CVE/LaTeX/main.Rnw

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\renewcommand{\t}[1]{{#1}^{T}}
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\renewcommand{\E}{\operatorname{\mathbb{E}}}
\newcommand{\var}{\operatorname{Var}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Meta Information %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\author{Daniel Kapla\\TU Wien
    \And Lukas Fertl\\TU Wien
    \And Efstathia Bura\\TU Wien}
\Plainauthor{Daniel Kapla, Lukas Fertl, Efstathia Bura}

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\title{Conditional Variance Estimation With the \pkg{CVE} Package in \proglang{R}}
\Plaintitle{Conditional Variance Estimation With the CVE Package in R}
\Shorttitle{The \pkg{CVE} Package}

\Abstract{
    Conditional variance estimation (CVE) is a novel sufficient dimension
    reduction (SDR) method for regressions satisfying $\E(Y | X) = \E(Y | \t{B} X)$,
    where $\t{B}X$ 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. Let $Y$ be a real univariate response and $X$
    a real $p$-dimensional covariate vector. We assume that the dependence of
    $Y$ and $X$ is modelled by
    \begin{displaymath}
        Y = g(\t{B}X) + \epsilon
    \end{displaymath}
    where $X$ is independent of $\epsilon$ with positive definite variance-covariance
    matrix $\var(X) = \Sigma_X$. $\epsilon$ is a mean zero random variable with
    finite $\var(\epsilon) = \E(\epsilon^2)$, $g$ is an unknown, continuous
    non-constant function, and $B = (b_1 , ..., b_k)$ is a real $p \times k$
    matrix of rank $k \leq p$. Without loss of generality $B$ is assumed to be
    orthonormal.
}

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\Keywords{Dimension reduction, \proglang{R}}
\Plainkeywords{Dimension reduction, R}

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\Address{
    Daniel Kapla\\
    Technische Universit\"at Wien\\
    Institute of Statistics and Mathematical Methods in Economics\\
    Faculty of Mathematics and Geoinformation\\
    TU Wien, Vienna, Austria\\
    E-mail: \email{daniel.kapla@tuwien.ac.at}\\
    URL: \url{https://kapla.at}\\
    \\
    Lukas Fertl\\
    Technische Universit\"at Wien\\
    Institute of Statistics and Mathematical Methods in Economics\\
    Faculty of Mathematics and Geoinformation\\
    TU Wien, Vienna, Austria\\
    E-mail: \email{lukas.fertl@tuwien.ac.at}\\
    \\
    Efstathia Bura\\
    Technische Universit\"at Wien\\
    Institute of Statistics and Mathematical Methods in Economics\\
    Faculty of Mathematics and Geoinformation\\
    TU Wien, Vienna, Austria
}

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\begin{document}

\section[Introduction: Sufficient dimension reduction in R]{Introduction: Sufficient dimension reduction in \proglang{R}}\label{sec:intro}

A bit of text
<<>>=
library(CVE)

dataset <- function(n, p = 20, p.mix = 0.5, lambda = 1, sd = 0.5) {
    B <- rep(1 / sqrt(p), p)
    # B <- c(rep(1 / sqrt(p), p / 2), rep(-1 / sqrt(p), p / 2))
    X <- matrix(rnorm(n * p), n, p)
    X <- X + lambda * (2 * rbinom(n, 1, p.mix) - 1)
    Y <- abs(X %*% B) + rnorm(n, 0, sd)
    list(B = B, X = X, Y = Y)
}

ds <- dataset(100)
ds.test <- dataset(100)
@
Then we apply both methods, the \code{CVE} and the \code{MAVE} methods 
<<>>=
fit.cve <- with(ds, cve(Y ~ X, k = 1))
fit.mave <- with(ds, MAVE::mave(Y ~ X, max.dim = 1, method = "meanMAVE"))
@
Get the estimated reduction matrices
<<>>=
B.cve <- coef(fit.cve, 1)
B.mave <- coef(fit.mave, 1)
@
and compute the prediction errors
<<>>=
Y.hat.cve <- with(ds.test, predict(fit.cve, X, 1))
Y.hat.mave <- with(ds.test, predict(fit.mave, X, 1))
# MSE - cve
mean((ds.test$Y - Y.hat.cve)^2)
# MSE - mave
mean((ds.test$Y - Y.hat.mave)^2)
@
and another bit of text


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\end{document}