#' Predicts SDR dimension using \code{\link[mda]{mars}} via a Cross-Validation.
#'
#' @param object instance of class \code{cve} (result of \code{cve},
#'  \code{cve.call}).
#' @param ... ignored.
#'
#' @return list with
#' \itemize{
#'    \item MSE: Mean Square Error,
#'    \item k: predicted dimensions.
#' }
#'
#' @examples
#' # create B for simulation
#' B <- rep(1, 5) / sqrt(5)
#' 
#' set.seed(21)
#' # creat predictor data x ~ N(0, I_p)
#' x <- matrix(rnorm(500), 100)
#' 
#' # simulate response variable
#' #    y = f(B'x) + err
#' # with f(x1) = x1 and err ~ N(0, 0.25^2)
#' y <- x %*% B + 0.25 * rnorm(100)
#'
#' # Calculate cve for unknown k between min.dim and max.dim.
#' cve.obj.simple <- cve(y ~ x) 
#'
#' predict_dim(cve.obj.simple)
#'
#' @export
predict_dim <- function(object, ...) {
    # Get centered training data and dimensions
    X <- scale(object$X, center = TRUE, scale = FALSE)
    n <- nrow(object$X) # umber of training data samples
    Sigma <- (1 / n) * crossprod(X, X)
    eig <- eigen(Sigma)
    Sigma_root <- eig$vectors %*% tcrossprod(diag(sqrt(eig$values)), eig$vectors)
    X <- X %*% solve(Sigma_root)

    pred <- matrix(0, n, length(object$res))
    colnames(pred) <- names(object$res)
    for (dr.k in object$res) {
        # get "name" of current dimension
        k <- as.character(dr.k$k)
        # Project dataset with current SDR basis
        X.proj <- X %*% dr.k$B

        for (i in 1:n) {
            model <- mda::mars(X.proj[-i, ], object$Y[-i])
            pred[i, k] <- predict(model, X.proj[i, , drop = F])
        }

    }
    MSE <- colMeans((pred - object$Y)^2)

    return(list(
        MSE = MSE,
        k = as.integer(names(which.min(MSE)))
    ))
}