Confidence regions for the location of response surface optima: the R package OptimaRegion

Abstract

Statistical inference on the location of the optima (global maxima or minima) is one of the main goals in the area of Response Surface Methodology, with many applications in engineering and science. While there exist previous methods for computing confidence regions on the location of optima, these are for linear models based on a Normal distribution assumption, and do not address specifically the difficulties associated with guaranteeing global optimality. This paper describes distribution-free methods for the computation of confidence regions on the location of the global optima of response surface models. The methods are based on bootstrapping and Tukey’s data depth, and therefore their performance does not rely on distributional assumptions about the errors affecting the response. An R language implementation, the package OptimaRegion, is described. Both parametric (quadratic and cubic polynomials in up to 5 covariates) and nonparametric models (thin plate splines in 2 covariates) are supported. A coverage analysis is presented demonstrating the quality of the regions found. The package also contains an R implementation of the Gloptipoly algorithm for the global optimization of polynomial responses subject to bounds.

Keywords:

• Response surface methods
• Optimization
• Nonparametric regression
• Data-depth

Notes

1 The estimated standard error of the estimated coverage $\widehat{p}$ is given by $\sqrt{\widehat{p}(1-\widehat{p})/n},$ so all the estimated coverages presented in this paper are very precise.

2 See Lasserre (2001) for the statement of the assumption.

3 The reader is referred to Henrion and Lasserre (2003) for additional details.