update: documentation
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*.pdf filter=lfs diff=lfs merge=lfs -text
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simulations/results/*
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literature/*
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CVE/src/*.o
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CVE/src/*.so
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CVE/src/*.dll
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*.tgz
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*.tar.xz
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*.tar.gz
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*.zip
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*.so
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*.Rcheck/*
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tmp/*
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wip/*
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Package: CVE
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Type: Package
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Title: Conditional Variance Estimator for Sufficient Dimension Reduction
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Version: 0.2
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Date: 2019-12-20
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Version: 0.3
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Date: 2021-03-04
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Author: Daniel Kapla <daniel@kapla.at>, Lukas Fertl <lukas.fertl@chello.at>
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Maintainer: Daniel Kapla <daniel@kapla.at>
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Description: Implementation of the Conditional Variance Estimation (CVE) method.
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License: GPL-3
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Encoding: UTF-8
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Imports: stats,graphics,mda
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RoxygenNote: 6.1.1
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Imports: stats,mda
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RoxygenNote: 7.0.2
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@ -2,7 +2,6 @@
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S3method(coef,cve)
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S3method(directions,cve)
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S3method(plot,cve)
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S3method(predict,cve)
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S3method(summary,cve)
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export(cve)
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@ -14,10 +13,6 @@ export(null)
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export(predict_dim)
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export(rStiefel)
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import(stats)
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importFrom(graphics,boxplot)
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importFrom(graphics,lines)
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importFrom(graphics,plot)
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importFrom(graphics,points)
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importFrom(mda,mars)
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importFrom(stats,model.frame)
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importFrom(stats,rbinom)
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171
CVE/R/CVE.R
171
CVE/R/CVE.R
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@ -3,7 +3,7 @@
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#' Conditional Variance Estimation (CVE) is a novel sufficient dimension
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#' reduction (SDR) method for regressions satisfying \eqn{E(Y|X) = E(Y|B'X)},
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#' where \eqn{B'X} is a lower dimensional projection of the predictors and
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#' \eqn{Y} is a univariate responce. CVE,
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#' \eqn{Y} is a univariate response. CVE,
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#' similarly to its main competitor, the mean average variance estimation
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#' (MAVE), is not based on inverse regression, and does not require the
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#' restrictive linearity and constant variance conditions of moment based SDR
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@ -11,7 +11,9 @@
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#' continuous predictors and link function. Let \eqn{X} be a real
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#' \eqn{p}-dimensional covariate vector. We assume that the dependence of
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#' \eqn{Y} and \eqn{X} is modelled by
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#'
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#' \deqn{Y = g(B'X) + \epsilon}
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#'
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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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@ -20,9 +22,28 @@
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#' a real \eqn{p \times k}{p x k} matrix of rank \eqn{k \leq p}{k <= p}.
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#' Without loss of generality \eqn{B} is assumed to be orthonormal.
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#'
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#' Further, the extended Ensemble Conditional Variance Estimation (ECVE) is
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#' implemented which is a SDR method in regressions with continuous response and
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#' predictors. ECVE applies to general non-additive error regression models.
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#'
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#' \deqn{Y = g(B'X, \epsilon)}
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#'
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#' It operates under the assumption that the predictors can be replaced by a
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#' lower dimensional projection without loss of information.It is a
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#' semiparametric forward regression model based exhaustive sufficient dimension
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#' reduction estimation method that is shown to be consistent under mild
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#' assumptions.
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#'
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#' @author Daniel Kapla, Lukas Fertl, Bura Efstathia
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#' @references Fertl, L. and Bura, E. (2019), Conditional Variance
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#' Estimation for Sufficient Dimension Reduction. Working Paper.
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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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#'
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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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#' @docType package
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#' @useDynLib CVE, .registration = TRUE
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@ -30,11 +51,52 @@
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#' Conditional Variance Estimator (CVE).
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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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#'
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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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#'
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#' \deqn{E(Y|X) = E(Y|B'X)}
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#'
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#' or, equivalently,
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#'
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#' \deqn{Y = g(B'X) + \epsilon}
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#'
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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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#'
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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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#'
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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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#'
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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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#'
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#' \deqn{F(Y|X) = F(Y|B'X)}
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#'
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#' or, equivalently,
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#'
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#' \deqn{Y = g(B'X, \epsilon)}
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#'
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#' where \eqn{F} is the conditional cumulative distribution function.
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#'
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#' @param 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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#' @param 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 "simple" implementation,
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#' \item "weighted" variation with adaptive weighting of slices.
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#' \item \code{"mean"} method to estimate the mean subspace, see [1].
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#' \item \code{"central"} ensemble method to estimate the central subspace,
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#' see [2].
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#' \item \code{"weighted.mean"} variation of \code{"mean"} method with
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#' adaptive weighting of slices, see [1].
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#' \item \code{"weighted.central"} variation of \code{"central"} method with
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#' adaptive weighting of slices, see [2].
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#' }
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#' see Fertl, L. and Bura, E. (2019).
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#' @param max.dim upper bounds for \code{k}, (ignored if \code{k} is supplied).
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#' @param ... optional parameters passed on to \code{cve.call}.
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#' @param ... optional parameters passed on to \code{\link{cve.call}}.
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#'
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#'
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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{%
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#' E(Y|X) = E(Y|B'X) %
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#' }
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#' or, equivalently,
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#' \deqn{%
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#' Y = g(B'X) + \epsilon %
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#' }
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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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#'
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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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#'
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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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#'
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#' @return 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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#'
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#' @examples
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#' # set dimensions for simulation model
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#' p <- 8
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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 <- 200
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#' n <- 100
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#' set.seed(21)
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#'
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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(100)
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#' # calculate cve with method 'mean' for k unknown in 1, ..., 4
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#' cve.obj.s <- cve(y ~ x, max.dim = 4) # default method 'mean'
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#' y <- (x %*% b1)^2 + 2 * (x %*% b2) + 0.25 * rnorm(n)
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#'
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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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#' # estimate dimension from cve.obj.s
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#' khat <- predict_dim(cve.obj.s)$k
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#' # get cve-estimate for B with dimensions (p, k = khat)
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#' B2 <- coef(cve.obj.s, k = khat)
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#' B2 <- coef(cve.obj.s, k = 2)
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#'
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#' # get projected X data (same as cve.obj.s$X %*% B2)
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#' proj.X <- directions(cve.obj.s, k = khat)
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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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#'
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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, khat)
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#' yhat <- predict(cve.obj.s, x.new, 2)
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#' plot(y.new, yhat)
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#'
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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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#'
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#' @seealso For a detailed description of \code{formula} see
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#' \code{\link{formula}}.
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#' @references Fertl, L. and Bura, E. (2019), Conditional Variance
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#' Estimation for Sufficient Dimension Reduction. Working Paper.
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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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#'
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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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#' @importFrom stats model.frame
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#' @export
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#' @inherit cve description
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#'
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#' @param X Design predictor matrix.
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#' @param Y \eqn{n}-dimensional vector of responces.
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#' @param Y \eqn{n}-dimensional vector of responses.
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#' @param h bandwidth or function to estimate bandwidth, defaults to internaly
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#' estimated bandwidth.
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#' @param nObs parameter for choosing bandwidth \code{h} using
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#' \code{\link{estimate.bandwidth}} (ignored if \code{h} is supplied).
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#' @param method specifies the CVE method variation as one of
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#' @param 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 "simple" exact implementation as described in the paper listed
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#' below.
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#' \item "weighted" variation with addaptive weighting of slices.
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#' \item \code{"mean"} method to estimate the mean subspace, see [1].
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#' \item \code{"central"} ensemble method to estimate the central subspace,
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#' see [2].
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#' \item \code{"weighted.mean"} variation of \code{"mean"} method with
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#' adaptive weighting of slices, see [1].
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#' \item \code{"weighted.central"} variation of \code{"central"} method with
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#' adaptive weighting of slices, see [2].
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#' }
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#' @param func_list a list of functions applied to \code{Y} to form the ensamble
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#' CVE for central sub-space estimation.
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#' @param func_list a list of functions applied to \code{Y} used by ECVE
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#' (see [2]) for central subspace estimation. The default ensemble are
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#' indicator functions of the \eqn{[0, 10], (10, 20], ..., (90, 100]}
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#' percent response quantiles. (only relevant if \code{method} is
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#' \code{"central"} or \code{"weighted.central"}, ignored otherwise)
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#' @param k Dimension of lower dimensional projection, if \code{k} is given
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#' only the specified dimension \code{B} matrix is estimated.
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#' @param min.dim lower bounds for \code{k}, (ignored if \code{k} is supplied).
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@ -214,6 +269,9 @@ cve <- function(formula, data, method = "mean", max.dim = 10L, ...) {
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#' @param attempts If \code{V.init} not supplied, the optimization is carried
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#' out \code{attempts} times with starting values drawn from the invariant
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#' measure on the Stiefel manifold (see \code{\link{rStiefel}}).
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#' @param nr.proj The number of projection used for projective resampling for
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#' multivariate response \eqn{Y} (under active development, ignored for
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#' univariate response).
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#' @param momentum number of \eqn{[0, 1)} giving the ration of momentum for
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#' eucledian gradient update with a momentum term. \code{momentum = 0}
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#' corresponds to normal gradient descend.
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@ -230,6 +288,7 @@ cve <- function(formula, data, method = "mean", max.dim = 10L, ...) {
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#' computation).
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#'
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#' @inherit cve return
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#' @inherit cve references
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#'
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#' @examples
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#' # create B for simulation (k = 1)
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@ -262,7 +321,7 @@ cve.call <- function(X, Y,
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momentum = 0.0, tau = 1.0, tol = 1e-3,
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slack = 0.0, gamma = 0.5,
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V.init = NULL,
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max.iter = 50L, attempts = 10L, nr.proj = 500L,
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max.iter = 50L, attempts = 10L, nr.proj = 1L,
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logger = NULL
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) {
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# Determine method with partial matching (shortcuts: "Weight" -> "weighted")
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10
CVE/R/coef.R
10
CVE/R/coef.R
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#' # set dimensions for simulation model
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#' p <- 8 # sample dimension
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#' k <- 2 # real dimension of SDR subspace
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#' n <- 200 # samplesize
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#' n <- 100 # samplesize
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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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#' 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(100)
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#' # calculate cve for k = 1, ..., 5
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#' cve.obj <- cve(y ~ x, max.dim = 5)
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#' # with f(x1, x2) = x1^2 + 2 * x2 and err ~ N(0, 0.1^2)
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#' y <- (x %*% b1)^2 + 2 * (x %*% b2) + 0.1 * rnorm(100)
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#' # calculate cve for k = 2, 3
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#' cve.obj <- cve(y ~ x, min.dim = 2, max.dim = 3)
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#' # get cve-estimate for B with dimensions (p, k = 2)
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#' B2 <- coef(cve.obj, k = 2)
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#'
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@ -200,8 +200,10 @@ rlaplace <- function(n = 1, mu = 0, sd = 1) {
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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 Fertl, L. and Bura, E. (2019), Conditional Variance
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#' Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
#' @references
|
||||
#' Fertl, L. and Bura, E. (2021), Conditional Variance
|
||||
#' Estimation for Sufficient Dimension Reduction.
|
||||
#' arXiv:2102.08782
|
||||
#'
|
||||
#' @import stats
|
||||
#' @importFrom stats rnorm rbinom
|
||||
|
|
|
|||
62
CVE/R/plot.R
62
CVE/R/plot.R
|
|
@ -1,62 +0,0 @@
|
|||
#' Elbow plot of the loss function.
|
||||
#'
|
||||
#' Boxplots of the output \code{L} from \code{\link{cve}} over \code{k} from
|
||||
#' \code{min.dim} to \code{max.dim}. For given \code{k}, \code{L} corresponds
|
||||
#' to \eqn{L_n(V, X_i)} where \eqn{V} is the minimizer of \eqn{L_n(V)} where
|
||||
#' \eqn{V} is an element of a Stiefel manifold (see
|
||||
#' Fertl, L. and Bura, E. (2019)).
|
||||
#'
|
||||
#' @param x an object of class \code{"cve"}, usually, a result of a call to
|
||||
#' \code{\link{cve}} or \code{\link{cve.call}}.
|
||||
#' @param ... Pass through parameters to [\code{\link{plot}}] and
|
||||
#' [\code{\link{lines}}]
|
||||
#'
|
||||
#' @examples
|
||||
#' # create B for simulation
|
||||
#' B <- cbind(rep(1, 6), (-1)^seq(6)) / sqrt(6)
|
||||
#'
|
||||
#' set.seed(21)
|
||||
#' # creat predictor data x ~ N(0, I_p)
|
||||
#' X <- matrix(rnorm(600), 100)
|
||||
#'
|
||||
#' # 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 %*% B[, 1])^2 + 2 * X %*% B[, 2] + rnorm(100, 0, .1)
|
||||
#'
|
||||
#' # Create bandwidth estimation function
|
||||
#' estimate.bandwidth <- function(X, k, nObs) {
|
||||
#' n <- nrow(X)
|
||||
#' p <- ncol(X)
|
||||
#' X_c <- scale(X, center = TRUE, scale = FALSE)
|
||||
#' 2 * qchisq((nObs - 1) / (n - 1), k) * sum(X_c^2) / (n * p)
|
||||
#' }
|
||||
#' # calculate cve with method 'simple' for k = min.dim,...,max.dim
|
||||
#' cve.obj.simple <- cve(Y ~ X, h = estimate.bandwidth, nObs = sqrt(nrow(X)))
|
||||
#'
|
||||
#' # elbow plot
|
||||
#' plot(cve.obj.simple)
|
||||
#'
|
||||
#' @references Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
#' Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
#'
|
||||
#' @seealso see \code{\link{par}} for graphical parameters to pass through
|
||||
#' as well as \code{\link{plot}}, the standard plot utility.
|
||||
#' @method plot cve
|
||||
#' @importFrom graphics plot lines points boxplot
|
||||
#' @export
|
||||
plot.cve <- function(x, ...) {
|
||||
L <- c()
|
||||
k <- c()
|
||||
for (dr.k in x$res) {
|
||||
if (class(dr.k) == 'cve.k') {
|
||||
k <- c(k, as.character(dr.k$k))
|
||||
L <- c(L, dr.k$L)
|
||||
}
|
||||
}
|
||||
L <- matrix(L, ncol = length(k)) / var(x$Y)
|
||||
boxplot(L, main = "elbow plot",
|
||||
xlab = "SDR dimension",
|
||||
ylab = "Sample loss distribution",
|
||||
names = k)
|
||||
}
|
||||
|
|
@ -1,7 +1,7 @@
|
|||
#' Predict method for CVE Fits.
|
||||
#'
|
||||
#' Predict response using projected data. The forward model \eqn{g(B' X)} is
|
||||
#' estimated with \code{\link{mars}} in the \code{\pkg{mda}} package.
|
||||
#' estimated with \code{\link{mars}} in the \pkg{mda} package.
|
||||
#'
|
||||
#' @param object an object of class \code{"cve"}, usually, a result of a call to
|
||||
#' \code{\link{cve}} or \code{\link{cve.call}}.
|
||||
|
|
|
|||
Binary file not shown.
|
|
@ -9,7 +9,7 @@
|
|||
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 and
|
||||
\eqn{Y} is a univariate responce. CVE,
|
||||
\eqn{Y} is a univariate response. 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
|
||||
|
|
@ -17,7 +17,10 @@ methods. CVE is data-driven and applies to additive error regressions with
|
|||
continuous predictors and link function. Let \eqn{X} be a real
|
||||
\eqn{p}-dimensional covariate vector. We assume that the dependence of
|
||||
\eqn{Y} and \eqn{X} is modelled by
|
||||
}
|
||||
\details{
|
||||
\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}
|
||||
|
|
@ -25,10 +28,27 @@ 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}.
|
||||
Without loss of generality \eqn{B} is assumed to be orthonormal.
|
||||
|
||||
Further, the extended Ensemble Conditional Variance Estimation (ECVE) is
|
||||
implemented which is a SDR method in regressions with continuous response and
|
||||
predictors. ECVE applies to general non-additive error regression models.
|
||||
|
||||
\deqn{Y = g(B'X, \epsilon)}
|
||||
|
||||
It operates under the assumption that the predictors can be replaced by a
|
||||
lower dimensional projection without loss of information.It is a
|
||||
semiparametric forward regression model based exhaustive sufficient dimension
|
||||
reduction estimation method that is shown to be consistent under mild
|
||||
assumptions.
|
||||
}
|
||||
\references{
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
[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
|
||||
}
|
||||
\author{
|
||||
Daniel Kapla, Lukas Fertl, Bura Efstathia
|
||||
|
|
|
|||
|
|
@ -25,7 +25,7 @@ cve-estimate of \eqn{B} with dimension \eqn{p\times k}{p x k}.
|
|||
# set dimensions for simulation model
|
||||
p <- 8 # sample dimension
|
||||
k <- 2 # real dimension of SDR subspace
|
||||
n <- 200 # samplesize
|
||||
n <- 100 # samplesize
|
||||
# create B for simulation
|
||||
b1 <- rep(1 / sqrt(p), p)
|
||||
b2 <- (-1)^seq(1, p) / sqrt(p)
|
||||
|
|
@ -36,10 +36,10 @@ set.seed(21)
|
|||
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 for k = 1, ..., 5
|
||||
cve.obj <- cve(y ~ x, max.dim = 5)
|
||||
# with f(x1, x2) = x1^2 + 2 * x2 and err ~ N(0, 0.1^2)
|
||||
y <- (x \%*\% b1)^2 + 2 * (x \%*\% b2) + 0.1 * rnorm(100)
|
||||
# calculate cve for k = 2, 3
|
||||
cve.obj <- cve(y ~ x, min.dim = 2, max.dim = 3)
|
||||
# get cve-estimate for B with dimensions (p, k = 2)
|
||||
B2 <- coef(cve.obj, k = 2)
|
||||
|
||||
|
|
|
|||
120
CVE/man/cve.Rd
120
CVE/man/cve.Rd
|
|
@ -4,7 +4,7 @@
|
|||
\alias{cve}
|
||||
\title{Conditional Variance Estimator (CVE).}
|
||||
\usage{
|
||||
cve(formula, data, method = "simple", max.dim = 10L, ...)
|
||||
cve(formula, data, method = "mean", max.dim = 10L, ...)
|
||||
}
|
||||
\arguments{
|
||||
\item{formula}{an object of class \code{"formula"} which is a symbolic
|
||||
|
|
@ -20,42 +20,17 @@ 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,
|
||||
\item "weighted" variation with adaptive weighting of slices.
|
||||
}
|
||||
see Fertl, L. and Bura, E. (2019).}
|
||||
\item "mean" method to estimate the mean subspace, see [1].
|
||||
\item "central" ensemble method to estimate the central subspace, see [2].
|
||||
\item "weighted.mean" variation of `"mean"` method with adaptive weighting
|
||||
of slices, see [1].
|
||||
\item "weighted.central" variation of `"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{cve.call}.
|
||||
|
||||
|
||||
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)}.}
|
||||
\item{...}{optional parameters passed on to \code{\link{cve.call}}.}
|
||||
}
|
||||
\value{
|
||||
an S3 object of class \code{cve} with components:
|
||||
|
|
@ -88,61 +63,108 @@ 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 <- 8
|
||||
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 <- 200
|
||||
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(100)
|
||||
# calculate cve with method 'simple' for k unknown in 1, ..., 4
|
||||
cve.obj.s <- cve(y ~ x, max.dim = 4) # default method 'simple'
|
||||
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')
|
||||
# estimate dimension from cve.obj.s
|
||||
khat <- predict_dim(cve.obj.s)$k
|
||||
# get cve-estimate for B with dimensions (p, k = khat)
|
||||
B2 <- coef(cve.obj.s, k = khat)
|
||||
B2 <- coef(cve.obj.s, k = 2)
|
||||
|
||||
# get projected X data (same as cve.obj.s$X \%*\% B2)
|
||||
proj.X <- directions(cve.obj.s, k = khat)
|
||||
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, khat)
|
||||
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 simple and weighted method
|
||||
# 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 simple and weighted cve estimate by
|
||||
# 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{
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
[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
|
||||
|
|
|
|||
|
|
@ -7,8 +7,8 @@
|
|||
cve.call(
|
||||
X,
|
||||
Y,
|
||||
method = "simple",
|
||||
func_list = list(function(x) x),
|
||||
method = c("mean", "weighted.mean", "central", "weighted.central"),
|
||||
func_list = NULL,
|
||||
nObs = sqrt(nrow(X)),
|
||||
h = NULL,
|
||||
min.dim = 1L,
|
||||
|
|
@ -22,23 +22,31 @@ cve.call(
|
|||
V.init = NULL,
|
||||
max.iter = 50L,
|
||||
attempts = 10L,
|
||||
nr.proj = 1L,
|
||||
logger = NULL
|
||||
)
|
||||
}
|
||||
\arguments{
|
||||
\item{X}{Design predictor matrix.}
|
||||
|
||||
\item{Y}{\eqn{n}-dimensional vector of responces.}
|
||||
\item{Y}{\eqn{n}-dimensional vector of responses.}
|
||||
|
||||
\item{method}{specifies the CVE method variation as one of
|
||||
\item{method}{This character string specifies the method of fitting. The
|
||||
options are
|
||||
\itemize{
|
||||
\item "simple" exact implementation as described in the paper listed
|
||||
below.
|
||||
\item "weighted" variation with addaptive weighting of slices.
|
||||
\item "mean" method to estimate the mean subspace, see [1].
|
||||
\item "central" ensemble method to estimate the central subspace, see [2].
|
||||
\item "weighted.mean" variation of `"mean"` method with adaptive weighting
|
||||
of slices, see [1].
|
||||
\item "weighted.central" variation of `"central"` method with adaptive
|
||||
weighting of slices, see [2].
|
||||
}}
|
||||
|
||||
\item{func_list}{a list of functions applied to `Y` to form the ensamble
|
||||
CVE for central sub-space estimation.}
|
||||
\item{func_list}{a list of functions applied to \code{Y} used by ECVE
|
||||
(see [2]) for central subspace estimation. The default ensemble are
|
||||
indicator functions of the \eqn{[0, 10], (10, 20], ..., (90, 100]}
|
||||
percent response quantiles. (only relevant if \code{method} is
|
||||
\code{"central"} or \code{"weighted.central"}, ignored otherwise)}
|
||||
|
||||
\item{nObs}{parameter for choosing bandwidth \code{h} using
|
||||
\code{\link{estimate.bandwidth}} (ignored if \code{h} is supplied).}
|
||||
|
|
@ -79,6 +87,10 @@ used as starting value in the optimization. (If supplied,
|
|||
out \code{attempts} times with starting values drawn from the invariant
|
||||
measure on the Stiefel manifold (see \code{\link{rStiefel}}).}
|
||||
|
||||
\item{nr.proj}{The number of projection used for projective resampling for
|
||||
multivariate response \eqn{Y} (under active development, ignored for
|
||||
univariate response).}
|
||||
|
||||
\item{logger}{a logger function (only for advanced users, slows down the
|
||||
computation).}
|
||||
}
|
||||
|
|
@ -113,6 +125,46 @@ 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{
|
||||
# create B for simulation (k = 1)
|
||||
|
|
@ -138,3 +190,12 @@ cve.obj.simple2 <- cve.call(X, Y, k = 1)
|
|||
coef(cve.obj.simple1, k = 1)
|
||||
coef(cve.obj.simple2, k = 1)
|
||||
}
|
||||
\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
|
||||
}
|
||||
|
|
|
|||
|
|
@ -122,6 +122,7 @@ location 0, shape-parameter 1, and the scale-parameter is chosen such that
|
|||
}
|
||||
|
||||
\references{
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
Fertl, L. and Bura, E. (2021), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction.
|
||||
arXiv:2102.08782
|
||||
}
|
||||
|
|
|
|||
|
|
@ -33,11 +33,11 @@ x <- matrix(rnorm(500), 100, 5)
|
|||
# y = f(B'x) + err
|
||||
# with f(x1) = x1 and err ~ N(0, 0.25^2)
|
||||
y <- x \%*\% B + 0.25 * rnorm(100)
|
||||
# calculate cve with method 'simple' for k = 1
|
||||
# calculate cve with method 'mean' for k = 1
|
||||
set.seed(21)
|
||||
cve.obj.simple <- cve(y ~ x, k = 1, method = 'mean')
|
||||
cve.obj.mean <- cve(y ~ x, k = 1, method = 'mean')
|
||||
# get projected data for k = 1
|
||||
x.proj <- directions(cve.obj.simple, k = 1)
|
||||
x.proj <- directions(cve.obj.mean, k = 1)
|
||||
# plot y against projected data
|
||||
plot(x.proj, y)
|
||||
|
||||
|
|
|
|||
|
|
@ -1,57 +0,0 @@
|
|||
% Generated by roxygen2: do not edit by hand
|
||||
% Please edit documentation in R/plot.R
|
||||
\name{plot.cve}
|
||||
\alias{plot.cve}
|
||||
\title{Elbow plot of the loss function.}
|
||||
\usage{
|
||||
\method{plot}{cve}(x, ...)
|
||||
}
|
||||
\arguments{
|
||||
\item{x}{an object of class \code{"cve"}, usually, a result of a call to
|
||||
\code{\link{cve}} or \code{\link{cve.call}}.}
|
||||
|
||||
\item{...}{Pass through parameters to [\code{\link{plot}}] and
|
||||
[\code{\link{lines}}]}
|
||||
}
|
||||
\description{
|
||||
Boxplots of the output \code{L} from \code{\link{cve}} over \code{k} from
|
||||
\code{min.dim} to \code{max.dim}. For given \code{k}, \code{L} corresponds
|
||||
to \eqn{L_n(V, X_i)} where \eqn{V} is the minimizer of \eqn{L_n(V)} where
|
||||
\eqn{V} is an element of a Stiefel manifold (see
|
||||
Fertl, L. and Bura, E. (2019)).
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation
|
||||
B <- cbind(rep(1, 6), (-1)^seq(6)) / sqrt(6)
|
||||
|
||||
set.seed(21)
|
||||
# creat predictor data x ~ N(0, I_p)
|
||||
X <- matrix(rnorm(600), 100)
|
||||
|
||||
# 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 \%*\% B[, 1])^2 + 2 * X \%*\% B[, 2] + rnorm(100, 0, .1)
|
||||
|
||||
# Create bandwidth estimation function
|
||||
estimate.bandwidth <- function(X, k, nObs) {
|
||||
n <- nrow(X)
|
||||
p <- ncol(X)
|
||||
X_c <- scale(X, center = TRUE, scale = FALSE)
|
||||
2 * qchisq((nObs - 1) / (n - 1), k) * sum(X_c^2) / (n * p)
|
||||
}
|
||||
# calculate cve with method 'simple' for k = min.dim,...,max.dim
|
||||
cve.obj.simple <- cve(Y ~ X, h = estimate.bandwidth, nObs = sqrt(nrow(X)))
|
||||
|
||||
# elbow plot
|
||||
plot(cve.obj.simple)
|
||||
|
||||
}
|
||||
\references{
|
||||
Fertl, L. and Bura, E. (2019), Conditional Variance
|
||||
Estimation for Sufficient Dimension Reduction. Working Paper.
|
||||
}
|
||||
\seealso{
|
||||
see \code{\link{par}} for graphical parameters to pass through
|
||||
as well as \code{\link{plot}}, the standard plot utility.
|
||||
}
|
||||
|
|
@ -21,7 +21,7 @@ prediced respone(s) for \code{newdata}.
|
|||
}
|
||||
\description{
|
||||
Predict response using projected data. The forward model \eqn{g(B' X)} is
|
||||
estimated with \code{\link{mars}} in the \code{\pkg{mda}} package.
|
||||
estimated with \code{\link{mars}} in the \pkg{mda} package.
|
||||
}
|
||||
\examples{
|
||||
# create B for simulation
|
||||
|
|
|
|||
|
|
@ -51,7 +51,7 @@ x <- matrix(rnorm(500), 100)
|
|||
y <- x \%*\% B + 0.25 * rnorm(100)
|
||||
|
||||
# Calculate cve for unknown k between min.dim and max.dim.
|
||||
cve.obj.simple <- cve(y ~ x)
|
||||
cve.obj.simple <- cve(y ~ x)
|
||||
|
||||
predict_dim(cve.obj.simple)
|
||||
|
||||
|
|
|
|||
235
cve_tensorflow.R
235
cve_tensorflow.R
|
|
@ -1,235 +0,0 @@
|
|||
library(CVE)
|
||||
library(reticulate)
|
||||
library(tensorflow)
|
||||
|
||||
#' Null space basis of given matrix `V`
|
||||
#'
|
||||
#' @param V `(p, q)` matrix
|
||||
#' @return Semi-orthogonal `(p, p - q)` matrix spaning the null space of `V`.
|
||||
#' @keywords internal
|
||||
#' @export
|
||||
null <- function(V) {
|
||||
tmp <- qr(V)
|
||||
set <- if(tmp$rank == 0L) seq_len(ncol(V)) else -seq_len(tmp$rank)
|
||||
return(qr.Q(tmp, complete = TRUE)[, set, drop = FALSE])
|
||||
}
|
||||
|
||||
subspace_dist <- function(A, B) {
|
||||
P <- A %*% solve(t(A) %*% A, t(A))
|
||||
Q <- B %*% solve(t(B) %*% B, t(B))
|
||||
norm(P - Q, 'F') / sqrt(ncol(A) + ncol(B))
|
||||
}
|
||||
|
||||
estimate.bandwidth <- function (X, k, nObs = sqrt(nrow(X)), version = 1L) {
|
||||
n <- nrow(X)
|
||||
p <- ncol(X)
|
||||
X_c <- scale(X, center = TRUE, scale = FALSE)
|
||||
|
||||
if (version == 1) {
|
||||
(2 * sum(X_c^2) / (n * p)) * (1.2 * n^(-1 / (4 + k)))^2
|
||||
} else if (version == 2) {
|
||||
2 * qchisq((nObs - 1) / (n - 1), k) * sum(X_c^2) / (n * p)
|
||||
} else {
|
||||
stop("Unknown version.")
|
||||
}
|
||||
}
|
||||
|
||||
tf_Variable <- function(obj, dtype = "float32", ...) {
|
||||
tf$Variable(obj, dtype = dtype, ...)
|
||||
}
|
||||
tf_constant <- function(obj, dtype = "float32", ...) {
|
||||
tf$constant(obj, dtype = dtype, ...)
|
||||
}
|
||||
|
||||
cve.tf <- function(X, Y, k, h = estimate.bandwidth(X, k, sqrt(nrow(X))),
|
||||
V.init = NULL, optimizer_initialier = tf$optimizers$RMSprop, attempts = 10L,
|
||||
nr.projections = nrow(X)^(3 / 2),
|
||||
sd_noise = 0, method = c("simple", "weighted")
|
||||
) {
|
||||
method <- match.arg(method)
|
||||
|
||||
`-0.5` <- tf_constant(-0.5)
|
||||
`1` <- tf_constant(1)
|
||||
`2` <- tf_constant(2)
|
||||
n <- nrow(X)
|
||||
p <- ncol(X)
|
||||
k <- as.integer(k)
|
||||
q <- p - k
|
||||
|
||||
if (!is.matrix(Y))
|
||||
Y <- as.matrix(Y)
|
||||
|
||||
# Projective resampling.
|
||||
if (ncol(Y) > 1L) {
|
||||
R <- matrix(rnorm(ncol(Y) * nr.projections), ncol(Y))
|
||||
R <- t(t(R) / sqrt(colSums(R^2)))
|
||||
Y <- Y %*% R
|
||||
}
|
||||
|
||||
X <- tf_constant(scale(X))
|
||||
Y <- tf_constant(scale(Y))
|
||||
I <- tf_constant(diag(1, p))
|
||||
h <- tf_Variable(h)
|
||||
|
||||
loss <- tf_function(function(V) {
|
||||
Q <- I - tf$matmul(V, V, transpose_b = TRUE)
|
||||
if (sd_noise > 0)
|
||||
XQ <- tf$matmul(X + tf$random$normal(list(n, p), stddev = sd_noise), Q)
|
||||
else
|
||||
XQ <- tf$matmul(X, Q)
|
||||
S <- tf$matmul(XQ, XQ, transpose_b = TRUE)
|
||||
d <- tf$linalg$diag_part(S)
|
||||
D <- tf$reshape(d, list(n, 1L)) + tf$reshape(d, list(1L, n)) - `2` * S
|
||||
K <- tf$exp((`-0.5` / h) * tf$pow(D, 2L))
|
||||
w <- tf$reduce_sum(K, 1L, keepdims = TRUE)
|
||||
y1 <- tf$divide(tf$matmul(K, Y), w)
|
||||
y2 <- tf$divide(tf$matmul(K, tf$pow(Y, 2L)), w)
|
||||
if (method == "simple") {
|
||||
l <- tf$reduce_mean(y2 - tf$pow(y1, 2L))
|
||||
} else {# weighted
|
||||
w <- w - `1`
|
||||
w <- w / tf$reduce_sum(w)
|
||||
l <- tf$reduce_sum(w * (y2 - tf$pow(y1, 2L)))
|
||||
l <- l / tf$cast(tf$shape(Y)[2], "float32")
|
||||
}
|
||||
l
|
||||
})
|
||||
|
||||
if (is.null(V.init))
|
||||
V.init <- qr.Q(qr(matrix(rnorm(p * q), p, q)))
|
||||
else
|
||||
attempts <- 1L
|
||||
V <- tf_Variable(V.init, constraint = function(w) { tf$linalg$qr(w)$q })
|
||||
|
||||
min.loss <- Inf
|
||||
for (attempt in seq_len(attempts)) {
|
||||
optimizer = optimizer_initialier()
|
||||
|
||||
out <- tf$while_loop(
|
||||
cond = tf_function(function(i, L) i < 400L),
|
||||
body = tf_function(function(i, L) {
|
||||
with(tf$GradientTape() %as% tape, {
|
||||
tape$watch(V)
|
||||
L <- loss(V)
|
||||
})
|
||||
grad <- tape$gradient(L, V)
|
||||
optimizer$apply_gradients(list(list(grad, V)))
|
||||
|
||||
list(i + 1L, L)
|
||||
}),
|
||||
loop_vars = list(tf_constant(0L, "int32"), tf_constant(Inf))
|
||||
)
|
||||
|
||||
if (as.numeric(out[[2]]) < min.loss) {
|
||||
min.loss <- as.numeric(out[[2]])
|
||||
min.V <- as.matrix(V)
|
||||
}
|
||||
V$assign(qr.Q(qr(matrix(rnorm(p * q), p, q))))
|
||||
}
|
||||
|
||||
list(B = null(min.V), V = min.V, loss = min.loss)
|
||||
}
|
||||
# ds <- dataset(1)
|
||||
# out <- cve.call2(ds$X, ds$Y, ncol(ds$B))
|
||||
|
||||
plot.sim <- function(sim) {
|
||||
name <- deparse(substitute(sim))
|
||||
ssd <- sapply(sim, function(s) subspace_dist(s$B.true, s$B.est))
|
||||
print(summary(ssd))
|
||||
h <- hist(ssd, freq = FALSE, breaks = seq(0, 1, 0.1), main = name,
|
||||
xlab = "Subspace Distance")
|
||||
lines(density(ssd, from = 0, to = 1))
|
||||
stat <- c(Median = median(ssd), Mean = mean(ssd))
|
||||
abline(v = stat, lty = 2)
|
||||
text(stat, max(h$density), names(stat),
|
||||
pos = if(diff(stat) > 0) c("2", "4") else c("4", "2"))
|
||||
}
|
||||
|
||||
multivariate.dataset <- function(dataset = 1, n = 100, p = 6, q = 4) {
|
||||
CVE <- getNamespace('CVE')
|
||||
X <- matrix(rnorm(n * p), n, p)
|
||||
|
||||
if (dataset == 1) {
|
||||
Delta <- diag(1, q, q)
|
||||
Delta[1, 2] <- Delta[2, 1] <- -0.5
|
||||
epsilon <- CVE$rmvnorm(n, sigma = Delta)
|
||||
B <- matrix(0, p, q)
|
||||
B[1, 1] <- 1
|
||||
B[2, 2] <- 2 / sqrt(5)
|
||||
B[3, 2] <- 1 / sqrt(5)
|
||||
|
||||
Y <- X %*% B + epsilon
|
||||
B <- B[, 1:2]
|
||||
}
|
||||
if (dataset == 2) {
|
||||
B <- matrix(c(0.8, 0.6, 0, 0, 0, 0))
|
||||
eps <- matrix(0, n, q)
|
||||
Delta <- diag(1, q, q)
|
||||
for(i in 1:n) {
|
||||
Delta[1, 2] <- Delta[2, 1] <- sin(X[i, ] %*% B)
|
||||
eps[i, ] <- CVE$rmvnorm(1, sigma = Delta)
|
||||
}
|
||||
Y<-cbind(exp(eps[, 1]), eps[, 2:4])
|
||||
}
|
||||
list(X = X, Y = Y, B = B)
|
||||
}
|
||||
|
||||
|
||||
set.seed(42)
|
||||
reps <- 5L
|
||||
sim.cve.m <- vector("list", reps)
|
||||
sim.cve.c <- vector("list", reps)
|
||||
sim.cve.wm <- vector("list", reps)
|
||||
sim.cve.wc <- vector("list", reps)
|
||||
sim.tf1 <- vector("list", reps)
|
||||
sim.tf2 <- vector("list", reps)
|
||||
|
||||
start <- Sys.time()
|
||||
for (i in 1:reps) {
|
||||
# ds <- dataset(1)
|
||||
ds <- multivariate.dataset(2, n = 400)
|
||||
|
||||
sim.cve.m[[i]] <- list(
|
||||
B.est = coef(CVE::cve.call(ds$X, ds$Y, k = ncol(ds$B), method = "mean"), ncol(ds$B)),
|
||||
B.true = ds$B
|
||||
)
|
||||
sim.cve.c[[i]] <- list(
|
||||
B.est = coef(CVE::cve.call(ds$X, ds$Y, k = ncol(ds$B), method = "central"), ncol(ds$B)),
|
||||
B.true = ds$B
|
||||
)
|
||||
sim.cve.wm[[i]] <- list(
|
||||
B.est = coef(CVE::cve.call(ds$X, ds$Y, k = ncol(ds$B), method = "weighted.mean"), ncol(ds$B)),
|
||||
B.true = ds$B
|
||||
)
|
||||
sim.cve.wc[[i]] <- list(
|
||||
B.est = coef(CVE::cve.call(ds$X, ds$Y, k = ncol(ds$B), method = "weighted.central"), ncol(ds$B)),
|
||||
B.true = ds$B
|
||||
)
|
||||
|
||||
# sim.tf1[[i]] <- list(
|
||||
# B.est = cve.tf(ds$X, ds$Y, ncol(ds$B),
|
||||
# optimizer_initialier = tf$optimizers$Adam)$B,
|
||||
# B.true = ds$B
|
||||
# )
|
||||
# sim.tf2[[i]] <- list(
|
||||
# B.est = cve.tf(ds$X, ds$Y, ncol(ds$B),
|
||||
# optimizer_initialier = tf$optimizers$Adam,
|
||||
# method = "weighted")$B,
|
||||
# B.true = ds$B
|
||||
# )
|
||||
|
||||
cat(sprintf("\r%4d/%d -", i, reps), format(Sys.time() - start), '\n')
|
||||
}
|
||||
|
||||
# pdf('subspace_comp.pdf')
|
||||
plot.sim(sim.cve)
|
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plot.sim(sim.tf1)
|
||||
plot.sim(sim.tf2)
|
||||
|
||||
par(mfrow = c(2, 2))
|
||||
plot.sim(sim.cve.m)
|
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plot.sim(sim.cve.c)
|
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plot.sim(sim.cve.wm)
|
||||
plot.sim(sim.cve.wc)
|
||||
|
||||
# dev.off()
|
||||
Loading…
Reference in New Issue