="EM" or training_method="numerical", respectively.

list_of_all_logLs_for_each_HMM_with_m_states

a list object containing all logarithmized Likelihoods of each trained HMM.

list_of_all_AICs_for_each_HMM_with_m_states

a list object containing the AIC values of all trained HMMs.

list_of_all_BICs_for_each_HMM_with_m_states

a list object containing the BIC values of all trained HMMs.

model_selection_over_AIC

is logical. TRUE, if model selection was based on AIC and FALSE, if model selection was based on BIC.

Arguments

x

a vector object of length T containing observations of a time-series x, which are assumed to be realizations of the (hidden Markov state dependent) observation process of the HMM.

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, only available for training_method="numerical"); normal (norm)).

min_m

minimum number of hidden states in the hidden Markov chain. Default value is 2.

max_m

maximum number of hidden states in the hidden Markov chain. Default value is 6.

n

a single numerical value specifying the number of samples to find the best starting values for the training algorithm. Default value is n=100.

training_method

a logical value indicating whether the Baum-Welch algorithm ("EM") or the method of direct numerical maximization ("numerical") should be applied for estimating the model specific parameters. See Baum_Welch_algorithm and direct_numerical_maximization for further details.

discr_logL

a logical object. Default is FALSE for the general log-likelihood, TRUE for the discrete log-likelihood (for distribution_class = "norm").

discr_logL_eps

a single numerical value, used to approximate the discrete log-likelihood for a hidden Markov model based on nomal distributions (for "norm"). The default value is 0.5.

Mstep_numerical

a logical object indicating whether the Maximization Step of the Baum-Welch algorithm should be performed by numerical maximization. Default is FALSE.

dynamical_selection

a logical value indicating whether the method of dynamical initial parameter selection should be applied (see Details). Default is TRUE.

BW_max_iter

a single numerical value representing the maximum number of iterations in the Baum-Welch algorithm. Default value is 50.

BW_limit_accuracy

a single numerical value representing the convergence criterion of the Baum-Welch algorithm. Default value is is 0.001.

BW_print

a logical object indicating whether the log-likelihood at each iteration-step shall be printed. Default is TRUE.

DNM_max_iter

a single numerical value representing the maximum number of iterations of the numerical maximization using the nlm-function (used to perform the Maximization Step of the Baum-Welch-algorithm). Default value is 50.

DNM_limit_accuracy

a single numerical value representing the convergence criterion of the numerical maximization algorithm using the nlm function (used to perform the Maximization Step of the Baum-Welch- algorithm). Default value is 0.001.

DNM_print

a single numerical value to determine the level of printing of the nlm-function. See nlm-function for further informations. The value 0 suppresses, that no printing will be outputted. Default value is 2 for full printing.

Author

Vitali Witowski (2013)

Details

More precisely, the function works as follows:
Step 1: In a first step, the algorithm estimates the model specific parameters for different values of m (indeed for min_m,...,max_m) using either the function Baum_Welch_algorithm or
direct_numerical_maximization. Therefore, the function first searches for plausible starting values by using the function initial_parameter_training. Step 2: In a second step, this function evaluates the AIC and BIC values for each HMM (built in Step 1) using the functions AIC_HMM and BIC_HMM. Then, based on that values, this function decides for the most plausible number of states m (respectively for the most appropriate HMM for the given time-series of observations). In case when AIC and BIC claim for a different m, the algorithm decides for the smaller value for m (with the background to have a more simplistic model). If the user is intereseted in having a HMM with a fixed number for m, min_m and max_m have to be chosen equally (for instance min_m=4 = max_m for a HMM with m=4 hidden states). To speed up the parameter estimation for each \(m > m_min\), the user can choose the method of dynamical initial parameter selection. If the method of dynamical intial parameter selection is not applied, the function initial_parameter_training will be called to find plausible starting values for each state \( m \in \{min_m, \ldots, max_m\}\).
If the method of dynamical intial parameter selection is applied, then starting parameter values using the function initial_parameter_training will be found only for the first HMM (respectively the HMM with m_min states). The further starting parameter values for the next HMM (with m+1 states and so on) are retained from the trained parameter values of the last HMM (with m states and so on).

References

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

See Also

initial_parameter_training, Baum_Welch_algorithm, direct_numerical_maximization, AIC_HMM, BIC_HMM

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 a poisson hidden Markov model using the Baum-Welch 
# algorithm for different number of states m=2,...,6
# \donttest{
 trained_HMMs <- 
  HMM_training(x = x, 
           min_m = 2, 
           max_m = 6, 
     distribution_class = "pois", 
     training_method = "EM")
     
# Various output values for the HMM 
names(trained_HMMs)

# Print details of the most plausible HMM for the given 
# time-series of observations
print(trained_HMMs$trained_HMM_with_selected_m)

# Print details of all trained HMMs (by this function) 
# for the given time-series of observations
print(trained_HMMs$list_of_all_trained_HMMs)

# Print the BIC-values of all trained HMMs for the given 
# time-series of observations  
print(trained_HMMs$list_of_all_BICs_for_each_HMM_with_m_states)

# Print the logL-values of all trained HMMs for the 
# given time-series of observations  
print(trained_HMMs$list_of_all_logLs_for_each_HMM_with_m_states)
# }