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

Xuan Ma, Jianhua Zhao, Changchun Shang, Fen Jiang, Philip L. H. Yu

TL;DR

The paper addresses robustness for matrix-valued data by deriving tBFA, a robust bilinear factor analysis based on the matrix-variate $t$ distribution. It develops two EM-type ML estimation algorithms (ECME and AECM) and introduces parameter-expanded variants (PX-ECME and PX-AECM) to accelerate convergence, along with a closed-form Fisher information matrix to quantify estimator precision. Empirical results on synthetic and real data demonstrate superior robustness and interpretability of tBFA compared with vector-based $t$FA and Gaussian matrix-factor models, including a higher breakdown point. The work advances robust matrix modeling by preserving matrix structure, enabling simultaneous row and column factor extraction, and offering practical tools for reliable factor analysis in heavy-tailed settings. It also lays groundwork for extensions to tensors and mixtures, broadening applicability to complex multiway data.

Abstract

Factor Analysis based on multivariate $t$ distribution ($t$fa) is a useful robust tool for extracting common factors on heavy-tailed or contaminated data. However, $t$fa is only applicable to vector data. When $t$fa is applied to matrix data, it is common to first vectorize the matrix observations. This introduces two challenges for $t$fa: (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 $t$fa. 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 ($t$bfa), is proposed in this paper. The novelty is that it is capable to simultaneously extract 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 $t$bfa are developed. Closed-form expression for the Fisher information matrix to calculate the accuracy of parameter estimates are derived. Empirical studies are conducted to understand the proposed $t$bfa model and compare with related competitors. The results demonstrate the superiority and practicality of $t$bfa. Importantly, $t$bfa exhibits a significantly higher breakdown point than $t$fa, making it more suitable for matrix data.

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

TL;DR

The paper addresses robustness for matrix-valued data by deriving tBFA, a robust bilinear factor analysis based on the matrix-variate distribution. It develops two EM-type ML estimation algorithms (ECME and AECM) and introduces parameter-expanded variants (PX-ECME and PX-AECM) to accelerate convergence, along with a closed-form Fisher information matrix to quantify estimator precision. Empirical results on synthetic and real data demonstrate superior robustness and interpretability of tBFA compared with vector-based FA and Gaussian matrix-factor models, including a higher breakdown point. The work advances robust matrix modeling by preserving matrix structure, enabling simultaneous row and column factor extraction, and offering practical tools for reliable factor analysis in heavy-tailed settings. It also lays groundwork for extensions to tensors and mixtures, broadening applicability to complex multiway data.

Abstract

Factor Analysis based on multivariate distribution (fa) is a useful robust tool for extracting common factors on heavy-tailed or contaminated data. However, fa is only applicable to vector data. When fa is applied to matrix data, it is common to first vectorize the matrix observations. This introduces two challenges for fa: (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 fa. 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 distribution (bfa), is proposed in this paper. The novelty is that it is capable to simultaneously extract 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 bfa are developed. Closed-form expression for the Fisher information matrix to calculate the accuracy of parameter estimates are derived. Empirical studies are conducted to understand the proposed bfa model and compare with related competitors. The results demonstrate the superiority and practicality of bfa. Importantly, bfa exhibits a significantly higher breakdown point than fa, making it more suitable for matrix data.
Paper Structure (28 sections, 2 theorems, 44 equations, 5 figures, 9 tables)

This paper contains 28 sections, 2 theorems, 44 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Related works
  3. Multivariate-t factor analysis (tFA)
  4. Bilinear factor analysis (BFA)
  5. Robust BFA based on a matrix-variate t distribution (tBFA)
  6. Matrix-variate t (Mt) distribution
  7. Model formulation
  8. Model identifiability
  9. Maximum likelihood inference of tBFA
  10. Maximum likelihood (ML) estimation of tBFA
  11. The ECME algorithm
  12. The AECM algorithm
  13. Acceleration via Parameter Expansion
  14. Matrix factor score
  15. Asymptotic properties and estimating precision of parameter estimates
  16. ...and 13 more sections

Key Result

Theorem 1

For tBFA model (eqn:tbfa), we have: where $\boldsymbol{\epsilon}_n=\hbox{vec} \left( \mathbf{X}_n-\mathbf{W} \right)$, $\dot{\boldsymbol{\Sigma}}_c^i =\partial \boldsymbol{\Sigma}_c/ \partial \boldsymbol{\theta}_c^i$ with $\boldsymbol{\theta}_c$ representing the i-th entry of $\boldsymbol{\theta}_c$, for each $i \in\{1, \ldots, d_c(d_

Figures (5)

  • Figure 1: Changes in log-likelihood $\mathcal{L}$ for PX-ECME (solid), ECME (dotted), PX-AECM (dashdot), and AECM (dashed) algorithms with (a) number of iterations on Data1; (b) CPU time on Data1; (c) number of iterations on Data2; (d) CPU time on Data2; (e) number of iterations on Data3; (f) CPU time on Data3.
  • Figure 2: The two-dimensional subspace plot of the three kinds of outliers in an asymmetric case, where the observations and outliers are shown as black points and blue circles, respectively. First column: FC outliers in the (a) FC subspace, and (b) OC subspace; Second column: OC outliers in the (c) FC subspace, and (d) OC subspace; Third column: FC+OC outliers in the (e) FC subspace, and (f) OC subspace.
  • Figure 3: (a) BIC values versus various $q$ for FA and tFA; (b) BIC values versus various $(q_c,q_r)$ for BFA and tBFA.
  • Figure 4: BIC values versus various $(q_c,q_r)$ for (a) tBFA and (b) BFA models.
  • Figure 5: Fama–French series: Estimated factors over time by tBFA.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof