# fitting exponential versus exponential w/ power

I have two models which I would like to investigate for my data. One form is: $$\label{one} f(r) = A e^{-B r}$$ and the second is: $$\label{two} g(r) = \frac{A}{r^{C}} e^{-B r}$$

Does anyone have any advice on how to tell the difference between these two? I typically have about ~ seven points that I can fit in all my situations. The problem is these seven points don't span a particularly large range of $r$ and so it's hard to see the difference between $g$ and $f$. The least-squares fits in the case of $g$ are more unstable, but I assume it's because it's very similar to $f$ and the parameters have more freedom...

Any advise or wisdom would be welcome, and I can clarify if there are any questions.

• Are you computing any figure of merit for the fits? Mar 20 '18 at 19:22
• @nicoguaro I have been computing the average error: $$\frac{1}{N} \sum_{i = 1}^{N} \left( \frac{|y_{i} - f(x_{i})|}{y_{i}} \right)^{2}$$ Mar 20 '18 at 19:27
• And which one is "better"? Also, I think that both models can be rewritten as linear regression problems. Mar 20 '18 at 19:32
• What is the range of $r$? And what values of $C$ are you obtaining? Mar 20 '18 at 19:36
• Do you care about the model? What I mean is: do the parameters $A$, $B$ and $C$ have physical meaning and are you trying to derive values for them using the fit? Or are you just trying to fit a function in order to interpolate/extrapolate your data? In the latter case, I would stick to the 'simplest' model, the one with as little parameters as possible. Certainly if you only have seven data points to do your fit... Mar 22 '18 at 14:15