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Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series

Jonghyeon Lee, Edward De Brouwer, Boumediene Hamzi, Houman Owhadi

TL;DR

This work proposes to address the problem of breaking down the vector field of the dynamical system by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the data-adapted kernels of Kernel Flows.

Abstract

A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a kernel. In particular, this strategy is highly efficient (both in terms of accuracy and complexity) when the kernel is data-adapted using Kernel Flows (KF)\cite{Owhadi19} (which uses gradient-based optimization to learn a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used for interpolation). Despite its previous successes, this strategy (based on interpolating the vector field driving the dynamical system) breaks down when the observed time series is not regularly sampled in time. In this work, we propose to address this problem by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the (KF) data-adapted kernels. We compare our approach with the classical one over different benchmark dynamical systems and show that it significantly improves the forecasting accuracy while remaining simple, fast, and robust.

Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Series

TL;DR

This work proposes to address the problem of breaking down the vector field of the dynamical system by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the data-adapted kernels of Kernel Flows.

Abstract

A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a kernel. In particular, this strategy is highly efficient (both in terms of accuracy and complexity) when the kernel is data-adapted using Kernel Flows (KF)\cite{Owhadi19} (which uses gradient-based optimization to learn a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used for interpolation). Despite its previous successes, this strategy (based on interpolating the vector field driving the dynamical system) breaks down when the observed time series is not regularly sampled in time. In this work, we propose to address this problem by directly approximating the vector field of the dynamical system by incorporating time differences between observations in the (KF) data-adapted kernels. We compare our approach with the classical one over different benchmark dynamical systems and show that it significantly improves the forecasting accuracy while remaining simple, fast, and robust.
Paper Structure (27 sections, 3 theorems, 23 equations, 11 figures, 7 tables)

This paper contains 27 sections, 3 theorems, 23 equations, 11 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Statement of the problem and proposed solution
  3. The problem
  4. A reminder on kernel methods for regularly sampled time series
  5. A reminder on the Kernel Flows (KF) algorithm
  6. The problem with irregularly sampled time series
  7. The proposed solution
  8. Experiments
  9. Multi-step predictions:
  10. Overview.
  11. Hénon map
  12. Van der Pol oscillator
  13. Lorenz
  14. Remark:
  15. Conclusion
  16. ...and 12 more sections

Key Result

Proposition 5.1

If $K$ is a reproducing kernel of a Hilbert space ${\mathcal{H}}$, then i. $K(x,y)$ is unique. ii. $\forall x,y \in {\mathcal{X}}$, $K(x,y)=K(y,x)$ (symmetry). iii. $\sum_{i,j=1}^q\beta_i\beta_jK(x_i,x_j) \ge 0$ for $\beta_i \in \mathbb{R}$, $x_i \in {\mathcal{X}}$ and $q\in\mathbb{N}_+$ (positive d

Figures (11)

  • Figure 1: Irregularly sampled training data for Hénon map
  • Figure 2: Hénon map reconstructions when the kernel parameters are not learnt (regression of model \ref{['eqjhdbdjehbd']} without learning the kernel).
  • Figure 3: Hénon map attractor reconstructions with learnt kernels.
  • Figure 4: Reconstruction (prediction) of the test time series of the Hénon map.
  • Figure 5: Van der Pol oscillator without learning the kernel (horizon has been reduced to 1).
  • ...and 6 more figures

Theorems & Definitions (4)

  • Definition 5.1
  • Proposition 5.1
  • Theorem 5.1
  • Theorem 5.2