Optimal Designs for Rational Function Regression
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有理函数回归的最优设计
- 研究问题
- 对于具有多项式或有理函数均值与异方差权重的回归模型，如何确定不同最优性准则下非序贯设计的支持点？
- 方法与思路
- 作者将多种最优设计准则纳入数学优化模型，生成一个以最优近似设计支持点为零点的多项式，再用数值优化求解。
- 材料陈述的贡献
- 摘要称统一处理多种设计准则，提供算法效率的理论保证，并得到最少支持点最优设计的支持集大小上界。
- 阅读关注点
- 应核对适用的回归与权重函数范围、多项式零点如何对应支持点，以及算法复杂度保证和支持集上界的条件。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；未提供上界表达式或复杂度条件，数值例子的短运行时间不能泛化。
原文摘要
We consider the problem of finding optimal nonsequential designs for a large class of regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. Since the design weights can be found easily by existing methods once the support is known, we concentrate on determining the support of the optimal design. The proposed method treats D-, E-, A-, and Φ p -optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well-established numerical optimization methods. In contrast to optimization-based methods previously proposed for the solution of similar design problems, our method also has theoretical guarantee of its algorithmic efficiency; in concordance with the theory, the actual running times of all numerical examples considered in the paper are negligible. The numerical stability of the method is demonstrated in an example involving high-degree polynomials. As a corollary, an upper bound on the size of the support set of the minimally supported optimal designs is also found.