R/dsp.R

In itdr: Integral Transformation Methods for SDR in Regression

#' Distance Between Two Subspaces.
#'
#' The ``\emph{dsp()}'' function calculates the distance between two subspaces, which are spanned by the columns of two matrices.
#' @param A A matrix with dimension p-by-d.
#' @param B A matrix with dimension p-by-d.
#' @return
#' Outputs are the following scale values.
#' \item{r}{One mines the trace correlation. That is, \eqn{r=1-\gamma} }
#'
#' \item{q}{One mines the vector correlation. That is, \eqn{q=1-\theta}}
#'
#' @export
#' @details
#' Let \bold{A} and \bold{B} be two full rank matrices of size
#' \eqn{p \times q}. Suppose \eqn{\mathcal{S}(\textbf{A})} and
#' \eqn{\mathcal{S}(\textbf{B})} are the column subspaces of matrices
#' \bold{A} and \bold{B}, respectively.
#' And, let \eqn{\lambda_i} 's with
#' \eqn{1 \geq \lambda_1^2 \geq \lambda_2^2 \geq,\cdots,\lambda_p^2\geq 0},
#' be the eigenvalues of the matrix \eqn{\textbf{B}^T\textbf{A}\textbf{A}^T\textbf{B}}.
#'
#' 1.Trace correlation, (Hotelling, 1936): \deqn{\gamma=\sqrt{\frac{1}{p}\sum_{i=1}^{p}\lambda_i^2}}
#'
#' 2.Vector correlation, (Hooper, 1959): \deqn{\theta=\sqrt{\prod_{i=1}^{p}\lambda_i^2}}
#' @references
#' Hooper J. (1959). Simultaneous Equations and Canonical Correlation Theory. \emph{Econometrica} 27, 245-256.
#'
#' Hotelling H. (1936). Relations Between Two Sets of Variates. \emph{Biometrika} 28, 321-377.

dsp <- function(A, B) {
  A.orth <- qr.Q(qr(A))
  B.orth <- qr.Q(qr(B))
  p <- nrow(B.orth)
  d <- ncol(B.orth)

  BAAB <- t(B.orth) %*% A.orth %*% t(A.orth) %*% B.orth
  BAAB.eig <- eigen(BAAB, only.values = T)$values

  rohat <- sqrt(abs(det(t(A.orth) %*% B %*% t(B) %*% A.orth)))
  rohat <- sqrt(abs(det(t(B) %*% A.orth %*% t(A.orth) %*% B)))

  h <- (diag(p) - B.orth %*% t(B.orth)) %*% A.orth
  msq <- apply(h, 2, function(x) sum(x^2))

  list(r = 1 - sqrt(mean((BAAB.eig))), q = 1 - sqrt(prod((BAAB.eig))), ro = rohat, msq = msq)
}