# Solved – Confidence limits for a nonlinear regression

confidence intervalleast squares

I've fitted some weighted data to a nonlinear model (namely a Sèrsic light profile), using least squares.

That is, I have fitted a weighted nonlinear least squares model of the form:

$$I_r = I_0 \exp\left[-\left(\frac{r}{\alpha}\right)^{1/n}\right]+\epsilon_r$$

where $I_0$, $α$ and $n$ are parameters.

How do I then find confidence limits for each parameter, i.e. the 1,2,3 sigmas?

You can always fallback on bootstrapped confidence intervals.

# Bootstrap resampling:

1. Let $$X$$ denote your training dataset. Let $$n$$ denote the number of samples in your training data. Let $$k$$ denote the number of resampling iterations you want to perform. The more the better, but $$k$$ should probably be no fewer than $$1000$$.

2. for $$i=1,2,\dots k$$, take a random sample $$\tilde{X}_i$$ (with replacement) of size $$n$$ from $$X$$. Train your model and calculate your model paramters. Let $$\tilde{\theta}_i$$ denote your fitted parameters trained on the $$i^{th}$$ resampled data set.

3. You can now calculate confidence intervals by determining the quantiles of $$\tilde{\theta} = [\tilde{\theta}_1, \tilde{\theta}_2,\dots \tilde{\theta}_k]$$. For example, to obtain a $$95\%$$ confidence interval, calculate the $$2.5\%$$ and $$97.5\%$$ quantiles of $$\tilde{\theta}$$.