I don't understand why I can't have a nls function for these data. I have tried with a lot of different start values and I have always the same error.
Here is what I have been doing:
expFct2 = function (x, a, b,c)
{
a*(1-exp(-x/b)) + c
}
vec_x <- c(77.87,87.76,68.6,66.29)
vec_y <- c(1,1,0.8,0.6)
dt <- data.frame(vec_x=vec_x,vec_y=vec_y)
ggplot(data = dt,aes(x = vec_x, y = vec_y)) + geom_point() +
geom_smooth(data=dt, method="nls", formula=y~expFct2(x, a, b, c),
se=F, start=list(a=1, b=75, c=-5)
I have always this error:
Error in method(formula, data = data, weights = weight, ...) :
singular gradient
This can be written with two linear parameters (.lin1 and .lin2) and one nonlinear parameter (b) like this:
a*(1-exp(-x/b)) + c
= (a+c) - a * exp(-x/b)
= .lin1 + .lin2 * exp(-x/b)
where .lin1 = a+c and .lin2 = -a (so a = - .lin2 and c = .lin1 + .lin2) This lets us use "plinear" which only requires specification of a starting value for the single nonlinear parameter (eliminating the problem of how to set the starting values for the other parameters) and which converges despite the starting value of b=75 being far from that of the solution:
nls(y ~ cbind(1, exp(-x/b)), start = list(b = 75), alg = "plinear")
Here is the result of a run from which we can see from the size of .lin2 that the problem is badly scaled:
> x <- c(77.87,87.76,68.6,66.29)
> y <- c(1,1,0.8,0.6)
> nls(y ~ cbind(1, exp(-x/b)), start = list(b = 75), alg = "plinear")
Nonlinear regression model
model: y ~ cbind(1, exp(-x/b))
data: parent.frame()
b .lin1 .lin2
3.351e+00 1.006e+00 -1.589e+08
residual sum-of-squares: 7.909e-05
Number of iterations to convergence: 9
Achieved convergence tolerance: 9.887e-07
> R.version.string
[1] "R version 2.14.2 Patched (2012-02-29 r58660)"
> win.version()
[1] "Windows Vista (build 6002) Service Pack 2"
EDIT: added sample run and comment on scaling.