add: CVE,
add: R document files, add: CVE_paper, add: package-doc, fix: doc typo
This commit is contained in:
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4bc9ca2f58
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export(cve)
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export(cve_cpp)
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export(dataset)
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export(estimate.bandwidth)
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export(index_test)
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export(kron_test)
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export(estimateBandwidth)
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export(rStiefel)
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export(test1)
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export(test2)
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export(test3)
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export(test4)
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import(Rcpp)
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importFrom(Rcpp,evalCpp)
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useDynLib(CVE)
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44
CVE/R/CVE.R
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44
CVE/R/CVE.R
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#' Conditional Variance Estimator
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#'
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#' Conditional Variance Estimator (CVE) is a novel sufficient dimension
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#' reduction (SDR) method for regressions satisfying E(Y|X) = E(Y|B'X),
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#' where B'X is a lower dimensional projection of the predictors.
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#'
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#' @param X A matrix of type numeric of dimensions N times dim where N is the number
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#' of entries with dim as data dimension.
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#' @param Y A vector of type numeric of length N (coresponds to \code{x}).
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#' @param k Guess for rank(B).
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#' @param nObs As describet in the paper.
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#'
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#' @param tol Tolerance for optimization stopping creteria.
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#' @export
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#'
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#' @seealso TODO: \code{vignette("CVE_paper", package="CVE")}.
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#'
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#' @references Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
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cve <- function(X, Y, k,
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nObs = sqrt(nrow(X)),
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tauInitial = 1.0,
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tol = 1e-3,
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slack = -1e-10,
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maxIter = 50L,
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attempts = 10L
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) {
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# check data parameter types
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stopifnot(
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is.matrix(X),
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is.vector(Y),
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typeof(X) == 'double',
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typeof(Y) == 'double'
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)
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# call CVE C++ implementation
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return(cve_cpp(X, Y, k,
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nObs,
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tauInitial,
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tol,
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slack,
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maxIter,
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attempts
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))
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}
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95
CVE/R/RcppExports.R
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95
CVE/R/RcppExports.R
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# Generated by using Rcpp::compileAttributes() -> do not edit by hand
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# Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393
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#' Stiefel Optimization.
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#'
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#' Stiefel Optimization for \code{V} given a dataset \code{X} and responces
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#' \code{Y} for the model \deqn{Y\sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
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#' to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%
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#' span(B) = orth(span(B))}.
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#'
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#' @param X data points
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#' @param Y response
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#' @param k assumed \eqn{rank(B)}
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#' @param nObs parameter for bandwidth estimation, typical value
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#' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
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#' @param tau Initial step size
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#' @param tol Tolerance for update error used for stopping criterion
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#' \eqn{|| V(j) V(j)' - V(j+1) V(j+1)' ||_2 < tol}{%
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#' \| V_j V_j' - V_{j+1} V_{j+1}' \|_2 < tol}.
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#' @param maxIter Upper bound of optimization iterations
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#'
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#' @return List containing the bandwidth \code{h}, optimization objective \code{V}
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#' and the matrix \code{B} estimated for the model as a orthogonal basis of the
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#' orthogonal space spaned by \code{V}.
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#'
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#' @rdname optStiefel
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#' @name optStiefel
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#' @keywords internal
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NULL
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#' Estimated bandwidth for CVE.
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#'
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#' Estimates a propper bandwidth \code{h} according
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#' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
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#' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
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#'
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#' @param X data matrix of dimension (n x p) with n data points X_i of dimension
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#' q. Therefor each row represents a datapoint of dimension p.
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#' @param k Guess for rank(B).
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#' @param nObs Ether numeric of a function. If specified as numeric value
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#' its used in the computation of the bandwidth directly. If its a function
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#' `nObs` is evaluated as \code{nObs(nrow(x))}. The default behaviou if not
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#' supplied at all is to use \code{nObs <- nrow(x)^0.5}.
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#'
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#' @seealso [qchisq()]
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#'
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#' @export
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estimateBandwidth <- function(X, k, nObs) {
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.Call('_CVE_estimateBandwidth', PACKAGE = 'CVE', X, k, nObs)
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}
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#' Random element from Stiefel Manifold `S(p, q)`.
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#'
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#' Draws an element of \eqn{S(p, q)} which is the Stiefel Manifold.
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#' This is done by taking the Q-component of the QR decomposition
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#' from a `(p, q)` Matrix with independent standart normal entries.
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#' As a semi-orthogonal Matrix the result `V` satisfies \eqn{V'V = I_q}.
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#'
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#' @param p Row dimension
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#' @param q Column dimension
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#'
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#' @return Matrix of dim `(p, q)`.
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#'
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#' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
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#'
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#' @export
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rStiefel <- function(p, q) {
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.Call('_CVE_rStiefel', PACKAGE = 'CVE', p, q)
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}
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#' Conditional Variance Estimation (CVE) method.
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#'
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#' This version uses a "simple" stiefel optimization schema.
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#'
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#' @param X data points
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#' @param Y response
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#' @param k assumed \eqn{rank(B)}
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#' @param nObs parameter for bandwidth estimation, typical value
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#' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
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#' @param tau Initial step size (default 1)
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#' @param tol Tolerance for update error used for stopping criterion (default 1e-5)
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#' @param slack Ratio of small negative error allowed in loss optimization (default -1e-10)
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#' @param maxIter Upper bound of optimization iterations (default 50)
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#' @param attempts Number of tryes with new random optimization starting points (default 10)
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#'
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#' @return List containing the bandwidth \code{h}, optimization objective \code{V}
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#' and the matrix \code{B} estimated for the model as a orthogonal basis of the
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#' orthogonal space spaned by \code{V}.
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#'
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#' @rdname cve_cpp_V1
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#' @export
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cve_cpp <- function(X, Y, k, nObs, tauInitial = 1., tol = 1e-5, slack = -1e-10, maxIter = 50L, attempts = 10L) {
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.Call('_CVE_cve_cpp', PACKAGE = 'CVE', X, Y, k, nObs, tauInitial, tol, slack, maxIter, attempts)
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}
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13
CVE/R/package.R
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CVE/R/package.R
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#' Conditional Variance Estimator (CVE)
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#'
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#' Conditional Variance Estimator for Sufficient Dimension
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#' Reduction
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#'
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#' TODO: And some details
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#'
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#' @docType package
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#' @author Loki
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#' @import Rcpp
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#' @importFrom Rcpp evalCpp
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#' @useDynLib CVE
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"_PACKAGE"
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BIN
CVE/inst/doc/CVE_paper.pdf
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BIN
CVE/inst/doc/CVE_paper.pdf
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CVE/man/CVE-package.Rd
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CVE/man/CVE-package.Rd
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% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/package.R
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\docType{package}
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\name{CVE-package}
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\alias{CVE}
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\alias{CVE-package}
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\title{Conditional Variance Estimator (CVE)}
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\description{
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Conditional Variance Estimator for Sufficient Dimension
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Reduction
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}
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\details{
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TODO: And some details
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}
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\author{
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Loki
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}
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32
CVE/man/cve.Rd
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32
CVE/man/cve.Rd
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% 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}
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\usage{
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cve(X, Y, k, nObs = sqrt(nrow(X)), tauInitial = 1, tol = 0.001,
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slack = -1e-10, maxIter = 50L, attempts = 10L)
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}
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\arguments{
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\item{X}{A matrix of type numeric of dimensions N times dim where N is the number
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of entries with dim as data dimension.}
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\item{Y}{A vector of type numeric of length N (coresponds to \code{x}).}
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\item{k}{Guess for rank(B).}
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\item{nObs}{As describet in the paper.}
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\item{tol}{Tolerance for optimization stopping creteria.}
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}
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\description{
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Conditional Variance Estimator (CVE) is a novel sufficient dimension
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reduction (SDR) method for regressions satisfying E(Y|X) = E(Y|B'X),
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where B'X is a lower dimensional projection of the predictors.
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}
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\references{
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Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
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}
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\seealso{
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TODO: \code{vignette("CVE_paper", package="CVE")}.
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}
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37
CVE/man/cve_cpp_V1.Rd
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CVE/man/cve_cpp_V1.Rd
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% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/RcppExports.R
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\name{cve_cpp}
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\alias{cve_cpp}
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\title{Conditional Variance Estimation (CVE) method.}
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\usage{
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cve_cpp(X, Y, k, nObs, tauInitial = 1, tol = 1e-05, slack = -1e-10,
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maxIter = 50L, attempts = 10L)
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}
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\arguments{
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\item{X}{data points}
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\item{Y}{response}
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\item{k}{assumed \eqn{rank(B)}}
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\item{nObs}{parameter for bandwidth estimation, typical value
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\code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].}
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\item{tol}{Tolerance for update error used for stopping criterion (default 1e-5)}
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\item{slack}{Ratio of small negative error allowed in loss optimization (default -1e-10)}
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\item{maxIter}{Upper bound of optimization iterations (default 50)}
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\item{attempts}{Number of tryes with new random optimization starting points (default 10)}
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\item{tau}{Initial step size (default 1)}
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}
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\value{
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List containing the bandwidth \code{h}, optimization objective \code{V}
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and the matrix \code{B} estimated for the model as a orthogonal basis of the
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orthogonal space spaned by \code{V}.
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}
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\description{
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This version uses a "simple" stiefel optimization schema.
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}
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65
CVE/man/dataset.Rd
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65
CVE/man/dataset.Rd
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% 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, B, p.mix = 0.3, lambda = 1)
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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"} or \code{"M5"}}
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\item{n}{nr samples}
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\item{p.mix}{Only for \code{"M4"}, see: below.}
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\item{lambda}{Only for \code{"M4"}, see: below.}
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\item{p}{Dim. of random variable \code{X}.}
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}
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\value{
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List with elements
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\item{X}{data}
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\item{Y}{response}
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\item{B}{Used dim-reduction matrix}
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\item{name}{Name of the dataset (name parameter)}
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}
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\description{
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Provides sample datasets. There are 5 different datasets named
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M1, M2, M3, M4 and M5 describet in the paper references below.
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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 data follows \eqn{X\sim N_p(0, \Sigma)}{X ~ N_p(0, Sigma)} 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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The link function \eqn{g} is given as
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\deqn{g(x) = \frac{x_1}{0.5 + (x_2 + 1.5)^2} + 0.5\epsilon}{g(x) = x_1 / (0.5 + (x_2 + 1.5)^2) + 0.5 epsilon}
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}
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\section{M2}{
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\eqn{X\sim N_p(0, \Sigma)}{X ~ N_p(0, Sigma)} with \eqn{k = 2} with a default of \eqn{n = 200} data points.
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The link function \eqn{g} is given as
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\deqn{g(x) = x_1 x_2^2 + 0.5\epsilon}{g(x) = x_1 x_2^2 + 0.5 epsilon}
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}
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\section{M3}{
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TODO:
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}
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\section{M4}{
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TODO:
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}
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\section{M5}{
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TODO:
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}
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\references{
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Fertl Likas, Bura Efstathia. Conditional Variance Estimation for Sufficient Dimension Reduction, 2019
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}
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27
CVE/man/estimateBandwidth.Rd
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CVE/man/estimateBandwidth.Rd
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% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/RcppExports.R
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\name{estimateBandwidth}
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\alias{estimateBandwidth}
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\title{Estimated bandwidth for CVE.}
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\usage{
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estimateBandwidth(X, k, nObs)
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}
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\arguments{
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\item{X}{data matrix of dimension (n x p) with n data points X_i of dimension
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q. Therefor each row represents a datapoint of dimension p.}
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\item{k}{Guess for rank(B).}
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\item{nObs}{Ether numeric of a function. If specified as numeric value
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its used in the computation of the bandwidth directly. If its a function
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`nObs` is evaluated as \code{nObs(nrow(x))}. The default behaviou if not
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supplied at all is to use \code{nObs <- nrow(x)^0.5}.}
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}
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\description{
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Estimates a propper bandwidth \code{h} according
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\deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
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h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
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}
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\seealso{
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[qchisq()]
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}
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35
CVE/man/optStiefel.Rd
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35
CVE/man/optStiefel.Rd
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% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/RcppExports.R
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\name{optStiefel}
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\alias{optStiefel}
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\title{Stiefel Optimization.}
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\arguments{
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\item{X}{data points}
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\item{Y}{response}
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\item{k}{assumed \eqn{rank(B)}}
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\item{nObs}{parameter for bandwidth estimation, typical value
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\code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].}
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\item{tau}{Initial step size}
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\item{tol}{Tolerance for update error used for stopping criterion
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\eqn{|| V(j) V(j)' - V(j+1) V(j+1)' ||_2 < tol}{%
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\| V_j V_j' - V_{j+1} V_{j+1}' \|_2 < tol}.}
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\item{maxIter}{Upper bound of optimization iterations}
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}
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\value{
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List containing the bandwidth \code{h}, optimization objective \code{V}
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and the matrix \code{B} estimated for the model as a orthogonal basis of the
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orthogonal space spaned by \code{V}.
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}
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\description{
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Stiefel Optimization for \code{V} given a dataset \code{X} and responces
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\code{Y} for the model \deqn{Y\sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
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to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%
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span(B) = orth(span(B))}.
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}
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\keyword{internal}
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25
CVE/man/rStiefel.Rd
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CVE/man/rStiefel.Rd
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% Generated by roxygen2: do not edit by hand
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% Please edit documentation in R/RcppExports.R
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\name{rStiefel}
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\alias{rStiefel}
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\title{Random element from Stiefel Manifold `S(p, q)`.}
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\usage{
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rStiefel(p, q)
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}
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\arguments{
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\item{p}{Row dimension}
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\item{q}{Column dimension}
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}
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\value{
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Matrix of dim `(p, q)`.
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}
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\description{
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Draws an element of \eqn{S(p, q)} which is the Stiefel Manifold.
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This is done by taking the Q-component of the QR decomposition
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from a `(p, q)` Matrix with independent standart normal entries.
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As a semi-orthogonal Matrix the result `V` satisfies \eqn{V'V = I_q}.
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}
|
||||
\seealso{
|
||||
<https://en.wikipedia.org/wiki/Stiefel_manifold>
|
||||
}
|
||||
285
CVE/src/CVE.cpp
Normal file
285
CVE/src/CVE.cpp
Normal file
@ -0,0 +1,285 @@
|
||||
//
|
||||
// Usage:
|
||||
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V1.cpp')"
|
||||
//
|
||||
|
||||
// only `RcppArmadillo.h` which includes `Rcpp.h`
|
||||
#include <RcppArmadillo.h>
|
||||
|
||||
// through the depends attribute `Rcpp` is tolled to create
|
||||
// hooks for `RcppArmadillo` needed by the build process.
|
||||
//
|
||||
// [[Rcpp::depends(RcppArmadillo)]]
|
||||
|
||||
// required for `R::qchisq()` used in `estimateBandwidth()`
|
||||
#include <Rmath.h>
|
||||
|
||||
//' Estimated bandwidth for CVE.
|
||||
//'
|
||||
//' Estimates a propper bandwidth \code{h} according
|
||||
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
|
||||
//' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
|
||||
//'
|
||||
//' @param X data matrix of dimension (n x p) with n data points X_i of dimension
|
||||
//' q. Therefor each row represents a datapoint of dimension p.
|
||||
//' @param k Guess for rank(B).
|
||||
//' @param nObs Ether numeric of a function. If specified as numeric value
|
||||
//' its used in the computation of the bandwidth directly. If its a function
|
||||
//' `nObs` is evaluated as \code{nObs(nrow(x))}. The default behaviou if not
|
||||
//' supplied at all is to use \code{nObs <- nrow(x)^0.5}.
|
||||
//'
|
||||
//' @seealso [qchisq()]
|
||||
//'
|
||||
//' @export
|
||||
// [[Rcpp::export]]
|
||||
double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs) {
|
||||
using namespace arma;
|
||||
|
||||
uword n = X.n_rows; // nr samples
|
||||
uword p = X.n_cols; // dimension of rand. var. `X`
|
||||
|
||||
// column mean
|
||||
mat M = mean(X);
|
||||
// center `X`
|
||||
mat C = X.each_row() - M;
|
||||
// trace of covariance matrix, `traceSigma = Tr(C' C)`
|
||||
double traceSigma = accu(C % C);
|
||||
// compute estimated bandwidth
|
||||
double qchi2 = R::qchisq((nObs - 1.0) / (n - 1), static_cast<double>(k), 1, 0);
|
||||
|
||||
return 2.0 * qchi2 * traceSigma / (p * n);
|
||||
}
|
||||
|
||||
//' Random element from Stiefel Manifold `S(p, q)`.
|
||||
//'
|
||||
//' Draws an element of \eqn{S(p, q)} which is the Stiefel Manifold.
|
||||
//' This is done by taking the Q-component of the QR decomposition
|
||||
//' from a `(p, q)` Matrix with independent standart normal entries.
|
||||
//' As a semi-orthogonal Matrix the result `V` satisfies \eqn{V'V = I_q}.
|
||||
//'
|
||||
//' @param p Row dimension
|
||||
//' @param q Column dimension
|
||||
//'
|
||||
//' @return Matrix of dim `(p, q)`.
|
||||
//'
|
||||
//' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
|
||||
//'
|
||||
//' @export
|
||||
// [[Rcpp::export]]
|
||||
arma::mat rStiefel(arma::uword p, arma::uword q) {
|
||||
arma::mat Q, R;
|
||||
arma::qr_econ(Q, R, arma::randn<arma::mat>(p, q));
|
||||
return Q;
|
||||
}
|
||||
|
||||
double gradient(const arma::mat& X,
|
||||
const arma::mat& X_diff,
|
||||
const arma::mat& Y,
|
||||
const arma::mat& Y_rep,
|
||||
const arma::mat& V,
|
||||
const double h,
|
||||
arma::mat* G = 0
|
||||
) {
|
||||
using namespace arma;
|
||||
|
||||
uword n = X.n_rows;
|
||||
uword p = X.n_cols;
|
||||
|
||||
// orthogonal projection matrix `Q = I - VV'` for dist computation
|
||||
mat Q = -(V * V.t());
|
||||
Q.diag() += 1;
|
||||
// calc pairwise distances as `D` with `D(i, j) = d_i(V, X_j)`
|
||||
vec D_vec = sum(square(X_diff * Q), 1);
|
||||
mat D = reshape(D_vec, n, n);
|
||||
// calc weights as `W` with `W(i, j) = w_i(V, X_j)`
|
||||
mat W = exp(D / (-2.0 * h));
|
||||
// column-wise normalization via 1-norm
|
||||
W = normalise(W, 1);
|
||||
vec W_vec = vectorise(W);
|
||||
// weighted `Y` means (first and second order)
|
||||
vec y1 = W.t() * Y;
|
||||
vec y2 = W.t() * square(Y);
|
||||
// loss for each `X_i`, meaning `L(i) = L_n(V, X_i)`
|
||||
vec L = y2 - square(y1);
|
||||
// `loss = L_n(V)`
|
||||
double loss = mean(L);
|
||||
// check if gradient as output variable is set
|
||||
if (G != 0) {
|
||||
// `G = grad(L_n(V))` a.k.a. gradient of `L` with respect to `V`
|
||||
vec scale = (repelem(L, n, 1) - square(Y_rep - repelem(y1, n, 1))) % W_vec % D_vec;
|
||||
mat X_diff_scale = X_diff.each_col() % scale;
|
||||
(*G) = X_diff_scale.t() * X_diff * V;
|
||||
(*G) *= -2.0 / (h * h * n);
|
||||
}
|
||||
|
||||
return loss;
|
||||
}
|
||||
|
||||
//' Stiefel Optimization.
|
||||
//'
|
||||
//' Stiefel Optimization for \code{V} given a dataset \code{X} and responces
|
||||
//' \code{Y} for the model \deqn{Y\sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
|
||||
//' to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%
|
||||
//' span(B) = orth(span(B))}.
|
||||
//'
|
||||
//' @param X data points
|
||||
//' @param Y response
|
||||
//' @param k assumed \eqn{rank(B)}
|
||||
//' @param nObs parameter for bandwidth estimation, typical value
|
||||
//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
|
||||
//' @param tau Initial step size
|
||||
//' @param tol Tolerance for update error used for stopping criterion
|
||||
//' \eqn{|| V(j) V(j)' - V(j+1) V(j+1)' ||_2 < tol}{%
|
||||
//' \| V_j V_j' - V_{j+1} V_{j+1}' \|_2 < tol}.
|
||||
//' @param maxIter Upper bound of optimization iterations
|
||||
//'
|
||||
//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
|
||||
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
|
||||
//' orthogonal space spaned by \code{V}.
|
||||
//'
|
||||
//' @rdname optStiefel
|
||||
//' @name optStiefel
|
||||
//' @keywords internal
|
||||
double optStiefel(
|
||||
const arma::mat& X,
|
||||
const arma::vec& Y,
|
||||
const int k,
|
||||
const double h,
|
||||
const double tauInitial,
|
||||
const double tol,
|
||||
const double slack,
|
||||
const int maxIter,
|
||||
arma::mat& V, // out
|
||||
arma::vec& history // out
|
||||
) {
|
||||
using namespace arma;
|
||||
|
||||
// get dimensions
|
||||
const uword n = X.n_rows; // nr samples
|
||||
const uword p = X.n_cols; // dim of random variable `X`
|
||||
const uword q = p - k; // rank(V) e.g. dim of ortho space of span{B}
|
||||
|
||||
// all `X_i - X_j` differences, `X_diff.row(i * n + j) = X_i - X_j`
|
||||
mat X_diff(n * n, p);
|
||||
for (uword i = 0, k = 0; i < n; ++i) {
|
||||
for (uword j = 0; j < n; ++j) {
|
||||
X_diff.row(k++) = X.row(i) - X.row(j);
|
||||
}
|
||||
}
|
||||
const vec Y_rep = repmat(Y, n, 1);
|
||||
const mat I_p = eye<mat>(p, p);
|
||||
|
||||
// initial start value for `V`
|
||||
V = rStiefel(p, q);
|
||||
|
||||
// init optimization `loss`es, `error` and stepsize `tau`
|
||||
// double loss_next = datum::inf;
|
||||
double loss;
|
||||
double error = datum::inf;
|
||||
double tau = tauInitial;
|
||||
int count;
|
||||
// main optimization loop
|
||||
for (count = 0; count < maxIter && error > tol; ++count) {
|
||||
// calc gradient `G = grad_V(L)(V)`
|
||||
mat G;
|
||||
loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
|
||||
// matrix `A` for colescy-transform of the gradient
|
||||
mat A = tau * ((G * V.t()) - (V * G.t()));
|
||||
// next iteration step of `V`
|
||||
mat V_tau = inv(I_p + A) * (I_p - A) * V;
|
||||
// loss after step `L(V(tau))`
|
||||
double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h);
|
||||
|
||||
// store `loss` in `history` and increase `count`
|
||||
history(count) = loss;
|
||||
|
||||
// validate if loss decreased
|
||||
if ((loss_tau - loss) > slack * loss) {
|
||||
// ignore step, retry with half the step size
|
||||
tau = tau / 2.;
|
||||
error = datum::inf;
|
||||
} else {
|
||||
// compute step error (break condition)
|
||||
error = norm((V * V.t()) - (V_tau * V_tau.t()), 2) / (2 * q);
|
||||
// shift for next iteration
|
||||
V = V_tau;
|
||||
loss = loss_tau;
|
||||
}
|
||||
}
|
||||
|
||||
// store final `loss`
|
||||
history(count) = loss;
|
||||
|
||||
return loss;
|
||||
}
|
||||
|
||||
//' Conditional Variance Estimation (CVE) method.
|
||||
//'
|
||||
//' This version uses a "simple" stiefel optimization schema.
|
||||
//'
|
||||
//' @param X data points
|
||||
//' @param Y response
|
||||
//' @param k assumed \eqn{rank(B)}
|
||||
//' @param nObs parameter for bandwidth estimation, typical value
|
||||
//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
|
||||
//' @param tau Initial step size (default 1)
|
||||
//' @param tol Tolerance for update error used for stopping criterion (default 1e-5)
|
||||
//' @param slack Ratio of small negative error allowed in loss optimization (default -1e-10)
|
||||
//' @param maxIter Upper bound of optimization iterations (default 50)
|
||||
//' @param attempts Number of tryes with new random optimization starting points (default 10)
|
||||
//'
|
||||
//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
|
||||
//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
|
||||
//' orthogonal space spaned by \code{V}.
|
||||
//'
|
||||
//' @rdname cve_cpp_V1
|
||||
//' @export
|
||||
// [[Rcpp::export]]
|
||||
Rcpp::List cve_cpp(
|
||||
const arma::mat& X,
|
||||
const arma::vec& Y,
|
||||
const int k,
|
||||
const double nObs,
|
||||
const double tauInitial = 1.,
|
||||
const double tol = 1e-5,
|
||||
const double slack = -1e-10,
|
||||
const int maxIter = 50,
|
||||
const int attempts = 10
|
||||
) {
|
||||
using namespace arma;
|
||||
|
||||
// tracker of current best results
|
||||
mat V_best;
|
||||
double loss_best = datum::inf;
|
||||
// estimate bandwidth
|
||||
double h = estimateBandwidth(X, k, nObs);
|
||||
|
||||
// loss history for each optimization attempt
|
||||
// each column contaions the iteration history for the loss
|
||||
mat history = mat(maxIter + 1, attempts);
|
||||
|
||||
// multiple stiefel optimization attempts
|
||||
for (int i = 0; i < attempts; ++i) {
|
||||
// declare output variables
|
||||
mat V; // estimated `V` space
|
||||
vec hist = vec(history.n_rows, fill::zeros); // optimization history
|
||||
double loss = optStiefel(X, Y, k, h, tauInitial, tol, slack, maxIter, V, hist);
|
||||
if (loss < loss_best) {
|
||||
loss_best = loss;
|
||||
V_best = V;
|
||||
}
|
||||
// write history to history collection
|
||||
history.col(i) = hist;
|
||||
}
|
||||
|
||||
// get `B` as kernal of `V.t()`
|
||||
mat B = null(V_best.t());
|
||||
|
||||
return Rcpp::List::create(
|
||||
Rcpp::Named("history") = history,
|
||||
Rcpp::Named("loss") = loss_best,
|
||||
Rcpp::Named("h") = h,
|
||||
Rcpp::Named("V") = V_best,
|
||||
Rcpp::Named("B") = B
|
||||
);
|
||||
}
|
||||
64
CVE/src/RcppExports.cpp
Normal file
64
CVE/src/RcppExports.cpp
Normal file
@ -0,0 +1,64 @@
|
||||
// Generated by using Rcpp::compileAttributes() -> do not edit by hand
|
||||
// Generator token: 10BE3573-1514-4C36-9D1C-5A225CD40393
|
||||
|
||||
#include <RcppArmadillo.h>
|
||||
#include <Rcpp.h>
|
||||
|
||||
using namespace Rcpp;
|
||||
|
||||
// estimateBandwidth
|
||||
double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs);
|
||||
RcppExport SEXP _CVE_estimateBandwidth(SEXP XSEXP, SEXP kSEXP, SEXP nObsSEXP) {
|
||||
BEGIN_RCPP
|
||||
Rcpp::RObject rcpp_result_gen;
|
||||
Rcpp::RNGScope rcpp_rngScope_gen;
|
||||
Rcpp::traits::input_parameter< const arma::mat& >::type X(XSEXP);
|
||||
Rcpp::traits::input_parameter< arma::uword >::type k(kSEXP);
|
||||
Rcpp::traits::input_parameter< double >::type nObs(nObsSEXP);
|
||||
rcpp_result_gen = Rcpp::wrap(estimateBandwidth(X, k, nObs));
|
||||
return rcpp_result_gen;
|
||||
END_RCPP
|
||||
}
|
||||
// rStiefel
|
||||
arma::mat rStiefel(arma::uword p, arma::uword q);
|
||||
RcppExport SEXP _CVE_rStiefel(SEXP pSEXP, SEXP qSEXP) {
|
||||
BEGIN_RCPP
|
||||
Rcpp::RObject rcpp_result_gen;
|
||||
Rcpp::RNGScope rcpp_rngScope_gen;
|
||||
Rcpp::traits::input_parameter< arma::uword >::type p(pSEXP);
|
||||
Rcpp::traits::input_parameter< arma::uword >::type q(qSEXP);
|
||||
rcpp_result_gen = Rcpp::wrap(rStiefel(p, q));
|
||||
return rcpp_result_gen;
|
||||
END_RCPP
|
||||
}
|
||||
// cve_cpp
|
||||
Rcpp::List cve_cpp(const arma::mat& X, const arma::vec& Y, const int k, const double nObs, const double tauInitial, const double tol, const double slack, const int maxIter, const int attempts);
|
||||
RcppExport SEXP _CVE_cve_cpp(SEXP XSEXP, SEXP YSEXP, SEXP kSEXP, SEXP nObsSEXP, SEXP tauInitialSEXP, SEXP tolSEXP, SEXP slackSEXP, SEXP maxIterSEXP, SEXP attemptsSEXP) {
|
||||
BEGIN_RCPP
|
||||
Rcpp::RObject rcpp_result_gen;
|
||||
Rcpp::RNGScope rcpp_rngScope_gen;
|
||||
Rcpp::traits::input_parameter< const arma::mat& >::type X(XSEXP);
|
||||
Rcpp::traits::input_parameter< const arma::vec& >::type Y(YSEXP);
|
||||
Rcpp::traits::input_parameter< const int >::type k(kSEXP);
|
||||
Rcpp::traits::input_parameter< const double >::type nObs(nObsSEXP);
|
||||
Rcpp::traits::input_parameter< const double >::type tauInitial(tauInitialSEXP);
|
||||
Rcpp::traits::input_parameter< const double >::type tol(tolSEXP);
|
||||
Rcpp::traits::input_parameter< const double >::type slack(slackSEXP);
|
||||
Rcpp::traits::input_parameter< const int >::type maxIter(maxIterSEXP);
|
||||
Rcpp::traits::input_parameter< const int >::type attempts(attemptsSEXP);
|
||||
rcpp_result_gen = Rcpp::wrap(cve_cpp(X, Y, k, nObs, tauInitial, tol, slack, maxIter, attempts));
|
||||
return rcpp_result_gen;
|
||||
END_RCPP
|
||||
}
|
||||
|
||||
static const R_CallMethodDef CallEntries[] = {
|
||||
{"_CVE_estimateBandwidth", (DL_FUNC) &_CVE_estimateBandwidth, 3},
|
||||
{"_CVE_rStiefel", (DL_FUNC) &_CVE_rStiefel, 2},
|
||||
{"_CVE_cve_cpp", (DL_FUNC) &_CVE_cve_cpp, 9},
|
||||
{NULL, NULL, 0}
|
||||
};
|
||||
|
||||
RcppExport void R_init_CVE(DllInfo *dll) {
|
||||
R_registerRoutines(dll, NULL, CallEntries, NULL, NULL);
|
||||
R_useDynamicSymbols(dll, FALSE);
|
||||
}
|
||||
13
cve_V0.R
13
cve_V0.R
@ -2,14 +2,14 @@
|
||||
#' Euclidean vector norm (2-norm)
|
||||
#'
|
||||
#' @param x Numeric vector
|
||||
#' @returns Numeric
|
||||
#' @return Numeric
|
||||
norm2 <- function(x) { return(sum(x^2)) }
|
||||
|
||||
#' Samples uniform from the Stiefel Manifold
|
||||
#'
|
||||
#' @param p row dim.
|
||||
#' @param q col dim.
|
||||
#' @returns `(p, q)` semi-orthogonal matrix
|
||||
#' @return `(p, q)` semi-orthogonal matrix
|
||||
rStiefl <- function(p, q) {
|
||||
return(qr.Q(qr(matrix(rnorm(p * q, 0, 1), p, q))))
|
||||
}
|
||||
@ -17,7 +17,7 @@ rStiefl <- function(p, q) {
|
||||
#' Matrix Trace
|
||||
#'
|
||||
#' @param M Square matrix
|
||||
#' @returns Trace \eqn{Tr(M)}
|
||||
#' @return Trace \eqn{Tr(M)}
|
||||
Tr <- function(M) {
|
||||
return(sum(diag(M)))
|
||||
}
|
||||
@ -25,7 +25,7 @@ Tr <- function(M) {
|
||||
#' Null space basis of given matrix `B`
|
||||
#'
|
||||
#' @param B `(p, q)` matrix
|
||||
#' @returns Semi-orthogonal `(p, p - q)` matrix `Q` spaning the null space of `B`
|
||||
#' @return Semi-orthogonal `(p, p - q)` matrix `Q` spaning the null space of `B`
|
||||
null <- function(M) {
|
||||
tmp <- qr(M)
|
||||
set <- if(tmp$rank == 0L) seq_len(ncol(M)) else -seq_len(tmp$rank)
|
||||
@ -60,8 +60,7 @@ estimateBandwidth<-function(X, k, nObs) {
|
||||
# if grad=T, gradient of L(V) also returned
|
||||
LV <- function(V, Xl, dtemp, h, q, Y, grad = TRUE) {
|
||||
N <- length(Y)
|
||||
if (is.vector(V)) { k <- 1 }
|
||||
else { k <- length(V[1,]) }
|
||||
k <- if (is.vector(V)) { 1 } else { ncol(V) }
|
||||
Xlv <- Xl %*% V
|
||||
d <- dtemp - ((Xlv^2) %*% rep(1, k))
|
||||
w <- dnorm(d / h) / dnorm(0)
|
||||
@ -108,7 +107,7 @@ LV <- function(V, Xl, dtemp, h, q, Y, grad = TRUE) {
|
||||
#aov_dat... (L_tilde_n(Vhat_k,X_i))_{i=1,..,N}
|
||||
#count...number of iterations
|
||||
#h...bandwidth
|
||||
cve_legacy <- function(
|
||||
cve_R <- function(
|
||||
X, Y, k,
|
||||
nObs = sqrt(nrow(X)),
|
||||
tauInitial = 1.0,
|
||||
|
||||
@ -1,4 +1,6 @@
|
||||
//
|
||||
// Standalone implementation for development.
|
||||
//
|
||||
// Usage:
|
||||
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V1.cpp')"
|
||||
//
|
||||
@ -17,7 +19,7 @@
|
||||
//' Estimated bandwidth for CVE.
|
||||
//'
|
||||
//' Estimates a propper bandwidth \code{h} according
|
||||
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)}\frac{2 tr(\Sigma)}{p}}{%
|
||||
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
|
||||
//' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
|
||||
//'
|
||||
//' @param X data matrix of dimension (n x p) with n data points X_i of dimension
|
||||
@ -60,7 +62,7 @@ double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs) {
|
||||
//' @param p Row dimension
|
||||
//' @param q Column dimension
|
||||
//'
|
||||
//' @returns Matrix of dim `(p, q)`.
|
||||
//' @return Matrix of dim `(p, q)`.
|
||||
//'
|
||||
//' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
|
||||
//'
|
||||
@ -138,6 +140,7 @@ double gradient(const arma::mat& X,
|
||||
//' orthogonal space spaned by \code{V}.
|
||||
//'
|
||||
//' @rdname optStiefel
|
||||
//' @keywords internal
|
||||
double optStiefel(
|
||||
const arma::mat& X,
|
||||
const arma::vec& Y,
|
||||
|
||||
@ -1,4 +1,5 @@
|
||||
// -*- mode: C++; c-indent-level: 4; c-basic-offset: 4; indent-tabs-mode: nil; -*-
|
||||
//
|
||||
// Standalone implementation for development.
|
||||
//
|
||||
// Usage:
|
||||
// ~$ R -e "library(Rcpp); Rcpp::sourceCpp('cve_V2.cpp')"
|
||||
@ -18,7 +19,7 @@
|
||||
//' Estimated bandwidth for CVE.
|
||||
//'
|
||||
//' Estimates a propper bandwidth \code{h} according
|
||||
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)}\frac{2 tr(\Sigma)}{p}}{%
|
||||
//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
|
||||
//' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
|
||||
//'
|
||||
//' @param X data matrix of dimension (n x p) with n data points X_i of dimension
|
||||
@ -61,7 +62,7 @@ double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs) {
|
||||
//' @param p Row dimension
|
||||
//' @param q Column dimension
|
||||
//'
|
||||
//' @returns Matrix of dim `(p, q)`.
|
||||
//' @return Matrix of dim `(p, q)`.
|
||||
//'
|
||||
//' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
|
||||
//'
|
||||
@ -141,6 +142,7 @@ double gradient(const arma::mat& X,
|
||||
//' orthogonal space spaned by \code{V}.
|
||||
//'
|
||||
//' @rdname optStiefel
|
||||
//' @keywords internal
|
||||
double optStiefel(
|
||||
const arma::mat& X,
|
||||
const arma::vec& Y,
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user