% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/CVE.R
\name{cve}
\alias{cve}
\title{Conditional Variance Estimator (CVE).}
\usage{
cve(formula, data, method = "simple", max.dim = 10L, ...)
}
\arguments{
\item{formula}{an object of class \code{"formula"} which is a symbolic
description of the model to be fitted like \eqn{Y\sim X}{Y ~ X} where
\eqn{Y} is a \eqn{n}-dimensional vector of the response variable and
\eqn{X} is a \eqn{n\times p}{n x p} matrix of the predictors.}

\item{data}{an optional data frame, containing the data for the formula if
supplied like \code{data <- data.frame(Y, X)} with dimension
\eqn{n \times (p + 1)}{n x (p + 1)}. By default the variables are taken from
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 "weighted" variation with adaptive weighting of slices.
}
see paper.}

\item{max.dim}{upper bounds for \code{k}, (ignored if \code{k} is supplied).}

\item{...}{optional parameters passed on to \code{cve.call}.}
}
\value{
an S3 object of class \code{cve} with components:
\describe{
   \item{X}{design matrix of predictor vector used for calculating
     cve-estimate,}
   \item{Y}{\eqn{n}-dimensional vector of responses used for calculating
       cve-estimate,}
   \item{method}{Name of used method,}
   \item{call}{the matched call,}
   \item{res}{list of components \code{V, L, B, loss, h} for
      each \code{k = min.dim, ..., max.dim}. If \code{k} was supplied in the
      call \code{min.dim = max.dim = k}.
      \itemize{
          \item \code{B} is the cve-estimate with dimension
              \eqn{p\times k}{p x k}.
          \item \code{V} is the orthogonal complement of \eqn{B}.
          \item \code{L} is the loss for each sample seperatels such that
              it's mean is \code{loss}.
          \item \code{loss} is the value of the target function that is 
              minimized, evaluated at \eqn{V}.
          \item \code{h} bandwidth parameter used to calculate
              \code{B, V, loss, L}.
      }
   }
}
}
\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.
}
\examples{
# set dimensions for simulation model
p <- 8
k <- 2
# create B for simulation
b1 <- rep(1 / sqrt(p), p)
b2 <- (-1)^seq(1, p) / sqrt(p)
B <- cbind(b1, b2)
# sample size
n <- 200
set.seed(21)
# creat predictor data x ~ N(0, I_p)
x <- matrix(rnorm(n * p), n, p)
# simulate response variable
#     y = f(B'x) + err
# with f(x1, x2) = x1^2 + 2 * x2 and err ~ N(0, 0.25^2)
y <- (x \%*\% b1)^2 + 2 * (x \%*\% b2) + 0.25 * rnorm(100)
# calculate cve with method 'simple' for k unknown in 1, ..., 4
cve.obj.s <- cve(y ~ x, max.dim = 4) # default method 'simple'
# calculate cve with method 'weighed' for k = 2
cve.obj.w <- cve(y ~ x, k = 2, method = 'weighted')
# estimate dimension from cve.obj.s
khat <- predict_dim(cve.obj.s)$k
# get cve-estimate for B with dimensions (p, k = khat)
B2 <- coef(cve.obj.s, k = khat)
# get projected X data (same as cve.obj.s$X \%*\% B2)
proj.X <- directions(cve.obj.s, k = khat)
#  plot y against projected data
plot(proj.X[, 1], y)
plot(proj.X[, 2], y)
# creat 10 new x points and y according to model
x.new <- matrix(rnorm(10 * p), 10, p)
y.new <- (x.new \%*\% b1)^2 + 2 * (x.new \%*\% b2) + 0.25 * rnorm(10)
# predict y.new
yhat <- predict(cve.obj.s, x.new, khat)
plot(y.new, yhat)
# projection matrix on span(B)
# same as B \%*\% t(B) since B is semi-orthogonal
PB <- B \%*\% solve(t(B) \%*\% B) \%*\% t(B)
# cve estimates for B with simple and weighted method
B.s <- coef(cve.obj.s, k = 2)
B.w <- coef(cve.obj.w, k = 2)
# same as B.s \%*\% t(B.s) since B.s is semi-orthogonal (same vor B.w)
PB.s <- B.s \%*\% solve(t(B.s) \%*\% B.s) \%*\% t(B.s)
PB.w <- B.w \%*\% solve(t(B.w) \%*\% B.w) \%*\% t(B.w)
# compare estimation accuracy of simple and weighted cve estimate by
# Frobenius norm of difference of projections.
norm(PB - PB.s, type = 'F')
norm(PB - PB.w, type = 'F')

}
\references{
Fertl Lukas, Bura Efstathia. (2019), Conditional Variance
Estimation for Sufficient Dimension Reduction. Working Paper.
}
\seealso{
For a detailed description of \code{formula} see
     \code{\link{formula}}.
}