I have $23$ data points at evenly spaced $x$ intervals with the $y$ values falling between $500$ and $18000.$ Are there rules of thumb regarding what could be the best curve to fit the data to?
If I fit it with a cubic polynomial, I get $R^2=.99$. I have not been able to get better than $R^2=.98$ with an exponential curve, $f(x)=Ae^{px}$.
On a small enough interval there wouldn't be much difference between a cubic polynomial and an exponential curve just as $e^x \approx 1+x+\frac{x^2}{2}+\frac{x^3}{6}$ on a small enough interval where "small enough" can be determined by the Taylor Remainder.
Is there a way to tell whether I have enough data to differentiate between whether I'm dealing with an exponential or a polynomial curve? I'm thinking spanning a few orders of magnitude might tell. Is there too little information?
I have good reason to believe the actual curve is a sigmoid of some kind.