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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (1): 123-134.doi: 10.3969/j.issn.1005-3085.2023.01.009

• • 上一篇    下一篇

混料试验设计中置换阵的性质及其应用

李俊鹏1,2,   李光辉2,   张崇岐1   

  1. 1. 广州大学经济与统计学院，广州 510006；2. 凯里学院理学院，凯里 556011
  • 出版日期:2023-02-15 发布日期:2023-04-11
  • 通讯作者: 张崇岐 E-mail: cqzhang@gzhu.edu.cn
  • 基金资助:
    国家自然科学基金(11901260; 12071096)；贵州省科学技术基金([2020]1Y010).

Applications and Properties of Permutation Matrix for Mixture Experimental Designs

LI Junpeng1,2,   LI Guanghui2,   ZHANG Chongqi1   

  1. 1. School of Economics and Statistics, Guangzhou University, Guangzhou 510006;
    2. School of Science, Kaili University, Kaili 556011
  • Online:2023-02-15 Published:2023-04-11
  • Contact: C. Zhang. E-mail address: cqzhang@gzhu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11901260; 12071096); the Foundation of Guizhou Science and Technology Department ([2020]1Y010).

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摘要：

为解决由于高维混料试验的设计阵的形式复杂且计算难度大的问题，使用高阶混料格点作为支撑集，提出了置换阵的定义。首先讨论了由置换阵构造出设计阵的方法；进而由此推导出信息矩阵与相关矩阵的显示表达；然后探索了置换阵的行列式、迹和特征值等方面的性质；最后通过实例验证置换阵用于设计分析的有效性，并进一步分析了利用置换阵构造高阶混料格点集和最优设计的必要性和可行性。

关键词: 置换阵, 信息矩阵, 混料试验设计, $D$-最优设计

Abstract:

In order to solve the problem of the complex form and complicated calculation of the design matrix for high-dimensional mixture experiments, the definition of the permutation matrix is proposed by using high-order mixture lattice points as the support set. Firstly, the method to construct the design matrix from the permutation matrix is discussed, and the representation of the information matrix and the correlation matrix are derived. Then, the properties of the determinant, trace, and eigenvalue of the permutation matrix are studied. Finally, the effectiveness of the permutation matrix for the analysis of mixture design is verified, and the necessity and feasibility of using the permutation matrix to construct a high-order lattice point set and an optimal design are further analyzed.

Key words: permutation matrix, information matrix, mixture experimental design, $D$-optimal design

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