Optimal designs for rational function regression

TL;DR

This paper introduces a unified, efficient optimization method for designing optimal experiments in polynomial and rational function regression models, applicable to both linear and nonlinear cases with heteroscedastic noise.

Contribution

It generalizes existing design methods by providing a unified optimization framework with theoretical efficiency guarantees and broad applicability to various regression models.

Findings

Supports D-, E-, A-, and $oldsymbol{ ext{Φ}_p}$-optimal designs

Demonstrates computational efficiency and stability in high-degree polynomial examples

Extends to nonlinear, robust, and trigonometric regression models

Abstract

We consider optimal non-sequential designs for a large class of (linear and nonlinear) regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. The proposed method treats D-, E-, A-, and $\Phi_p$-optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well established numerical optimization methods. In contrast to previous optimization-based methods proposed for similar design problems, it also has theoretical guarantee of its algorithmic efficiency; in fact, the running times of all numerical examples considered in the paper are negligible. The stability of the method is demonstrated in an example involving high degree polynomials. After discussing linear models, applications for finding locally optimal designs for nonlinear regression models involving rational functions are presented, then extensions to robust regression designs, and trigonometric regression are shown. As a corollary, an upper bound on the size of the support set of the minimally-supported optimal designs is also found. The method is of considerable practical importance, with the potential for instance to impact design software development. Further study of the optimality conditions of the main optimization model might also yield new theoretical insights.