Minimizing the Condition Number to Construct Design Points for Polynomial Regression Models

Abstract

In this paper we study a new optimality criterion, the $K$-optimality criterion, for constructing optimal experimental designs for polynomial regression models. We focus on the $p$th order polynomial regression model with symmetric design space $[-1,1]$. For this model, we show that there is always a symmetric $K$-optimal design with exactly $p+1$ support points including the boundary points $-1$ and $1$. It is well known that the condition number for a positive definite matrix as the ratio of the maximum eigenvalue to the minimum eigenvalue is usually nonsmooth. We show that for our model, the condition number of the information matrix is continuously differentiable. Theoretical $K$-optimal designs are derived for $p=1$ and $2$. Numerical results are presented for $3 \le p \le 10$.