R/emMRHLP.R

In fchamroukhi/MRHLP: Multiple Regression Model with a Hidden Logistic Process ('MRHLP') for Joint Segmentation of Multivariate Time Series

#' emMRHLP implemens the EM algorithm to fit a MRHLP model.
#'
#' emMRHLP implements the maximum-likelihood parameter estimation of the MRHLP
#' model by the Expectation-Maximization (EM) algorithm.
#'
#' @details emMRHLP function implements the EM algorithm of the MRHLP model.
#'   This function starts with an initialization of the parameters done by the
#'   method `initParam` of the class [ParamMRHLP][ParamMRHLP], then it
#'   alternates between the E-Step (method of the class [StatMRHLP][StatMRHLP])
#'   and the M-Step (method of the class [ParamMRHLP][ParamMRHLP]) 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{(m, d)} representing a \eqn{d} dimension
#'   function of `X` observed at points \eqn{1,\dots,m}. `Y` is the observed
#'   response/output.
#' @param K The number of regimes (MRHLP 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 [ModelMRHLP][ModelMRHLP].
#' @seealso [ModelMRHLP], [ParamMRHLP], [StatMRHLP]
#' @export
#'
#' @examples
#' data(toydataset)
#' x <- toydataset$x
#' y <- toydataset[,c("y1", "y2", "y3")]
#'
#' mrhlp <- emMRHLP(X = x, Y = y, K = 5, p = 1, verbose = TRUE)
#'
#' mrhlp$summary()
#'
#' mrhlp$plot()
emMRHLP <- 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) {

  if (is.vector(Y)) { # Univariate time series
    Y <- as.matrix(Y)
  }
  mData <- MData(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 <- ParamMRHLP$new(mData = mData, K = K, p = p, q = q, variance_type = variance_type)
    param$initParam(try_EM)
    iter <- 0
    converge <- FALSE
    prev_loglik <- -Inf

    stat <- StatMRHLP(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) {
        message("EM - MRHLP: 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
      } # 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$loglik
    }
    if (n_tries > 1 && verbose) {
      message("Max value of the log-likelihood: ", stat$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) {
    message("Max value of the log-likelihood: ", statSolution$loglik, "\n")
  }

  # Finish the computation of statistics
  statSolution$computeStats(paramSolution)

  return(ModelMRHLP$new(param = paramSolution, stat = statSolution))
}