#' emMixRHLP implements the EM algorithm to fit a mixture of RHLP models.
#'
#' emMixRHLP implements the maximum-likelihood parameter estimation of a mixture
#' of RHLP models by the Expectation-Maximization (EM) algorithm.
#'
#' @details emMixRHLP function implements the EM algorithm. This function starts
#' with an initialization of the parameters done by the method `initParam` of
#' the class [ParamMixRHLP][ParamMixRHLP], then it alternates between the
#' E-Step (method of the class [StatMixRHLP][StatMixRHLP]) and the M-Step
#' (method of the class [ParamMixRHLP][ParamMixRHLP]) until convergence (until
#' the relative variation of log-likelihood between two steps of the EM
#' algorithm is less than the `threshold` parameter).
#'
#' @param X Numeric vector of length \emph{m} representing the covariates/inputs
#' \eqn{x_{1},\dots,x_{m}}.
#' @param Y Matrix of size \eqn{(n, m)} representing the observed
#' responses/outputs. `Y` consists of \emph{n} functions of `X` observed at
#' points \eqn{1,\dots,m}.
#' @param K The number of clusters (Number of RHLP models).
#' @param R The number of regimes (RHLP components) for each cluster.
#' @param p Optional. The order of the polynomial regression. By default, `p` is
#' set at 3.
#' @param q Optional. The dimension of the logistic regression. For the purpose
#' of segmentation, it must be set to 1 (which is the default value).
#' @param variance_type Optional character indicating if the model is
#' "homoskedastic" or "heteroskedastic". By default the model is
#' "heteroskedastic".
#' @param init_kmeans Optional. A logical indicating whether or not the curve
#' partition should be initialized by the K-means algorithm. Otherwise the
#' curve partition is initialized randomly.
#' @param n_tries Optional. Number of runs of the EM algorithm. The solution
#' providing the highest log-likelihood will be returned.
#'
#' If `n_tries` > 1, then for the first run, parameters are initialized by
#' uniformly segmenting the data into R segments, and for the next runs,
#' parameters are initialized by randomly segmenting the data into R
#' contiguous segments.
#' @param max_iter Optional. The maximum number of iterations for the EM
#' algorithm.
#' @param threshold Optional. A numeric value specifying the threshold for the
#' relative difference of log-likelihood between two steps of the EM as
#' stopping criteria.
#' @param verbose Optional. A logical value indicating whether or not values of
#' the log-likelihood should be printed during EM iterations.
#' @param verbose_IRLS Optional. A logical value indicating whether or not
#' values of the criterion optimized by IRLS should be printed at each step of
#' the EM algorithm.
#' @return EM returns an object of class [ModelMixRHLP][ModelMixRHLP].
#' @seealso [ModelMixRHLP], [ParamMixRHLP], [StatMixRHLP]
#' @export
#'
#' @examples
#' data(toydataset)
#'
#' # Let's fit a mixRHLP model on a dataset containing 2 clusters:
#' data <- toydataset[1:190,1:21]
#' x <- data$x
#' Y <- t(data[,2:ncol(data)])
#'
#' mixrhlp <- emMixRHLP(X = x, Y = Y, K = 2, R = 2, p = 1, verbose = TRUE)
#'
#' mixrhlp$summary()
#'
#' mixrhlp$plot()
emMixRHLP <- function(X, Y, K, R, p = 3, q = 1, variance_type = c("heteroskedastic", "homoskedastic"), init_kmeans = TRUE, n_tries = 1, max_iter = 1000, threshold = 1e-5, verbose = FALSE, verbose_IRLS = FALSE) {
fData <- FData(X, Y)
top <- 0
try_EM <- 0
best_loglik <- -Inf
while (try_EM < n_tries) {
try_EM <- try_EM + 1
if (n_tries > 1 && verbose) {
message("EM try number: ", try_EM, "\n")
}
# Initialization
variance_type <- match.arg(variance_type)
param <- ParamMixRHLP(fData = fData, K = K, R = R, p = p, q = q, variance_type = variance_type)
param$initParam(init_kmeans, try_EM)
iter <- 0
converge <- FALSE
prev_loglik <- -Inf
stat <- StatMixRHLP(param)
while (!converge && (iter <= max_iter)) {
stat$EStep(param)
param$MStep(stat, verbose_IRLS)
iter <- iter + 1
if (verbose) {
message("EM - mixRHLP: Iteration: ", iter, " | log-likelihood: " , stat$loglik)
}
if (prev_loglik - stat$loglik > 1e-5) {
warning("EM log-likelihood is decreasing from ", prev_loglik, "to ", stat$loglik, "!")
top <- top + 1
if (top > 20) {
break
}
}
# Test of convergence
converge <- abs((stat$loglik - prev_loglik) / prev_loglik) <= threshold
if (is.na(converge)) {
converge <- FALSE
}
prev_loglik <- stat$loglik
stat$stored_loglik <- c(stat$stored_loglik, stat$loglik)
} # End of EM loop
if (stat$loglik > best_loglik) {
statSolution <- stat$copy()
paramSolution <- param$copy()
best_loglik <- stat$loglik
}
if (n_tries > 1 && verbose) {
message("Max value of the log-likelihood: ", stat$loglik, "\n\n")
}
}
# Computation of c_ig the hard partition of the curves and the cluster labels klas
statSolution$MAP()
if (n_tries > 1 && verbose) {
message("Best value of the log-likelihood: ", statSolution$loglik, "\n")
}
statSolution$computeStats(paramSolution)
return(ModelMixRHLP(param = paramSolution, stat = statSolution))
}
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