Recursive construction of biorthogonal polynomials for handling polynomial regression

TL;DR

This paper introduces a recursive method to construct biorthogonal polynomials for polynomial regression, improving stability and flexibility by leveraging existing orthogonal polynomials like Legendre, Laguerre, and Chebyshev.

Contribution

It presents a novel recursive approach for building biorthogonal polynomial bases that enhances stability and allows easy basis modification in polynomial regression.

Findings

Reduces instability in polynomial regression.

Enables easy basis upgrading and downgrading.

Demonstrates effectiveness with classical orthogonal polynomials.

Abstract

An adaptive procedure for constructing polynomials which are biorthogonal to the basis of monomials in the same finite-dimensional inner product space is proposed. By taking advantage of available orthogonal polynomials, the proposed methodology reduces the well-known instability problem arising from the matrix inversion involved in classical polynomial regression. The recurrent generation of the biorthogonal basis facilitates the upgrading of all its members to include an additional one. Moreover, it allows for a natural downgrading of the basis. This convenient feature leads to a straightforward approach for reducing the number of terms in the polynomial regression approximation. The merit of this approach is illustrated through a series of examples where the resulting biorthogonal basis is derived from Legendre, Laguerre, and Chebyshev orthogonal polynomials.