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Spatiotemporal Covariance Neural Networks

Andrea Cavallo, Mohammad Sabbaqi, Elvin Isufi

TL;DR

This work addresses learning from irregular, streaming multivariate time series by exploiting the covariance structure as a graph. It introduces STVNN, a layered architecture built from SpatioTemporal coVariance Filters (STVF) that operate on online covariance estimates and recent time-window samples to produce joint spatiotemporal embeddings. The authors prove stability guarantees showing the output perturbations due to online covariance and parameter estimation decay with time, with a dominant covariance-uncertainty term that scales as $O(1/\sqrt{t})$ and a parameter term scaling as $O(1/t)$, and they discuss the impact of the memory size $T$. Empirical results on synthetic and real data demonstrate robust adaptation to distribution shifts and competitive forecasting performance, with code release.

Abstract

Modeling spatiotemporal interactions in multivariate time series is key to their effective processing, but challenging because of their irregular and often unknown structure. Statistical properties of the data provide useful biases to model interdependencies and are leveraged by correlation and covariance-based networks as well as by processing pipelines relying on principal component analysis (PCA). However, PCA and its temporal extensions suffer instabilities in the covariance eigenvectors when the corresponding eigenvalues are close to each other, making their application to dynamic and streaming data settings challenging. To address these issues, we exploit the analogy between PCA and graph convolutional filters to introduce the SpatioTemporal coVariance Neural Network (STVNN), a relational learning model that operates on the sample covariance matrix of the time series and leverages joint spatiotemporal convolutions to model the data. To account for the streaming and non-stationary setting, we consider an online update of the parameters and sample covariance matrix. We prove the STVNN is stable to the uncertainties introduced by these online estimations, thus improving over temporal PCA-based methods. Experimental results corroborate our theoretical findings and show that STVNN is competitive for multivariate time series processing, it adapts to changes in the data distribution, and it is orders of magnitude more stable than online temporal PCA.

Spatiotemporal Covariance Neural Networks

TL;DR

This work addresses learning from irregular, streaming multivariate time series by exploiting the covariance structure as a graph. It introduces STVNN, a layered architecture built from SpatioTemporal coVariance Filters (STVF) that operate on online covariance estimates and recent time-window samples to produce joint spatiotemporal embeddings. The authors prove stability guarantees showing the output perturbations due to online covariance and parameter estimation decay with time, with a dominant covariance-uncertainty term that scales as and a parameter term scaling as , and they discuss the impact of the memory size . Empirical results on synthetic and real data demonstrate robust adaptation to distribution shifts and competitive forecasting performance, with code release.

Abstract

Modeling spatiotemporal interactions in multivariate time series is key to their effective processing, but challenging because of their irregular and often unknown structure. Statistical properties of the data provide useful biases to model interdependencies and are leveraged by correlation and covariance-based networks as well as by processing pipelines relying on principal component analysis (PCA). However, PCA and its temporal extensions suffer instabilities in the covariance eigenvectors when the corresponding eigenvalues are close to each other, making their application to dynamic and streaming data settings challenging. To address these issues, we exploit the analogy between PCA and graph convolutional filters to introduce the SpatioTemporal coVariance Neural Network (STVNN), a relational learning model that operates on the sample covariance matrix of the time series and leverages joint spatiotemporal convolutions to model the data. To account for the streaming and non-stationary setting, we consider an online update of the parameters and sample covariance matrix. We prove the STVNN is stable to the uncertainties introduced by these online estimations, thus improving over temporal PCA-based methods. Experimental results corroborate our theoretical findings and show that STVNN is competitive for multivariate time series processing, it adapts to changes in the data distribution, and it is orders of magnitude more stable than online temporal PCA.
Paper Structure (24 sections, 3 theorems, 48 equations, 5 figures, 2 tables)

This paper contains 24 sections, 3 theorems, 48 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Formulation
  4. Spatiotemporal Covariance Neural Networks
  5. Stability Analysis
  6. Stability of spatiotemporal covariance filter
  7. Stability of STVNN
  8. Numerical Results
  9. Model analysis
  10. Forecasting
  11. Conclusion
  12. Acknowledgments.
  13. Disclosure of Interests.
  14. Temporal PCA
  15. Proof of \ref{['th:stability_filter']}
  16. ...and 9 more sections

Key Result

theorem thmcountertheorem

Consider a multivariate time series $\mathbf{x}_t\in\mathbb{R}^N$ with underlying covariance matrix $\mathbf{C}$ and let the sample covariance matrix estimated from $t$ samples be $\mathbf{\hat{C}}_{t}$. Additionally let the instances satisfy w.l.o.g. $\|\mathbf{x}_t\|\leq 1$, let As. as_lipschitz-a where $Q$ is an absolute constant, $k_{\textnormal{max}} = \max_jk_j$, and $k_j=\left( \mathbb{E}[\

Figures (5)

  • Figure 1: Spatiotemporal covariance filter pipeline. We observe a new time series sample $\mathbf{x}_t$ and update the covariance estimate $\mathbf{\hat{C}}_{t}$, which the STVF uses as a weighted graph adjacency matrix to model spatial interactions. Then, we sum the embeddings of the last $T$ samples to model temporal interactions.
  • Figure 2: Embedding difference of the STVNN (i.e., $\| \mathsf{\Phi}(\mathbf{\hat{C}}_{t},\mathbf{h}^*, \mathbf{x}_{T:t}) - \mathsf{\Phi}(\mathbf{C},\mathbf{h}^*, \mathbf{x}_{T:t}) \|$) and TPCA with estimated and optimal covariance and parameters on different datasets for distinct observation windows $T$ and covariance eigenvalues distribution tails on synthetic datasets. Larger tails imply closer eigenvalues and higher kurtosis, leading to less distinguishable principal components.
  • Figure 3: Loss evolution on the synthetic non-stationary dataset.
  • Figure 4: Ablation study on non-stationary datasets.
  • Figure 5: Eigenvalue distribution for different tail sizes.

Theorems & Definitions (9)

  • remark thmcounterremark
  • definition thmcounterdefinition: Spatiotemporal covariance filter (STVF)
  • definition thmcounterdefinition: SpatioTemporal coVariance Neural Networks (STVNN)
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proposition thmcounterproposition
  • proof
  • remark thmcounterremark