I would like to fit a model to data that has the form,
$$f(x)=e^{a+bx+cx^2...}$$
The data is Gaussian distributed, and I was going to use a nonlinear least squares method (probably something in scipy). Each $y(x)$ has and associated $\sigma$ so I would like to do a weighted nonlinear least-squares.
The standard method seems to be to take the logarithm of the data and fit a polynomial to it using linear least-squares. However, taking the logarithm of the data would do something to the uncertainties (I haven't even really looked into how the uncertainties would transform under logarithm). Additionally, maybe I want to play around with my model a bit at fit something more of the form
$$f(x)=(a+bx)e^{cx^2}$$
or whatever.
Thus I would prefer to keep everything as general as possible and just fit the model to the data directly.
Linear least-squares involves minimizing the $L^2$ norm of $\mathbf{b}-A\mathbf{x}$, and, if you have a poorly conditioned model ($A$), then you can get garbage results from numerical methods. I assume that matrix conditioning is also important in nonlinear least squares?
I have a range questions. Is there anything inherently poorly conditioned about exponential models in general? Are nonlinear least squares methods inherently worse than linear least squares methods? The linear least squares method minimizes the $\chi^2$ statistic which maximizes the likelihood if the data is Gaussian distributed. Does the nonlinear least squares method still give a maximum likelihood model fit if the data is Gaussian?