CVE/CVE_C/R/CVE.R

#' Conditional Variance Estimator (CVE) Package.
#'
#' Conditional Variance Estimation (CVE) is a novel sufficient dimension
#' reduction (SDR) method for regressions satisfying \eqn{E(Y|X) = E(Y|B'X)},
#' where \eqn{B'X} is a lower dimensional projection of the predictors. CVE,
#' similarly to its main competitor, the mean average variance estimation
#' (MAVE), is not based on inverse regression, and does not require the
#' restrictive linearity and constant variance conditions of moment based SDR
#' methods. CVE is data-driven and applies to additive error regressions with
#' continuous predictors and link function. The effectiveness and accuracy of 
#' CVE compared to MAVE and other SDR techniques is demonstrated in simulation
#' studies. CVE is shown to outperform MAVE in some model set-ups, while it
#' remains largely on par under most others.
#'
#' Let \eqn{Y} be real denotes a univariate response and \eqn{X} a real
#' \eqn{p}-dimensional covariate vector. We assume that the dependence of
#' \eqn{Y} and \eqn{X} is modelled by
#' \deqn{Y = g(B'X) + \epsilon}
#' where \eqn{X} is independent of \eqn{\epsilon} with positive definite
#' variance-covariance matrix \eqn{Var(X) = \Sigma_X}. \eqn{\epsilon} is a mean
#' zero random variable with finite \eqn{Var(\epsilon) = E(\epsilon^2)}, \eqn{g}
#' is an unknown, continuous non-constant function,
#' and \eqn{B = (b_1, ..., b_k)} is
#' a real \eqn{p \times k}{p x k} of rank \eqn{k <= p}{k \leq p}. 
#' Without loss of generality \eqn{B} is assumed to be orthonormal.
#'
#' @author Daniel Kapla, Lukas Fertl, Bura Efstathia
#' @references Fertl Lukas, Bura Efstathia. Conditional Variance Estimation for
#' Sufficient Dimension Reduction, 2019
#'
#' @importFrom stats model.frame
#' @docType package
#' @useDynLib CVE, .registration = TRUE
"_PACKAGE"

#' Conditional Variance Estimator (CVE).
#'
#' TODO: reuse of package description and details!!!!
#'
#' @param formula an object of class \code{"formula"} which is a symbolic
#' description of the model to be fitted.
#' @param data an optional data frame, containing the data for the formula if
#' supplied.
#' @param method specifies the CVE method variation as one of
#' \itemize{
#'      \item "simple" exact implementation as describet in the paper listed 
#'          below.
#'      \item "weighted" variation with addaptive weighting of slices.
#' }
#' @param ... Parameters passed on to \code{cve.call}.
#' @examples
#' library(CVE)
#'
#' # create dataset
#' n <- 200
#' p <- 12
#' X <- matrix(rnorm(n * p), n, p)
#' B <- cbind(c(1, rep(0, p - 1)), c(0, 1, rep(0, p - 2)))
#' Y <- X %*% B
#' Y <- Y[, 1L]^2 + Y[, 2L]^2 + rnorm(n, 0, 0.3)
#'
#' # Call the CVE method.
#' dr <- cve(Y ~ X)
#' round(dr[[2]]$B, 1)
#'
#' @seealso For a detailed description of the formula parameter see
#'      [\code{\link{formula}}].
#' @export
cve <- function(formula, data, method = "simple", max.dim = 10L, ...) {
    # check for type of `data` if supplied and set default
    if (missing(data)) {
        data <- environment(formula)
    } else if (!is.data.frame(data)) {
        stop("Parameter 'data' must be a 'data.frame' or missing.")
    }

    # extract `X`, `Y` from `formula` with `data`
    model <- stats::model.frame(formula, data)
    X <- as.matrix(model[ ,-1L, drop = FALSE])
    Y <- as.double(model[ , 1L])

    # pass extracted data on to [cve.call()]
    dr <- cve.call(X, Y, method = method, max.dim = max.dim, ...)

    # overwrite `call` property from [cve.call()]
    dr$call <- match.call()
    return(dr)
}

#' @param nObs parameter for choosing bandwidth \code{h} using
#'   \code{\link{estimate.bandwidth}} (ignored if \code{h} is supplied).
#' @param X data matrix with samples in its rows.
#' @param Y Responces (1 dimensional).
#' @param k Dimension of lower dimensional projection, if given only the
#'      specified dimension is estimated.
#' @param min.dim lower bounds for \code{k}, (ignored if \code{k} is supplied).
#' @param max.dim upper bounds for \code{k}, (ignored if \code{k} is supplied).
#' @param tau Initial step-size.
#' @param tol Tolerance for break condition.
#' @param epochs maximum number of optimization steps.
#' @param attempts number of arbitrary different starting points.
#' @param logger a logger function (only for addvanced user).
#' @rdname cve
#' @export
cve.call <- function(X, Y, method = "simple",
                     nObs = sqrt(nrow(X)), h = NULL,
                     min.dim = 1L, max.dim = 10L, k = NULL,
                     tau = 1.0, tol = 1e-3,
                     epochs = 50L, attempts = 10L,
                     logger = NULL) {

    # parameter checking
    if (!(is.matrix(X) && is.numeric(X))) {
        stop("Parameter 'X' should be a numeric matrices.")
    }
    if (!is.numeric(Y)) {
        stop("Parameter 'Y' must be numeric.")
    }
    if (is.matrix(Y) || !is.double(Y)) {
        Y <- as.double(Y)
    }
    if (nrow(X) != length(Y)) {
        stop("Rows of 'X' and 'Y' elements are not compatible.")
    }
    if (ncol(X) < 2) {
        stop("'X' is one dimensional, no need for dimension reduction.")
    }

    if (missing(k) || is.null(k)) {
        min.dim <- as.integer(min.dim)
        max.dim <- as.integer(min(max.dim, ncol(X) - 1L))
    } else {
        min.dim <- as.integer(k)
        max.dim <- as.integer(k)
    }
    if (min.dim > max.dim) {
        stop("'min.dim' bigger 'max.dim'.")
    }
    if (max.dim >= ncol(X)) {
        stop("'max.dim' (or 'k') must be smaller than 'ncol(X)'.")
    }

    if (missing(h) || is.null(h)) {
        estimate <- TRUE
    } else if (is.function(h)) {
        estimate <- TRUE
        estimate.bandwidth <- h
    } else if (is.numeric(h) && h > 0.0) {
        estimate <- FALSE
        h <- as.double(h)
    } else {
        stop("Bandwidth 'h' must be positive numeric.")
    }

    if (!is.numeric(tau) || length(tau) > 1L || tau <= 0.0) {
        stop("Initial step-width 'tau' must be positive number.")
    } else {
        tau <- as.double(tau)
    }
    if (!is.numeric(tol) || length(tol) > 1L || tol < 0.0) {
        stop("Break condition tolerance 'tol' must be not negative number.")
    } else {
        tol <- as.double(tol)
    }

    if (!is.numeric(epochs) || length(epochs) > 1L) {
        stop("Parameter 'epochs' must be positive integer.")
    } else if (!is.integer(epochs)) {
        epochs <- as.integer(epochs)
    }
    if (epochs < 1L) {
        stop("Parameter 'epochs' must be at least 1L.")
    }
    if (!is.numeric(attempts) || length(attempts) > 1L) {
        stop("Parameter 'attempts' must be positive integer.")
    } else if (!is.integer(attempts)) {
        attempts <- as.integer(attempts)
    }
    if (attempts < 1L) {
        stop("Parameter 'attempts' must be at least 1L.")
    }

    if (is.function(logger)) {
        loggerEnv <- environment(logger)
    } else {
        loggerEnv <- NULL
    }

    # Call specified method.
    method <- tolower(method)
    call <- match.call()
    dr <- list()
    for (k in min.dim:max.dim) {

        if (estimate) {
            h <- estimate.bandwidth(X, k, nObs)
        }

        if (method == 'simple') {
            dr.k <- .Call('cve_simple', PACKAGE = 'CVE',
                          X, Y, k, h,
                          tau, tol,
                          epochs, attempts,
                          logger, loggerEnv)
        #     dr.k <- cve_simple(X, Y, k, nObs = nObs, ...)
        # } else if (method == 'linesearch') {
        #     dr.k <- cve_linesearch(X, Y, k, nObs = nObs, ...)
        # } else if (method == 'rcg') {
        #     dr.k <- cve_rcg(X, Y, k, nObs = nObs, ...)
        # } else if (method == 'momentum') {
        #     dr.k <- cve_momentum(X, Y, k, nObs = nObs, ...)
        # } else if (method == 'rmsprob') {
        #     dr.k <- cve_rmsprob(X, Y, k, nObs = nObs, ...)
        # } else if (method == 'sgdrmsprob') {
        #     dr.k <- cve_sgdrmsprob(X, Y, k, nObs = nObs, ...)
        # } else if (method == 'sgd') {
        #     dr.k <- cve_sgd(X, Y, k, nObs = nObs, ...)
        } else {
            stop('Got unknown method.')
        }
        dr.k$B <- null(dr.k$V)
        dr.k$loss <- mean(dr.k$L)
        dr.k$h <- h
        dr.k$k <- k
        class(dr.k) <- "cve.k"
        dr[[k]] <- dr.k
    }

    # augment result information
    dr$X <- X
    dr$Y <- Y
    dr$method <- method
    dr$call <- call
    class(dr) <- "cve"
    return(dr)
}

#' Loss distribution kink plot.
#'
#' @param x Object of class \code{"cve"} (result of [\code{\link{cve}}]).
#' @param ... Pass through parameters to [\code{\link{plot}}] and
#'      [\code{\link{lines}}]
#'
#' @seealso see \code{\link{par}} for graphical parameters to pass through
#'      as well as \code{\link{plot}}, the standard plot utility.
#' @importFrom graphics plot lines points
#' @method plot cve
#' @export
plot.cve <- function(x, ...) {
    L <- c()
    k <- c()
    for (dr.k in x) {
        if (class(dr.k) == 'cve.k') {
            k <- c(k, paste0(dr.k$k))
            L <- c(L, dr.k$L)
        }
    }
    L <- matrix(L, ncol = length(k)) / var(x$Y)
    boxplot(L, main = "Kink plot",
            xlab = "SDR dimension",
            ylab = "Sample loss distribution",
            names = k)
    # lines(apply(L, 2, mean)) # TODO: ?
}

#' Prints a summary of a \code{cve} result.
#' @param object Instance of 'cve' as return of \code{cve}.
#' @method summary cve
#' @export
summary.cve <- function(object, ...) {
    cat('Summary of CVE result - Method: "', object$method, '"\n',
        '\n',
        'Dataset size:   ', nrow(object$X), '\n',
        'Data Dimension: ', ncol(object$X), '\n',
        'SDR Dimension:  ', object$k, '\n',
        'loss:           ', object$loss, '\n',
        '\n',
        'Called via:\n',
        '    ',
        sep='')
    print(object$call)
}