Wigglyness is Diagonalizable - A Novel Approach to Polynomial and Rational Approximation

Linear regression is a classical framework for modeling linear relationships between variables. To model nonlinear relationships, though, a typical statistics course quickly introduces polynomial regressions for the job. However, they are dismissed just as quickly due to producing unwanted wigglyness in the model. One way to measure a model’s wigglyness is by the l2-norm on its second derivative, ||f”(x)||. In this talk, I will show how orthogonal polynomials linearize a generalized version of this norm, making this wigglyness measure linear with respect to the polynomial coefficient. I will also apply this idea to rational polynomials, an alternate way to model nonlinearity, providing better approximations than polynomial and spline-based methods.

Oct 20, 2023