Spatial Regression Models, Response Surfaces, and Process Optimization

Abstract

Spatial regression models are developed as a complementary alternative to second-order polynomial response surfaces in the context of process optimization. These models provide estimates of design variable effects and smooth, data-faithful approximations to the unknown response function over the design space. The predicted response surfaces are driven by the covariance structures of the models. Several structures, isotropic and anisotropic, are considered and connections with thin plate splines are reviewed. Estimation of covariance parameters is achieved via maximum likelihood and residual maximum likelihood. A feature of the spatial regression approach is the visually appealing graphical summaries that are produced. These allow rapid and intuitive identification of process windows on the design space for which the response achieves target performance. Relevant design issues are briefly discussed and spatial designs, such as the packing designs available in Gosset, are suggested as a suitable design complement. The spatial regression models also perform well with no global design, for example with data obtained from series of designs on the same space of design variables. The approach is illustrated with an example involving the optimization of components in a DNA amplification assay. A Monte Carlo comparison of the spatial models with both thin plate splines and second-order polynomial response surfaces for a scenario motivated by the example is also given. This shows superior performance of the spatial models to the second-order polynomials with respect to both prediction over the complete design space and for cross-validation prediction error in the region of the optimum. An anisotropic spatial regression model performs best for a high noise case and both this model and the thin plate spline for a low noise case. Spatial regression is recommended for construction of response surfaces in all process optimization applications.

Key Words:

• Maximin and minimax design
• Mixed models
• Response surface methods
• Thin plate splines
• Universal kriging