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Autoregressive with Slack Time Series Model for Forecasting a Partially-Observed Dynamical Time Series

Akifumi Okuno, Yuya Morishita, Yoh-ichi Mototake

TL;DR

The paper tackles forecasting partially observed dynamical time series defined by an evolution function, where some state variables are unobserved. It proposes the autoregressive with slack time Series (ARS) model, which jointly estimates the evolution matrix $\boldsymbol B$ and imputes missing variables as slack time series $\{\boldsymbol z^{\dagger}_j\}$, leveraging a time-invariant linear dynamical assumption. A theoretical guarantee in the 2D case shows ARS can reconstruct the underlying true dynamics from a single observed dimension, and numerical experiments on circular motion and Lorenz-like dynamics demonstrate improved forecasting over conventional AR. The work clarifies connections to state-space formulations and higher-order AR models, discusses limitations such as overparameterization and optimization efficiency, and outlines future directions including extensions to interaction terms and practical applications in plasma physics.

Abstract

This study delves into the domain of dynamical systems, specifically the forecasting of dynamical time series defined through an evolution function. Traditional approaches in this area predict the future behavior of dynamical systems by inferring the evolution function. However, these methods may confront obstacles due to the presence of missing variables, which are usually attributed to challenges in measurement and a partial understanding of the system of interest. To overcome this obstacle, we introduce the autoregressive with slack time series (ARS) model, that simultaneously estimates the evolution function and imputes missing variables as a slack time series. Assuming time-invariance and linearity in the (underlying) entire dynamical time series, our experiments demonstrate the ARS model's capability to forecast future time series. From a theoretical perspective, we prove that a 2-dimensional time-invariant and linear system can be reconstructed by utilizing observations from a single, partially observed dimension of the system.

Autoregressive with Slack Time Series Model for Forecasting a Partially-Observed Dynamical Time Series

TL;DR

The paper tackles forecasting partially observed dynamical time series defined by an evolution function, where some state variables are unobserved. It proposes the autoregressive with slack time Series (ARS) model, which jointly estimates the evolution matrix and imputes missing variables as slack time series , leveraging a time-invariant linear dynamical assumption. A theoretical guarantee in the 2D case shows ARS can reconstruct the underlying true dynamics from a single observed dimension, and numerical experiments on circular motion and Lorenz-like dynamics demonstrate improved forecasting over conventional AR. The work clarifies connections to state-space formulations and higher-order AR models, discusses limitations such as overparameterization and optimization efficiency, and outlines future directions including extensions to interaction terms and practical applications in plasma physics.

Abstract

This study delves into the domain of dynamical systems, specifically the forecasting of dynamical time series defined through an evolution function. Traditional approaches in this area predict the future behavior of dynamical systems by inferring the evolution function. However, these methods may confront obstacles due to the presence of missing variables, which are usually attributed to challenges in measurement and a partial understanding of the system of interest. To overcome this obstacle, we introduce the autoregressive with slack time series (ARS) model, that simultaneously estimates the evolution function and imputes missing variables as a slack time series. Assuming time-invariance and linearity in the (underlying) entire dynamical time series, our experiments demonstrate the ARS model's capability to forecast future time series. From a theoretical perspective, we prove that a 2-dimensional time-invariant and linear system can be reconstructed by utilizing observations from a single, partially observed dimension of the system.
Paper Structure (20 sections, 2 theorems, 29 equations, 6 figures, 1 table)

This paper contains 20 sections, 2 theorems, 29 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symbols and notations
  3. Preliminaries
  4. Dynamical system
  5. Autoregressive model
  6. Problem setting
  7. Autoregressive with Slack Time Series (ARS) Model
  8. ARS model
  9. Estimation of the ARS model
  10. Numerical Experiments
  11. Settings
  12. Results
  13. Discussions and Conclusion
  14. Discussion 1: ARS model can recover the underlying true dynamical time series
  15. Discussion 2: Relation to state-space model
  16. ...and 5 more sections

Key Result

Proposition 1

Let $d=2,r=s=1$ and assume the identity eq:underlying_true_dynamics for the underlying true dynamics. With sufficiently large $n$, the future prediction via the ARS model: $\hat{z}_{n+k}=(1,0)\hat{B}^k\hat{\boldsymbol x}^{\ddagger}_{n}$ coincides with the underlying true $z_{n+k}$, for any $k \in \m

Figures (6)

  • Figure 1: Time series forecasting with (red) conventional AR model shown in Equation \ref{['eq:AR_observed']} and (blue) proposed ARS model shown in Equation \ref{['eq:ARS']}. See Section \ref{['subsec:ARS']} for further details of the experiment.
  • Figure 2: Circular motion. While (left) the entire dynamics $\boldsymbol x(t)=(\cos t,\sin t)$ is time-invariant (as $\boldsymbol x(t+h)$ can be identified as a function of $\boldsymbol x(t)$), (right) the partial observation $z(t)=x_1(t)=\cos t$ is time-variant (as the next state of $x(t)=0$ can be both $x(t+h)>0$ and $x(t+h)<0$ depending on the current time $t$).
  • Figure 3: (i) Circular motion, $n=100, \sigma=0$
  • Figure 4: (i) Circular motion, $n=100, \sigma=0.01$
  • Figure 5: (ii) Lorenz, $n=100, \sigma=0$
  • ...and 1 more figures

Theorems & Definitions (3)

  • Example 1
  • Proposition 1
  • Proposition 2