Is a Relationship Quadratic?

Question

I have a relationship $y=f(x)$ for which I can obtain data through simulation.

I have good reason to suspect that this relationship is quadratic (rather than, say, exponential), and would like to provide evidence for this.

I was thinking of the following method, and I would like to ask if there is anything wrong with it:

a) Obtain $y$ data for a limited interval of $x$, say $x\in (0, x_a)$.

b) Fit a quadratic function to this data using least squares regression.

c) Obtain data for a larger value of $x$ (say $10x_a$) and check the percentage error that the model leads to when extrapolated to this value.

Does this prove anything?

Answer 1

A better method is to plot the points $(\log x,\log y)$ for all data points $(x,y)$. If the relationship is roughly quadratic, $y\approx Cx^2$, then $\log y\approx 2\log x+\log C$, which is a linear equation in $\log x$ and $\log y$. Straight lines are easier to detect in a point plot, and if you get a straight line with slope $2$, then you can be pretty sure your relationship is roughly quadratic.

Answer 2

This is a problem of model comparison. A standard approach would be to fit the two competing models with same data and compare their performance on a test set.

You could do least squares fit, or if you have better idea about the noise distribution, you can use maximum likelihood estimation.