双线性因子分析  

Bilinear Factor Analysis

作　　者：尚长春 马璇 蒋芬 赵建华[1] SHANG Changchun;MA Xuan;JIANG Fen;ZHAO Jianhua(School of Statistics and Mathematics,Yunnan University of Finance and Economics,Kunming 650221;School of Mathematics and Statistics,Guilin University of Technology,Guilin 541004)

机构地区：[1]云南财经大学统计与数学学院,昆明650221 [2]桂林理工大学数学与统计学院,桂林541004

出　　处：《系统科学与数学》2024年第4期1159-1188,共30页Journal of Systems Science and Mathematical Sciences

基　　金：国家自然科学基金(12161089,11761076);云南省教育厅科学研究基金(2022Y488);云南省科技厅科学研究基金(202201AU070105);云南财经大学科学研究基金(2021D10,2024YUFEYC013);广西中青年教师基础能力提升项目(2021KY0256)资助课题。

摘　　要：因子分析(factor analysis,FA)是一种流行的从多变量中提取公因子的统计技术,但它仅适用于向量值数据(每个数据点为一向量).当FA应用于矩阵值数据(每个数据点为一矩阵)时,一种常用的做法是首先将矩阵值观测向量化.然而,向量化使得因子分析面临两个问题:可解释性变差,容易陷入维数灾难.为了解决这两个问题,文章从矩阵值数据本身固有的矩阵结构出发,提出双线性因子分析(bilinear FA,BFA).新颖性在于BFA采用双线性变换,模型参数大大减少,有效克服了维数灾难问题,同时提取感兴趣的行变量和列变量公因子.文章开发了两种有效算法用于BFA模型参数的极大似然估计,讨论了估计的理论性质并明确地求出Fisher信息矩阵的解析表达式来计算参数估计的准确度,研究了BFA的模型选择问题.与传统因子得分为一向量不同,BFA的因子得分为一矩阵,文章为矩阵因子得分提供了计算方法以及可视化方法.最后,构建实证研究来理解提出的BFA模型并与相关方法进行比较.结果表明了BFA在矩阵值数据分析上的优越性和实用性.Factor analysis(FA) is a popular statistical technique that is used to identify the latent common factors among a set of variables.Nevertheless,it is only applicable to vector-valued data,where observations are vectors.To apply FA to matrix-valued data,where observations are matrices,one common solution is to first vectorize the matrix observations.However,the vectorization may cause FA to suffer from two problems:Poor interpretability and curse of dimensionality.To solve the two problems,the authors utilize the inherent matrix data structure and propose bilinear factor analysis(BFA) in this paper.The novelties are that BFA uses a bilinear transformation,which greatly reduces the model parameters and thus can overcome the curse of dimensionality;Moreover,it can simultaneously identify the interesting common row,column factors among the row,column variables,respectively.The authors develop two efficient algorithms for finding the maximum likelihood(ML)estimates.The authors give the theoretical property of the ML estimator and derive explicitly the closed-form expression of Fisher information matrix to evaluate the estimator's accuracy.The authors then discuss the model selection issue.Unlike the traditional FA,where the factor score is a vector,the factor score in BFA is a matrix.The authors further develop the approaches for calculating the matrix factor scores and visualizing them.Empirical studies are constructed to understand the proposed BFA model and compare with relevant methods.The results reveal the superiority and practicability of BFA in matrix-valued data analysis.

关 键 词：因子分析 矩阵值数据 极大似然估计 EM算法 模型选择 

分 类 号：O212.1[理学—概率论与数理统计]