What can be measured on a curve?

Question

I am doing a maths problem in which I must mathematically compare models when the assumption that polynomial was the ideal fit is changed. I then used a cosine function to fit it. What can be measured on a curve to compare and contrast them? I've had it suggested to find the second derivative but what will this show me? Will it compare how tight the curves are?

Answer 1

Take the integral of the square of the difference between the actual curve and your model, across an appropriate interval: $$\int_a^b \left(f(x)-m(x) \right)^2 dx$$

This is the limit of "accumulated errors" of your model $m(x)$ compared to the actual function $f(x)$, weighted by a squares approach, across the interval you are interested in.

A model with close fit in most of the interval but large deviance in some of it will be weighted unfavourably compared to a model with consistent "okay" deviance. If you really want to hammer down a large deviance, you can use higher powers, but this will require a modulus correction, i.e. the absolute value of the difference (a problem which is negated when using even integer exponents).

A similar approach is used in the least squares linear approximation method, where the discrete equivalent of the above expression is minimised, and the resulting curve used as the model.