Polynomial Regression - Alternative Approaches

Alternative Approaches

Polynomial regression is one example of regression analysis using basis functions to model a functional relationship between two quantities. A drawback of polynomial bases is that the basis functions are "non-local", meaning that the fitted value of y at a given value x = x₀ depends strongly on data values with x far from x₀. In modern statistics, polynomial basis-functions are used along with new basis functions, such as splines, radial basis functions, and wavelets. These families of basis functions offer a more parsimonious fit for many types of data.

The goal of polynomial regression is to model a non-linear relationship between the independent and dependent variables (technically, between the independent variable and the conditional mean of the dependent variable). This is similar to the goal of nonparametric regression, which aims to capture non-linear regression relationships. Therefore, non-parametric regression approaches such as smoothing can be useful alternatives to polynomial regression. Some of these methods make use of a localized form of classical polynomial regression. An advantage of traditional polynomial regression is that the inferential framework of multiple regression can be used (this also holds when using other families of basis functions such as splines).