141 lines
5.5 KiB
R
141 lines
5.5 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 like \eqn{Y\sim X}{Y ~ X} where
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\eqn{Y} is a \eqn{n}-dimensional vector of the response variable and
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\eqn{X} is a \eqn{n\times p}{n x p} matrix of the predictors.}
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\item{data}{an optional data frame, containing the data for the formula if
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supplied like \code{data <- data.frame(Y, X)} with dimension
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\eqn{n \times (p + 1)}{n x (p + 1)}. By default the variables are taken from
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the environment from which \code{cve} is called.}
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\item{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 "weighted" variation with adaptive weighting of slices.
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}
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see paper.}
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\item{max.dim}{upper bounds for \code{k}, (ignored if \code{k} is supplied).}
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\item{...}{optional 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}{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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# 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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# sample size
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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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\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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