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### oscorespls.fit.R: The multiresponse orthogonal scores algorithm
###
### Implements an adapted version of the `orthogonal scores' algorithm as
### described in Martens and Naes, pp. 121--122 and 157--158.
#' @title Orthogonal scores PLSR
#'
#' @description Fits a PLSR model with the orthogonal scores algorithm (aka the NIPALS
#' algorithm).
#'
#' @details This function should not be called directly, but through the generic
#' functions \code{plsr} or \code{mvr} with the argument
#' \code{method="oscorespls"}. It implements the orthogonal scores algorithm,
#' as described in \cite{Martens and Næs (1989)}. This is one of the two
#' \dQuote{classical} PLSR algorithms, the other being the orthogonal loadings
#' algorithm.
#'
#' @param X a matrix of observations. \code{NA}s and \code{Inf}s are not
#' allowed.
#' @param Y a vector or matrix of responses. \code{NA}s and \code{Inf}s are
#' not allowed.
#' @param ncomp the number of components to be used in the modelling.
#' @param center logical, determines if the \eqn{X} and \eqn{Y} matrices are
#' mean centered or not. Default is to perform mean centering.
#' @param stripped logical. If \code{TRUE} the calculations are stripped as
#' much as possible for speed; this is meant for use with cross-validation or
#' simulations when only the coefficients are needed. Defaults to
#' \code{FALSE}.
#' @param tol numeric. The tolerance used for determining convergence in
#' multi-response models.
#' @param maxit positive integer. The maximal number of iterations used in the
#' internal Eigenvector calculation.
#' @param \dots other arguments. Currently ignored.
#' @return A list containing the following components is returned:
#' \item{coefficients}{an array of regression coefficients for 1, \ldots{},
#' \code{ncomp} components. The dimensions of \code{coefficients} are
#' \code{c(nvar, npred, ncomp)} with \code{nvar} the number of \code{X}
#' variables and \code{npred} the number of variables to be predicted in
#' \code{Y}.} \item{scores}{a matrix of scores.} \item{loadings}{a matrix of
#' loadings.} \item{loading.weights}{a matrix of loading weights.}
#' \item{Yscores}{a matrix of Y-scores.} \item{Yloadings}{a matrix of
#' Y-loadings.} \item{projection}{the projection matrix used to convert X to
#' scores.} \item{Xmeans}{a vector of means of the X variables.}
#' \item{Ymeans}{a vector of means of the Y variables.} \item{fitted.values}{an
#' array of fitted values. The dimensions of \code{fitted.values} are
#' \code{c(nobj, npred, ncomp)} with \code{nobj} the number samples and
#' \code{npred} the number of Y variables.} \item{residuals}{an array of
#' regression residuals. It has the same dimensions as \code{fitted.values}.}
#' \item{Xvar}{a vector with the amount of X-variance explained by each
#' component.} \item{Xtotvar}{Total variance in \code{X}.}
#'
#' If \code{stripped} is \code{TRUE}, only the components \code{coefficients},
#' \code{Xmeans} and \code{Ymeans} are returned.
#' @author Ron Wehrens and Bjørn-Helge Mevik
#' @seealso \code{\link{mvr}} \code{\link{plsr}} \code{\link{pcr}}
#' \code{\link{kernelpls.fit}} \code{\link{widekernelpls.fit}}
#' \code{\link{simpls.fit}}
#' @references Martens, H., Næs, T. (1989) \emph{Multivariate calibration.}
#' Chichester: Wiley.
#' @keywords regression multivariate
#' @export
oscorespls.fit <- function(X, Y, ncomp, center = TRUE, stripped = FALSE,
tol = .Machine$double.eps^0.5, maxit = 100, ...)
{
## Initialise
Y <- as.matrix(Y)
if (!stripped) {
## Save dimnames
dnX <- dimnames(X)
dnY <- dimnames(Y)
}
## Remove dimnames for performance (doesn't seem to matter; in fact,
## as far as it has any effect, it hurts a tiny bit in most situations.
dimnames(X) <- dimnames(Y) <- NULL
nobj <- dim(X)[1]
npred <- dim(X)[2]
nresp <- dim(Y)[2]
W <- P <- matrix(0, nrow = npred, ncol = ncomp)
tQ <- matrix(0, nrow = ncomp, ncol = nresp) # Y loadings; transposed
B <- array(0, dim = c(npred, nresp, ncomp))
if (!stripped) {
TT <- U <- matrix(0, nrow = nobj, ncol = ncomp)
tsqs <- numeric(ncomp) # t't
fitted <- residuals <- array(0, dim = c(nobj, nresp, ncomp))
}
## C1
if (center) {
Xmeans <- colMeans(X)
X <- X - rep(Xmeans, each = nobj)
Ymeans <- colMeans(Y)
Y <- Y - rep(Ymeans, each = nobj)
} else {
## Set means to zero. Will ensure that predictions do not take the
## mean into account.
Xmeans <- rep_len(0, npred)
Ymeans <- rep_len(0, nresp)
}
## Must be done here due to the deflation of X
if (!stripped) Xtotvar <- sum(X * X)
for (a in 1:ncomp) {
## Initial values:
if (nresp == 1) { # pls1
u.a <- Y # FIXME: scale?
} else { # pls2
## The coloumn of Y with largest sum of squares:
u.a <- Y[,which.max(colSums(Y * Y))]
t.a.old <- 0
}
nit <- 0
repeat {
nit <- nit + 1 # count the iterations
## C2.1
w.a <- crossprod(X, u.a)
w.a <- w.a / sqrt(c(crossprod(w.a)))
## C2.2
t.a <- X %*% w.a
## C2.3
tsq <- c(crossprod(t.a))
t.tt <- t.a / tsq
## C2.4
q.a <- crossprod(Y, t.tt)
if (nresp == 1)
break # pls1: no iteration
## C2.4b-c
## Convergence check for pls2:
if (sum(abs((t.a - t.a.old) / t.a), na.rm = TRUE) < tol)
break
else {
u.a <- Y %*% q.a / c(crossprod(q.a))
t.a.old <- t.a # Save for comparison
}
if (nit >= maxit) {
warning("No convergence in ", maxit, " iterations\n")
break
}
}
## C2.3 contd.
p.a <- crossprod(X, t.tt)
## C2.5
X <- X - t.a %*% t(p.a)
Y <- Y - t.a %*% t(q.a)
## Save scores etc:
W[,a] <- w.a
P[,a] <- p.a
tQ[a,] <- q.a
if (!stripped) {
TT[,a] <- t.a
U[,a] <- u.a
tsqs[a] <- tsq
## (For very tall, slim X and Y, X0 %*% B[,,a] is slightly faster,
## due to less overhead.)
fitted[,,a] <- TT[,1:a] %*% tQ[1:a,, drop=FALSE]
residuals[,,a] <- Y
}
}
## Calculate rotation matrix:
if (ncomp == 1) {
## For 1 component, R == W:
R <- W
} else {
PW <- crossprod(P, W)
## It is known that P^tW is right bi-diagonal (one response) or upper
## triangular (multiple responses), with all diagonal elements equal to 1.
if (nresp == 1) {
## For single-response models, direct calculation of (P^tW)^-1 is
## simple, and faster than using backsolve.
PWinv <- diag(ncomp)
bidiag <- - PW[row(PW) == col(PW)-1]
for (a in 1:(ncomp - 1))
PWinv[a,(a+1):ncomp] <- cumprod(bidiag[a:(ncomp-1)])
} else {
PWinv <- backsolve(PW, diag(ncomp))
}
R <- W %*% PWinv
}
## Calculate regression coefficients:
for (a in 1:ncomp) {
B[,,a] <- R[,1:a, drop=FALSE] %*% tQ[1:a,, drop=FALSE]
}
if (stripped) {
## Return as quickly as possible
list(coefficients = B, Xmeans = Xmeans, Ymeans = Ymeans)
} else {
fitted <- fitted + rep(Ymeans, each = nobj) # Add mean
## Add dimnames and classes:
objnames <- dnX[[1]]
if (is.null(objnames)) objnames <- dnY[[1]]
prednames <- dnX[[2]]
respnames <- dnY[[2]]
compnames <- paste("Comp", 1:ncomp)
nCompnames <- paste(1:ncomp, "comps")
dimnames(TT) <- dimnames(U) <- list(objnames, compnames)
dimnames(R) <- dimnames(W) <- dimnames(P) <-
list(prednames, compnames)
dimnames(tQ) <- list(compnames, respnames)
dimnames(B) <- list(prednames, respnames, nCompnames)
dimnames(fitted) <- dimnames(residuals) <-
list(objnames, respnames, nCompnames)
class(TT) <- class(U) <- "scores"
class(P) <- class(W) <- class(tQ) <- "loadings"
list(coefficients = B,
scores = TT, loadings = P,
loading.weights = W,
Yscores = U, Yloadings = t(tQ),
projection = R,
Xmeans = Xmeans, Ymeans = Ymeans,
fitted.values = fitted, residuals = residuals,
Xvar = colSums(P * P) * tsqs,
Xtotvar = Xtotvar)
}
}
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