Robust bilinear factor analysis based on the matrix-variate t distribution

Abstract

Factor Analysis based on the multivariate t distribution (tFA) is a useful robust tool for extracting common factors on heavy-tailed or contaminated data. However, tFA is only applicable to vector data. When tFA is applied to matrix data, it is common to first vectorize the matrix observations. This introduces two challenges for tFA: (i) the inherent matrix structure of the data is broken, and (ii) robustness may be lost, as vectorized matrix data typically results in a high data dimension, which could easily lead to the breakdown of tFA. To address these issues, starting from the intrinsic matrix structure of matrix data, a novel robust factor analysis model, namely bilinear factor analysis built on the matrix-variate t distribution (tBFA), is proposed in this paper. The novelty is that it is capable of simultaneously extracting common factors for both row and column variables of interest on heavy-tailed or contaminated matrix data. Two efficient algorithms for maximum likelihood estimation of tBFA are developed. Closed-form expressions for the Fisher information matrix to calculate the accuracy of parameter estimates are derived. Empirical studies are conducted to understand the proposed tBFA model and compare it with related competitors. The results demonstrate the superiority and practicality of tBFA. Importantly, tBFA exhibits a significantly higher breakdown point than tFA, making it more suitable for matrix data.