Simplifying multivariate second-order response surfaces by fitting constrained models using automatic differentiation.

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.