365 lines
13 KiB
R
365 lines
13 KiB
R
#' Conditional Variance Estimator (CVE) Package.
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#'
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#' Conditional Variance Estimation (CVE) is a novel sufficient dimension
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#' reduction (SDR) method for regressions satisfying \eqn{E(Y|X) = E(Y|B'X)},
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#' where \eqn{B'X} is a lower dimensional projection of the predictors. CVE,
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#' similarly to its main competitor, the mean average variance estimation
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#' (MAVE), is not based on inverse regression, and does not require the
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#' restrictive linearity and constant variance conditions of moment based SDR
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#' methods. CVE is data-driven and applies to additive error regressions with
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#' continuous predictors and link function. The effectiveness and accuracy of
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#' CVE compared to MAVE and other SDR techniques is demonstrated in simulation
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#' studies. CVE is shown to outperform MAVE in some model set-ups, while it
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#' remains largely on par under most others.
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#' Let \eqn{Y} be real denotes a univariate response and \eqn{X} a real
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#' \eqn{p}-dimensional covariate vector. We assume that the dependence of
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#' \eqn{Y} and \eqn{X} is modelled by
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#' \deqn{Y = g(B'X) + \epsilon}
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#' where \eqn{X} is independent of \eqn{\epsilon} with positive definite
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#' variance-covariance matrix \eqn{Var(X) = \Sigma_X}. \eqn{\epsilon} is a mean
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#' zero random variable with finite \eqn{Var(\epsilon) = E(\epsilon^2)}, \eqn{g}
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#' is an unknown, continuous non-constant function,
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#' and \eqn{B = (b_1, ..., b_k)} is
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#' a real \eqn{p \times k}{p x k} of rank \eqn{k <= p}{k \leq p}.
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#' Without loss of generality \eqn{B} is assumed to be orthonormal.
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#'
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#' @author Daniel Kapla, Lukas Fertl, Bura Efstathia
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#' @references Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
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#' Estimation for Sufficient Dimension Reduction. Working Paper.
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#'
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#' @docType package
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#' @useDynLib CVE, .registration = TRUE
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"_PACKAGE"
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#' Conditional Variance Estimator (CVE).
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#'
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#' @inherit CVE-package description
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#'
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#' @param formula an object of class \code{"formula"} which is a symbolic
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#' description of the model to be fitted.
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#' @param data an optional data frame, containing the data for the formula if
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#' supplied.
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#' @param method specifies the CVE method variation as one of
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#' \itemize{
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#' \item "simple" exact implementation as described in the paper listed
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#' below.
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#' \item "weighted" variation with addaptive weighting of slices.
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#' }
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#' @param max.dim upper bounds for \code{k}, (ignored if \code{k} is supplied).
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#' @param ... Parameters passed on to \code{cve.call}.
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#'
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#' @return an S3 object of class \code{cve} with components:
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#' \describe{
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#' \item{X}{Original training data,}
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#' \item{Y}{Responce of original training data,}
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#' \item{method}{Name of used method,}
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#' \item{call}{the matched call,}
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#' \item{res}{list of components \code{V, L, B, loss, h} and \code{k} for
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#' each \eqn{k=min.dim,...,max.dim} (dimension).}
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#' }
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#'
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#' @examples
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#' # set dimensions for simulation model
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#' p <- 8
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#' k <- 2
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#' # create B for simulation
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#' b1 <- rep(1 / sqrt(p), p)
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#' b2 <- (-1)^seq(1, p) / sqrt(p)
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#' B <- cbind(b1, b2)
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#' # samplsize
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#' n <- 200
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#' set.seed(21)
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#' # creat predictor data x ~ N(0, I_p)
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#' x <- matrix(rnorm(n * p), n, p)
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#' # simulate response variable
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#' # y = f(B'x) + err
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#' # with f(x1, x2) = x1^2 + 2 * x2 and err ~ N(0, 0.25^2)
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#' y <- (x %*% b1)^2 + 2 * (x %*% b2) + 0.25 * rnorm(100)
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#' # calculate cve with method 'simple' for k unknown in 1, ..., 4
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#' cve.obj.s <- cve(y ~ x, max.dim = 4) # default method 'simple'
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#' # calculate cve with method 'weighed' for k = 2
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#' cve.obj.w <- cve(y ~ x, k = 2, method = 'weighted')
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#' # estimate dimension from cve.obj.s
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#' khat <- predict_dim(cve.obj.s)$k
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#' # get cve-estimate for B with dimensions (p, k = khat)
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#' B2 <- coef(cve.obj.s, k = khat)
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#' # get projected X data (same as cve.obj.s$X %*% B2)
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#' proj.X <- directions(cve.obj.s, k = khat)
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#' # plot y against projected data
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#' plot(proj.X[, 1], y)
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#' plot(proj.X[, 2], y)
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#' # creat 10 new x points and y according to model
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#' x.new <- matrix(rnorm(10 * p), 10, p)
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#' y.new <- (x.new %*% b1)^2 + 2 * (x.new %*% b2) + 0.25 * rnorm(10)
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#' # predict y.new
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#' yhat <- predict(cve.obj.s, x.new, khat)
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#' plot(y.new, yhat)
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#' # projection matrix on span(B)
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#' # same as B %*% t(B) since B is semi-orthogonal
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#' PB <- B %*% solve(t(B) %*% B) %*% t(B)
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#' # cve estimates for B with simple and weighted method
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#' B.s <- coef(cve.obj.s, k = 2)
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#' B.w <- coef(cve.obj.w, k = 2)
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#' # same as B.s %*% t(B.s) since B.s is semi-orthogonal (same vor B.w)
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#' PB.s <- B.s %*% solve(t(B.s) %*% B.s) %*% t(B.s)
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#' PB.w <- B.w %*% solve(t(B.w) %*% B.w) %*% t(B.w)
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#' # compare estimation accuracy of simple and weighted cve estimate by
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#' # Frobenius norm of difference of projections.
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#' norm(PB - PB.s, type = 'F')
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#' norm(PB - PB.w, type = 'F')
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#'
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#' @seealso For a detailed description of \code{formula} see
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#' \code{\link{formula}}.
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#' @references Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
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#' Estimation for Sufficient Dimension Reduction. Working Paper.
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#'
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#' @importFrom stats model.frame
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#' @export
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cve <- function(formula, data, method = "simple", max.dim = 10L, ...) {
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# check for type of `data` if supplied and set default
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if (missing(data)) {
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data <- environment(formula)
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} else if (!is.data.frame(data)) {
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stop("Parameter 'data' must be a 'data.frame' or missing.")
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}
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# extract `X`, `Y` from `formula` with `data`
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model <- stats::model.frame(formula, data)
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X <- as.matrix(model[ ,-1L, drop = FALSE])
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Y <- as.double(model[ , 1L])
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# pass extracted data on to [cve.call()]
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dr <- cve.call(X, Y, method = method, max.dim = max.dim, ...)
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# overwrite `call` property from [cve.call()]
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dr$call <- match.call()
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return(dr)
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}
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#' @inherit cve title
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#' @inherit cve description
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#'
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#' @param nObs parameter for choosing bandwidth \code{h} using
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#' \code{\link{estimate.bandwidth}} (ignored if \code{h} is supplied).
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#' @param X data matrix with samples in its rows.
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#' @param Y Responses (1 dimensional).
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#' @param method specifies the CVE method variation as one of
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#' \itemize{
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#' \item "simple" exact implementation as described in the paper listed
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#' below.
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#' \item "weighted" variation with addaptive weighting of slices.
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#' }
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#' @param k Dimension of lower dimensional projection, if \code{k} is given
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#' only the specified dimension \code{B} matrix is estimated.
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#' @param min.dim lower bounds for \code{k}, (ignored if \code{k} is supplied).
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#' @param max.dim upper bounds for \code{k}, (ignored if \code{k} is supplied).
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#' @param tau Initial step-size.
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#' @param tol Tolerance for break condition.
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#' @param max.iter maximum number of optimization steps.
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#' @param attempts number of arbitrary different starting points.
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#' @param logger a logger function (only for advanced user, significantly slows
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#' down the computation).
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#' @param h bandwidth or function to estimate bandwidth, defaults to internaly
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#' estimated bandwidth.
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#' @param momentum number of [0, 1) giving the ration of momentum for eucledian
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#' gradient update with a momentum term.
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#' @param slack Positive scaling to allow small increases of the loss while
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#' optimizing.
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#' @param gamma step-size reduction multiple.
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#' @param V.init Semi-orthogonal matrix of dimensions `(ncol(X), ncol(X) - k)
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#' as optimization starting value. (If supplied, \code{attempts} is
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#' set to 1 and \code{k} to match dimension)
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#'
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#' @inherit cve return
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#'
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#' @examples
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#' # create B for simulation (k = 1)
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#' B <- rep(1, 5) / sqrt(5)
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#'
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#' set.seed(21)
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#' # creat predictor data X ~ N(0, I_p)
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#' X <- matrix(rnorm(500), 100, 5)
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#' # simulate response variable
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#' # Y = f(B'X) + err
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#' # with f(x1) = x1 and err ~ N(0, 0.25^2)
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#' Y <- X %*% B + 0.25 * rnorm(100)
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#'
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#' # calculate cve with method 'simple' for k = 1
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#' set.seed(21)
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#' cve.obj.simple1 <- cve(Y ~ X, k = 1)
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#'
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#' # same as
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#' set.seed(21)
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#' cve.obj.simple2 <- cve.call(X, Y, k = 1)
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#'
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#' # extract estimated B's.
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#' coef(cve.obj.simple1, k = 1)
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#' coef(cve.obj.simple2, k = 1)
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#' @export
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cve.call <- function(X, Y, method = "simple",
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nObs = sqrt(nrow(X)), h = NULL,
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min.dim = 1L, max.dim = 10L, k = NULL,
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momentum = 0.0, tau = 1.0, tol = 1e-3,
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slack = 0.0, gamma = 0.5,
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V.init = NULL,
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max.iter = 50L, attempts = 10L,
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logger = NULL) {
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# Determine method with partial matching (shortcuts: "Weight" -> "weighted")
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methods <- list(
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"simple" = 0L,
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"weighted" = 1L
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)
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method_nr <- methods[[tolower(method), exact = FALSE]]
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if (!is.integer(method_nr)) {
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stop('Unable to determine method.')
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}
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method <- names(which(method_nr == methods))
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# parameter checking
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if (!is.numeric(momentum) || length(momentum) > 1L) {
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stop("Momentum must be a number.")
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}
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if (!is.double(momentum)) {
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momentum <- as.double(momentum)
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}
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if (momentum < 0.0 || momentum >= 1.0) {
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stop("Momentum must be in [0, 1).")
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}
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if (!(is.matrix(X) && is.numeric(X))) {
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stop("Parameter 'X' should be a numeric matrix.")
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}
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if (!is.numeric(Y)) {
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stop("Parameter 'Y' must be numeric.")
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}
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if (is.matrix(Y) || !is.double(Y)) {
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Y <- as.double(Y)
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}
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if (nrow(X) != length(Y)) {
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stop("Rows of 'X' and 'Y' elements are not compatible.")
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}
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if (ncol(X) < 2) {
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stop("'X' is one dimensional, no need for dimension reduction.")
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}
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if (!is.null(V.init)) {
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if (!is.matrix(V.init)) {
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stop("'V.init' must be a matrix.")
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}
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if (!all.equal(crossprod(V.init), diag(1, ncol(V.init)))) {
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stop("'V.init' must be semi-orthogonal.")
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}
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if (ncol(X) != nrow(V.init) || ncol(X) <= ncol(V.init)) {
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stop("Dimension missmatch of 'V.init' and 'X'")
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}
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min.dim <- max.dim <- ncol(X) - ncol(V.init)
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attempts <- 0L
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} else if (missing(k) || is.null(k)) {
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min.dim <- as.integer(min.dim)
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max.dim <- as.integer(min(max.dim, ncol(X) - 1L))
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} else {
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min.dim <- max.dim <- as.integer(k)
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}
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if (min.dim > max.dim) {
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stop("'min.dim' bigger 'max.dim'.")
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}
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if (max.dim >= ncol(X)) {
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stop("'max.dim' (or 'k') must be smaller than 'ncol(X)'.")
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}
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if (missing(h) || is.null(h)) {
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estimate <- TRUE
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} else if (is.function(h)) {
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estimate <- TRUE
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estimate.bandwidth <- h
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} else if (is.numeric(h) && h > 0.0) {
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estimate <- FALSE
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h <- as.double(h)
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} else {
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stop("Bandwidth 'h' must be positive numeric.")
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}
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if (!is.numeric(tau) || length(tau) > 1L || tau <= 0.0) {
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stop("Initial step-width 'tau' must be positive number.")
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} else {
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tau <- as.double(tau)
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}
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if (!is.numeric(tol) || length(tol) > 1L || tol < 0.0) {
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stop("Break condition tolerance 'tol' must be not negative number.")
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} else {
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tol <- as.double(tol)
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}
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if (!is.numeric(slack) || length(slack) > 1L) {
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stop("Break condition slack 'slack' must be not negative number.")
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} else {
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slack <- as.double(slack)
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}
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if (!is.numeric(gamma) || length(gamma) > 1L || gamma <= 0.0 || gamma >= 1.0) {
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stop("Stepsize reduction 'gamma' must be between 0 and 1.")
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} else {
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gamma <- as.double(gamma)
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}
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if (!is.numeric(max.iter) || length(max.iter) > 1L) {
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stop("Parameter 'max.iter' must be positive integer.")
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} else if (!is.integer(max.iter)) {
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max.iter <- as.integer(max.iter)
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}
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if (max.iter < 1L) {
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stop("Parameter 'max.iter' must be at least 1L.")
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}
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if (is.null(V.init)) {
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if (!is.numeric(attempts) || length(attempts) > 1L) {
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stop("Parameter 'attempts' must be positive integer.")
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} else if (!is.integer(attempts)) {
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attempts <- as.integer(attempts)
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}
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if (attempts < 1L) {
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stop("Parameter 'attempts' must be at least 1L.")
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}
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}
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if (is.function(logger)) {
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loggerEnv <- environment(logger)
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} else {
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loggerEnv <- NULL
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}
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# Call specified method.
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method <- tolower(method)
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call <- match.call()
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dr <- list()
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dr$res <- list()
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for (k in min.dim:max.dim) {
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if (estimate) {
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h <- estimate.bandwidth(X, k, nObs)
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}
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dr.k <- .Call('cve_export', PACKAGE = 'CVE',
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X, Y, k, h,
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method_nr,
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V.init,
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momentum, tau, tol,
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slack, gamma,
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max.iter, attempts,
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logger, loggerEnv)
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dr.k$B <- null(dr.k$V)
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dr.k$loss <- mean(dr.k$L)
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dr.k$h <- h
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dr.k$k <- k
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class(dr.k) <- "cve.k"
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dr$res[[as.character(k)]] <- dr.k
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}
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# augment result information
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dr$X <- X
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dr$Y <- Y
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dr$method <- method
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dr$call <- call
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class(dr) <- "cve"
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return(dr)
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}
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