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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): 41-54.doi: 10.3969/j.issn.1005-3085.2023.01.003

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基于 2DPCA 的矩阵时间序列统计监控及推断

高钰乔,   夏志明,   王  丹   

  1. 西北大学数学学院，西安 710127
  • 出版日期:2023-02-15 发布日期:2023-04-11
  • 通讯作者: 夏志明 E-mail: statxzm@nwu.edu.cn
  • 基金资助:
    国家自然科学基金 (11771353; 12171391).

Statistical Monitoring and Inference of Matrix Time Series Based on 2DPCA

GAO Yuqiao,   XIA Zhiming,   WANG Dan   

  1. School of Mathematics, Northwest University, Xi'an 710127
  • Online:2023-02-15 Published:2023-04-11
  • Contact: Z. Xia. E-mail address: statxzm@nwu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11771353; 12171391).

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摘要：

在多元统计过程控制的研究中，越来越多的学者开始关注到矩阵数据的在线监控问题。矩阵数据通常可被拉直为向量数据再进行监控，但拉直操作破坏了矩阵数据的原有结构信息。而2DPCA方法直接对矩阵数据进行特征提取，可保留矩阵的结构特征。因此，利用2DPCA方法研究矩阵值时间序列的统计监控及推断是有意义的。首先基于2DPCA方法对矩阵数据进行正交投影获取特征，通过融合这些特征构造监控统计量；其次证明了该监控统计量的极限分布为卡方分布，并利用该分布进行统计推断。模拟实验表明：该方法理论正确；当样本容量较大时，该方法相对于同类方法表现更优。

关键词: 变点, 主成分分析, 2DPCA, 矩阵正态分布, 卡方分布

Abstract:

In the field of multivariate statistical process control, more and more scholars begin to pay attention to the online monitoring of matrix data. Matrix data can usually be reshaped into vector data and then monitored, but the reshape operation destroys the original structure information of matrix data. The 2DPCA method directly extracts the features of the matrix data and can retain the structural features of the matrix. Therefore, it is meaningful to use the 2DPCA method to study the statistically monitoring and inference of the matrix data time series. Based on the 2DPCA method, an orthogonal projection is performed on the matrix data to obtain features, and the monitoring statistics are constructed by using these features. Finally, it is proved that the limit distribution of the monitoring statistics is Chi-square distribution, and the statistical inference is carried out by using this distribution. Simulation experiments show that the method is theoretically correct, and when the sample size is large, the proposed method performs better than similar methods.

Key words: change point, PCA, 2DPCA, matrix normal distribution, Chi-square distribution

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