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.

摘要

对最优值（全局最大值或最小值）位置的统计推断是响应曲面法领域的主要目标之一，在工程和科学领域有许多应用。虽然以前存在计算最优位置置信区域的方法，但这些方法适用于基于正态分布假设的线性模型，并没有具体解决与保证全局最优性相关的困难。本文介绍了在响应曲面模型的全局最优位置上计算置信区域的无分布方法。这些方法基于引导程序和 Tukey 的数据深度，因此它们的性能不依赖于关于影响响应的错误的分布假设。R 语言实现，包 OptimaRegion，被描述。支持参数（最多 5 个协变量的二次和三次多项式）和非参数模型（2 个协变量的薄板样条）。提供了覆盖分析，以证明所发现区域的质量。该包还包含 Gloptipoly 算法的 R 实现，用于全局优化受边界约束的多项式响应。