129 lines
5.1 KiB
R
129 lines
5.1 KiB
R
% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/datasets.R
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\name{dataset}
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\alias{dataset}
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\title{Generates test datasets.}
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\usage{
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dataset(name = "M1", n = NULL, p = 20, sd = 0.5, ...)
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}
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\arguments{
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\item{name}{One of \code{"M1"}, \code{"M2"}, \code{"M3"}, \code{"M4",}
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\code{"M5"}, \code{"M6"} or \code{"M7"}. Alternative just the dataset number
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1-7.}
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\item{n}{number of samples.}
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\item{p}{Dimension of random variable \eqn{X}.}
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\item{sd}{standard diviation for error term \eqn{\epsilon}.}
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\item{...}{Additional parameters only for "M2" (namely \code{pmix} and
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\code{lambda}), see: below.}
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}
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\value{
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List with elements
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\itemize{
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\item{X}{data, a \eqn{n\times p}{n x p} matrix.}
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\item{Y}{response.}
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\item{B}{the dim-reduction matrix}
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\item{name}{Name of the dataset (name parameter)}
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}
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}
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\description{
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Provides sample datasets M1-M7 used in the paper Conditional variance
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estimation for sufficient dimension reduction, Lukas Fertl, Efstathia Bura.
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The general model is given by:
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\deqn{Y = g(B'X) + \epsilon}
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}
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\section{M1}{
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The predictors are distributed as
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\eqn{X\sim N_p(0, \Sigma)}{X ~ N_p(0, \Sigma)} with
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\eqn{\Sigma_{i, j} = 0.5^{|i - j|}}{\Sigma_ij = 0.5^|i - j|} for
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\eqn{i, j = 1,..., p} for a subspace dimension of \eqn{k = 1} with a default
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of \eqn{n = 100} data points. \eqn{p = 20},
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\eqn{b_1 = (1,1,1,1,1,1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_1 = (1,1,1,1,1,1,0,...,0)' / sqrt(6)}, and \eqn{Y} is
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given as \deqn{Y = cos(b_1'X) + \epsilon} where \eqn{\epsilon} is
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distributed as generalized normal distribution with location 0,
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shape-parameter 0.5, and the scale-parameter is chosen such that
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\eqn{Var(\epsilon) = 0.5}.
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}
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\section{M2}{
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The predictors are distributed as \eqn{X \sim Z 1_p \lambda + N_p(0, I_p)}{X ~ Z 1_p \lambda + N_p(0, I_p)}. with
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\eqn{Z \sim 2 Binom(p_{mix}) - 1\in\{-1, 1\}}{Z~2Binom(pmix)-1} where
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\eqn{1_p} is the \eqn{p}-dimensional vector of one's, for a subspace
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dimension of \eqn{k = 1} with a default of \eqn{n = 100} data points.
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\eqn{p = 20}, \eqn{b_1 = (1,1,1,1,1,1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_1 = (1,1,1,1,1,1,0,...,0)' / sqrt(6)},
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and \eqn{Y} is \deqn{Y = cos(b_1'X) + 0.5\epsilon} where \eqn{\epsilon} is
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standard normal.
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Defaults for \code{pmix} is 0.3 and \code{lambda} defaults to 1.
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}
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\section{M3}{
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The predictors are distributed as \eqn{X\sim N_p(0, I_p)}{X~N_p(0, I_p)}
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for a subspace
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dimension of \eqn{k = 1} with a default of \eqn{n = 100} data points.
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\eqn{p = 20}, \eqn{b_1 = (1,1,1,1,1,1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_1 = (1,1,1,1,1,1,0,...,0)' / sqrt(6)},
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and \eqn{Y} is
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\deqn{Y = 2 log(|b_1'X| + 2) + 0.5\epsilon} where \eqn{\epsilon} is
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standard normal.
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}
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\section{M4}{
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The predictors are distributed as \eqn{X\sim N_p(0,\Sigma)}{X~N_p(0,\Sigma)}
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with \eqn{\Sigma_{i, j} = 0.5^{|i - j|}}{\Sigma_ij = 0.5^|i - j|} for
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\eqn{i, j = 1,..., p} for a subspace dimension of \eqn{k = 2} with a default
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of \eqn{n = 100} data points. \eqn{p = 20},
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\eqn{b_1 = (1,1,1,1,1,1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_1 = (1,1,1,1,1,1,0,...,0)' / sqrt(6)},
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\eqn{b_2 = (1,-1,1,-1,1,-1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_2 = (1,-1,1,-1,1,-1,0,...,0)' / sqrt(6)}
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and \eqn{Y} is given as \deqn{Y = \frac{b_1'X}{0.5 + (1.5 + b_2'X)^2} + 0.5\epsilon}{Y = (b_1'X) / (0.5 + (1.5 + b_2'X)^2) + 0.5\epsilon}
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where \eqn{\epsilon} is standard normal.
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}
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\section{M5}{
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The predictors are distributed as \eqn{X\sim U([0,1]^p)}{X~U([0, 1]^p)}
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where \eqn{U([0, 1]^p)} is the uniform distribution with
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independent components on the \eqn{p}-dimensional hypercube for a subspace
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dimension of \eqn{k = 2} with a default of \eqn{n = 200} data points.
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\eqn{p = 20},
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\eqn{b_1 = (1,1,1,1,1,1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_1 = (1,1,1,1,1,1,0,...,0)' / sqrt(6)},
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\eqn{b_2 = (1,-1,1,-1,1,-1,0,...,0)' / \sqrt{6}\in\mathcal{R}^p}{b_2 = (1,-1,1,-1,1,-1,0,...,0)' / sqrt(6)}
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and \eqn{Y} is given as \deqn{Y = cos(\pi b_1'X)(b_2'X + 1)^2 + 0.5\epsilon}
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where \eqn{\epsilon} is standard normal.
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}
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\section{M6}{
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The predictors are distributed as \eqn{X\sim N_p(0, I_p)}{X~N_p(0, I_p)}
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for a subspace dimension of \eqn{k = 3} with a default of \eqn{n = 200} data
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point. \eqn{p = 20, b_1 = e_1, b_2 = e_2}, and \eqn{b_3 = e_p}, where
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\eqn{e_j} is the \eqn{j}-th unit vector in the \eqn{p}-dimensional space.
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\eqn{Y} is given as \deqn{Y = (b_1'X)^2+(b_2'X)^2+(b_3'X)^2+0.5\epsilon}
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where \eqn{\epsilon} is standard normal.
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}
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\section{M7}{
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The predictors are distributed as \eqn{X\sim t_3(I_p)}{X~t_3(I_p)} where
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\eqn{t_3(I_p)} is the standard multivariate t-distribution with 3 degrees of
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freedom, for a subspace dimension of \eqn{k = 4} with a default of
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\eqn{n = 200} data points.
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\eqn{p = 20, b_1 = e_1, b_2 = e_2, b_3 = e_3}, and \eqn{b_4 = e_p}, where
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\eqn{e_j} is the \eqn{j}-th unit vector in the \eqn{p}-dimensional space.
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\eqn{Y} is given as \deqn{Y = (b_1'X)(b_2'X)^2+(b_3'X)(b_4'X)+0.5\epsilon}
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where \eqn{\epsilon} is distributed as generalized normal distribution with
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location 0, shape-parameter 1, and the scale-parameter is chosen such that
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\eqn{Var(\epsilon) = 0.25}.
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}
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\references{
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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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}
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