- wip: reviewing and fixing doc.
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@ -2,7 +2,7 @@ Package: CVE
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Type: Package
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Title: Conditional Variance Estimator for Sufficient Dimension Reduction
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Version: 0.2
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Date: 2019-11-13
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Date: 2019-12-20
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Author: Daniel Kapla <daniel@kapla.at>, Lukas Fertl <lukas.fertl@chello.at>
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Maintainer: Daniel Kapla <daniel@kapla.at>
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Description: Implementation of the Conditional Variance Estimation (CVE) method.
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62
CVE/R/CVE.R
62
CVE/R/CVE.R
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@ -2,16 +2,13 @@
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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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#' where \eqn{B'X} is a lower dimensional projection of the predictors and
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#' \eqn{Y} is a univariate responce. 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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#' continuous predictors and link function. Let \eqn{X} be 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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@ -20,11 +17,11 @@
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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 \leq p}{k <= p}.
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#' a real \eqn{p \times k}{p x k} matrix of rank \eqn{k \leq p}{k <= 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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#' @references Fertl, L. and Bura, E. (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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@ -33,7 +30,10 @@
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#' Conditional Variance Estimator (CVE).
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#'
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#' @inherit CVE-package description
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#' This is the main function in the \code{CVE} package. It creates objects of
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#' class \code{"cve"} to estimate the mean subspace. Helper functions that
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#' require a \code{"cve"} object can then be applied to the output from this
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#' function.
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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 like \eqn{Y\sim X}{Y ~ X} where
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@ -46,13 +46,41 @@
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#' @param method This character string specifies the method of fitting. The
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#' options are
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#' \itemize{
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#' \item "simple" implementation as described in the paper.
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#' \item "simple" implementation,
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#' \item "weighted" variation with adaptive weighting of slices.
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#' }
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#' see paper.
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#' see Fertl, L. and Bura, E. (2019).
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#' @param max.dim upper bounds for \code{k}, (ignored if \code{k} is supplied).
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#' @param ... optional parameters passed on to \code{cve.call}.
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#'
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#'
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#' Conditional Variance Estimation (CVE) is a sufficient dimension reduction
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#' (SDR) method for regressions studying \eqn{E(Y|X)}, the conditional
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#' expectation of a response \eqn{Y} given a set of predictors \eqn{X}. This
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#' function provides methods for estimating the dimension and the subspace
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#' spanned by the columns of a \eqn{p\times k}{p x k} matrix \eqn{B} of minimal
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#' rank \eqn{k} such that
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#' \deqn{%
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#' E(Y|X) = E(Y|B'X) %
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#' }
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#' or, equivalently,
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#' \deqn{%
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#' Y = g(B'X) + \epsilon %
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#' }
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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, and \eqn{B = (b_1,..., b_k)}
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#' is a real \eqn{p \times k}{p x k} matrix of rank \eqn{k \leq p}{k <= p}.
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#'
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#' Both the dimension \eqn{k} and the subspace \eqn{span(B)} are unknown. The
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#' CVE method makes very few assumptions.
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#'
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#' A kernel matrix \eqn{\hat{B}}{Bhat} is estimated such that the column space
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#' of \eqn{\hat{B}}{Bhat} should be close to the mean subspace \eqn{span(B)}.
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#' The primary output from this method is a set of orthonormal vectors,
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#' \eqn{\hat{B}}{Bhat}, whose span estimates \eqn{span(B)}.
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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}{design matrix of predictor vector used for calculating
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@ -130,7 +158,7 @@
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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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#' @references Fertl, L. and Bura, E. (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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@ -159,8 +187,8 @@ cve <- function(formula, data, method = "simple", max.dim = 10L, ...) {
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#' @inherit cve title
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#' @inherit cve description
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#'
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#' @param X Design matrix with dimension \eqn{n\times p}{n x p}.
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#' @param Y numeric array of length \eqn{n} of Responses.
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#' @param X Design predictor matrix.
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#' @param Y \eqn{n}-dimensional vector of responces.
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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 nObs parameter for choosing bandwidth \code{h} using
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@ -193,7 +221,7 @@ cve <- function(formula, data, method = "simple", max.dim = 10L, ...) {
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#' @param V.init Semi-orthogonal matrix of dimensions `(ncol(X), ncol(X) - k)
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#' used as starting value in the optimization. (If supplied,
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#' \code{attempts} is set to 0 and \code{k} to match dimension).
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#' @param logger a logger function (only for advanced user, slows down the
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#' @param logger a logger function (only for advanced users, slows down the
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#' computation).
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#'
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#' @inherit cve return
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@ -209,11 +237,11 @@ cve <- function(formula, data, method = "simple", max.dim = 10L, ...) {
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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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#'
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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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#'
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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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16
CVE/R/coef.R
16
CVE/R/coef.R
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@ -1,14 +1,14 @@
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#' Gets estimated SDR basis.
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#' Extracts estimated SDR basis.
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#'
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#' Returns the SDR basis matrix for dimension \code{k}, i.e. returns the
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#' cve-estimate with dimension \eqn{p\times k}{p x k}.
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#' cve-estimate of \eqn{B} with dimension \eqn{p\times k}{p x k}.
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#'
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#' @param object instance of \code{cve} as output from \code{\link{cve}} or
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#' \code{\link{cve.call}}.
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#' @param object an object of class \code{"cve"}, usually, a result of a call to
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#' \code{\link{cve}} or \code{\link{cve.call}}.
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#' @param k the SDR dimension.
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#' @param ... ignored.
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#' @param ... ignored (no additional arguments).
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#'
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#' @return dir the matrix of CS or CMS of given dimension
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#' @return The matrix \eqn{B} of dimensions \eqn{p\times k}{p x k}.
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#'
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#' @examples
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#' # set dimensions for simulation model
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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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#'
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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(n * p), n, p)
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@ -31,7 +31,7 @@
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#' cve.obj <- cve(y ~ x, max.dim = 5)
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#' # get cve-estimate for B with dimensions (p, k = 2)
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#' B2 <- coef(cve.obj, k = 2)
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#'
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#'
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#' # Projection matrix on span(B)
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#' # equivalent to `B %*% t(B)` since B is semi-orthonormal
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#' PB <- B %*% solve(t(B) %*% B) %*% t(B)
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@ -1,15 +1,17 @@
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#' @export
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directions <- function(dr, k) {
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directions <- function(object, k, ...) {
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UseMethod("directions")
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}
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#' Computes projected training data \code{X} for given dimension `k`.
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#'
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#' Projects the dimensional design matrix \eqn{X} on the columnspace of the
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#' cve-estimate for given dimension \eqn{k}.
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#' Returns \eqn{B'X}. That is the dimensional design matrix \eqn{X} on the
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#' columnspace of the cve-estimate for given dimension \eqn{k}.
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#'
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#' @param dr Instance of \code{'cve'} as returned by \code{\link{cve}}.
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#' @param object an object of class \code{"cve"}, usually, a result of a call to
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#' \code{\link{cve}} or \code{\link{cve.call}}.
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#' @param k SDR dimension to use for projection.
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#' @param ... ignored (no additional arguments).
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#'
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#' @return the \eqn{n\times k}{n x k} dimensional matrix \eqn{X B} where \eqn{B}
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#' is the cve-estimate for dimension \eqn{k}.
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@ -32,12 +34,14 @@ directions <- function(dr, k) {
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#' # plot y against projected data
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#' plot(x.proj, y)
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#'
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#' @seealso \code{\link{cve}}
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#'
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#' @method directions cve
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#' @aliases directions directions.cve
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#' @export
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directions.cve <- function(dr, k) {
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if (!(k %in% names(dr$res))) {
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directions.cve <- function(object, k, ...) {
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if (!(k %in% names(object$res))) {
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stop("SDR directions for requested dimension `k` not computed.")
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}
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return(dr$X %*% dr$res[[as.character(k)]]$B)
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return(object$X %*% object$res[[as.character(k)]]$B)
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}
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#' h = (2 * tr(\Sigma) / p) * (1.2 * n^(-1 / (4 + k)))^2}
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#' Alternative version 2 is used for dimension prediction which is given by
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#' \deqn{%
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#' h = (2 * tr(\Sigma) / p) * \chi_k^{-1}(\frac{nObs - 1}{n - 1})}{%
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#' h = \frac{2 tr(\Sigma)}{p} \chi_k^{-1}(\frac{nObs - 1}{n - 1})}{%
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#' h = (2 * tr(\Sigma) / p) * \chi_k^-1((nObs - 1) / (n - 1))}
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#' with \eqn{n} the sample size, \eqn{p} its dimension and the
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#' covariance-matrix \eqn{\Sigma}, which is \code{(n-1)/n} times the sample
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#' covariance estimate.
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#' with \eqn{n} the sample size, \eqn{p} the dimension of \eqn{X} and
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#' \eqn{\Sigma} is \eqn{(n - 1) / n} times the sample covariance matrix of
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#' \eqn{X}.
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#'
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#' @param X a \eqn{n\times p}{n x p} matrix with samples in its rows.
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#' @param k Dimension of lower dimensional projection.
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#' @param X the \eqn{n\times p}{n x p} matrix of predictor values.
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#' @param k the SDR dimension.
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#' @param nObs number of points in a slice, only for version 2.
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#' @param version either \code{1} or \code{2}.
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#'
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13
CVE/R/plot.R
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CVE/R/plot.R
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@ -1,13 +1,16 @@
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#' Loss distribution elbow plot.
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#' Elbow plot of the loss function.
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#'
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#' Boxplots of the output \code{L} from \code{\link{cve}} over \code{k} from
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#' \code{min.dim} to \code{max.dim}. For given \code{k}, \code{L} corresponds
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#' to \eqn{L_n(V, X_i)} where \eqn{V \in S(p, p - k)}{V} is the minimizer of
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#' \eqn{L_n(V)}, for further details see the paper.
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#' to \eqn{L_n(V, X_i)} where \eqn{V} is a stiefel manifold element as
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#' minimizer of
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#' \eqn{L_n(V)}, for further details see Fertl, L. and Bura, E. (2019).
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#'
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#' @param x Object of class \code{"cve"} (result of [\code{\link{cve}}]).
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#' @param x an object of class \code{"cve"}, usually, a result of a call to
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#' \code{\link{cve}} or \code{\link{cve.call}}.
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#' @param ... Pass through parameters to [\code{\link{plot}}] and
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#' [\code{\link{lines}}]
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#'
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#' @examples
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#' # create B for simulation
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#' B <- cbind(rep(1, 6), (-1)^seq(6)) / sqrt(6)
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#' # elbow plot
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#' plot(cve.obj.simple)
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#'
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#' @references Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
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#' @references Fertl, L. and Bura, E. (2019), Conditional Variance
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#' Estimation for Sufficient Dimension Reduction. Working Paper.
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#'
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#' @seealso see \code{\link{par}} for graphical parameters to pass through
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#' Predict method for CVE Fits.
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#'
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#' Predict response using projected data where the forward model \eqn{g(B'X)}
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#' is estimated using \code{\link{mars}}.
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#' Predict response using projected data \eqn{B'C} by fitting
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#' \eqn{g(B'C) + \epsilon} using \code{\link{mars}}.
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#'
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#' @param object instance of class \code{cve} (result of \code{cve},
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#' \code{cve.call}).
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#' @param newdata Matrix of the new data to be predicted.
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#' @param dim dimension of SDR space to be used for data projecition.
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#' @param object an object of class \code{"cve"}, usually, a result of a call to
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#' \code{\link{cve}} or \code{\link{cve.call}}.
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#' @param newdata Matrix of new predictor values, \eqn{C}.
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#' @param k dimension of SDR space to be used for data projection.
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#' @param ... further arguments passed to \code{\link{mars}}.
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#'
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#' @return prediced response of data \code{newdata}.
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#' @return prediced response at \code{newdata}.
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#'
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#' @examples
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#' # create B for simulation
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#' @importFrom mda mars
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#' @method predict cve
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#' @export
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predict.cve <- function(object, newdata, dim, ...) {
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predict.cve <- function(object, newdata, k, ...) {
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if (missing(newdata)) {
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stop("No data supplied.")
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}
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if (missing(dim)) {
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if (missing(k)) {
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stop("No dimension supplied.")
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}
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newdata <- matrix(newdata, nrow = 1L)
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}
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B <- object$res[[as.character(dim)]]$B
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B <- object$res[[as.character(k)]]$B
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model <- mda::mars(object$X %*% B, object$Y)
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predict(model, newdata %*% B)
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#'
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#' This function estimates the dimension of the mean dimension reduction space,
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#' i.e. number of columns of \eqn{B} matrix. The default method \code{'CV'}
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#' performs cross-validation using \code{mars}. Given
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#' performs l.o.o cross-validation using \code{mars}. Given
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#' \code{k = min.dim, ..., max.dim} a cross-validation via \code{mars} is
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#' performed on the dataset \eqn{(Y i, B_k' X_i)_{i = 1, ..., n}} where
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#' \eqn{B_k} is the \eqn{p \times k}{p x k} dimensional CVE estimate given
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#' \eqn{k}. The estimated SDR dimension is the \eqn{k} where the
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#' cross-validation mean squared error is the lowest. The method \code{'elbow'}
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#' estimates the dimension via \eqn{k = argmin_k L_n(V_{p − k})} where
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#' \eqn{V_{p − k}} is the CVE estimate of the orthogonal columnspace of
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#' \eqn{B_k}. Method \code{'wilcoxon'} is similar to \code{'elbow'} but finds
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#' the minimum using the wilcoxon-test.
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#' performed on the dataset \eqn{(Y_i, B_k' X_i)_{i = 1, ..., n}} where
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#' \eqn{B_k} is the \eqn{p \times k}{p x k} dimensional CVE estimate. The
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#' estimated SDR dimension is the \eqn{k} where the
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#' cross-validation mean squared error is minimal. The method \code{'elbow'}
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#' estimates the dimension via \eqn{k = argmin_k L_n(V_{p - k})} where
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#' \eqn{V_{p - k}} is space that is orthogonal to the columns-space of the CVE estimate of \eqn{B_k}. Method \code{'wilcoxon'} is similar to \code{'elbow'}
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#' but finds the minimum using the wilcoxon-test.
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#'
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#' @param object instance of class \code{cve} (result of \code{\link{cve}},
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#' \code{\link{cve.call}}).
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#' @param object an object of class \code{"cve"}, usually, a result of a call to
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#' \code{\link{cve}} or \code{\link{cve.call}}.
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#' @param method This parameter specify which method will be used in dimension
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#' estimation. It provides three methods \code{'CV'} (default), \code{'elbow'},
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#' and \code{'wilcoxon'} to estimate the dimension of the SDR.
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@ -8,16 +8,13 @@
|
|||
\description{
|
||||
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,
|
||||
where \eqn{B'X} is a lower dimensional projection of the predictors and
|
||||
\eqn{Y} is a univariate responce. 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
|
||||
continuous predictors and link function. Let \eqn{X} be 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}
|
||||
|
|
@ -26,11 +23,11 @@ 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 \leq p}{k <= p}.
|
||||
a real \eqn{p \times k}{p x k} matrix of rank \eqn{k \leq p}{k <= p}.
|
||||
Without loss of generality \eqn{B} is assumed to be orthonormal.
|
||||
}
|
||||
\references{
|
||||
Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
}
|
||||
\author{
|
||||
|
|
|
|||
|
|
@ -2,24 +2,24 @@
|
|||
% Please edit documentation in R/coef.R
|
||||
\name{coef.cve}
|
||||
\alias{coef.cve}
|
||||
\title{Gets estimated SDR basis.}
|
||||
\title{Extracts estimated SDR basis.}
|
||||
\usage{
|
||||
\method{coef}{cve}(object, k, ...)
|
||||
}
|
||||
\arguments{
|
||||
\item{object}{instance of \code{cve} as output from \code{\link{cve}} or
|
||||
\code{\link{cve.call}}.}
|
||||
\item{object}{an object of class \code{"cve"}, usually, a result of a call to
|
||||
\code{\link{cve}} or \code{\link{cve.call}}.}
|
||||
|
||||
\item{k}{the SDR dimension.}
|
||||
|
||||
\item{...}{ignored.}
|
||||
\item{...}{ignored (no additional arguments).}
|
||||
}
|
||||
\value{
|
||||
dir the matrix of CS or CMS of given dimension
|
||||
The matrix \eqn{B} of dimensions \eqn{p\times k}{p x k}.
|
||||
}
|
||||
\description{
|
||||
Returns the SDR basis matrix for dimension \code{k}, i.e. returns the
|
||||
cve-estimate with dimension \eqn{p\times k}{p x k}.
|
||||
cve-estimate of \eqn{B} with dimension \eqn{p\times k}{p x k}.
|
||||
}
|
||||
\examples{
|
||||
# set dimensions for simulation model
|
||||
|
|
@ -30,7 +30,7 @@ n <- 200 # samplesize
|
|||
b1 <- rep(1 / sqrt(p), p)
|
||||
b2 <- (-1)^seq(1, p) / sqrt(p)
|
||||
B <- cbind(b1, b2)
|
||||
|
||||
|
||||
set.seed(21)
|
||||
# creat predictor data x ~ N(0, I_p)
|
||||
x <- matrix(rnorm(n * p), n, p)
|
||||
|
|
@ -42,7 +42,7 @@ y <- (x \%*\% b1)^2 + 2 * (x \%*\% b2) + 0.25 * rnorm(100)
|
|||
cve.obj <- cve(y ~ x, max.dim = 5)
|
||||
# get cve-estimate for B with dimensions (p, k = 2)
|
||||
B2 <- coef(cve.obj, k = 2)
|
||||
|
||||
|
||||
# Projection matrix on span(B)
|
||||
# equivalent to `B \%*\% t(B)` since B is semi-orthonormal
|
||||
PB <- B \%*\% solve(t(B) \%*\% B) \%*\% t(B)
|
||||
|
|
|
|||
|
|
@ -20,14 +20,42 @@ the environment from which \code{cve} is called.}
|
|||
\item{method}{This character string specifies the method of fitting. The
|
||||
options are
|
||||
\itemize{
|
||||
\item "simple" implementation as described in the paper.
|
||||
\item "simple" implementation,
|
||||
\item "weighted" variation with adaptive weighting of slices.
|
||||
}
|
||||
see paper.}
|
||||
see Fertl, L. and Bura, E. (2019).}
|
||||
|
||||
\item{max.dim}{upper bounds for \code{k}, (ignored if \code{k} is supplied).}
|
||||
|
||||
\item{...}{optional parameters passed on to \code{cve.call}.}
|
||||
\item{...}{optional parameters passed on to \code{cve.call}.
|
||||
|
||||
|
||||
Conditional Variance Estimation (CVE) is a sufficient dimension reduction
|
||||
(SDR) method for regressions studying \eqn{E(Y|X)}, the conditional
|
||||
expectation of a response \eqn{Y} given a set of predictors \eqn{X}. This
|
||||
function provides methods for estimating the dimension and the subspace
|
||||
spanned by the columns of a \eqn{p\times k}{p x k} matrix \eqn{B} of minimal
|
||||
rank \eqn{k} such that
|
||||
\deqn{%
|
||||
E(Y|X) = E(Y|B'X) %
|
||||
}
|
||||
or, equivalently,
|
||||
\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} matrix of rank \eqn{k \leq p}{k <= p}.
|
||||
|
||||
Both the dimension \eqn{k} and the subspace \eqn{span(B)} are unknown. The
|
||||
CVE method makes very few assumptions.
|
||||
|
||||
A kernel matrix \eqn{\hat{B}}{Bhat} is estimated such that the column space
|
||||
of \eqn{\hat{B}}{Bhat} should be close to the mean subspace \eqn{span(B)}.
|
||||
The primary output from this method is a set of orthonormal vectors,
|
||||
\eqn{\hat{B}}{Bhat}, whose span estimates \eqn{span(B)}.}
|
||||
}
|
||||
\value{
|
||||
an S3 object of class \code{cve} with components:
|
||||
|
|
@ -56,28 +84,10 @@ an S3 object of class \code{cve} with components:
|
|||
}
|
||||
}
|
||||
\description{
|
||||
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 \leq p}{k <= p}.
|
||||
Without loss of generality \eqn{B} is assumed to be orthonormal.
|
||||
This is the main function in the \code{CVE} package. It creates objects of
|
||||
class \code{"cve"} to estimate the mean subspace. Helper functions that
|
||||
require a \code{"cve"} object can then be applied to the output from this
|
||||
function.
|
||||
}
|
||||
\examples{
|
||||
# set dimensions for simulation model
|
||||
|
|
@ -131,7 +141,7 @@ norm(PB - PB.w, type = 'F')
|
|||
|
||||
}
|
||||
\references{
|
||||
Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
}
|
||||
\seealso{
|
||||
|
|
|
|||
|
|
@ -10,9 +10,9 @@ cve.call(X, Y, method = "simple", nObs = sqrt(nrow(X)), h = NULL,
|
|||
max.iter = 50L, attempts = 10L, logger = NULL)
|
||||
}
|
||||
\arguments{
|
||||
\item{X}{Design matrix with dimension \eqn{n\times p}{n x p}.}
|
||||
\item{X}{Design predictor matrix.}
|
||||
|
||||
\item{Y}{numeric array of length \eqn{n} of Responses.}
|
||||
\item{Y}{\eqn{n}-dimensional vector of responces.}
|
||||
|
||||
\item{method}{specifies the CVE method variation as one of
|
||||
\itemize{
|
||||
|
|
@ -60,7 +60,7 @@ used as starting value in the optimization. (If supplied,
|
|||
out \code{attempts} times with starting values drawn from the invariant
|
||||
measure on the Stiefel manifold (see \code{\link{rStiefel}}).}
|
||||
|
||||
\item{logger}{a logger function (only for advanced user, slows down the
|
||||
\item{logger}{a logger function (only for advanced users, slows down the
|
||||
computation).}
|
||||
}
|
||||
\value{
|
||||
|
|
@ -90,28 +90,10 @@ an S3 object of class \code{cve} with components:
|
|||
}
|
||||
}
|
||||
\description{
|
||||
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 \leq p}{k <= p}.
|
||||
Without loss of generality \eqn{B} is assumed to be orthonormal.
|
||||
This is the main function in the \code{CVE} package. It creates objects of
|
||||
class \code{"cve"} to estimate the mean subspace. Helper functions that
|
||||
require a \code{"cve"} object can then be applied to the output from this
|
||||
function.
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation (k = 1)
|
||||
|
|
|
|||
|
|
@ -5,20 +5,23 @@
|
|||
\alias{directions}
|
||||
\title{Computes projected training data \code{X} for given dimension `k`.}
|
||||
\usage{
|
||||
\method{directions}{cve}(dr, k)
|
||||
\method{directions}{cve}(object, k, ...)
|
||||
}
|
||||
\arguments{
|
||||
\item{dr}{Instance of \code{'cve'} as returned by \code{\link{cve}}.}
|
||||
\item{object}{an object of class \code{"cve"}, usually, a result of a call to
|
||||
\code{\link{cve}} or \code{\link{cve.call}}.}
|
||||
|
||||
\item{k}{SDR dimension to use for projection.}
|
||||
|
||||
\item{...}{ignored (no additional arguments).}
|
||||
}
|
||||
\value{
|
||||
the \eqn{n\times k}{n x k} dimensional matrix \eqn{X B} where \eqn{B}
|
||||
is the cve-estimate for dimension \eqn{k}.
|
||||
}
|
||||
\description{
|
||||
Projects the dimensional design matrix \eqn{X} on the columnspace of the
|
||||
cve-estimate for given dimension \eqn{k}.
|
||||
Returns \eqn{B'X}. That is the dimensional design matrix \eqn{X} on the
|
||||
columnspace of the cve-estimate for given dimension \eqn{k}.
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation (k = 1)
|
||||
|
|
@ -39,3 +42,6 @@ x.proj <- directions(cve.obj.simple, k = 1)
|
|||
plot(x.proj, y)
|
||||
|
||||
}
|
||||
\seealso{
|
||||
\code{\link{cve}}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -7,9 +7,9 @@
|
|||
estimate.bandwidth(X, k, nObs, version = 1L)
|
||||
}
|
||||
\arguments{
|
||||
\item{X}{a \eqn{n\times p}{n x p} matrix with samples in its rows.}
|
||||
\item{X}{the \eqn{n\times p}{n x p} matrix of predictor values.}
|
||||
|
||||
\item{k}{Dimension of lower dimensional projection.}
|
||||
\item{k}{the SDR dimension.}
|
||||
|
||||
\item{nObs}{number of points in a slice, only for version 2.}
|
||||
|
||||
|
|
@ -26,11 +26,11 @@ defaults to using the following formula (version 1)
|
|||
h = (2 * tr(\Sigma) / p) * (1.2 * n^(-1 / (4 + k)))^2}
|
||||
Alternative version 2 is used for dimension prediction which is given by
|
||||
\deqn{%
|
||||
h = (2 * tr(\Sigma) / p) * \chi_k^{-1}(\frac{nObs - 1}{n - 1})}{%
|
||||
h = \frac{2 tr(\Sigma)}{p} \chi_k^{-1}(\frac{nObs - 1}{n - 1})}{%
|
||||
h = (2 * tr(\Sigma) / p) * \chi_k^-1((nObs - 1) / (n - 1))}
|
||||
with \eqn{n} the sample size, \eqn{p} its dimension and the
|
||||
covariance-matrix \eqn{\Sigma}, which is \code{(n-1)/n} times the sample
|
||||
covariance estimate.
|
||||
with \eqn{n} the sample size, \eqn{p} the dimension of \eqn{X} and
|
||||
\eqn{\Sigma} is \eqn{(n - 1) / n} times the sample covariance matrix of
|
||||
\eqn{X}.
|
||||
}
|
||||
\examples{
|
||||
# set dimensions for simulation model
|
||||
|
|
|
|||
|
|
@ -2,12 +2,13 @@
|
|||
% Please edit documentation in R/plot.R
|
||||
\name{plot.cve}
|
||||
\alias{plot.cve}
|
||||
\title{Loss distribution elbow plot.}
|
||||
\title{Elbow plot of the loss function.}
|
||||
\usage{
|
||||
\method{plot}{cve}(x, ...)
|
||||
}
|
||||
\arguments{
|
||||
\item{x}{Object of class \code{"cve"} (result of [\code{\link{cve}}]).}
|
||||
\item{x}{an object of class \code{"cve"}, usually, a result of a call to
|
||||
\code{\link{cve}} or \code{\link{cve.call}}.}
|
||||
|
||||
\item{...}{Pass through parameters to [\code{\link{plot}}] and
|
||||
[\code{\link{lines}}]}
|
||||
|
|
@ -15,8 +16,9 @@
|
|||
\description{
|
||||
Boxplots of the output \code{L} from \code{\link{cve}} over \code{k} from
|
||||
\code{min.dim} to \code{max.dim}. For given \code{k}, \code{L} corresponds
|
||||
to \eqn{L_n(V, X_i)} where \eqn{V \in S(p, p - k)}{V} is the minimizer of
|
||||
\eqn{L_n(V)}, for further details see the paper.
|
||||
to \eqn{L_n(V, X_i)} where \eqn{V} is a stiefel manifold element as
|
||||
minimizer of
|
||||
\eqn{L_n(V)}, for further details see Fertl, L. and Bura, E. (2019).
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation
|
||||
|
|
@ -46,7 +48,7 @@ plot(cve.obj.simple)
|
|||
|
||||
}
|
||||
\references{
|
||||
Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
}
|
||||
\seealso{
|
||||
|
|
|
|||
|
|
@ -4,24 +4,24 @@
|
|||
\alias{predict.cve}
|
||||
\title{Predict method for CVE Fits.}
|
||||
\usage{
|
||||
\method{predict}{cve}(object, newdata, dim, ...)
|
||||
\method{predict}{cve}(object, newdata, k, ...)
|
||||
}
|
||||
\arguments{
|
||||
\item{object}{instance of class \code{cve} (result of \code{cve},
|
||||
\code{cve.call}).}
|
||||
\item{object}{an object of class \code{"cve"}, usually, a result of a call to
|
||||
\code{\link{cve}} or \code{\link{cve.call}}.}
|
||||
|
||||
\item{newdata}{Matrix of the new data to be predicted.}
|
||||
\item{newdata}{Matrix of new predictor values, \eqn{C}.}
|
||||
|
||||
\item{dim}{dimension of SDR space to be used for data projecition.}
|
||||
\item{k}{dimension of SDR space to be used for data projection.}
|
||||
|
||||
\item{...}{further arguments passed to \code{\link{mars}}.}
|
||||
}
|
||||
\value{
|
||||
prediced response of data \code{newdata}.
|
||||
prediced response at \code{newdata}.
|
||||
}
|
||||
\description{
|
||||
Predict response using projected data where the forward model \eqn{g(B'X)}
|
||||
is estimated using \code{\link{mars}}.
|
||||
Predict response using projected data \eqn{B'C} by fitting
|
||||
\eqn{g(B'C) + \epsilon} using \code{\link{mars}}.
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation
|
||||
|
|
|
|||
|
|
@ -7,8 +7,8 @@
|
|||
predict_dim(object, ..., method = "CV")
|
||||
}
|
||||
\arguments{
|
||||
\item{object}{instance of class \code{cve} (result of \code{\link{cve}},
|
||||
\code{\link{cve.call}}).}
|
||||
\item{object}{an object of class \code{"cve"}, usually, a result of a call to
|
||||
\code{\link{cve}} or \code{\link{cve.call}}.}
|
||||
|
||||
\item{...}{ignored.}
|
||||
|
||||
|
|
@ -26,16 +26,15 @@ list with
|
|||
\description{
|
||||
This function estimates the dimension of the mean dimension reduction space,
|
||||
i.e. number of columns of \eqn{B} matrix. The default method \code{'CV'}
|
||||
performs cross-validation using \code{mars}. Given
|
||||
performs l.o.o cross-validation using \code{mars}. Given
|
||||
\code{k = min.dim, ..., max.dim} a cross-validation via \code{mars} is
|
||||
performed on the dataset \eqn{(Y i, B_k' X_i)_{i = 1, ..., n}} where
|
||||
\eqn{B_k} is the \eqn{p \times k}{p x k} dimensional CVE estimate given
|
||||
\eqn{k}. The estimated SDR dimension is the \eqn{k} where the
|
||||
cross-validation mean squared error is the lowest. The method \code{'elbow'}
|
||||
estimates the dimension via \eqn{k = argmin_k L_n(V_{p − k})} where
|
||||
\eqn{V_{p − k}} is the CVE estimate of the orthogonal columnspace of
|
||||
\eqn{B_k}. Method \code{'wilcoxon'} is similar to \code{'elbow'} but finds
|
||||
the minimum using the wilcoxon-test.
|
||||
performed on the dataset \eqn{(Y_i, B_k' X_i)_{i = 1, ..., n}} where
|
||||
\eqn{B_k} is the \eqn{p \times k}{p x k} dimensional CVE estimate. The
|
||||
estimated SDR dimension is the \eqn{k} where the
|
||||
cross-validation mean squared error is minimal. The method \code{'elbow'}
|
||||
estimates the dimension via \eqn{k = argmin_k L_n(V_{p - k})} where
|
||||
\eqn{V_{p - k}} is space that is orthogonal to the columns-space of the CVE estimate of \eqn{B_k}. Method \code{'wilcoxon'} is similar to \code{'elbow'}
|
||||
but finds the minimum using the wilcoxon-test.
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation
|
||||
|
|
|
|||
Loading…
Reference in New Issue