84 lines
3.0 KiB
R
84 lines
3.0 KiB
R
% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/CVE.R
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\name{cve}
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\alias{cve}
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\title{Conditional Variance Estimator (CVE).}
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\usage{
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cve(formula, data, method = "simple", max.dim = 10L, ...)
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}
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\arguments{
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\item{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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\item{data}{an optional data frame, containing the data for the formula if
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supplied.}
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\item{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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\item{...}{Parameters passed on to \code{cve.call}.}
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}
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\value{
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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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\description{
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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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\examples{
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# create dataset
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x <- matrix(rnorm(400), 100, 4)
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y <- x[, 1] + x[, 2] + as.matrix(rnorm(100))
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# Call CVE.
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dr <- cve(y ~ x)
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# Call weighted CVE.
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dr.weighted <- cve(y ~ x, method = "weighted")
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# Training data responces of reduced data.
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y.est <- directions(dr, 1)
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# Extract SDR subspace basis of dimension 1.
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B <- coef(dr.momentum, 1)
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}
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\references{
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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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\seealso{
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For a detailed description of \code{formula} see
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\code{\link{formula}}.
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}
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