R: Self-Starting Nls Asymptotic Regression Model

SSasymp {stats} | R Documentation

## 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

`input` | a numeric vector of values at which to evaluate the model.
`Asym` | a numeric parameter representing the horizontal asymptote on the right side (very large values of `input`).
`R0` | a numeric parameter representing the response when `input` is zero.
`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+(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)

[Package stats version 3.5.1 Index]