PROJECTED PRINCIPAL COMPONENT ANALYSIS IN FACTOR MODELS.

Area of Science:

• Statistics
• Machine Learning
• Econometrics

Background:

• Conventional Principal Component Analysis (PCA) can be sensitive to noise in high-dimensional data.
• Factor analysis is crucial for uncovering latent structures in complex datasets.
• Estimating latent factors accurately is challenging, particularly with limited sample sizes.

Purpose of the Study:

• To introduce Projected Principal Component Analysis (Projected-PCA) for improved factor analysis.
• To enhance the estimation of unobserved latent factors by leveraging covariate information.
• To develop a flexible semi-parametric factor model incorporating subject-specific covariates.

Main Methods:

• Projecting the data matrix onto a linear space spanned by covariates before applying PCA.
• Decomposing factor loading matrices into covariate-explained components and residual components.
• Utilizing sieve approximations within an additive model to capture covariate effects on factor loadings.

Main Results:

• Projected-PCA demonstrates more accurate estimation of latent factors compared to conventional PCA.
• Achieved significantly faster convergence rates for smooth factor loading matrices.
• Accurate factor estimation is possible even with large dimensionality and finite sample sizes.

Conclusions:

• Projected-PCA offers substantial improvements in factor analysis, particularly for high-dimensional, low-sample data.
• The proposed semi-parametric model provides a flexible framework for incorporating covariate effects.
• The method facilitates nonparametric tests for covariate influence on factor loadings.