Non-linear fit in R

I have a set of data:

x<- c(0.00100000, 0.08977778, 0.17855556, 0.26733333, 0.35611111, 0.44488889, 0.53366667, 0.62244444, 0.71122222)
y <- c(0.094982692, 0.074529874, 0.051782003, 0.026081288, -0.003576486, -0.038845028, -0.082764061, -0.141998524, -0.141998524)

and I wanted to fit a logarithmic curve to the data (y = c*naturalLog(x + k) + d). I tried nls in R ( which was as useful as tits on a fish). I also tried to fit exponential, with no sucess (fit was very bad). My question is, can you see a logarithmic or exponential correlation in the data? My end goal is to find the correlation coefficients, so I can transform y and make the correlation linear.

Any help would be appreciated.

This Weibull-like curve fits in only 5 iterations:

fm <- nls(y ~ a + (1-a) * (1 - exp(b * x^c)), 
          data.frame(x, y), 
          start = list(a = 0.5, b = 1, c = 1))
fm

giving:

Nonlinear regression model
  model: y ~ a + (1 - a) * (1 - exp(b * x^c))
   data: data.frame(x, y)
      a       b       c 
0.09653 0.37163 1.17432 
 residual sum-of-squares: 0.0009496

Number of iterations to convergence: 5 
Achieved convergence tolerance: 3.901e-06

The fit looks good:

plot(y ~ x)
lines(fitted(fm) ~ x, col = "red")

You can use following code

library(minpack.lm)

x <- c(0.00100000, 0.08977778, 0.17855556, 0.26733333, 0.35611111, 0.44488889, 0.53366667, 0.62244444, 0.71122222)
y <- c(0.094982692, 0.074529874, 0.051782003, 0.026081288, -0.003576486, -0.038845028, -0.082764061, -0.141998524, -0.141998524)

#Create a data frame
df <- data.frame(x, y)

#Write the equation
nls.eq <- as.formula(y ~ (c*log(x + k) + d))

#Fitting equation using minpack.lm package
nls.out <- nlsLM(nls.eq, 
                 data = df,
                 start=list(c=1, k=1, d=1),
                 control  = nls.lm.control(maxiter = 500),
                 algorithm = "LM")

You've said that you used this code:

m <- nls(1-x ~ a * log(y + b) , data = df, start = list(a = 1, b = 1), nls.control(maxiter = 500)) 

> Instead of x (death rate), I used 1-x (birth rate).

Normally we use y as the response, not x. Also be careful that birth_rate = 1 - death_rate is not generally true, rather survival rate is 1 minus death rate.

An alternative method to fit the logarithmic function :

The general principle of the method used is explained in : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales The fitting isn't wrt the equation model but wrt a linear integral equation to which the equation model is solution. In the present case the linear integral equation is : . This avoid iterative calculus and no need for "guessed" initial values of the parameters.