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Mining of Switching Sparse Networks for Missing Value Imputation in Multivariate Time Series

Kohei Obata, Koki Kawabata, Yasuko Matsubara, Yasushi Sakurai

TL;DR

MissNet addresses missing-value imputation in multivariate time series by jointly learning regime-switching dynamics, a state-space latent representation, and switching sparse networks via graphical lasso. It integrates a discrete Markov process for regime changes, a linear-Gaussian SSM for temporal dependencies, and network inference that updates per regime, enabling interpretable regime-specific relationships. The method demonstrates improved imputations over state-of-the-art baselines, scales linearly with time length, and provides visualizable networks that illuminate how features relate under different regimes. Empirical results on synthetic and real datasets confirm both accuracy gains and interpretability, offering a practical approach for complex, non-stationary time-series data with missing blocks.

Abstract

Multivariate time series data suffer from the problem of missing values, which hinders the application of many analytical methods. To achieve the accurate imputation of these missing values, exploiting inter-correlation by employing the relationships between sequences (i.e., a network) is as important as the use of temporal dependency, since a sequence normally correlates with other sequences. Moreover, exploiting an adequate network depending on time is also necessary since the network varies over time. However, in real-world scenarios, we normally know neither the network structure nor when the network changes beforehand. Here, we propose a missing value imputation method for multivariate time series, namely MissNet, that is designed to exploit temporal dependency with a state-space model and inter-correlation by switching sparse networks. The network encodes conditional independence between features, which helps us understand the important relationships for imputation visually. Our algorithm, which scales linearly with reference to the length of the data, alternatively infers networks and fills in missing values using the networks while discovering the switching of the networks. Extensive experiments demonstrate that MissNet outperforms the state-of-the-art algorithms for multivariate time series imputation and provides interpretable results.

Mining of Switching Sparse Networks for Missing Value Imputation in Multivariate Time Series

TL;DR

MissNet addresses missing-value imputation in multivariate time series by jointly learning regime-switching dynamics, a state-space latent representation, and switching sparse networks via graphical lasso. It integrates a discrete Markov process for regime changes, a linear-Gaussian SSM for temporal dependencies, and network inference that updates per regime, enabling interpretable regime-specific relationships. The method demonstrates improved imputations over state-of-the-art baselines, scales linearly with time length, and provides visualizable networks that illuminate how features relate under different regimes. Empirical results on synthetic and real datasets confirm both accuracy gains and interpretability, offering a practical approach for complex, non-stationary time-series data with missing blocks.

Abstract

Multivariate time series data suffer from the problem of missing values, which hinders the application of many analytical methods. To achieve the accurate imputation of these missing values, exploiting inter-correlation by employing the relationships between sequences (i.e., a network) is as important as the use of temporal dependency, since a sequence normally correlates with other sequences. Moreover, exploiting an adequate network depending on time is also necessary since the network varies over time. However, in real-world scenarios, we normally know neither the network structure nor when the network changes beforehand. Here, we propose a missing value imputation method for multivariate time series, namely MissNet, that is designed to exploit temporal dependency with a state-space model and inter-correlation by switching sparse networks. The network encodes conditional independence between features, which helps us understand the important relationships for imputation visually. Our algorithm, which scales linearly with reference to the length of the data, alternatively infers networks and fills in missing values using the networks while discovering the switching of the networks. Extensive experiments demonstrate that MissNet outperforms the state-of-the-art algorithms for multivariate time series imputation and provides interpretable results.
Paper Structure (34 sections, 2 theorems, 22 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 34 sections, 2 theorems, 22 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Problem definition
  5. Graphical lasso
  6. Proposed MissNet
  7. Optimization model
  8. Algorithm
  9. E-step
  10. M-step
  11. Update $\dot{\bm{S}}$
  12. Overall algorithm
  13. Complexity analysis
  14. Experiments
  15. Experimental setup
  16. ...and 19 more sections

Key Result

Lemma 1

The time complexity of MissNet is $O( \#iter \cdot ( K^{2}\sum_{t=1}^T(L^{3} + L^{2}N_{t} + LN_{t}^{2} + N_{t}^{3}) + KL^{2}N^{2} + KTL^{2}N + KN^{3} ) )$.

Figures (8)

  • Figure 1: An illustrative example of a multivariate time series including missing blocks, where each time point of the data is allocated to two regimes, each with a distinct network, where the edges show relationships between features.
  • Figure 2: Graphical model of MissNet at each iteration.
  • Figure 3: RMSE of (a), (b) Synthetic ($N = 50$), (c) $\sim$ (k) MotionCapture ($N = 123$) and (l) Motes ($N = 54$) datasets.
  • Figure 4: Critical difference diagram of real-world datasets.
  • Figure 5: Case study on MotionCapture Run dataset.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2