Simultaneous variable and factor selection via sparse group lasso in factor analysis

This paper considers variable and factor selection in factor analysis. We treat the factor loadings for each observable variable as a group, and introduce a weighted sparse group lasso penalty to the complete log-likelihood. The proposal simultaneously selects observable variables and latent factors of a factor analysis model in a data-driven fashion; it produces a more flexible and sparse factor loading structure than existing methods. For parameter estimation, we derive an expectation-maximization algorithm that optimizes the penalized log-likelihood. The tuning parameters of the procedure are selected by a likelihood cross-validation criterion that yields satisfactory results in various simulation settings. Simulation results reveal that the proposed method can better identify the possibly sparse structure of the true factor loading matrix with higher estimation accuracy than existing methods. A real data example is also presented to demonstrate its performance in practice.

本文针对因子分析中的变量与因子选择问题展开研究。我们将每个可观测变量的因子载荷视作一个组，并针对完整对数似然（complete log-likelihood）引入加权稀疏组套索（sparse group lasso）惩罚项。所提方法以数据驱动的方式同时完成因子分析模型的可观测变量与潜在因子选择，相较于现有方法，能够生成更具灵活性且更为稀疏的因子载荷结构。针对参数估计任务，我们推导了可优化惩罚对数似然的期望-最大化（expectation-maximization）算法。该方法的调优参数通过似然交叉验证准则选取，该准则在各类模拟场景中均可取得令人满意的效果。模拟结果显示，相较于现有方法，所提方法能够更精准地识别真实因子载荷矩阵潜在的稀疏结构，同时具备更高的估计精度。本文还通过一则实际数据示例验证了该方法在实际应用中的表现。