173 lines
6.2 KiB
R
173 lines
6.2 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 = "mean", 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 "mean" method to estimate the mean subspace, see [1].
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\item "central" ensemble method to estimate the central subspace, see [2].
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\item "weighted.mean" variation of `"mean"` method with adaptive weighting
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of slices, see [1].
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\item "weighted.central" variation of `"central"` method with adaptive
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weighting of slices, see [2].
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}}
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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{\link{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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This is the main function in the \code{CVE} package. It creates objects of
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class \code{"cve"} to estimate the mean subspace. Helper functions that
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require a \code{"cve"} object can then be applied to the output from this
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function.
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Conditional Variance Estimation (CVE) is a sufficient dimension reduction
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(SDR) method for regressions studying \eqn{E(Y|X)}, the conditional
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expectation of a response \eqn{Y} given a set of predictors \eqn{X}. This
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function provides methods for estimating the dimension and the subspace
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spanned by the columns of a \eqn{p\times k}{p x k} matrix \eqn{B} of minimal
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rank \eqn{k} such that
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\deqn{E(Y|X) = E(Y|B'X)}
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or, equivalently,
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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, and \eqn{B = (b_1,..., b_k)}
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is a real \eqn{p \times k}{p x k} matrix of rank \eqn{k \leq p}{k <= p}.
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Both the dimension \eqn{k} and the subspace \eqn{span(B)} are unknown. The
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CVE method makes very few assumptions.
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A kernel matrix \eqn{\hat{B}}{Bhat} is estimated such that the column space
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of \eqn{\hat{B}}{Bhat} should be close to the mean subspace \eqn{span(B)}.
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The primary output from this method is a set of orthonormal vectors,
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\eqn{\hat{B}}{Bhat}, whose span estimates \eqn{span(B)}.
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The method central implements the Ensemble Conditional Variance Estimation
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(ECVE) as described in [2]. It augments the CVE method by applying an
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ensemble of functions (parameter \code{func_list}) to the response to
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estimate the central subspace. This corresponds to the generalization
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\deqn{F(Y|X) = F(Y|B'X)}
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or, equivalently,
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\deqn{Y = g(B'X, \epsilon)}
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where \eqn{F} is the conditional cumulative distribution function.
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}
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\examples{
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# set dimensions for simulation model
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p <- 5
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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 <- 100
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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(n)
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# calculate cve with method 'mean' for k unknown in 1, ..., 3
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cve.obj.s <- cve(y ~ x, max.dim = 2) # default method 'mean'
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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.mean')
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B2 <- coef(cve.obj.s, k = 2)
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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 = 2)
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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, 2)
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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 mean 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 mean 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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[1] Fertl, L. and Bura, E. (2021), Conditional Variance
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Estimation for Sufficient Dimension Reduction.
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arXiv:2102.08782
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[2] Fertl, L. and Bura, E. (2021), Ensemble Conditional Variance
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Estimation for Sufficient Dimension Reduction.
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arXiv:2102.13435
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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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