MATH

Introducing Koopman Operator for Time Series Forecasting

1. Introduction

The Koopman operator provides a powerful lens for analyzing nonlinear dynamical systems by lifting them into an infinite-dimensional linear space. This tutorial explains its mathematical foundations, numerical implementation, and applications in time series forecasting, where it enables linear prediction of complex nonlinear trends.

2. Dynamical Systems & the Koopman Perspective

2.1 Problem Setup

Consider a discrete-time nonlinear system:
\(\mathbf{x}_{t+1} = \mathbf{F}(\mathbf{x}_t)\)
where \(\mathbf{x}_t \in \mathbb{R}^n\) is the system state (e.g., stock prices, sensor readings). Directly modeling \(\mathbf{F}\) is challenging due to nonlinearity.

2.2 Koopman Operator Definition

The Koopman operator \(\mathcal{K}\) acts on observables \(g: \mathbb{R}^n \to \mathbb{C}\):
\(\mathcal{K} g(\mathbf{x}_t) = g(\mathbf{F}(\mathbf{x}_t)) = g(\mathbf{x}_{t+1})\)
Critically, \(\mathcal{K}\) is linear even if \(\mathbf{F}\) is nonlinear. This allows spectral analysis and global linear prediction.

3. Spectral Decomposition & Forecasting

3.1 Eigenfunctions and Modes

Let \(\phi_j(\mathbf{x})\) be eigenfunctions of \(\mathcal{K}\) with eigenvalues \(\lambda_j\):
\(\mathcal{K} \phi_j = \lambda_j \phi_j\)
Any observable \(g(\mathbf{x})\) can be expanded as:
\(g(\mathbf{x}) = \sum_{j=1}^\infty v_j \phi_j(\mathbf{x})\)
where \(v_j\) are Koopman modes. The time evolution becomes:
\(g(\mathbf{x}_t) = \sum_{j=1}^\infty v_j \lambda_j^t \phi_j(\mathbf{x}_0)\)

3.2 Forecasting with Koopman Modes

For the identity observable \(\mathbf{g}(\mathbf{x}) = \mathbf{x}\):
\(\mathbf{x}_t = \sum_{j=1}^\infty \mathbf{v}_j \lambda_j^t \phi_j(\mathbf{x}_0)\)
This provides a linear predictive model in Koopman space, even for nonlinear systems.

7. Challenges in Time Series Forecasting

  1. Observable Selection: Poor observables lead to biased predictions.
  2. Mode Truncation: Finite-dimensional \(\mathbf{K}\) loses long-term accuracy.
  3. Non-Stationarity: \(\mathcal{K}\) may drift for time-varying systems.

8. Applications

  1. Finance: Predict stock volatility with nonlinear Koopman modes.
  2. Energy: Forecast electricity demand from temperature/social trends.
  3. Epidemiology: Model disease spread with linearized dynamics.

9. Further Reading

The Koopman operator transforms nonlinear forecasting into a linear problem in a learned latent space, blending dynamical systems theory with modern machine learning. By mastering its spectral properties and approximation techniques, you can tackle complex temporal patterns with rigor and efficiency.