# Is there any objective criterion to determine the adequate degree of polynomial approximations?

I carry out a polynomial function approximation to a load duration curve (monotone decreasing function). I can approximate the load duration curve by using polynomial degrees from 4 to 12. I need an objective criterion to select the degree of polynomial considering a tradeoff between accuracy and included number of parameters in the function. Statistically, one can use adjusted R2 or information criterion. But these statistical criteria do not work because the residuals are nonlinear (not suitable for adjusted R2) and also I use least squares (do not have maximum likelihood value). Is there any other criterion to choose the adequate degree of polynomial from 4th to 12th degree?

• The best criterion is not to use polynomials unless an underlying theory indicates they are appropriate (in which case the theory almost always tells you what degree to use), especially for monotonic regression. It's unclear what "nonlinear" residuals could be, but it seems to mean they contra-indicate using least squares to make the fit. If you could explain why you're trying to fit functions to your load duration, your readers might be able to make suggestions that help you improve your overall analysis. – whuber Jul 26 '15 at 21:49
• SInce what is adequate is inherently subjective (it depends on your situation/purpose and your preferences), I'd have to say "no, there's no objective criterion". At least not without specifying more things up front. – Glen_b Jul 27 '15 at 0:06
• @whuber According to Weierstrass Approximaton Theory, any continuous function can be approximated to any accuracy by a polynomial of high enough degree. I came up with the idea of using mean absolute percentage error, the deviation of area under the curve from the actual area (corresponds to that year's electricity consumption) and the error in approximating highest demand. The best result is using 11th degree polynomial; however compared to a 5th degree polynomial the gain is not that much with respect to the number of parameters used. But this is an arbitrary decision not objective. – Dirk Jul 27 '15 at 20:11
• @Glen_b Although I can have a basis to compare the accuracy of approximations, I dont have a measure to compare gained accuracy w.r.t number of parameters used in polynomials. – Dirk Jul 27 '15 at 20:18
• @Dirk That is fine in theory and useless in practice. The problem with polynomial approximations is that they can appear to fit data well but they change wildly in between the data points. At the very least, use a maximum error (the $L_\infty$ norm) rather than the mean so that you can get some control over those wild swings. Consider low-degree polynomial splines instead of polynomials. – whuber Jul 27 '15 at 20:21