% File src/library/stats/man/SSasympOff.Rd
% Part of the R package, http://www.R-project.org
% Copyright 1995-2007 R Core Development Team
% Distributed under GPL 2 or later

\name{SSasympOff}
\title{Asymptotic Regression Model with an Offset}
\usage{
SSasympOff(input, Asym, lrc, c0)
}
\alias{SSasympOff}
\arguments{
 \item{input}{a numeric vector of values at which to evaluate the model.}
 \item{Asym}{a numeric parameter representing the horizontal asymptote on
   the right side (very large values of \code{input}).}
 \item{lrc}{a numeric parameter representing the natural logarithm of
   the rate constant.}
 \item{c0}{a numeric parameter representing the \code{input} for which the
   response is zero.}
}
\description{
  This \code{selfStart} model evaluates an alternative parametrization
  of the asymptotic
  regression function and the gradient with respect to those parameters.
  It has an \code{initial}
  attribute that creates initial estimates of the parameters
  \code{Asym}, \code{lrc}, and \code{c0}.
}
\value{
  a numeric vector of the same length as \code{input}.  It is the value of
  the expression \code{Asym*(1 - exp(-exp(lrc)*(input - c0)))}.  If all of
  the arguments \code{Asym}, \code{lrc}, and \code{c0} 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}};
  \code{example(SSasympOff)} gives graph showing the \code{SSasympOff}
  parametrization, where \eqn{\phi_1} is \code{Asymp},
  \eqn{\phi_3} is \code{c0}, and \eqn{t_{0.5}} is % FIXME
}
\examples{
CO2.Qn1 <- CO2[CO2$Plant == "Qn1", ]
SSasympOff( CO2.Qn1$conc, 32, -4, 43 )  # response only
Asym <- 32; lrc <- -4; c0 <- 43
SSasympOff( CO2.Qn1$conc, Asym, lrc, c0 ) # response and gradient
getInitial(uptake ~ SSasympOff( conc, Asym, lrc, c0), data = CO2.Qn1)
## Initial values are in fact the converged values
fm1 <- nls(uptake ~ SSasympOff( conc, Asym, lrc, c0), data = CO2.Qn1)
summary(fm1)
\dontshow{
require(graphics)

  xx <- seq(0.5, 5, len = 101)
  yy <- 5 * (1 -  exp(-(xx - 0.5)/(2*log(2))))
  par(mar = c(0, 0, 4.0, 0))
  plot(xx, yy, type = "l", axes = FALSE, ylim = c(0,6), xlim = c(-1, 5),
       xlab = "", ylab = "", lwd = 2,
       main = "Parameters in the SSasympOff model")
  usr <- par("usr")
  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))
  abline(h = 5, lty = 2, lwd = 0)
  arrows(-0.8, 2.1, -0.8, 0, length = 0.1, angle = 25)
  arrows(-0.8, 2.9, -0.8, 5, length = 0.1, angle = 25)
  text(-0.8, 2.5, expression(phi[1]), adj = c(0.5, 0.5))
  segments(0.5, 0, 0.5, 3, lty = 2, lwd = 0.75)
  text(0.5, 3.1, expression(phi[3]), adj = c(0.5, 0))
  segments(1.5, 2.525, 1.5, 3, lty = 2, lwd = 0.75)
  arrows(0.7, 2.65, 0.5, 2.65, length = 0.08, angle = 25)
  arrows(1.3, 2.65, 1.5, 2.65, length = 0.08, angle = 25)
  text(1.0, 2.65, expression(t[0.5]), adj = c(0.5, 0.5))
}
}
\keyword{models}
