HMM_training: Training of Hidden Markov Models
In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models
HMM_training R Documentation
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
)
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.
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).
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.
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
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
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)
HMMpa documentation built on April 3, 2025, 10:29 p.m.
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| HMM_training | R Documentation |
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
)
Arguments
x |
a vector object of length |
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 ( |
min_m |
minimum number of hidden states in the hidden Markov chain.
Default value is |
max_m |
maximum number of hidden states in the hidden Markov chain.
Default value is |
n |
a single numerical value specifying the number of samples to find the best
starting values for the training algorithm. Default value is |
training_method |
a logical value indicating whether the Baum-Welch algorithm
( |
discr_logL |
a logical object. Default is |
discr_logL_eps |
a single numerical value, used to approximate the discrete
log-likelihood for a hidden Markov model based on nomal distributions
(for |
Mstep_numerical |
a logical object indicating whether the Maximization Step of
the Baum-Welch algorithm should be performed by numerical maximization.
Default is |
dynamical_selection |
a logical value indicating whether the method of dynamical
initial parameter selection should be applied (see Details). Default is |
BW_max_iter |
a single numerical value representing the maximum number of
iterations in the Baum-Welch algorithm. Default value is |
BW_limit_accuracy |
a single numerical value representing the convergence
criterion of the Baum-Welch algorithm. Default value is is |
BW_print |
a logical object indicating whether the log-likelihood at each
iteration-step shall be printed. Default is |
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 |
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 |
DNM_print |
a single numerical value to determine the level of printing of the
|
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).
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 theBaum_Welch_algorithmor
direct_numerical_maximizationalgorithm, 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_algorithmordirect_numerical_maximizationfortraining_method="EM"ortraining_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 andFALSE, if model selection was based on BIC.
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
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
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)
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