predict_dim_cv <- 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)))
    ))
}
# TODO: write doc
predict_dim_elbow <- function(object) {
    # extract original data from object (cve result)
    X <- object$X
    Y <- object$Y
    # Get dimensions
    n <- nrow(X)
    p <- ncol(X)

    losses <- vector("double", length(object$res))
    names(losses) <- names(object$res)
    # Compute per sample losses with alternative bandwidth for each dimension.
    for (dr.k in object$res) {
        # extract dimension specific estimates and dimensions.
        k <- dr.k$k
        V <- dr.k$V
        # estimate bandwidth according alternative formula.
        h <- estimate.bandwidth(X, k, sqrt(n), version = 2L)
        # Projected `X`
        XQ <- X %*% (diag(1, p) - tcrossprod(V)) # X (I - V V')
        # Compute distances
        d2 <- tcrossprod(XQ) # XQ XQ'
        d1 <- matrix(diag(d2), n, n)
        D <- d1 - 2 * d2 + t(d1)
        # Apply kernel
        # Note: CVE uses for d = ||Q(X_i - X_j)|| the kernel exp(-d^4 / (2 h^2))
        K <- exp((-0.5 / h^2) * D^2)
        # sum columns
        colSumsK <- colSums(K)
        # compute weighted and square meighted reponses
        y1 <- (K %*% Y) / colSumsK
        y2 <- (K %*% Y^2) / colSumsK
        # total loss
        losses[[as.character(k)]] <- mean(y2 - y1^2)
    }

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

predict_dim_wilcoxon <- function(object, p.value = 0.05) {
    # extract original data from object (cve result)
    X <- object$X
    Y <- object$Y
    # Get dimensions
    n <- nrow(X)
    p <- ncol(X)
    
    L <- matrix(NA, n, length(object$res))
    colnames(L) <- names(object$res)
    # Compute per sample losses with alternative bandwidth for each dimension.
    for (dr.k in object$res) {
        # extract dimension specific estimates and dimensions.
        k <- dr.k$k
        V <- dr.k$V
        # estimate bandwidth according alternative formula.
        h <- estimate.bandwidth(X, k, sqrt(n), version = 2L)
        # Projected `X`
        XQ <- X %*% (diag(1, p) - tcrossprod(V)) # X (I - V V')
        # Compute distances
        d2 <- tcrossprod(XQ) # XQ XQ'
        d1 <- matrix(diag(d2), n, n)
        D <- d1 - 2 * d2 + t(d1)
        # Apply kernel
        # Note: CVE uses for d = ||Q(X_i - X_j)|| the kernel exp(-d^4 / (2 h^2))
        K <- exp((-0.5 / h^2) * D^2)
        # sum columns
        colSumsK <- colSums(K)
        # compute weighted and square meighted reponses
        y1 <- (K %*% Y) / colSumsK
        y2 <- (K %*% Y^2) / colSumsK
        # element-wise L for dim. k
        L[, as.character(k)] <- y2 - y1^2
    }

    for (ind in seq_len(length(object$res) - 1L)) {
        p.test <- wilcox.test(L[, ind], L[, ind + 1L],
                              alternative = "less")$p.value
        if (p.test < p.value) {
            return(list(
                p.value = p.test,
                k = object$res[[ind]]$k
            ))
        }
    }

    return(list(
        p.value = NA,
        k = object$res[[length(object$res)]]$k
    ))
}

#' \code{"TODO: @Lukas"}
#' 
#' @param object instance of class \code{cve} (result of \code{\link{cve}},
#'  \code{\link{cve.call}}).
#' @param method one of \code{"CV"}, \code{"elbow"} or \code{"wilcoxon"}.
#' @param ... ignored.
#'
#' @return list with \code{"k"} the predicted dimension and method dependent 
#'      informatoin.
#'
#' @section Method cv:
#' TODO: \code{"TODO: @Lukas"}.
#'
#' @section Method elbow:
#' TODO: \code{"TODO: @Lukas"}.
#'
#' @section Method wilcoxon:
#' TODO: \code{"TODO: @Lukas"}.
#'
#' @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, ..., method = "CV") {
    # Check if there are dimensions to select.
    if (length(object$res) == 1L) {
        return(list(
            message = "Only one dim. estimated.",
            k = as.integer(names(object$res))
        ))
    }

    # Determine method "fuzzy".
    methods <- c("cv", "elbow", "wilcoxon")
    names(methods) <- methods
    method <- methods[[tolower(method), exact = FALSE]]
    if (is.null(method)) {
        stop('Unable to determine method.')
    }

    if (method == "cv") {
        return(predict_dim_cv(object))
    } else if (method == "elbow") {
        return(predict_dim_elbow(object))
    } else if (method == "wilcoxon") {
        return(predict_dim_wilcoxon(object))
    } else {
        stop("Unable to determine method.")
    }
}