Training of Hidden Markov Models
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.
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 )
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 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.
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.
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 , 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 .
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).
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 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)
MacDonald, I. L., Zucchini, W. (2009) Hidden Markov Models for Time Series: An Introduction Using R, Boca Raton: Chapman & Hall.
initial_parameter_training, Baum_Welch_algorithm, direct_numerical_maximization, AIC_HMM, BIC_HMM
Vitali Witowski (2013)