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Koopman Analysis of Sea Surface Temperature with a Signature Kernel

Nozomi Sugiura

Abstract

We develop a trajectory-based Koopman method for sea surface temperature (SST) that lifts annual SST segments using a signature kernel -- a reproducing kernel Hilbert space (RKHS) kernel that compares paths via iterated-integral features -- and learns the one-year shift operator. By operating on annual trajectory segments rather than instantaneous fields, the method encodes finite-time history, which helps capture memory effects in SST-only evolution. The resulting operator improves out-of-sample multi-year forecast skill relative to a climatology baseline and reveals coherent spectral modes. We implement the approach via kernel extended dynamic mode decomposition (EDMD) on signature-kernel Gram matrices, yielding a single pipeline for forecasting and spectral diagnostics of high-dimensional SST dynamics.

Koopman Analysis of Sea Surface Temperature with a Signature Kernel

Abstract

We develop a trajectory-based Koopman method for sea surface temperature (SST) that lifts annual SST segments using a signature kernel -- a reproducing kernel Hilbert space (RKHS) kernel that compares paths via iterated-integral features -- and learns the one-year shift operator. By operating on annual trajectory segments rather than instantaneous fields, the method encodes finite-time history, which helps capture memory effects in SST-only evolution. The resulting operator improves out-of-sample multi-year forecast skill relative to a climatology baseline and reveals coherent spectral modes. We implement the approach via kernel extended dynamic mode decomposition (EDMD) on signature-kernel Gram matrices, yielding a single pipeline for forecasting and spectral diagnostics of high-dimensional SST dynamics.
Paper Structure (77 sections, 50 equations, 6 figures)

This paper contains 77 sections, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Koopman operators and data-driven approximations.
  4. Ocean and SST dynamics: from linear Markov models to Koopman.
  5. Neural operators as predictive baselines and their limitations for mode diagnostics.
  6. Why pathwise representations are needed.
  7. Gap addressed by this work.
  8. Methods
  9. Pipeline overview
  10. Annual path segments from monthly anomalies
  11. Signature-kernel lifting on path space
  12. Signature kernel.
  13. Gram matrices.
  14. Koopman matrix estimation by kEDMD
  15. Orthogonalization and Koopman matrix.
  16. ...and 62 more sections

Figures (6)

  • Figure 1: Overview of the trajectory-based Koopman pipeline via signature-kernel kEDMD.
  • Figure 2: Leave-future-out (LFO) forecast skill as a function of lead time (August-start). Left: kPC. Right: RMSE.
  • Figure 3: Spatial distribution of RMSE difference relative to climatology at $s=5$ years (August-start, LFO). We plot $\Delta\mathrm{RMSE}=\mathrm{RMSE}_{\text{clim}}-\mathrm{RMSE}_{\text{method}}$ (in $\celsius$); positive values indicate improvement over climatology. Left: SigK-EDMD. Right: SPK.
  • Figure 4: Spectral diagnostics under LSO (August-start). Left: Koopman eigenvalues of $K$ (unit circle as reference). Right: heatmap of $K^\ast K$.
  • Figure 5: Selected Koopman modes under LSO (August-start, $s=5$): mode #12 (period $\approx 20$ yr; KOE-like), #22 (period $\approx 9.1$ yr; PDO-like), and #60 (period $\approx 2.9$ yr; CP-ENSO-like). These labels indicate qualitative resemblance to commonly discussed SST-variability patterns. Right: corresponding mode coordinate along the record, shown in a de-amplified form for readability. The plotted Koopman time series shows eigenfunction values $a_k(t)=\psi_k(X_t)$ with eigenvalue $\mu_k$; any decay when $|\mu_k|<1$ reflects the fitted finite-dimensional approximation (finite samples and truncation), not physical dissipation.
  • ...and 1 more figures