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#' emRHLP implements the EM algorithm to fit a RHLP model.
#'
#' emRHLP implements the maximum-likelihood parameter estimation of the RHLP
#' model by the Expectation-Maximization (EM) algorithm.
#'
#' @details emRHLP function implements the EM algorithm for the RHLP model. This
#' function starts with an initialization of the parameters done by the method
#' `initParam` of the class [ParamRHLP][ParamRHLP], then it alternates between
#' the E-Step (method of the class [StatRHLP][StatRHLP]) and the M-Step
#' (method of the class [ParamRHLP][ParamRHLP]) 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 Numeric vector of length \emph{m} representing the observed
#' response/output \eqn{y_{1},\dots,y_{m}}.
#' @param K The number of regimes (RHLP components).
#' @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 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 K segments, and for the next runs,
#' parameters are initialized by randomly segmenting the data into K
#' 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 [ModelRHLP][ModelRHLP].
#' @seealso [ModelRHLP], [ParamRHLP], [StatRHLP]
#' @export
#'
#' @examples
#' data(univtoydataset)
#'
#' rhlp <- emRHLP(univtoydataset$x, univtoydataset$y, K = 3, p = 1, verbose = TRUE)
#'
#' rhlp$summary()
#'
#' rhlp$plot()
emRHLP <- function(X, Y, K, p = 3, q = 1, variance_type = c("heteroskedastic", "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-6, verbose = FALSE, verbose_IRLS = FALSE) {
top <- 0
try_EM <- 0
best_loglik <- -Inf
while (try_EM < n_tries) {
try_EM <- try_EM + 1
if (n_tries > 1 && verbose) {
cat(paste0("EM try number: ", try_EM, "\n\n"))
}
# Initialization
variance_type <- match.arg(variance_type)
param <- ParamRHLP$new(X = X, Y = Y, K = K, p = p, q = q, variance_type = variance_type)
param$initParam(try_EM)
iter <- 0
converge <- FALSE
prev_loglik <- -Inf
stat <- StatRHLP(paramRHLP = param)
while (!converge && (iter <= max_iter)) {
stat$EStep(param)
reg_irls <- param$MStep(stat, verbose_IRLS)
stat$computeLikelihood(reg_irls)
iter <- iter + 1
if (verbose) {
cat(paste0("EM: Iteration : ", iter, " || log-likelihood : " , stat$loglik, "\n"))
}
if (prev_loglik - stat$loglik > 1e-5) {
if (verbose) {
warning(paste0("EM log-likelihood is decreasing from ", prev_loglik, "to ", stat$loglik, " !"))
}
top <- top + 1
if (top > 20)
break
}
# Test of convergence
converge <- abs((stat$loglik - prev_loglik) / prev_loglik) <= threshold
if (is.na(converge)) {
converge <- FALSE
} # Basically for the first iteration when prev_loglik is Inf
prev_loglik <- stat$loglik
stat$stored_loglik <- c(stat$stored_loglik, stat$loglik)
} # End of the EM loop
if (stat$loglik > best_loglik) {
statSolution <- stat$copy()
paramSolution <- param$copy()
best_loglik <- stat$loglik
}
if (n_tries > 1 && verbose) {
cat(paste0("Max value of the log-likelihood: ", stat$loglik, "\n\n"))
}
}
# Computation of Z_ik the hard partition of the curves and klas (the estimated segment labels z_i)
statSolution$MAP()
if (n_tries > 1 && verbose) {
cat(paste0("Max value of the log-likelihood: ", statSolution$loglik, "\n"))
}
# End of the computation of the model statistics
statSolution$computeStats(paramSolution)
return(ModelRHLP$new(param = paramSolution, stat = statSolution))
}
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