基于Prenet惩罚的稀疏探索性因子分析

Sparse Exploratory Factor Analysis Based on Prenet Penalty

因子分析作为重要的统计降维技术,旨在通过探索变量之间的协方差结构,用以揭示显变量与潜变量之间的数量关系.因子载荷矩阵的稀疏估计是关系因子分析效果及解释的重要环节,主要有基于旋转的两步法和基于惩罚的一步法两种策略.为了获得易于解释的稀疏结构,文章提出了一种基于prenet惩罚的稀疏因子模型.具体来讲,将因子载荷A分解为正交矩阵Q和对角矩阵D乘积,并对Q施加prenet惩罚,提出求解稀疏因子载荷矩阵的估计方法,并给出了广义期望最大化的求解算法.模拟结果显示,与现有的估计方法相比,基于prenet惩罚的方法估计结果比较稳定,易于获得稀疏的因子载荷;通过实例数据分析表明:该方法在因子分析的稀疏处理方面具有明显的优势.

As an important statistical dimensionality reduction technique,factor analysis aims to reveal the quantitative relationship between explicit variables and latent variables by exploring the covariance structure among variables.The sparse estimation of factor loading matrix is an important part of determining the effect and interpretation of factor analysis,and there are two strategies:The two-step method based on rotation and the one-step method based on penalty.In order to obtain an easy-to-interpret sparse structure,this paper proposes a sparse factor model based on prenet penalty.Specifically,factor loading A is decomposed into the product of an orthogonal matrix Q and a diagonal matrix D.A prenet penalty is imposed on Q,an estimation method for solving the sparse factor loading matrix is proposed,and a solving algorithm for generalized expectation maximization is given.The simulation experiment results show that compared with the existing estimation method,the estimation results of the method based on the prenet penalty is relatively stable,and it is easy to obtain sparse factor loadings.The example data analysis shows that the method has obvious advantages in the sparse processing of factor analysis.