do a nonlinear least square fit in r - r

Question

I have two vectors:
y <- c(0.044924, 0.00564, 0.003848, 0.002385, 0.001448, 0.001138,
0.001025, 0.000983, 0.00079, 0.000765, 0.000721, 0.00061, 0.000606,
0.000699, 0.000883, 0.001069, 0.001226, 0.001433, 0.00162, 0.001685,
0.001604, 0.001674, 0.001706, 0.001683, 0.001505, 0.001497, 0.001416,
0.001449, 0.001494, 0.001544, 0.00142, 0.001458, 0.001544, 0.001279,
0.00159, 0.001756, 0.001749, 0.001909, 0.001885, 0.002063, 0.002265,
0.002137, 0.002391, 0.002619, 0.002733, 0.002957, 0.003244, 0.003407,
0.003563, 0.003889, 0.004312, 0.004459, 0.004946, 0.005248, 0.005302,
0.00574, 0.006141, 0.006977, 0.007386, 0.007843, 0.008473, 0.008949,
0.010164, 0.010625, 0.011279, 0.01191, 0.012762, 0.014539, 0.01477)
x <- 0:68
I am trying to use the non-linear least squares function to fit the data but I keep getting the error:
Error in nlsModel(formula, mf, start, wts) : singular gradient matrix at initial parameter estimates
My code is:
a=0.00012
b=0.08436
k=0.21108
fit = nls(y ~ (a*b*exp(b*x)*k)/((k*b)+(a*(exp(b*x)-1))), start=list(a=a,b=b,k=k))
The parameters I have entered are parameters that I know are close to the expected values. Does anyone know what am I doing wrong here?
I have tried various initial values for the parameters a, b and k, but I always get some kind of error.

Answer 1

Use optim() instead. You have to make a function which takes a,b and k as input (collected as a vector), and which returns the squared error as a result:
func <-function(pars) {
a <- pars["a"]
b <- pars["b"]
k <- pars["k"]
fitted <- (a*b*exp(b*x)*k)/((k*b)+(a*(exp(b*x)-1)))
sum((y-fitted)^2)
}
Then we run optim() using the initial values:
result <- optim(c(a=0.00012, b=0.08436, k=0.21108), func)
To test the resulting fit:
plot(x, y)
a <- result$par["a"]
b <- result$par["b"]
k <- result$par["k"]
lines((a*b*exp(b*x)*k)/((k*b)+(a*(exp(b*x)-1))), col = "blue")