The Method of Finite Difference Regression

Abstract

In this paper I present a novel polynomial regression method called Finite Difference Regression for a uniformly sampled sequence of noisy data points that determines the order of the best fitting polynomial and provides estimates of its coefficients. Unlike classical least-squares polynomial regression methods in the case where the order of the best fitting polynomial is unknown and must be determined from the R 2 value of the fit, I show how the t -test from statistics can be combined with the method of finite differences to yield a more sensitive and objective measu re of the order of the best fitting polynomial. Furthermore, it is shown how these finite differences used in the determination of the order, can be reemployed to produce excellent estimates of the coefficients of the best fitting polynomial. I show that not only are these coefficients unbiased and consistent, but also that the asymptotic properties of the fit get better with increasing degrees of the fitting polynomial.