# fitting exponential versus exponential w/ power

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

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? – nicoguaro Mar 20 '18 at 19:22
• @nicoguaro I have been computing the average error: $$\frac{1}{N} \sum_{i = 1}^{N} \left( \frac{|y_{i} - f(x_{i})|}{y_{i}} \right)^{2}$$ – kηives Mar 20 '18 at 19:27
• And which one is "better"? Also, I think that both models can be rewritten as linear regression problems. – nicoguaro Mar 20 '18 at 19:32
• What is the range of $r$? And what values of $C$ are you obtaining? – Charles 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... – GertVdE Mar 22 '18 at 14:15