Self-Starting Nls Asymptotic Regression Model

Description

This selfStart model evaluates the asymptotic regression function and its gradient. It has an initial attribute that will evaluate initial estimates of the parameters Asym, R0, and lrc for a given set of data.

Note that SSweibull() generalizes this asymptotic model with an extra parameter.

Usage

SSasymp(input, Asym, R0, lrc)

Arguments

Value

a numeric vector of the same length as input. It is the value of the expression Asym+(R0-Asym)*exp(-exp(lrc)*input). If all of the arguments Asym, R0, and lrc are names of objects, the gradient matrix with respect to these names is attached as an attribute named gradient.

Author(s)

José Pinheiro and Douglas Bates

See Also

nls, selfStart

Examples

Lob.329 <- Loblolly[ Loblolly$Seed == "329", ]
SSasymp( Lob.329$age, 100, -8.5, -3.2 )   # response only
local({
  Asym <- 100 ; resp0 <- -8.5 ; lrc <- -3.2
  SSasymp( Lob.329$age, Asym, resp0, lrc) # response _and_ gradient
})
getInitial(height ~ SSasymp( age, Asym, resp0, lrc), data = Lob.329)
## Initial values are in fact the converged values
fm1 <- nls(height ~ SSasymp( age, Asym, resp0, lrc), data = Lob.329)
summary(fm1)

## Visualize the SSasymp()  model  parametrization :

  xx <- seq(-.3, 5, len = 101)
  ##  Asym + (R0-Asym) * exp(-exp(lrc)* x) :
  yy <- 5 - 4 * exp(-xx / exp(3/4))
  stopifnot( all.equal(yy, SSasymp(xx, Asym = 5, R0 = 1, lrc = -3/4)) )
  require(graphics)
  op <- par(mar = c(0, .2, 4.1, 0))
  plot(xx, yy, type = "l", axes = FALSE, ylim = c(0,5.2), xlim = c(-.3, 5),
       xlab = "", ylab = "", lwd = 2,
       main = quote("Parameters in the SSasymp model " ~
                    {f[phi](x) == phi[1] + (phi[2]-phi[1])*~e^{-e^{phi[3]}*~x}}))
  mtext(quote(list(phi[1] == "Asym", phi[2] == "R0", phi[3] == "lrc")))
  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(c(0.35, 0.65), 1,
         c(0  ,  1   ), 1, length = 0.08, angle = 25); text(0.5, 1, quote(1))
  y0 <- 1 + 4*exp(-3/4) ; t.5 <- log(2) / exp(-3/4) ; AR2 <- 3 # (Asym + R0)/2
  segments(c(1, 1), c( 1, y0),
           c(1, 0), c(y0,  1),  lty = 2, lwd = 0.75)
  text(1.1, 1/2+y0/2, quote((phi[1]-phi[2])*e^phi[3]), adj = c(0,.5))
  axis(2, at = c(1,AR2,5), labels= expression(phi[2], frac(phi[1]+phi[2],2), phi[1]),
       pos=0, las=1)
  arrows(c(.6,t.5-.6), AR2,
         c(0, t.5   ), AR2, length = 0.08, angle = 25)
  text(   t.5/2,   AR2, quote(t[0.5]))
  text(   t.5 +.4, AR2,
       quote({f(t[0.5]) == frac(phi[1]+phi[2],2)}~{} %=>% {}~~
                {t[0.5] == frac(log(2), e^{phi[3]})}), adj = c(0, 0.5))
  par(op)