mHMM: Multilevel hidden Markov model using Bayesian estimation
In mHMMbayes: Multilevel Hidden Markov Models Using Bayesian Estimation
mHMM R Documentation
Multilevel hidden Markov model using Bayesian estimation
Description
mHMM fits a multilevel (also known as mixed or random effects)
hidden Markov model (HMM) to intense longitudinal data with categorical
or continuous (i.e., normally distributed) observations of multiple
subjects using Bayesian estimation, and creates an object of class mHMM.
By using a multilevel framework, we allow for heterogeneity in the model
parameters between subjects, while estimating one overall HMM. The function
includes the possibility to add covariates at level 2 (i.e., at the subject
level) and have varying observation lengths over subjects. For a short
description of the package see mHMMbayes. See
vignette("tutorial-mhmm") for an introduction to multilevel hidden
Markov models and the package, and see vignette("estimation-mhmm") for
an overview of the used estimation algorithms.
Usage
mHMM(
s_data,
data_distr = "categorical",
gen,
xx = NULL,
start_val,
mcmc,
return_path = FALSE,
show_progress = TRUE,
gamma_hyp_prior = NULL,
emiss_hyp_prior = NULL,
gamma_sampler = NULL,
emiss_sampler = NULL
)
Arguments
s_data
A matrix containing the observations to be modeled, where the
rows represent the observations over time. In s_data, the first
column indicates subject id number. Hence, the id number is repeated over
rows equal to the number of observations for that subject. The subsequent
columns contain the dependent variable(s). Note that in case of categorical
dependent variable(s), input are integers (i.e., whole numbers) that range
from 1 to q_emiss (see below) and is numeric (i.e., not be a (set
of) factor variable(s)). The total number of rows are equal to the sum over
the number of observations of each subject, and the number of columns are
equal to the number of dependent variables (n_dep) + 1. The number
of observations can vary over subjects.
data_distr
String vector with length 1 describing the
observation type of the data. Currently supported are 'categorical'
and 'continuous'. Note that for multivariate data, all dependent
variables are assumed to be of the same observation type. The default
equals to data_distr = 'categorical'.
gen
List containing the following elements denoting the general model
properties:
m: numeric vector with length 1 denoting the number
of hidden states
n_dep: numeric vector with length 1 denoting the
number of dependent variables
q_emiss: only to be specified if the data represents
categorical data. Numeric vector with length n_dep denoting the
number of observed categories for the categorical emission distribution
for each of the dependent variables.
xx
An optional list of (level 2) covariates to predict the transition
matrix and/or the emission probabilities. Level 2 covariate(s) means that
there is one observation per subject of each covariate. The first element
in the list xx is used to predict the transition matrix. Subsequent
elements in the list are used to predict the emission distribution of (each
of) the dependent variable(s). Each element in the list is a matrix, with
the number of rows equal to the number of subjects. The first column of
each matrix represents the intercept, that is, a column only consisting of
ones. Subsequent columns correspond to covariates used to predict the
transition matrix / emission distribution. See Details for more
information on the use of covariates.
If xx is omitted completely, xx defaults to NULL,
resembling no covariates. Specific elements in the list can also be left
empty (i.e., set to NULL) to signify that either the transition
probability matrix or a specific emission distribution is not predicted by
covariates.
start_val
List containing the start values for the transition
probability matrix gamma and the emission distribution(s). The first
element of the list contains a m by m matrix with the start
values for gamma. The subsequent n_dep elements each contain a
matrix with the start values for the emission distribution(s). In case of
categorical observations: a m by q_emiss[k] matrix denoting
the probability of observing each categorical outcome (columns) within each
state (rows). In case of continuous observations: a m by 2 matrix
denoting the mean (first column) and standard deviation (second column) of
the Normal emission distribution within each state (rows). Note that
start_val should not contain nested lists (i.e., lists within
lists).
mcmc
List of Markov chain Monte Carlo (MCMC) arguments, containing the
following elements:
J: numeric vector with length 1 denoting the number
of iterations of the MCMC algorithm
burn_in: numeric vector with length 1 denoting the
burn-in period for the MCMC algorithm.
return_path
A logical scalar. Should the sampled state sequence
obtained at each iteration and for each subject be returned by the function
(sample_path = TRUE) or not (sample_path = FALSE). Note that
the sampled state sequence is quite a large object, hence the default
setting is sample_path = FALSE. Can be used for local decoding
purposes.
show_progress
A logical scaler. Should the function show a text
progress bar in the R console to represent the progress of the
algorithm (show_progress = TRUE) or not (show_progress =
FALSE). Defaults to show_progress = TRUE.
gamma_hyp_prior
An optional object of class mHMM_prior_gamma
containing user specified parameter values for the hyper-prior distribution
on the transition probability matrix gamma, generated by the function
prior_gamma.
emiss_hyp_prior
An object of the class mHMM_prior_emiss
containing user specified parameter values for the hyper-prior distribution
on the categorical or Normal emission distribution(s), generated by the
function prior_emiss_cat or prior_emiss_cont.
Mandatory in case of continuous observations, optional in case of
categorical observations.
gamma_sampler
An optional object of the class mHMM_pdRW_gamma
containing user specified settings for the proposal distribution of the
random walk (RW) Metropolis sampler on the subject level transition
probability matrix parameters, generated by the function
pd_RW_gamma.
emiss_sampler
An optional object of the class mHMM_pdRW_emiss
containing user specified settings for the proposal distribution of the
random walk (RW) Metropolis sampler on the subject level emission
distribution(s) parameters, generated by the function
pd_RW_emiss_cat. Only applicable in case of categorical
observations.
Details
Covariates specified in xx can either be dichotomous or continuous
variables. Dichotomous variables have to be coded as 0/1 variables.
Categorical or factor variables can as yet not be used as predictor
covariates. The user can however break up the categorical variable in
multiple dummy variables (i.e., dichotomous variables), which can be used
simultaneously in the analysis. Continuous predictors are automatically
centered. That is, the mean value of the covariate is subtracted from all
values of the covariate such that the new mean equals zero. This is done such
that the presented probabilities in the output (i.e., for the population
transition probability matrix and population emission probabilities)
correspond to the predicted probabilities at the average value of the
covariate(s).
Value
mHMM returns an object of class mHMM, which has
print and summary methods to see the results.
The object contains always the following components:
PD_subjA list containing one list per subject with the
elements trans_prob, cat_emiss or cont_emiss in case
of categorical or continuous observations, respectively, and
log_likl, providing the subject parameter estimates over the
iterations of the MCMC sampler. trans_prob relates to the transition
probabilities gamma, cat_emiss to the categorical emission
distribution (emission probabilities), cont_emiss to the continuous
emission distributions (subsequently the the emission means and the (fixed
over subjects) emission standard deviation), and log_likl to the log
likelihood over the MCMC iterations. Iterations are contained in the rows,
the parameters in the columns.
gamma_prob_barA matrix containing the group level parameter
estimates of the transition probabilities over the iterations of the hybrid
Metropolis within Gibbs sampler. The iterations of the sampler are
contained in the rows, and the columns contain the group level parameter
estimates. If covariates were included in the analysis, the group level
probabilities represent the predicted probability given that the covariate
is at the average value for continuous covariates, or given that the
covariate equals zero for dichotomous covariates.
gamma_int_barA matrix containing the group level intercepts
of the Multinomial logistic regression modeling the transition
probabilities over the iterations of the hybrid Metropolis within Gibbs
sampler. The iterations of the sampler are contained in the rows, and the
columns contain the group level intercepts.
gamma_cov_barA matrix containing the group level regression
coefficients of the Multinomial logistic regression predicting the
transition probabilities over the iterations of the hybrid Metropolis within
Gibbs sampler. The iterations of the sampler are contained in the rows, and
the columns contain the group level regression coefficients.
gamma_int_subjA list containing one matrix per subject
denoting the subject level intercepts of the Multinomial logistic
regression modeling the transition probabilities over the iterations of the
hybrid Metropolis within Gibbs sampler. The iterations of the sampler are
contained in the rows, and the columns contain the subject level
intercepts.
gamma_nacceptA matrix containing the number of accepted
draws at the subject level RW Metropolis step for each set of parameters of
the transition probabilities. The subjects are contained in the rows, and
the columns contain the sets of parameters.
inputOverview of used input specifications: the distribution
type of the observations data_distr, the number of
states m, the number of used dependent variables n_dep, (in
case of categorical observations) the number of output categories for each
of the dependent variables q_emiss, the number of iterations
J and the specified burn in period burn_in of the hybrid
Metropolis within Gibbs sampler, the number of subjects n_subj, the
observation length for each subject n_vary, and the column names of
the dependent variables dep_labels.
sample_pathA list containing one matrix per subject with the
sampled hidden state sequence over the hybrid Metropolis within Gibbs
sampler. The time points of the dataset are contained in the rows, and the
sampled paths over the iterations are contained in the columns. Only
returned if return_path = TRUE.
Additionally, in case of categorical observations, the mHMM return object
contains:
emiss_prob_barA list containing one matrix per dependent
variable, denoting the group level emission probabilities of each dependent
variable over the iterations of the hybrid Metropolis within Gibbs sampler.
The iterations of the sampler are contained in the rows of the matrix, and
the columns contain the group level emission probabilities. If covariates
were included in the analysis, the group level probabilities represent the
predicted probability given that the covariate is at the average value for
continuous covariates, or given that the covariate equals zero for
dichotomous covariates.
emiss_int_barA list containing one matrix per dependent
variable, denoting the group level intercepts of each dependent variable of
the Multinomial logistic regression modeling the probabilities of the
emission distribution over the iterations of the hybrid Metropolis within
Gibbs sampler. The iterations of the sampler are contained in the rows of
the matrix, and the columns contain the group level intercepts.
emiss_cov_barA list containing one matrix per dependent
variable, denoting the group level regression coefficients of the
Multinomial logistic regression predicting the emission probabilities within
each of the dependent variables over the iterations of the hybrid
Metropolis within Gibbs sampler. The iterations of the sampler are
contained in the rows of the matrix, and the columns contain the group
level regression coefficients.
emiss_int_subjA list containing one list per subject denoting
the subject level intercepts of each dependent variable of the Multinomial
logistic regression modeling the probabilities of the emission distribution
over the iterations of the hybrid Metropolis within Gibbs sampler. Each
lower level list contains one matrix per dependent variable, in which
iterations of the sampler are contained in the rows, and the columns
contain the subject level intercepts.
emiss_nacceptA list containing one matrix per dependent
variable with the number of accepted draws at the subject level RW
Metropolis step for each set of parameters of the emission distribution.
The subjects are contained in the rows, and the columns of the matrix
contain the sets of parameters.
In case of continuous observations, the mHMM return object contains:
emiss_mu_barA list containing one matrix per dependent
variable, denoting the group level means of the Normal emission
distribution of each dependent variable over the iterations of the Gibbs
sampler. The iterations of the sampler are contained in the rows of the
matrix, and the columns contain the group level emission means. If
covariates were included in the analysis, the group level means represent
the predicted mean given that the covariate is at the average value for
continuous covariates, or given that the covariate equals zero for
dichotomous covariates.
emiss_varmu_barA list containing one matrix per dependent
variable, denoting the variance between the subject level means of the
Normal emission distributions. over the iterations of the Gibbs sampler. The
iterations of the sampler are contained in the rows of the matrix, and the
columns contain the group level variance in the mean.
emiss_sd_barA list containing one matrix per dependent
variable, denoting the (fixed over subjects) standard deviation of the
Normal emission distributions over the iterations of the Gibbs sampler. The
iterations of the sampler are contained in the rows of the matrix, and the
columns contain the group level emission variances.
emiss_cov_barA list containing one matrix per dependent
variable, denoting the group level regression coefficients predicting the
emission means within each of the dependent variables over the iterations
of the Gibbs sampler. The iterations of the sampler are contained in the
rows of the matrix, and the columns contain the group level regression
coefficients.
label_switchA matrix of m rows and n_dep
columns containing the percentage of times the group mean of the emission
distribution of state i was sampled to be a smaller value compared to
the group mean of of the emission distribution of state i-1. If the
state dependent means of the emission distributions were given in a ranked
order (low to high) to both the start values and hyper-priors, a high
percentage in label_switch indicates that label switching possibly
poses a problem in the analysis, and further diagnostics (e.g.,
traceplots and posterior distributions) should be inspected.
inputOverview of used input specifications: the number of
states m, the number of used dependent variables n_dep, the
number of iterations J and the specified burn in period
burn_in of the hybrid Metropolis within Gibbs sampler, the number of
subjects n_subj, the observation length for each subject
n_vary, and the column names of the dependent variables
dep_labels.
References
\insertRefrabiner1989mHMMbayes
\insertRefscott2002mHMMbayes
\insertRefaltman2007mHMMbayes
\insertRefrossi2012mHMMbayes
\insertRefzucchini2017mHMMbayes
See Also
sim_mHMM for simulating multilevel hidden Markov data,
vit_mHMM for obtaining the most likely hidden state sequence
for each subject using the Viterbi algorithm, obtain_gamma
and obtain_emiss for obtaining the transition or emission
distribution probabilities of a fitted model at the group or subject level,
and plot.mHMM for plotting the posterior densities of a
fitted model.
Examples
###### Example on package (categorical) example data, see ?nonverbal
# specifying general model properties:
m <- 2
n_dep <- 4
q_emiss <- c(3, 2, 3, 2)
# specifying starting values
start_TM <- diag(.8, m)
start_TM[lower.tri(start_TM) | upper.tri(start_TM)] <- .2
start_EM <- list(matrix(c(0.05, 0.90, 0.05,
0.90, 0.05, 0.05), byrow = TRUE,
nrow = m, ncol = q_emiss[1]), # vocalizing patient
matrix(c(0.1, 0.9,
0.1, 0.9), byrow = TRUE, nrow = m,
ncol = q_emiss[2]), # looking patient
matrix(c(0.90, 0.05, 0.05,
0.05, 0.90, 0.05), byrow = TRUE,
nrow = m, ncol = q_emiss[3]), # vocalizing therapist
matrix(c(0.1, 0.9,
0.1, 0.9), byrow = TRUE, nrow = m,
ncol = q_emiss[4])) # looking therapist
# Run a model without covariate(s):
# Note that for reasons of running time, J is set at a ridiculous low value.
# One would typically use a number of iterations J of at least 1000,
# and a burn_in of 200.
out_2st <- mHMM(s_data = nonverbal,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EM),
mcmc = list(J = 11, burn_in = 5))
out_2st
summary(out_2st)
# plot the posterior densities for the transition and emission probabilities
plot(out_2st, component = "gamma", col =c("darkslategray3", "goldenrod"))
# Run a model including a covariate (see ?nonverbal_cov) to predict the
# emission distribution for each of the 4 dependent variables:
n_subj <- 10
xx_emiss <- rep(list(matrix(c(rep(1, n_subj),nonverbal_cov$std_CDI_change),
ncol = 2, nrow = n_subj)), n_dep)
xx <- c(list(matrix(1, ncol = 1, nrow = n_subj)), xx_emiss)
out_2st_c <- mHMM(s_data = nonverbal, xx = xx,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EM),
mcmc = list(J = 11, burn_in = 5))
###### Example on categorical simulated data
# Simulate data for 10 subjects with each 100 observations:
n_t <- 100
n <- 10
m <- 2
n_dep <- 1
q_emiss <- 3
gamma <- matrix(c(0.8, 0.2,
0.3, 0.7), ncol = m, byrow = TRUE)
emiss_distr <- list(matrix(c(0.5, 0.5, 0.0,
0.1, 0.1, 0.8), nrow = m, ncol = q_emiss, byrow = TRUE))
data1 <- sim_mHMM(n_t = n_t, n = n, gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
gamma = gamma, emiss_distr = emiss_distr, var_gamma = .5, var_emiss = .5)
# Specify remaining required analysis input (for the example, we use simulation
# input as starting values):
n_dep <- 1
q_emiss <- 3
# Run the model on the simulated data:
out_2st_sim <- mHMM(s_data = data1$obs,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(gamma), emiss_distr),
mcmc = list(J = 11, burn_in = 5))
###### Example on continuous simulated data
# simulating multivariate continuous data
n_t <- 100
n <- 10
m <- 3
n_dep <- 2
gamma <- matrix(c(0.8, 0.1, 0.1,
0.2, 0.7, 0.1,
0.2, 0.2, 0.6), ncol = m, byrow = TRUE)
emiss_distr <- list(matrix(c( 50, 10,
100, 10,
150, 10), nrow = m, byrow = TRUE),
matrix(c(5, 2,
10, 5,
20, 3), nrow = m, byrow = TRUE))
data_cont <- sim_mHMM(n_t = n_t, n = n, data_distr = 'continuous', gen = list(m = m, n_dep = n_dep),
gamma = gamma, emiss_distr = emiss_distr, var_gamma = .1, var_emiss = c(.5, 0.01))
# Specify hyper-prior for the continuous emission distribution
manual_prior_emiss <- prior_emiss_cont(
gen = list(m = m, n_dep = n_dep),
emiss_mu0 = list(matrix(c(30, 70, 170), nrow = 1),
matrix(c(7, 8, 18), nrow = 1)),
emiss_K0 = list(1, 1),
emiss_V = list(rep(100, m), rep(25, m)),
emiss_nu = list(1, 1),
emiss_a0 = list(rep(1, m), rep(1, m)),
emiss_b0 = list(rep(1, m), rep(1, m)))
# Run the model on the simulated data:
# Note that for reasons of running time, J is set at a ridiculous low value.
# One would typically use a number of iterations J of at least 1000,
# and a burn_in of 200.
out_3st_cont_sim <- mHMM(s_data = data_cont$obs,
data_distr = 'continuous',
gen = list(m = m, n_dep = n_dep),
start_val = c(list(gamma), emiss_distr),
emiss_hyp_prior = manual_prior_emiss,
mcmc = list(J = 11, burn_in = 5))
mHMMbayes documentation built on Oct. 2, 2023, 5:06 p.m.
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| mHMM | R Documentation |
Multilevel hidden Markov model using Bayesian estimation
Description
mHMM fits a multilevel (also known as mixed or random effects)
hidden Markov model (HMM) to intense longitudinal data with categorical
or continuous (i.e., normally distributed) observations of multiple
subjects using Bayesian estimation, and creates an object of class mHMM.
By using a multilevel framework, we allow for heterogeneity in the model
parameters between subjects, while estimating one overall HMM. The function
includes the possibility to add covariates at level 2 (i.e., at the subject
level) and have varying observation lengths over subjects. For a short
description of the package see mHMMbayes. See
vignette("tutorial-mhmm") for an introduction to multilevel hidden
Markov models and the package, and see vignette("estimation-mhmm") for
an overview of the used estimation algorithms.
Usage
mHMM(
s_data,
data_distr = "categorical",
gen,
xx = NULL,
start_val,
mcmc,
return_path = FALSE,
show_progress = TRUE,
gamma_hyp_prior = NULL,
emiss_hyp_prior = NULL,
gamma_sampler = NULL,
emiss_sampler = NULL
)
Arguments
s_data |
A matrix containing the observations to be modeled, where the
rows represent the observations over time. In |
data_distr |
String vector with length 1 describing the
observation type of the data. Currently supported are |
gen |
List containing the following elements denoting the general model properties:
|
xx |
An optional list of (level 2) covariates to predict the transition
matrix and/or the emission probabilities. Level 2 covariate(s) means that
there is one observation per subject of each covariate. The first element
in the list If |
start_val |
List containing the start values for the transition
probability matrix gamma and the emission distribution(s). The first
element of the list contains a |
mcmc |
List of Markov chain Monte Carlo (MCMC) arguments, containing the following elements:
|
return_path |
A logical scalar. Should the sampled state sequence
obtained at each iteration and for each subject be returned by the function
( |
show_progress |
A logical scaler. Should the function show a text
progress bar in the |
gamma_hyp_prior |
An optional object of class |
emiss_hyp_prior |
An object of the class |
gamma_sampler |
An optional object of the class |
emiss_sampler |
An optional object of the class |
Details
Covariates specified in xx can either be dichotomous or continuous
variables. Dichotomous variables have to be coded as 0/1 variables.
Categorical or factor variables can as yet not be used as predictor
covariates. The user can however break up the categorical variable in
multiple dummy variables (i.e., dichotomous variables), which can be used
simultaneously in the analysis. Continuous predictors are automatically
centered. That is, the mean value of the covariate is subtracted from all
values of the covariate such that the new mean equals zero. This is done such
that the presented probabilities in the output (i.e., for the population
transition probability matrix and population emission probabilities)
correspond to the predicted probabilities at the average value of the
covariate(s).
Value
mHMM returns an object of class mHMM, which has
print and summary methods to see the results.
The object contains always the following components:
PD_subjA list containing one list per subject with the elements
trans_prob,cat_emissorcont_emissin case of categorical or continuous observations, respectively, andlog_likl, providing the subject parameter estimates over the iterations of the MCMC sampler.trans_probrelates to the transition probabilities gamma,cat_emissto the categorical emission distribution (emission probabilities),cont_emissto the continuous emission distributions (subsequently the the emission means and the (fixed over subjects) emission standard deviation), andlog_liklto the log likelihood over the MCMC iterations. Iterations are contained in the rows, the parameters in the columns.gamma_prob_barA matrix containing the group level parameter estimates of the transition probabilities over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows, and the columns contain the group level parameter estimates. If covariates were included in the analysis, the group level probabilities represent the predicted probability given that the covariate is at the average value for continuous covariates, or given that the covariate equals zero for dichotomous covariates.
gamma_int_barA matrix containing the group level intercepts of the Multinomial logistic regression modeling the transition probabilities over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows, and the columns contain the group level intercepts.
gamma_cov_barA matrix containing the group level regression coefficients of the Multinomial logistic regression predicting the transition probabilities over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows, and the columns contain the group level regression coefficients.
gamma_int_subjA list containing one matrix per subject denoting the subject level intercepts of the Multinomial logistic regression modeling the transition probabilities over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows, and the columns contain the subject level intercepts.
gamma_nacceptA matrix containing the number of accepted draws at the subject level RW Metropolis step for each set of parameters of the transition probabilities. The subjects are contained in the rows, and the columns contain the sets of parameters.
inputOverview of used input specifications: the distribution type of the observations
data_distr, the number of statesm, the number of used dependent variablesn_dep, (in case of categorical observations) the number of output categories for each of the dependent variablesq_emiss, the number of iterationsJand the specified burn in periodburn_inof the hybrid Metropolis within Gibbs sampler, the number of subjectsn_subj, the observation length for each subjectn_vary, and the column names of the dependent variablesdep_labels.sample_pathA list containing one matrix per subject with the sampled hidden state sequence over the hybrid Metropolis within Gibbs sampler. The time points of the dataset are contained in the rows, and the sampled paths over the iterations are contained in the columns. Only returned if
return_path = TRUE.
Additionally, in case of categorical observations, the mHMM return object
contains:
emiss_prob_barA list containing one matrix per dependent variable, denoting the group level emission probabilities of each dependent variable over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level emission probabilities. If covariates were included in the analysis, the group level probabilities represent the predicted probability given that the covariate is at the average value for continuous covariates, or given that the covariate equals zero for dichotomous covariates.
emiss_int_barA list containing one matrix per dependent variable, denoting the group level intercepts of each dependent variable of the Multinomial logistic regression modeling the probabilities of the emission distribution over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level intercepts.
emiss_cov_barA list containing one matrix per dependent variable, denoting the group level regression coefficients of the Multinomial logistic regression predicting the emission probabilities within each of the dependent variables over the iterations of the hybrid Metropolis within Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level regression coefficients.
emiss_int_subjA list containing one list per subject denoting the subject level intercepts of each dependent variable of the Multinomial logistic regression modeling the probabilities of the emission distribution over the iterations of the hybrid Metropolis within Gibbs sampler. Each lower level list contains one matrix per dependent variable, in which iterations of the sampler are contained in the rows, and the columns contain the subject level intercepts.
emiss_nacceptA list containing one matrix per dependent variable with the number of accepted draws at the subject level RW Metropolis step for each set of parameters of the emission distribution. The subjects are contained in the rows, and the columns of the matrix contain the sets of parameters.
In case of continuous observations, the mHMM return object contains:
emiss_mu_barA list containing one matrix per dependent variable, denoting the group level means of the Normal emission distribution of each dependent variable over the iterations of the Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level emission means. If covariates were included in the analysis, the group level means represent the predicted mean given that the covariate is at the average value for continuous covariates, or given that the covariate equals zero for dichotomous covariates.
emiss_varmu_barA list containing one matrix per dependent variable, denoting the variance between the subject level means of the Normal emission distributions. over the iterations of the Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level variance in the mean.
emiss_sd_barA list containing one matrix per dependent variable, denoting the (fixed over subjects) standard deviation of the Normal emission distributions over the iterations of the Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level emission variances.
emiss_cov_barA list containing one matrix per dependent variable, denoting the group level regression coefficients predicting the emission means within each of the dependent variables over the iterations of the Gibbs sampler. The iterations of the sampler are contained in the rows of the matrix, and the columns contain the group level regression coefficients.
label_switchA matrix of
mrows andn_depcolumns containing the percentage of times the group mean of the emission distribution of stateiwas sampled to be a smaller value compared to the group mean of of the emission distribution of statei-1. If the state dependent means of the emission distributions were given in a ranked order (low to high) to both the start values and hyper-priors, a high percentage inlabel_switchindicates that label switching possibly poses a problem in the analysis, and further diagnostics (e.g., traceplots and posterior distributions) should be inspected.inputOverview of used input specifications: the number of states
m, the number of used dependent variablesn_dep, the number of iterationsJand the specified burn in periodburn_inof the hybrid Metropolis within Gibbs sampler, the number of subjectsn_subj, the observation length for each subjectn_vary, and the column names of the dependent variablesdep_labels.
References
\insertRefrabiner1989mHMMbayes
\insertRefscott2002mHMMbayes
\insertRefaltman2007mHMMbayes
\insertRefrossi2012mHMMbayes
\insertRefzucchini2017mHMMbayes
See Also
sim_mHMM for simulating multilevel hidden Markov data,
vit_mHMM for obtaining the most likely hidden state sequence
for each subject using the Viterbi algorithm, obtain_gamma
and obtain_emiss for obtaining the transition or emission
distribution probabilities of a fitted model at the group or subject level,
and plot.mHMM for plotting the posterior densities of a
fitted model.
Examples
###### Example on package (categorical) example data, see ?nonverbal
# specifying general model properties:
m <- 2
n_dep <- 4
q_emiss <- c(3, 2, 3, 2)
# specifying starting values
start_TM <- diag(.8, m)
start_TM[lower.tri(start_TM) | upper.tri(start_TM)] <- .2
start_EM <- list(matrix(c(0.05, 0.90, 0.05,
0.90, 0.05, 0.05), byrow = TRUE,
nrow = m, ncol = q_emiss[1]), # vocalizing patient
matrix(c(0.1, 0.9,
0.1, 0.9), byrow = TRUE, nrow = m,
ncol = q_emiss[2]), # looking patient
matrix(c(0.90, 0.05, 0.05,
0.05, 0.90, 0.05), byrow = TRUE,
nrow = m, ncol = q_emiss[3]), # vocalizing therapist
matrix(c(0.1, 0.9,
0.1, 0.9), byrow = TRUE, nrow = m,
ncol = q_emiss[4])) # looking therapist
# Run a model without covariate(s):
# Note that for reasons of running time, J is set at a ridiculous low value.
# One would typically use a number of iterations J of at least 1000,
# and a burn_in of 200.
out_2st <- mHMM(s_data = nonverbal,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EM),
mcmc = list(J = 11, burn_in = 5))
out_2st
summary(out_2st)
# plot the posterior densities for the transition and emission probabilities
plot(out_2st, component = "gamma", col =c("darkslategray3", "goldenrod"))
# Run a model including a covariate (see ?nonverbal_cov) to predict the
# emission distribution for each of the 4 dependent variables:
n_subj <- 10
xx_emiss <- rep(list(matrix(c(rep(1, n_subj),nonverbal_cov$std_CDI_change),
ncol = 2, nrow = n_subj)), n_dep)
xx <- c(list(matrix(1, ncol = 1, nrow = n_subj)), xx_emiss)
out_2st_c <- mHMM(s_data = nonverbal, xx = xx,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EM),
mcmc = list(J = 11, burn_in = 5))
###### Example on categorical simulated data
# Simulate data for 10 subjects with each 100 observations:
n_t <- 100
n <- 10
m <- 2
n_dep <- 1
q_emiss <- 3
gamma <- matrix(c(0.8, 0.2,
0.3, 0.7), ncol = m, byrow = TRUE)
emiss_distr <- list(matrix(c(0.5, 0.5, 0.0,
0.1, 0.1, 0.8), nrow = m, ncol = q_emiss, byrow = TRUE))
data1 <- sim_mHMM(n_t = n_t, n = n, gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
gamma = gamma, emiss_distr = emiss_distr, var_gamma = .5, var_emiss = .5)
# Specify remaining required analysis input (for the example, we use simulation
# input as starting values):
n_dep <- 1
q_emiss <- 3
# Run the model on the simulated data:
out_2st_sim <- mHMM(s_data = data1$obs,
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(gamma), emiss_distr),
mcmc = list(J = 11, burn_in = 5))
###### Example on continuous simulated data
# simulating multivariate continuous data
n_t <- 100
n <- 10
m <- 3
n_dep <- 2
gamma <- matrix(c(0.8, 0.1, 0.1,
0.2, 0.7, 0.1,
0.2, 0.2, 0.6), ncol = m, byrow = TRUE)
emiss_distr <- list(matrix(c( 50, 10,
100, 10,
150, 10), nrow = m, byrow = TRUE),
matrix(c(5, 2,
10, 5,
20, 3), nrow = m, byrow = TRUE))
data_cont <- sim_mHMM(n_t = n_t, n = n, data_distr = 'continuous', gen = list(m = m, n_dep = n_dep),
gamma = gamma, emiss_distr = emiss_distr, var_gamma = .1, var_emiss = c(.5, 0.01))
# Specify hyper-prior for the continuous emission distribution
manual_prior_emiss <- prior_emiss_cont(
gen = list(m = m, n_dep = n_dep),
emiss_mu0 = list(matrix(c(30, 70, 170), nrow = 1),
matrix(c(7, 8, 18), nrow = 1)),
emiss_K0 = list(1, 1),
emiss_V = list(rep(100, m), rep(25, m)),
emiss_nu = list(1, 1),
emiss_a0 = list(rep(1, m), rep(1, m)),
emiss_b0 = list(rep(1, m), rep(1, m)))
# Run the model on the simulated data:
# Note that for reasons of running time, J is set at a ridiculous low value.
# One would typically use a number of iterations J of at least 1000,
# and a burn_in of 200.
out_3st_cont_sim <- mHMM(s_data = data_cont$obs,
data_distr = 'continuous',
gen = list(m = m, n_dep = n_dep),
start_val = c(list(gamma), emiss_distr),
emiss_hyp_prior = manual_prior_emiss,
mcmc = list(J = 11, burn_in = 5))
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