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\name{ace}
\alias{ace}
\title{Alternating Conditional Expectations}
\usage{
ace(x, y, wt = rep(1, nrow(x)), cat = NULL, mon = NULL, lin = NULL,
circ = NULL, delrsq = 0.01)
}
\arguments{
\item{x}{a matrix containing the independent variables.}
\item{y}{a vector containing the response variable.}
\item{wt}{an optional vector of weights.}
\item{cat}{an optional integer vector specifying which variables
assume categorical values. Positive values in \code{cat} refer
to columns of the \code{x} matrix and zero to the response
variable. Variables must be numeric, so a character variable
should first be transformed with as.numeric() and then specified
as categorical.}
\item{mon}{an optional integer vector specifying which variables are
to be transformed by monotone transformations. Positive values
in \code{mon} refer to columns of the \code{x} matrix and zero
to the response variable.}
\item{lin}{an optional integer vector specifying which variables are
to be transformed by linear transformations. Positive values in
\code{lin} refer to columns of the \code{x} matrix and zero to
the response variable.}
\item{circ}{an integer vector specifying which variables assume
circular (periodic) values. Positive values in \code{circ}
refer to columns of the \code{x} matrix and zero to the response
variable.}
\item{delrsq}{termination threshold. Iteration stops when R-squared
changes by less than \code{delrsq} in 3 consecutive iterations
(default 0.01).}
}
\description{
Uses the alternating conditional expectations algorithm to find the
transformations of y and x that maximise the proportion of variation
in y explained by x.
}
\value{
A structure with the following components:
\item{x}{the input x matrix.}
\item{y}{the input y vector.}
\item{tx}{the transformed x values.}
\item{ty}{the transformed y values.}
\item{rsq}{the multiple R-squared value for the transformed values.}
\item{l}{the codes for cat, mon, ...}
\item{m}{not used in this version of ace}
}
\references{
Breiman and Friedman, Journal of the American Statistical
Association (September, 1985).
The R code is adapted from S code for avas() by Tibshirani, in the
Statlib S archive; the FORTRAN is a double-precision version of
FORTRAN code by Friedman and Spector in the Statlib general
archive.
}
\examples{
TWOPI <- 8*atan(1)
x <- runif(200,0,TWOPI)
y <- exp(sin(x)+rnorm(200)/2)
a <- ace(x,y)
par(mfrow=c(3,1))
plot(a$y,a$ty) # view the response transformation
plot(a$x,a$tx) # view the carrier transformation
plot(a$tx,a$ty) # examine the linearity of the fitted model
# From D. Wang and M. Murphy (2005), Identifying nonlinear relationships
# regression using the ACE algorithm. Journal of Applied Statistics,
# 32, 243-258.
X1 <- runif(100)*2-1
X2 <- runif(100)*2-1
X3 <- runif(100)*2-1
X4 <- runif(100)*2-1
# Original equation of Y:
Y <- log(4 + sin(3*X1) + abs(X2) + X3^2 + X4 + .1*rnorm(100))
# Transformed version so that Y, after transformation, is a
# linear function of transforms of the X variables:
# exp(Y) = 4 + sin(3*X1) + abs(X2) + X3^2 + X4
a1 <- ace(cbind(X1,X2,X3,X4),Y)
par(mfrow=c(2,1))
# For each variable, show its transform as a function of
# the original variable and the of the transform that created it,
# showing that the transform is recovered.
plot(X1,a1$tx[,1])
plot(sin(3*X1),a1$tx[,1])
plot(X2,a1$tx[,2])
plot(abs(X2),a1$tx[,2])
plot(X3,a1$tx[,3])
plot(X3^2,a1$tx[,3])
plot(X4,a1$tx[,4])
plot(X4,a1$tx[,4])
plot(Y,a1$ty)
plot(exp(Y),a1$ty)
}
\keyword{models}
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