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I am reading a scientific paper that fits a numerical model to a timeseries (observations). The numerical model has a few free parameters, some of them linear and some are not, which the authors wish to fit to the observations.

The numerical model produces a timeseries for each set of input parameters. The authors use regression (I think, they don't explicitly say so) to produce a $\chi^2$ value for each model run. Then, they report the parameters that minimize the $\chi^2$ value as those that best fit the observed timeseries.

So, just as an example, if the observations timeseries are represented by $O(t)$, then the model produces $i={1..N}$ timeseries $M_i(t,a_i,b)$, where $a_i$ is a linear model parameter that changes for each timeseries and $b$ is a nonlinear parameter which the authors choose to hold constant and do not attempt to fit, but is a free parameter (so in theory it may be changed). The parameter $a_i$ producing model timeseries $M_i$ that has the lowest $\chi^2$ is selected as best fit parameter.

This confuses me, I admit. Here are my questions:

  1. As I mentioned, the authors do not explicitly say which method they used, but do report $\chi^2$ values. Is there any possibility they did not use regression? Is $\chi^2$ minimization used in any other goodness of fit method?
  2. If regression is the only option, can the authors even do that? As I mentioned, some of the parameters are nonlinear with the function. I mean, the model is linear with the one fit parameter $a$, but the for the other parameter $b$ (which they hold constant and do not attempt to fit) it is nonlinear. I am only mentioning it, since for the best of my knowledge you cannot trust $\chi^2$ for nonlinear regression since the number of degrees of freedom is unknown. I am not sure if this holds in this case, as they treat $b$ as a constant.
  3. If this approach does indeed make sense: the authors compare the $\chi^2$ of a few different models to say which model best fits the data (that is, after they found the best fit parameter $a_i$ for each timeseries). How do you estimate the errors in this case? Using cross validation? (they did not produce any errors with their $\chi^2$s).

I would appreciate any help.

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