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#' selectMHMMR implements a model selection procedure to select an optimal MHMMR
#' model with unknown structure.
#'
#' @details selectMHMMR selects the optimal MHMMR model among a set of model
#' candidates by optimizing a model selection criteria, including the Bayesian
#' Information Criterion (BIC). This function first fits the different MHMMR
#' model candidates by varying the number of regimes `K` from `Kmin` to `Kmax`
#' and the order of the polynomial regression `p` from `pmin` to `pmax`. The
#' model having the highest value of the chosen selection criterion is then
#' selected.
#'
#' @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 time
#' series observed at points \eqn{1,\dots,m}.
#' @param Kmin The minimum number of regimes (c components).
#' @param Kmax The maximum number of regimes (MHMMR components).
#' @param pmin The minimum order of the polynomial regression.
#' @param pmax The maximum order of the polynomial regression.
#' @param criterion The criterion used to select the MHMMR model ("BIC", "AIC").
#' @param verbose Optional. A logical value indicating whether or not a summary
#' of the selected model should be displayed.
#' @return selectMHMMR returns an object of class [ModelMHMMR][ModelMHMMR]
#' representing the selected MHMMR model according to the chosen `criterion`.
#' @seealso [ModelMHMMR]
#' @export
#'
#' @examples
#' data(multivtoydataset)
#' x <- multivtoydataset$x
#' y <- multivtoydataset[, c("y1", "y2", "y3")]
#'
#' selectedmhmmr <- selectMHMMR(X = x, Y = y, Kmin = 2, Kmax = 6,
#' pmin = 0, pmax = 2)
#'
#' selectedmhmmr$summary()
selectMHMMR <- function(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4, criterion = c("BIC", "AIC"), verbose = TRUE) {
criterion <- match.arg(criterion)
vmhmmr <- Vectorize(function(K, p, X1 = X, Y1 = Y) emMHMMR(X = X1, Y = Y1, K, p),
vectorize.args = c("K", "p"))
mhmmr <- outer(Kmin:Kmax, pmin:pmax, vmhmmr)
if (criterion == "BIC") {
results <- apply(mhmmr, 1:2, function(x) x[[1]]$stat$BIC)
} else {
results <- apply(mhmmr, 1:2, function(x) x[[1]]$stat$AIC)
}
rownames(results) <- sapply(Kmin:Kmax, function(x) paste0("(K = ", x, ")"))
colnames(results) <- sapply(pmin:pmax, function(x) paste0("(p = ", x, ")"))
selected <- mhmmr[which(results == max(results), arr.ind = T)][[1]]
if (verbose) {
cat(paste0("The MHMMR model selected via the \"", criterion, "\" has K = ",
selected$param$K, " regimes \n and the order of the ",
"polynomial regression is p = ", selected$param$p, "."))
cat("\n")
cat(paste0("BIC = ", selected$stat$BIC, "\n"))
cat(paste0("AIC = ", selected$stat$AIC, "\n"))
}
return(selected)
}
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