SSasympOrig Self-Starting Nls Asymptotic Regression Model through the Origin

Description

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

Usage

SSasympOrig(input, Asym, lrc)

Arguments

input	a numeric vector of values at which to evaluate the model.
Asym	a numeric parameter representing the horizontal asymptote.
lrc	a numeric parameter representing the natural logarithm of the rate constant.

Value

a numeric vector of the same length as input. It is the value of the expression Asym*(1 - exp(-exp(lrc)*input)). If all of the arguments Asym 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$Seed == "329", ]
SSasympOrig(Lob.329$age, 100, -3.2)  # response only
local({   Asym <- 100; lrc <- -3.2
  SSasympOrig(Lob.329$age, Asym, lrc) # response and gradient
})
getInitial(height ~ SSasympOrig(age, Asym, lrc), data = Lob.329)
## Initial values are in fact the converged values
fm1 <- nls(height ~ SSasympOrig(age, Asym, lrc), data = Lob.329)
summary(fm1)


## Visualize the SSasympOrig()  model  parametrization :

  xx <- seq(0, 5, length.out = 101)
  yy <- 5 * (1- exp(-xx * log(2)))
  stopifnot( all.equal(yy, SSasympOrig(xx, Asym = 5, lrc = log(log(2)))) )

  require(graphics)
  op <- par(mar = c(0, 0, 3.5, 0))
  plot(xx, yy, type = "l", axes = FALSE, ylim = c(0,5), xlim = c(-1/4, 5),
       xlab = "", ylab = "", lwd = 2,
       main = quote("Parameters in the SSasympOrig model"~~ f[phi](x)))
  mtext(quote(list(phi[1] == "Asym", phi[2] == "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)
  axis(2, at = 5*c(1/2,1), labels= expression(frac(phi[1],2), phi[1]), pos=0, las=1)
  arrows(c(.3,.7), 5/2,
         c(0, 1 ), 5/2, length = 0.08, angle = 25)
  text(   0.5,     5/2, quote(t[0.5]))
  text(   1 +.4,   5/2,
       quote({f(t[0.5]) == frac(phi[1],2)}~{} %=>% {}~~{t[0.5] == frac(log(2), e^{phi[2]})}),
       adj = c(0, 0.5))
  par(op)