Multivariate Time Series Data Imputation via Distributionally Robust Regularization

TL;DR

This work tackles imputation for multivariate time series under distributional shift caused by non-stationarity and missing-not-at-random patterns. It introduces DRIO, a distributionally robust imputation objective that balances point-wise reconstruction with worst-case distributional alignment within a Wasserstein ambiguity set, enabling hedge against biased observations. A tractable dual reduces the infinite-dimensional optimization over distributions to adversarial sample trajectories, and an alternating training algorithm enables end-to-end learning with flexible BRITS-like backbones. Empirical results across ten real-world datasets show DRIO achieving Pareto-optimal trade-offs between reconstruction accuracy and distributional alignment, with robustness particularly evident under MNAR and a reliable reconstruction-based cross-validation strategy for hyperparameter selection.

Abstract

Multivariate time series (MTS) imputation is often compromised by mismatch between observed and true data distributions -- a bias exacerbated by non-stationarity and systematic missingness. Standard methods that minimize reconstruction error or encourage distributional alignment risk overfitting these biased observations. We propose the Distributionally Robust Regularized Imputer Objective (DRIO), which jointly minimizes reconstruction error and the divergence between the imputer and a worst-case distribution within a Wasserstein ambiguity set. We derive a tractable dual formulation that reduces infinite-dimensional optimization over measures to adversarial search over sample trajectories, and propose an adversarial learning algorithm compatible with flexible deep learning backbones. Comprehensive experiments on diverse real-world datasets show DRIO consistently improves imputation under both missing-completely-at-random and missing-not-at-random settings, reaching Pareto-optimal trade-offs between reconstruction accuracy and distributional alignment.

Figures (5)

• Figure 1: Overview of Multivariate Time Series Imputation under $\textsf{DRIO}$.
• Figure 2: Trade-off Between MSE and W2 averaged across all datasets and missing ratios under MCAR (left) and MNAR (right). $\textsf{DRIO}$ and PSW lie at the Pareto front at both cases, with $\textsf{DRIO}$ offering lower MSE at comparable W2.
• Figure 3: MSE and standard deviation on artificial missing entries in testing set, calculated across all ten datasets, under MCAR and MNAR at various missing ratios.
• Figure 4: Validation MSE on artificial missing entries across the hyperparameter grid, averaged over all datasets. Values show mean $\pm$ standard deviation. Lower is better.
• Figure 5: Training curves showing reconstruction error (MSE) and worst-case Sinkhorn divergence over epochs. Left: Gait MCAR 50% with $\alpha=0.99$, $\gamma=5.0$. Right: Gait MNAR 90% with $\alpha=0.75$, $\gamma=5.0$.

Theorems & Definitions (4)

• Remark 3.1: Ambiguity Set Geometry
• Theorem 3.2: Tractable Relaxation of Robust Imputation
• proof : Proof Sketch
• proof : Proof of Theorem \ref{['thm:dual-formulation']}