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\name{sumouter}
\alias{sumouter}
\alias{quadform}
\alias{bilinearform}
\title{Compute Quadratic Forms}
\description{
Calculates certain quadratic forms of matrices.
}
\usage{
sumouter(x, w=NULL, y=x)
quadform(x, v)
bilinearform(x, v, y)
}
\arguments{
\item{x,y}{A matrix, whose rows are the vectors in the quadratic form.}
\item{w}{Optional vector of weights}
\item{v}{Matrix determining the quadratic form}
}
\value{
A vector or matrix.
}
\details{
The matrices \code{x} and \code{y} will be interpreted as
collections of row vectors. They must have the same number of rows.
The entries of \code{x} and \code{y} may be
numeric, integer, logical or complex values.
The command \code{sumouter} computes the sum of the outer
products of corresponding row vectors, weighted by the entries of \code{w}:
\deqn{
M = \sum_i w_i x_i^\top y_i
}{
M = sum[i] (w[i] * outer(x[i,], y[i,]))
}
where \eqn{x_i}{x[i,]} is the \code{i}-th row of \code{x}
and \eqn{y_i}{y[i,]} is the \code{i}-th row of \code{y}
(after removing any rows containing \code{NA} or other non-finite
values).
If \code{w} is missing or \code{NULL}, the weights will be taken as 1.
The result is a \eqn{p \times q}{p * q} matrix where
\code{p = ncol(x)} and \code{q = ncol(y)}.
The command \code{quadform} evaluates the quadratic form, defined by
the matrix \code{v}, for each of the row vectors of \code{x}:
\deqn{
y_i = x_i V x_i^\top
}{
y[i] = x[i,] \%*\% v \%*\% t(x[i,])
}
The result \code{y} is a numeric vector of length \code{n} where
\code{n = nrow(x)}. If \code{x[i,]} contains \code{NA} or
other non-finite values, then \code{y[i] = NA}.
If \code{v} is missing or \code{NULL},
it will be taken as the identity matrix, so
that the resulting values will be
\deqn{
y_i = x_i x_i^\top
}{
y[i] = x[i,] \%*\% t(x[i,])
}
The command \code{bilinearform} evaluates the more general bilinear
form defined by the matrix \code{v}. Here \code{x} and \code{y} must
be matrices of the same dimensions. For each row vector of
\code{x} and corresponding row vector of \code{y}, the bilinear form is
\deqn{
z_i = x_i V y_i^\top
}{
z[i] = x[i,] \%*\% v \%*\% t(y[i,])
}
The result \code{z} is a numeric vector of length \code{n} where
\code{n = nrow(x)}. If \code{x[i,]} or \code{y[i,]} contains \code{NA} or
other non-finite values, then \code{z[i] = NA}.
If \code{v} is missing or \code{NULL},
it will be taken as the identity matrix, so
that the resulting values will be
\deqn{
z_i = x_i y_i^\top
}{
z[i] = x[i,] \%*\% t(y[i,])
}
}
\examples{
x <- matrix(1:12, 4, 3)
dimnames(x) <- list(c("Wilma", "Fred", "Barney", "Betty"), letters[1:3])
x
sumouter(x)
w <- 4:1
sumouter(x, w)
v <- matrix(1, 3, 3)
quadform(x, v)
# should be the same as quadform(x, v)
bilinearform(x, v, x)
# See what happens with NA's
x[3,2] <- NA
sumouter(x, w)
quadform(x, v)
}
\author{
\adrian
and \rolf
}
\keyword{array}
|
