140 lines
5.3 KiB
R
140 lines
5.3 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.call}
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\alias{cve.call}
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\title{Conditional Variance Estimator (CVE).}
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\usage{
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cve.call(X, Y, method = "simple", nObs = sqrt(nrow(X)), h = NULL,
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min.dim = 1L, max.dim = 10L, k = NULL, momentum = 0, tau = 1,
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tol = 0.001, slack = 0, gamma = 0.5, V.init = NULL,
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max.iter = 50L, attempts = 10L, logger = NULL)
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}
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\arguments{
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\item{X}{Design matrix with dimension \eqn{n\times p}{n x p}.}
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\item{Y}{numeric array of length \eqn{n} of Responses.}
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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{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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\item{h}{bandwidth or function to estimate bandwidth, defaults to internaly
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estimated bandwidth.}
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\item{min.dim}{lower bounds for \code{k}, (ignored if \code{k} is supplied).}
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\item{max.dim}{upper bounds for \code{k}, (ignored if \code{k} is supplied).}
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\item{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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\item{momentum}{number of \eqn{[0, 1)} giving the ration of momentum for
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eucledian gradient update with a momentum term. \code{momentum = 0}
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corresponds to normal gradient descend.}
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\item{tau}{Initial step-size.}
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\item{tol}{Tolerance for break condition.}
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\item{slack}{Positive scaling to allow small increases of the loss while
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optimizing, i.e. \code{slack = 0.1} allows the target function to
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increase up to \eqn{10 \%} in one optimization step.}
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\item{gamma}{step-size reduction multiple. If gradient step with step size
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\code{tau} is not accepted \code{gamma * tau} is set to the next step
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size.}
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\item{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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\item{max.iter}{maximum number of optimization steps.}
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\item{attempts}{If \code{V.init} not supplied, the optimization is carried
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out \code{attempts} times with starting values drawn from the invariant
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measure on the Stiefel manifold (see \code{\link{rStiefel}}).}
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\item{logger}{a logger function (only for advanced user, slows down the
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computation).}
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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}{design matrix of predictor vector used for calculating
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cve-estimate,}
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\item{Y}{\eqn{n}-dimensional vector of responses used for calculating
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cve-estimate,}
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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} for
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each \code{k = min.dim, ..., max.dim}. If \code{k} was supplied in the
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call \code{min.dim = max.dim = k}.
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\itemize{
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\item \code{B} is the cve-estimate with dimension
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\eqn{p\times k}{p x k}.
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\item \code{V} is the orthogonal complement of \eqn{B}.
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\item \code{L} is the loss for each sample seperatels such that
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it's mean is \code{loss}.
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\item \code{loss} is the value of the target function that is
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minimized, evaluated at \eqn{V}.
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\item \code{h} bandwidth parameter used to calculate
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\code{B, V, loss, L}.
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}
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}
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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 \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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\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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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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# 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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# 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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# 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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}
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