# R/sim_mHMM.R In mHMMbayes: Multilevel Hidden Markov Models Using Bayesian Estimation

#### Documented in sim_mHMM

#' Simulate data using a multilevel hidden Markov model
#'
#' \code{sim_mHMM} simulates data for multiple subjects, for which the data have
#' categorical observations that follow a hidden Markov model (HMM) with an
#' multilevel structure. The multilevel structure implies that each subject is
#' allowed to have its own set of parameters, and that the parameters at the
#' subject level (level 1) are tied together by a population distribution at
#' level 2 for each of the corresponding parameters. The shape of the population
#' distribution for each of the parameters is a normal (i.e., Gaussian)
#' distribution. In addition to (natural and/or unexplained) heterogeneity
#' between subjects, the subjects parameters can also depend on a (set of)
#' covariate(s).
#'
#' In simulating the data, having a multilevel structure means that the
#' parameters for each subject are sampled from the population level
#' distribution of the corresponding parameter. The user specifies the
#' population distribution for each parameter: the average population transition
#' probability matrix and its variance, and the average population emission
#' distribution and its variance. For now, the variance is assumed fixed for all
#' components of the transition probability matrix and for all components of the
#' emission distribution, and the simulated data can only consist of one
#' dependent variable. In addition, at this point only one dependent variable can
#' be simulated. That is, the hidden Markov model is a univariate hidden Markov
#' model.
#'
#' Note: the subject specific) initial state distributions (i.e., the
#' probability of each of the states at the first time point) needed to simulate
#' the data are obtained from the stationary distributions of the subject
#' specific transition probability matrices gamma.
#'
#'
#' \code{beta}: As the first element in each row of \code{gamma} is used as reference category
#'   in the multinomial logistic regression, the first matrix in the list \code{beta} used to predict
#' transition probability matrix \code{gamma} has a number of rows equal to \code{m}
#' and the number of columns equal to \code{m} - 1. The first element in the
#' first row corresponds to the probability of switching from state one to state
#' two. The second element in the first row corresponds to the probability of
#' switching from state one to state three, and so on. The last element in the
#' first row corresponds to the probability of switching from state one to the
#' last state. The same principle holds for the second matrix in the list
#' \code{beta} used to predict the emission distribution \code{emiss_distr}: the first element
#' in the first row corresponds to the probability of observing category two in
#' state one. The second element in the first row corresponds to the probability
#' of observing category three is state one, and so on. The last element in the
#' first row corresponds to the probability of observing the last category in
#' state one.
#'
#'
#' @param n_t The length of the observed sequence to be simulated for each
#'   subject. To only simulate subject specific transition probability matrices
#'   gamma and emission distributions (and no data), set \code{t} to 0.
#' @param n The number of subjects for which data is simulated.
#' @param m The number of hidden states in the HMM used for
#'   simulating data.
#' @param q_emiss The number of categories of the simulated observations.
#' @param gamma A matrix with \code{m} rows and \code{m} columns containing the
#'   average population transition probability matrix used for simulating the
#'   data. That is, the probability to switch from hidden state \emph{i} (row
#'   \emph{i}) to hidden state \emph{j} (column  \emph{j}).
#' @param emiss_distr A matrix with \code{m} rows and \code{q_emiss} columns
#'   containing the average population emission distribution of the
#'   (categorical) observations given the hidden states. That is, the
#'   probability of observing category \emph{k} (column \emph{k}) in state
#'   \emph{i} (row \emph{i}).
#' @param beta List of two matrices containing the regression parameters to
#'   predict \code{gamma} and/or \code{emiss_distr} in combination with
#'   \code{xx_vec} using multinomial logistic regression. The first matrix is
#'   used to predict the transition probability matrix \code{gamma}. The
#'   second matrix is used to predict the emission distribution
#'   \code{emiss_distr} of the dependent variable. In both matrices, one
#'   regression parameter is specified for each element in \code{gamma} and
#'   \code{emiss_distr}, with the following exception. The first element in each
#'   row of \code{gamma} and/or \code{emiss_distr} is used as reference category
#'   in the multinomial logistic regression. As such, no regression parameters
#'   can be specified for these parameters. Hence, the first matrix in the list
#'   \code{beta} to predict \code{gamma} consist of a matrix with the number of
#'   rows equal to \code{m} and the number of columns equal to \code{m} - 1. The
#'   second matrix in the list \code{beta} to predict \code{emiss_distr} consist
#'   of a matrix with the number of rows equal to \code{m} and the number of
#'   columns equal to \code{q_emiss} - 1. See \emph{details} for more information.
#'   Note that if \code{beta} is specified, \code{xx_vec} has to be specified as
#'   well. If \code{beta} is omitted completely, \code{beta} defaults to NULL,
#'   resembling no prediction of \code{gamma} or \code{emiss_distr} using
#'   covariates. One of the two elements in the list can also be left empty
#'   (i.e., set to \code{NULL}) to signify that either the transition
#'   probability matrix or a specific emission distribution is not predicted by
#'   covariates.
#' @param xx_vec List of two vectors containing the covariate(s) to predict
#'   \code{gamma} and/or \code{emiss_distr} using the regression parameters
#'   specified in \code{beta}. The covariate used to predict \code{gamma} and
#'   \code{emiss_distr} can either be the same covariate, two different
#'   covariates, or a covariate for one element and none for the other. At this
#'   point, it is only possible to use one covariate for both \code{gamma} and
#'   \code{emiss_distr}. The first vector of the list \code{xx_vec} is used to
#'   predict the transition matrix. The second vector of the list \code{xx_vec}
#'   is used to predict the emission distribution of the dependent variable. For
#'   both vectors, the number of observations should be  equal to \code{n} the
#'   number of subjects to be simulated. If \code{xx_vec} is omitted completely,
#'   \code{xx_vec} defaults to NULL, resembling no covariates at all. One of the
#'   two elements in the list can also be left empty (i.e., set to \code{NULL})
#'   to signify that either the transition probability matrix or the emission
#'   distribution is not predicted by covariates.
#' @param var_gamma An integer denoting the variance between subjects in the
#'   transition probability matrix. Note that this value corresponds to the
#'   variance of the parameters of the multinomial distribution (i.e., the
#'   intercepts of the regression equation of the multinomial distribution used
#'   to sample the transition probability matrix), see details below. In
#'   addition, only one variance value can be specified for the complete
#'   transition probability matrix, hence the variance is assumed fixed across
#'   all components. The default equals 1, which corresponds to quite some
#'   variation between subjects. A less extreme value would be 0.5. If one wants
#'   to simulate data from exactly the same HMM for all subjects, var_gamma
#'   should be set to 0.
#' @param var_emiss An integer denoting the variance between subjects in the
#'   emission distribution. Note that this value corresponds to the variance of
#'   the parameters of the multinomial distribution (i.e., the intercepts of the
#'   regression equation of the multinomial distribution used to sample the
#'   components of the emission distribution), see details below.  In addition,
#'   only one variance value can be specified for the complete emission
#'   distribution, hence the variance is assumed fixed across all components.
#'   The default equals 1, which corresponds to quite some variation between
#'   subjects. A less extreme value would be 0.5. If one wants to simulate data
#'   from exactly the same HMM for all subjects, var_emiss should be set to
#'   0.
#' @param return_ind_par A logical scalar. Should the subject specific
#'   transition probability matrix \code{gamma} and emission probability matrix
#'   \code{emiss_distr} be returned by the function (\code{return_ind_par =
#'   TRUE}) or not (\code{return_ind_par = FALSE}). The default equals
#'   \code{return_ind_par = FALSE}.
#'
#'
#' @return The following components are returned by the function \code{sim_mHMM}:
#' \describe{
#'   \item{\code{states}}{A matrix containing the simulated hidden state
#'   sequences, with one row per hidden state per subject. The first column
#'   indicates subject id number. The second column contains the simulated
#'   hidden state sequence, consecutively for all subjects. Hence, the id number
#'   is repeated over the rows (with the number of repeats equal to the length
#'   of the simulated hidden state sequence \code{T} for each subject).}
#'   \item{\code{obs}}{A matrix containing the simulated observed outputs, with
#'   one row per simulated observation per subject. The first column indicates
#'   subject id number. The second column contains the simulated observation
#'   sequence, consecutively for all subjects. Hence, the id number is repeated
#'   over rows (with the number of repeats equal to the length of the simulated
#'   observation sequence \code{T} for each subject).}
#'   \item{\code{gamma}}{A list containing \code{n} elements with the simulated
#'   subject specific transition probability matrices \code{gamma}. Only
#'   returned if \code{return_ind_par} is set to \code{TRUE}.}
#'   \item{\code{emiss_distr}}{A list containing \code{n} elements with the
#'   simulated subject specific emission probability matrices
#'   \code{emiss_distr}. Only returned if \code{return_ind_par} is set to
#'   \code{TRUE}.}
#' }
#'
#'
#' @seealso \code{\link{mHMM}} for analyzing multilevel hidden Markov data.
#'
#'
#' @examples
#' # simulating data for 10 subjects with each 100 observations
#' n_t     <- 100
#' n       <- 10
#' m       <- 3
#' q_emiss <- 4
#' 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 <- matrix(c(0.5, 0.5, 0.0, 0.0,
#'                         0.1, 0.1, 0.8, 0.0,
#'                         0.0, 0.0, 0.1, 0.9), nrow = m, ncol = q_emiss, byrow = TRUE)
#' data1 <- sim_mHMM(n_t = n_t, n = n, m = m, q_emiss = q_emiss, gamma = gamma,
#'                   emiss_distr = emiss_distr, var_gamma = 1, var_emiss = 1)
#' head(data1$obs) #' head(data1$states)
#'
#' # including a covariate to predict (only) the transition probability matrix gamma
#' beta      <- rep(list(NULL), 2)
#' beta[[1]] <- matrix(c(0.5, 1.0,
#'                      -0.5, 0.5,
#'                       0.0, 1.0), byrow = TRUE, ncol = 2)
#' xx_vec      <- rep(list(NULL),2)
#' xx_vec[[1]] <-  c(rep(0,5), rep(1,5))
#' data2 <- sim_mHMM(n_t = n_t, n = n, m = m, q_emiss = q_emiss, gamma = gamma,
#'                   emiss_distr = emiss_distr, beta = beta, xx_vec = xx_vec,
#'                   var_gamma = 1, var_emiss = 1)
#'
#'
#' # simulating subject specific transition probability matrices and emission distributions only
#' n_t <- 0
#' n <- 5
#' m <- 3
#' q_emiss <- 4
#' 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 <- matrix(c(0.5, 0.5, 0.0, 0.0,
#'                         0.1, 0.1, 0.8, 0.0,
#'                         0.0, 0.0, 0.1, 0.9), nrow = m, ncol = q_emiss, byrow = TRUE)
#' data3 <- sim_mHMM(n_t = n_t, n = n, m = m, q_emiss = q_emiss, gamma = gamma,
#'                   emiss_distr = emiss_distr, var_gamma = 1, var_emiss = 1)
#' data3
#'
#' data4 <- sim_mHMM(n_t = n_t, n = n, m = m, q_emiss = q_emiss, gamma = gamma,
#'                   emiss_distr = emiss_distr, var_gamma = .5, var_emiss = .5)
#' data4
#' @export

sim_mHMM <- function(n_t, n, m, q_emiss, gamma, emiss_distr, beta = NULL, xx_vec = NULL,
var_gamma = 1, var_emiss = 1, return_ind_par = FALSE){

# Inbuild checks for correct specification of parameters ---------------------

if (dim(gamma)[1] != m){
stop(paste("The transiton probability matrix gamma should be a", m, "by", m, "matrix."))
}
if (dim(gamma)[2] != m){
stop(paste("The transiton probability matrix gamma should be a", m, "by", m, "matrix."))
}
if(!isTRUE(all.equal(apply(gamma,1,sum), rep(1,m)))){
stop("The elements in each row of the transition probability matrix gamma should sum up to 1")
}
if (dim(emiss_distr)[1] != m){
stop(paste("The number of rows of the emission distribution matrix should be
equal to the number of states, which is", m, "."))
}
if (dim(emiss_distr)[2] != q_emiss){
stop(paste("The number of columns of the emission distribution matrix should be
equal to the number of observable categories, which is", q_emiss, "."))
}
if(!isTRUE(all.equal(apply(emiss_distr, 1, sum), rep(1, m)))){
stop("The elements in each row of the emission distribution matrix should sum up to 1")
}
if((is.null(xx_vec) & !is.null(beta)) | (!is.null(xx_vec) & is.null(beta))){
stop("Either only xx_vec or only beta is specified. Please specify both 1) the values for the covariate
in xx_vec and 2) the values of the regression parameters in beta, to allow correct simulation of the
data.")
}
if(!is.null(xx_vec)){
if((!is.null(xx_vec[[1]]) & is.null(beta[[1]])) |
(!is.null(xx_vec[[2]]) & is.null(beta[[2]]))){
stop("Either only xx_vec or only beta is specified in one of the elements.
Please specify both 1) the values for the covariate in xx_vec and 2)
the values of the regression parameters in beta if either one is not
empty, to allow correct simulation of the data.")
}
}
if(!is.null(beta)){
if((!is.null(beta[[1]]) & is.null(xx_vec[[1]])) |
(!is.null(beta[[2]]) & is.null(xx_vec[[2]]))){
stop("Either only xx_vec or only beta is specified in one of the elements.
Please specify both 1) the values for the covariate in xx_vec and 2)
the values of the regression parameters in beta if either one is not
empty, to allow correct simulation of the data.")
}
}
if(!is.null(xx_vec)){
if((!is.null(xx_vec[[1]]) & length(xx_vec[[1]]) != n) |
(!is.null(xx_vec[[2]]) & length(xx_vec[[2]]) != n)){
stop("The length of the vectors in xx_vec should be equal to the number of subjects to be simulated,
set in n, if (the element in) xx_vec is not set to NULL.")
}
}
if (!is.null(beta)){
if (!is.null(beta[[1]])){
if ((dim(beta[[1]])[1] != (m)) | (dim(beta[[1]])[2] != (m-1))){
stop(paste("The first element of beta to predict the transiton probability matrix gamma should be a m (", m, " ) by m - 1 (", m - 1, ") matrix."))
}
}
if (!is.null(beta[[2]])){
if((dim(beta[[2]])[1] != (m)) | (dim(beta[[2]])[2] != (q_emiss-1))){
stop(paste("The second element of beta to predict the emission distribution should be a m (", m, ") by q_emiss - 1 (", q_emiss - 1, ") matrix."))
}
}
}
if(is.null(xx_vec)){
xx_vec <- rep(list(NULL), 2)
xx_vec[[1]] <- rep(1,n)
xx_vec[[2]] <- rep(1,n)
} else if(is.null(xx_vec[[1]])) {
xx_vec[[1]] <- rep(1,n)
} else if(is.null(xx_vec[[2]])) {
xx_vec[[2]] <- rep(1,n)
}
if(is.null(beta)){
beta <- rep(list(NULL), 2)
beta[[1]] <- matrix(0, ncol = m - 1, nrow = m)
beta[[2]] <- matrix(0, ncol = q_emiss - 1, nrow = m)
}
if(is.null(beta[[1]])) {
beta[[1]] <- matrix(0, ncol = m - 1, nrow = m)
} else if (is.null(beta[[2]])) {
beta[[2]] <- matrix(0, ncol = q_emiss - 1, nrow = m)
}

# Simulating the data ---------------------

states <- matrix(ncol = 2, nrow = n_t*n)
states[,1] <- rep(1:n, each = n_t)
obs <- matrix(ncol = 2, nrow = n_t*n)
obs[,1] <- rep(1:n, each = n_t)
sub_gamma <- rep(list(NULL), n)
sub_emiss <- rep(list(NULL), n)
mnl_gamma <- prob_to_int(gamma)
mnl_emiss <- prob_to_int(emiss_distr)
for(j in 1:n){
sub_gamma[[j]] <- int_to_prob(mnl_gamma + xx_vec[[1]][j] * beta[[1]] +
rnorm(n = m * (m-1), mean = 0, sd = sqrt(var_gamma)))
sub_emiss[[j]] <- int_to_prob(mnl_emiss + xx_vec[[2]][j] * beta[[2]] +
rnorm(n = m * (q_emiss-1), mean = 0, sd = sqrt(var_emiss)))
if(n_t != 0){
init <- solve(t(diag(m) - sub_gamma[[j]] + 1), rep(1, m))
states[((j-1) * n_t + 1), 2] <- sample(x = 1:m, size = 1, prob = init)
obs[((j-1) * n_t + 1), 2] <- sample(x = 1:q_emiss, size = 1, prob = sub_emiss[[j]][states[((j-1) * n_t + 1), 2],])
for(i in 2:n_t){
states[((j-1) * n_t + i), 2] <- sample(x = 1:m, size = 1, prob = sub_gamma[[j]][states[((j-1) * n_t + i - 1), 2],])
obs[((j-1) * n_t + i), 2] <- sample(x = 1:q_emiss, size = 1, prob = sub_emiss[[j]][states[((j-1) * n_t + i), 2],])
}
}
}
colnames(states) <- c("subj", "state")
colnames(obs)    <- c("subj", "observation")
if (return_ind_par == FALSE & n_t != 0){
return(list(states = states, obs = obs))
} else if (return_ind_par == TRUE & n_t != 0){
return(list(states = states, obs = obs, subject_gamma = sub_gamma, subject_emiss = sub_emiss))
} else if (n_t == 0){
return(list(subject_gamma = sub_gamma, subject_emiss = sub_emiss))
}
}


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mHMMbayes documentation built on Oct. 30, 2019, 5:05 p.m.