% 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 = "mean", 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 \code{"mean"} method to estimate the mean subspace, see [1].
   \item \code{"central"} ensemble method to estimate the central subspace,
     see [2].
   \item \code{"weighted.mean"} variation of \code{"mean"} method with
     adaptive weighting of slices, see [1].
   \item \code{"weighted.central"} variation of \code{"central"} method with
     adaptive weighting of slices, see [2].
}}

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

\item{...}{optional parameters passed on to \code{\link{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{
This is the main function in the \code{CVE} package. It creates objects of
class \code{"cve"} to estimate the mean subspace. Helper functions that
require a \code{"cve"} object can then be applied to the output from this
function.

Conditional Variance Estimation (CVE) is a sufficient dimension reduction
(SDR) method for regressions studying \eqn{E(Y|X)}, the conditional
expectation of a response \eqn{Y} given a set of predictors \eqn{X}. This
function provides methods for estimating the dimension and the subspace
spanned by the columns of a \eqn{p\times k}{p x k} matrix \eqn{B} of minimal
rank \eqn{k} such that

\deqn{E(Y|X) = E(Y|B'X)}

or, equivalently,

\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} matrix of rank \eqn{k \leq p}{k <= p}.

Both the dimension \eqn{k} and the subspace \eqn{span(B)} are unknown. The 
CVE method makes very few assumptions.

A kernel matrix \eqn{\hat{B}}{Bhat} is estimated such that the column space
of \eqn{\hat{B}}{Bhat} should be close to the mean subspace \eqn{span(B)}.
The primary output from this method is a set of orthonormal vectors,
\eqn{\hat{B}}{Bhat}, whose span estimates \eqn{span(B)}.

The method central implements the Ensemble Conditional Variance Estimation
(ECVE) as described in [2]. It augments the CVE method by applying an
ensemble of functions (parameter \code{func_list}) to the response to
estimate the central subspace. This corresponds to the generalization

\deqn{F(Y|X) = F(Y|B'X)}

or, equivalently,

\deqn{Y = g(B'X, \epsilon)}

where \eqn{F} is the conditional cumulative distribution function.
}
\examples{
# set dimensions for simulation model
p <- 5
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 <- 100
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(n)

# calculate cve with method 'mean' for k unknown in 1, ..., 3
cve.obj.s <- cve(y ~ x, max.dim = 2) # default method 'mean'
# calculate cve with method 'weighed' for k = 2
cve.obj.w <- cve(y ~ x, k = 2, method = 'weighted.mean')
B2 <- coef(cve.obj.s, k = 2)

# get projected X data (same as cve.obj.s$X \%*\% B2)
proj.X <- directions(cve.obj.s, k = 2)
#  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, 2)
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 mean 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 mean and weighted cve estimate by
# Frobenius norm of difference of projections.
norm(PB - PB.s, type = 'F')
norm(PB - PB.w, type = 'F')

}
\references{
[1] Fertl, L. and Bura, E. (2021), Conditional Variance
         Estimation for Sufficient Dimension Reduction.
         arXiv:2102.08782

   [2] Fertl, L. and Bura, E. (2021), Ensemble Conditional Variance
         Estimation for Sufficient Dimension Reduction.
         arXiv:2102.13435
}
\seealso{
For a detailed description of \code{formula} see
     \code{\link{formula}}.
}