% File src/library/stats/man/SSasympOff.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2017 R Core Team
% Distributed under GPL 2 or later

\name{SSasympOff}
\title{Self-Starting \code{nls} Asymptotic 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{\enc{José}{Jose} Pinheiro and Douglas Bates}
\seealso{\code{\link{nls}}, \code{\link{selfStart}};
  \code{example(SSasympOff)} gives graph showing the \code{SSasympOff}
  parametrization.
}
\examples{
CO2.Qn1 <- CO2[CO2$Plant == "Qn1", ]
SSasympOff(CO2.Qn1$conc, 32, -4, 43)  # response only
local({  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)

## Visualize the SSasympOff()  model  parametrization :

  xx <- seq(0.25, 8,  by=1/16)
  yy <- 5 * (1 -  exp(-(xx - 3/4)*0.4))
  stopifnot( all.equal(yy, SSasympOff(xx, Asym = 5, lrc = log(0.4), c0 = 3/4)) )
  require(graphics)
  op <- par(mar = c(0, 0, 4.0, 0))
  plot(xx, yy, type = "l", axes = FALSE, ylim = c(-.5,6), xlim = c(-1, 8),
       xlab = "", ylab = "", lwd = 2,
       main = "Parameters in the SSasympOff model")
  mtext(quote(list(phi[1] == "Asym", phi[2] == "lrc", phi[3] == "c0")))
  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 = 3)
  arrows(-0.8, c(2.1, 2.9),
         -0.8, c(0  , 5  ), length = 0.1, angle = 25)
  text  (-0.8, 2.5, quote(phi[1]))
  segments(3/4, -.2, 3/4, 1.6, lty = 2)
  text    (3/4,    c(-.3, 1.7), quote(phi[3]))
  arrows(c(1.1, 1.4), -.15,
         c(3/4, 7/4), -.15, length = 0.07, angle = 25)
  text    (3/4 + 1/2, -.15, quote(1))
  segments(c(3/4, 7/4, 7/4), c(0, 0, 2),   # 5 * exp(log(0.4)) = 2
           c(7/4, 7/4, 3/4), c(0, 2, 0),  lty = 2, lwd = 2)
  text(      7/4 +.1, 2./2, quote(phi[1]*e^phi[2]), adj = c(0, .5))
  par(op)
}
\keyword{models}
