Parameter uncertainty in curve fitting

Question

My real problem has a much more complexity and a different function than following. However, for the sake of simplicity assume I have a data that can be described as a one dimensional Gaussian function:

$$y = Ae\left(\frac{-(x-\mu)^2}{\sigma}\right)$$

The $x$ variable which is the input to this Gaussian function is not an independent variable, rather $x$ values could be time for example. The output of the function is only the intensity of a signal.

After I fit my Gaussian function to my data I will have three fitted parameters $A$, $\mu$, and $\sigma$.

If the problem was regression, that is having an independent variable $x$ and the dependent variable $y$, then I would use the covariance matrix.

However, in this case of curve fitting, $x$ is not an independent variable.

How can I estimate the uncertainty in the model fits?

Some references to learn about uncertainty in curve fitting is also appreciated.

Answer 1

OP I think your question about optimisation is better suited to the operations research site.

curve fitting does not require random variables.

1. you define an objective function
2. you find a (local?) minimum (where the gradient (of objective function wrt parameters) is zero
3. the "uncertainty" in the parameter estimates is related to the second derivative: the greater the curvature, the more sensitive the objective function is to the parameter and the more "certainty" you have

https://en.wikipedia.org/wiki/Mathematical_optimization

and see eg https://en.m.wikipedia.org/wiki/Non-linear_least_squares, for least squares objective function