```
###### example 1
## values over which to simulate data
x <- seq(0,5,length=100)
## model based on a list of parameters
getPred <- function(parS, xx) parS$a * exp(xx * parS$b) + parS$c
## parameter values used to simulate data
pp <- list(a=9,b=-1, c=6)
## simulated data, with noise
simDNoisy <- getPred(pp,x) + rnorm(length(x),sd=.1)
## plot data
plot(x,simDNoisy, main="data")
## residual function
residFun <- function(p, observed, xx) observed - getPred(p,xx)
## starting values for parameters
parStart <- list(a=3,b=-.001, c=1)
## perform fit
nls.out <- nls.lm(par=parStart, fn = residFun, observed = simDNoisy,
xx = x, control = nls.lm.control(nprint=1))
## plot model evaluated at final parameter estimates
lines(x,getPred(as.list(coef(nls.out)), x), col=2, lwd=2)
## summary information on parameter estimates
summary(nls.out)
###### example 2
## function to simulate data
f <- function(TT, tau, N0, a, f0) {
expr <- expression(N0*exp(-TT/tau)*(1 + a*cos(f0*TT)))
eval(expr)
}
## helper function for an analytical gradient
j <- function(TT, tau, N0, a, f0) {
expr <- expression(N0*exp(-TT/tau)*(1 + a*cos(f0*TT)))
c(eval(D(expr, "tau")), eval(D(expr, "N0" )),
eval(D(expr, "a" )), eval(D(expr, "f0" )))
}
## values over which to simulate data
TT <- seq(0, 8, length=501)
## parameter values underlying simulated data
p <- c(tau = 2.2, N0 = 1000, a = 0.25, f0 = 8)
## get data
Ndet <- do.call("f", c(list(TT = TT), as.list(p)))
## with noise
N <- Ndet + rnorm(length(Ndet), mean=Ndet, sd=.01*max(Ndet))
## plot the data to fit
par(mfrow=c(2,1), mar = c(3,5,2,1))
plot(TT, N, bg = "black", cex = 0.5, main="data")
## define a residual function
fcn <- function(p, TT, N, fcall, jcall)
(N - do.call("fcall", c(list(TT = TT), as.list(p))))
## define analytical expression for the gradient
fcn.jac <- function(p, TT, N, fcall, jcall)
-do.call("jcall", c(list(TT = TT), as.list(p)))
## starting values
guess <- c(tau = 2.2, N0 = 1500, a = 0.25, f0 = 10)
## to use an analytical expression for the gradient found in fcn.jac
## uncomment jac = fcn.jac
out <- nls.lm(par = guess, fn = fcn, jac = fcn.jac,
fcall = f, jcall = j,
TT = TT, N = N, control = nls.lm.control(nprint=1))
## get the fitted values
N1 <- do.call("f", c(list(TT = TT), out$par))
## add a blue line representing the fitting values to the plot of data
lines(TT, N1, col="blue", lwd=2)
## add a plot of the log residual sum of squares as it is made to
## decrease each iteration; note that the RSS at the starting parameter
## values is also stored
plot(1:(out$niter+1), log(out$rsstrace), type="b",
main="log residual sum of squares vs. iteration number",
xlab="iteration", ylab="log residual sum of squares", pch=21,bg=2)
## get information regarding standard errors
summary(out)
```

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