Simplifying Multivariate Second-Order Response Surfaces by Fitting Constrained Models Using Automatic Differentiation

• Published in: Technometrics Vol. 47; no. 3; pp. 249 - 259
• Main Authors: Ringrose, Trevor J, Forth, Shaun A
• Format: Journal Article
• Language: English
• Published: Alexandria, VI Taylor & Francis 01.08.2005; Milwaukee, WI The American Society for Quality and The American Statistical Association; American Society for Quality Control; American Statistical Association; American Society for Quality
• Subjects: Augmented information matrix; Canonical analysis; Canonical forms; Datasets; Eigenvalues; Eigenvectors; Exact sciences and technology; Mathematical differentiation; Mathematical models; Mathematics; Modeling; Multivariate analysis; Multivariate regression; Parametric models; Polynomials; Probability and statistics; Regression analysis; Response surface methodology; Sciences and techniques of general use; Simulations; Standard error; Statistics; Canonical analysis; Augmented information matrix; Unconstrained optimization; Response surface; Multivariate regression; Maximum likelihood; Response surface methodology; Principal component analysis
• ISSN: 0040-1706; 1537-2723
• DOI: 10.1198/004017005000000148

Multivariate regression models for second-order polynomial response surfaces are proposed. The fitted surfaces for each response variable are constrained so that when expressed in their canonical forms they have features in common, such as common stationary points or common sets of eigenvectors. This can greatly reduce the number of parameters required and make the set of surfaces easier to interpret together, at the cost of a greater computational burden. However, the use of automatic differentiation within the package Matlab is shown to be easy and to reduce this burden considerably. We describe the models and how to fit them and derive standard errors, and report a small simulation study and an application to a dataset.