Optimal Design of Experiments for Non-Linear Response Surface Models

Abstract

Summary Many chemical and biological experiments involve multiple treatment factors and often it is convenient to fit a non‐linear model in these factors. This non‐linear model can be mechanistic, empirical or a hybrid of the two. Motivated by experiments in chemical engineering, we focus on D‐optimal designs for multifactor non‐linear response surfaces in general. To find and study optimal designs, we first implement conventional point and co‐ordinate exchange algorithms. Next, we develop a novel multiphase optimization method to construct D‐optimal designs with improved properties. The benefits of this method are demonstrated by application to two experiments involving non‐linear regression models. The designs obtained are shown to be considerably more informative than designs obtained by using traditional design optimality algorithms.

Optimal Experimental Design in the Presence of Nested Factors

“…It is also unnecessary to discretize the range of continuous factors, since a one-dimensional continuous optimization algorithm such as the one from Brent (1973) can be used to determine optimal values for the levels of the continuous factors. Modern coordinate-exchange A c c e p t e d M a n u s c r i p t algorithms implementing this approach have been used by Jones and Goos (2012), Ruseckaite et al (2017) and Huang et al (2019). The use of a continuous optimizer is especially useful if there are factors that are ingredients of a mixture or if there are linear or nonlinear constraints on the experimental region .…”

“…Recently, to avoid the problems inherent to using a candidate set, Huang et al (2019) proposed a candidate-set-free point-exchange algorithm to optimally design experiments for non-linear models. Given a random starting design, the candidate-set-free algorithm uses a multi-dimensional continuous optimizer, such as the Nelder-Mead or quasi-Newton method, to move current design points to better positions.…”

“…This avoids the need to discretize the continuous factors. Huang et al (2019) state that their new algorithm is especially useful for non-linear regression models. This is because, in the absence of constraints on the factor levels, a candidate set with equally spaced levels will permit the construction of a design that is either optimal or very close to optimal for linear models, provided the number of runs is sufficiently large.…”

Induced Start Dynamic Sampling for Wafer Metrology Optimization

“…based on minimum correlation with the already selected sites is recommended. As future work, we plan to investigate the applicability of ISDS to the optimal design of experiments, a topic that is closely connected to the measurement plan optimization problem described here, and which is relevant in many domains, for example, in medical testing and in chemical manufacturing [31]- [34]. In particular, we note potential parallels that may exist between ISDS and optimal design of experiments with regard to incorporating prior knowledge via initial designs and employing a non-sequential exchange algorithm for targeted design point selection [35].…”

A note on the output of a coordinate-exchange algorithm for optimal experimental design

“…For the results we report in this short communication, we utilized an interior point algorithm as implemented in the Matlab function fmincon, and described in Byrd et al (1999Byrd et al ( , 2000 and Waltz et al (2006). The interior point method is a popular choice in recent implementations of the coordinate exchange algorithm (Ruseckaite et al;Huang et al;. The orignal coordinate-exchange algorithm of Meyer and Nachtsheim (1995), however, used grid optimization rather than a continuous optimizer.…”