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Modewise Additive Factor Model for Matrix Time Series

Elynn Chen, Yuefeng Han, Jiayu Li, Ke Xu

TL;DR

MAFM tackles the limitation of traditional multiplicative matrix factor models by introducing a modewise additive structure for matrix time series, decomposing $\boldsymbol{X}_t$ into row- and column-factor components via $\boldsymbol{X}_t = \boldsymbol{F}_t \boldsymbol{A}^\top + \boldsymbol{B} \boldsymbol{G}_t^\top + \boldsymbol{E}_t$ with a canonical form that identifies loading spaces up to rotation. The authors develop a two-stage estimation procedure, MINE for initialization and COMPAS for refinement using orthogonal-complement projections to eliminate cross-modal interference, and they establish non-asymptotic convergence rates and asymptotic normality for loading estimators under high-dimensional, dependent settings. A data-driven inference framework with plug-in covariance estimators enables valid confidence intervals and hypothesis tests for loading matrices, complemented by a Matrix Bernstein inequality for dependent matrix time series. Empirical validation includes simulations and an OECD macroeconomic application, where MAFM achieves better fit and forecasting performance than competing methods while providing interpretable, uncertainty-quantified loadings.

Abstract

We introduce a Modewise Additive Factor Model (MAFM) for matrix-valued time series that captures row-specific and column-specific latent effects through an additive structure, offering greater flexibility than multiplicative frameworks such as Tucker and CP factor models. In MAFM, each observation decomposes into a row-factor component, a column-factor component, and noise, allowing distinct sources of variation along different modes to be modeled separately. We develop a computationally efficient two-stage estimation procedure: Modewise Inner-product Eigendecomposition (MINE) for initialization, followed by Complement-Projected Alternating Subspace Estimation (COMPAS) for iterative refinement. The key methodological innovation is that orthogonal complement projections completely eliminate cross-modal interference when estimating each loading space. We establish convergence rates for the estimated factor loading matrices under proper conditions. We further derive asymptotic distributions for the loading matrix estimators and develop consistent covariance estimators, yielding a data-driven inference framework that enables confidence interval construction and hypothesis testing. As a technical contribution of independent interest, we establish matrix Bernstein inequalities for quadratic forms of dependent matrix time series. Numerical experiments on synthetic and real data demonstrate the advantages of the proposed method over existing approaches.

Modewise Additive Factor Model for Matrix Time Series

TL;DR

MAFM tackles the limitation of traditional multiplicative matrix factor models by introducing a modewise additive structure for matrix time series, decomposing into row- and column-factor components via with a canonical form that identifies loading spaces up to rotation. The authors develop a two-stage estimation procedure, MINE for initialization and COMPAS for refinement using orthogonal-complement projections to eliminate cross-modal interference, and they establish non-asymptotic convergence rates and asymptotic normality for loading estimators under high-dimensional, dependent settings. A data-driven inference framework with plug-in covariance estimators enables valid confidence intervals and hypothesis tests for loading matrices, complemented by a Matrix Bernstein inequality for dependent matrix time series. Empirical validation includes simulations and an OECD macroeconomic application, where MAFM achieves better fit and forecasting performance than competing methods while providing interpretable, uncertainty-quantified loadings.

Abstract

We introduce a Modewise Additive Factor Model (MAFM) for matrix-valued time series that captures row-specific and column-specific latent effects through an additive structure, offering greater flexibility than multiplicative frameworks such as Tucker and CP factor models. In MAFM, each observation decomposes into a row-factor component, a column-factor component, and noise, allowing distinct sources of variation along different modes to be modeled separately. We develop a computationally efficient two-stage estimation procedure: Modewise Inner-product Eigendecomposition (MINE) for initialization, followed by Complement-Projected Alternating Subspace Estimation (COMPAS) for iterative refinement. The key methodological innovation is that orthogonal complement projections completely eliminate cross-modal interference when estimating each loading space. We establish convergence rates for the estimated factor loading matrices under proper conditions. We further derive asymptotic distributions for the loading matrix estimators and develop consistent covariance estimators, yielding a data-driven inference framework that enables confidence interval construction and hypothesis testing. As a technical contribution of independent interest, we establish matrix Bernstein inequalities for quadratic forms of dependent matrix time series. Numerical experiments on synthetic and real data demonstrate the advantages of the proposed method over existing approaches.
Paper Structure (14 sections, 9 theorems, 37 equations, 11 figures, 3 tables, 2 algorithms)

This paper contains 14 sections, 9 theorems, 37 equations, 11 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Organization
  4. Identifiability of Modewise Additive Factor Model
  5. Estimation
  6. Theoretical Properties
  7. Convergence Rates for Factor Loading Estimation
  8. Statistical Inference for Factor Loadings
  9. Data-Driven Statistical Inference
  10. Matrix Bernstein Inequality for Quadratic Forms of Matrix Time Series
  11. Simulation
  12. Real Data Analysis
  13. Conclusion
  14. Additional Simulation Results

Key Result

Proposition 2.1

Suppose the matrix time series $\boldsymbol{X}_t \in \mathbb{R}^{d_1 \times d_2}$ follows the modewise additive factor model eqn: factor model with $\boldsymbol{F}_t$ and $\boldsymbol{G}_t$ satisfying Conditions (C1)--(C2). Then, the loading spaces spanned by $\boldsymbol{U}_{\boldsymbol{B}}$ and $\

Figures (11)

  • Figure 1: Estimation error $\log({\cal D}(\boldsymbol{U}_{\boldsymbol{A}}, \widehat{\boldsymbol{U}}_{\boldsymbol{A}}))$ under strong factor loading strength regime with $(\delta_0, \delta_1) = (0, 0)$ across dimensions $d$ and sample sizes $T$ for three estimation methods: MINE, P-COMPAS, and COMPAS.
  • Figure 2: Estimation error $\log({\cal D}(\boldsymbol{U}_{\boldsymbol{A}}, \widehat{\boldsymbol{U}}_{\boldsymbol{A}}))$ under weak factor loading strength regime with $(\delta_0, \delta_1) = (0.3, 0.5)$ across dimensions $d$ and sample sizes $T$ for three estimation methods: MINE, P-COMPAS, and COMPAS.
  • Figure 3: Normal QQ plots for the first coordinate of the standardized first row of $\widehat{\boldsymbol{U}}_{\boldsymbol{A}}-\boldsymbol{U}_{\boldsymbol{A}}\boldsymbol{R}_{\boldsymbol{A}}$ with $T=200$ under the strong loading strength regime. The top row uses the oracle inference procedure with population covariance matrices, and the bottom row uses the feasible data-driven inference procedure with plug-in covariance estimators. Columns correspond to $d\in\{20,50,100\}$.
  • Figure 4: Histograms for the first coordinate of the standardized first row of $\widehat{\boldsymbol{U}}_{\boldsymbol{A}}-\boldsymbol{U}_{\boldsymbol{A}}\boldsymbol{R}_{\boldsymbol{A}}$ with $T=200$ under the strong loading strength regime, overlaid with the ${\cal N}(0,1)$ density (red curve). The top row uses the oracle inference procedure with population covariance matrices, and the bottom row uses the feasible data-driven inference procedure with plug-in covariance estimators. Columns correspond to $d\in\{20,50,100\}$.
  • Figure 5: Eigenvalue proportion diagnostics for rank selection. (a) Eigenvalue proportions $\lambda_i / \sum_j \lambda_j$ of the row-factor component matrix $n^{-1}\sum_{t=1}^n \boldsymbol{X}_t^\top \boldsymbol{X}_t$; (b) Eigenvalue proportions of the column-factor component matrix $n^{-1}\sum_{t=1}^n \boldsymbol{X}_t \boldsymbol{X}_t^\top$, where $\lambda_i$ denotes the $i$th largest eigenvalue.
  • ...and 6 more figures

Theorems & Definitions (18)

  • Proposition 2.1
  • Remark 1
  • Remark 2
  • Remark 3: Variance Components
  • Theorem 4.1
  • Remark 4: Sample size requirement
  • Theorem 4.2
  • Remark 5: Comparison with other matrix factor models
  • Theorem 4.3: Partial complement projection
  • Remark 6
  • ...and 8 more