fitting function for a given data set - r

Question

I'm trying to fitting the following function y(x)=a*( 1 + (x^2)/(b^2) )^t to a particular set of data , where, a, b and t are constants that want to determine by fitting.
I try the following, for example
len <- 24
x = runif(len)
y = x^3 + runif(len, min = -0.1, max = 0.1)
plot(x, y)
s <- seq(from = 0, to = 1, length = 50)
lines(s, s^3, lty = 2)
df <- data.frame(x, y)
m <- nls(y~a*( 1 + (x^2)/(b^2) )^t, data = df, start = list(a=1,t=0, b=1), trace = T)
> Error in nlsModel(formula, mf, start, wts) :
singular gradient matrix at initial parameter estimates
Can someone help me to set this function to these points, even if the fitting becomes bad, the important is to get fit this function, ie that she run on the data
thanks everyone

Answer 1

Because your data are changing randomly, for some situations the value of a is close to zero and your function becomes zero. The curve fit procedure fails at that point. Randomizing the start parameters might work for some situations.
A slightly more stable output can be computed using the LM algorithm:
require("minpack.lm")
LMCurveFit <- function(df) {
# The function to be fit
FitFunction <- function(params, x) {
with (
as.list(params), {
a*(1 + x^2/b^2)^t
}
)
}
# Residual
Residual <- function(params, x, y) {
FitFunction(params, x) - y
}
# Sum of squares of residuals
ssqfun <- function(params, x, y) {
sum(Residual(params, x, y)^2)
}
# Normalize the data
x_max = max(x)
y_max = max(y)
df$x = df$x/x_max
df$y = df$y/y_max
# Define start parameters
a_start = 0.1
b_start = 1.0
t_start = 1.0
param_start = c(a = a_start,
b = b_start,
t = t_start)
# Do LM fit
nls.out <- nls.lm(par = param_start,
fn = Residual,
control = nls.lm.control(nprint=0,
ftol=.Machine$double.eps,
ptol=.Machine$double.eps,
maxfev=10000, maxiter=1024),
x = df$x,
y = df$y)
# Revert scaling
nls.out$par[1] = nls.out$par[1]*y_max
nls.out$par[2] = nls.out$par[2]*x_max
# Get the parameters
params_fit = coef(nls.out)
print(params_fit)
# Compute predicted values
predicted = FitFunction(as.list(params_fit), df$x*x_max)
}
# LM fit
pred_y = LMCurveFit(df)
lines(x, pred_y)