% $Id: SSbiexp.Rd,v 1.2.4.1 2001/08/07 14:05:15 ripley Exp $
\name{SSbiexp}
\title{Biexponential model}
\usage{
SSbiexp(input, A1, lrc1, A2, lrc2)
}
\alias{SSbiexp}
\arguments{
 \item{input}{a numeric vector of values at which to evaluate the model.}
 \item{A1}{a numeric parameter representing the multiplier of the first
   exponential.}
 \item{lrc1}{a numeric parameter representing the natural logarithm of
   the rate constant of the first exponential.}
 \item{A2}{a numeric parameter representing the multiplier of the second
   exponential.}
 \item{lrc2}{a numeric parameter representing the natural logarithm of
   the rate constant of the second exponential.}
}
\description{
  This \code{selfStart} model evaluates the biexponential model function
  and its gradient.  It has an \code{initial} attribute that 
  creates initial estimates of the parameters \code{A1}, \code{lrc1},
  \code{A2}, and \code{lrc2}.
}
\value{
  a numeric vector of the same length as \code{input}.  It is the value of
  the expression
  \code{A1*exp(-exp(lrc1)*input)+A2*exp(-exp(lrc2)*input)}.
  If all of the arguments \code{A1}, \code{lrc1}, \code{A2}, and
  \code{lrc2} are names of objects, the gradient matrix with respect to
  these names is attached as an attribute named \code{gradient}.
}
\author{Jose Pinheiro and Douglas Bates}
\seealso{\code{\link{nls}}, \code{\link{selfStart}}
}
\examples{
data( Indometh )
Indo.1 <- Indometh[Indometh$Subject == 1, ]
SSbiexp( Indo.1$time, 3, 1, 0.6, -1.3 )  # response only
A1 <- 3; lrc1 <- 1; A2 <- 0.6; lrc2 <- -1.3
SSbiexp( Indo.1$time, A1, lrc1, A2, lrc2 ) # response and gradient
getInitial(conc ~ SSbiexp(time, A1, lrc1, A2, lrc2), data = Indo.1)
## Initial values are in fact the converged values
fm1 <- nls(conc ~ SSbiexp(time, A1, lrc1, A2, lrc2), data = Indo.1)
summary(fm1)
\testonly{
  xx <- seq(0, 5, len = 101)
  y1 <- 3.5 * exp(-4*xx)
  y2 <- 1.5 * exp(-xx)
  yy <- y1 + y2
  par(mar = c(0, 0, 3.5, 0))
  plot(xx, yy, type = "n", axes = FALSE, ylim = c(-0.2,6), xlim = c(0, 5),
       xlab = "", ylab = "", main = "Components of the SSbiexp model")
  usr <- par("usr")
  lines(xx, y1, lty = 2, lwd = 0.75)
  lines(xx, y2, lty = 3, lwd = 0.75)
  lines(xx, yy, lwd = 2)
  segments(1:5, 0, 1:5, -0.15)
  segments(0, 1:5, -0.06, 1:5)
  text(-0.1, 1:5, 1:5, adj = c(1, 0.25), cex = 0.7)
  text(1:5, -0.15, 1:5, adj = c(0.5, 1), cex = 0.7)
  arrows(usr[1], 0, usr[2], 0, length = 0.1, angle = 25)
  arrows(0, usr[3], 0, usr[4], length = 0.1, angle = 25)
  text(usr[2] - 0.2, 0.1, "x", adj = c(1, 0))
  text(-0.1, usr[4], "y", adj = c(1, 1))
}
}
\keyword{models}
