Koopman Analysis of Sea Surface Temperature with a Signature Kernel

This paper presents a trajectory-based Koopman method that utilizes signature kernels to encode finite-time history in sea surface temperature data, enabling improved multi-year forecasting and the identification of coherent spectral modes through kernel extended dynamic mode decomposition.

The Gist

Imagine you are trying to predict the ocean's behavior, but you only have maps of the sea surface temperature taken every month. You stack them up in a notebook. If you try to guess tomorrow's temperature just by looking at today's number, you might get it wrong because the weather has a "memory." It depends on what happened last week, last month, or even last year.

This paper is about a new, clever way to predict Sea Surface Temperature (SST)—the temperature of the ocean's surface—by treating the data not as a series of isolated snapshots, but as a continuous, moving story.

Here is the breakdown of their method, using simple analogies:

1. The Problem: The "Snapshot" Trap

Most traditional methods look at the ocean like a photographer taking a picture every month. They say, "The ocean was 20°C in January, 21°C in February..." and try to predict March based only on February.

The Flaw: The ocean is a complex system. If you only look at the ocean's surface temperature without seeing the wind or the deep water, the system looks "forgetful" (non-Markovian). It's like trying to guess the ending of a movie by only looking at the current frame, without remembering the plot that came before.

2. The Solution: The "Movie Reel" Approach

Instead of looking at single frames, the authors treat the data as a movie reel.

• The Old Way: "What is the temperature right now?"
• The New Way: "What did the temperature do over the last year?"

They take a year's worth of temperature data and wrap it up into a single "path" or "trajectory." Think of this path as a unique fingerprint of how the ocean behaved that year. Did it heat up slowly? Did it spike suddenly? Did it oscillate? The shape of this "yearly path" contains the memory the old methods missed.

3. The Secret Sauce: The "Signature Kernel"

Now, how do you compare two different "yearly paths"? You can't just subtract one number from another. You need a way to measure the shape and the order of the path.

The authors use something called a Signature Kernel.

• The Analogy: Imagine two people walking through a park.
  • Old Method (Snapshot): You only check where they are standing at 1:00 PM.
  • New Method (Signature): You watch their entire walk. You notice: "Ah, this person walked in a circle, then went straight, then turned left."
  • The "Signature" is a mathematical way of encoding that entire journey (the loops, the turns, the speed) into a single, complex description. It captures the order of events. If you walk left-then-right, it's different from right-then-left, even if you end up in the same spot.

This "Signature" allows the computer to understand the history of the ocean's temperature, not just its current state.

4. The Magic Machine: The "Koopman Operator"

Once they have these "yearly paths," they need a machine to predict the next year's path based on the current one.

They use a technique called Koopman Analysis.

• The Analogy: Imagine the ocean's behavior is a chaotic dance. It's hard to predict the next move of a single dancer (the temperature). But, if you look at the music or the pattern of the dance, the rules become simple and linear.
• The Koopman Operator is like a "pattern translator." It takes the complex, messy, non-linear dance of the ocean and translates it into a simple, linear rule. Once translated, predicting the future becomes as easy as following a straight line on a graph.

5. The Results: Better Forecasts and Hidden Rhythms

The authors tested this new method against the old "snapshot" methods and simple climate averages.

• Better Prediction: Their "Movie Reel + Signature" method predicted future ocean temperatures much better, especially for long-term forecasts (5 to 10 years out). It showed meaningfully improved forecast skill relative to the baselines.
• Finding the Rhythm: Because their method is based on a "linear translator," they could easily see the hidden rhythms of the ocean. They found specific "modes" (repeating patterns) that look like famous climate phenomena:
  • A 20-year cycle (like a slow, deep breathing of the ocean).
  • A 9-year cycle (similar to the Pacific Decadal Oscillation).
  • A 3-year cycle (related to El Niño).

Summary

In short, this paper says: "Stop looking at the ocean one photo at a time. Watch the whole movie, understand the story of the path, and use a special mathematical translator to predict the next chapter."

By respecting the history and order of the data, they built a tool that doesn't just guess the future; it understands the deep, rhythmic memory of the Earth's oceans.

Technical Summary

1. Problem Statement

The paper addresses the challenge of modeling and forecasting Sea Surface Temperature (SST) dynamics using only SST observations, without access to the full coupled ocean-atmosphere state.

• Non-Markovian Nature: When observing only a subset of the system (SST alone), the dynamics appear history-dependent (non-Markovian) because the unobserved atmospheric and deep-ocean states influence the surface evolution.
• Nonlinearity: The underlying physics are inherently nonlinear, making simple linear models (like Linear Inverse Models) insufficient for capturing complex memory effects.
• Limitations of Existing Methods:
  • Standard Linear Models: Assume Markovianity and linearity, failing to capture long-term memory.
  • Neural Operators: While powerful for prediction, they are often "black boxes" that do not naturally provide interpretable spectral diagnostics (eigenmodes, frequencies, decay rates) essential for climate science.
  • Takens Embedding: Traditional delay-coordinate embeddings do encode history, though in a different form that does not explicitly preserve the path-level temporal order structure leveraged by signatures.

2. Methodology

The authors propose a Trajectory-based Koopman Analysis framework that lifts the dynamics from instantaneous fields to annual path segments, utilizing Signature Kernels to handle high-dimensional, non-Markovian data.

A. State Representation: Annual Path Segments

Instead of treating SST as a sequence of monthly snapshots, the method treats the state $X_t$ as a continuous annual trajectory (a path) derived from 12 months of SST anomalies.

• Preprocessing: Monthly SST fields are converted to anomalies using a strictly past-only rolling climatology to ensure no future data leakage.
• Path Construction: A year's anomalies are cumulatively summed to form a piecewise-linear path $X_t: [0, 1] \to \mathbb{R}^d$.
• Dynamics: The system evolution is modeled as a one-year shift operator: $F(X_t) = X_{t+1}$.

B. Lifting via Signature Kernels

To linearize the nonlinear dynamics on this path space, the authors use Koopman Operator Theory, which describes evolution as a linear operator acting on observables (functions of the state).

• Signature Features: They employ Path Signatures, a mathematical tool from rough path theory that encodes the temporal order of a trajectory through iterated integrals. This captures the "shape" and history of the path.
• Kernelization: Since computing explicit signature tensors for high-dimensional SST fields is computationally prohibitive, they use a Signature Kernel ($\kappa_{sig}$). This maps paths into a Reproducing Kernel Hilbert Space (RKHS) where inner products can be computed efficiently via Gram matrices.
• Comparison Baseline: They compare against a Sum-of-Pairs Kernel (SPK), which compares monthwise components independently (ignoring temporal order within the year).

C. Learning the Operator: Kernel EDMD

The authors implement Kernel Extended Dynamic Mode Decomposition (kEDMD):

1. Construct Gram matrices ($G$) and cross-Gram matrices ($A$) using the signature kernel on pairs of consecutive annual paths $(X_t, X_{t+1})$.
2. Solve the generalized eigenproblem $Av = \mu Gv$ to approximate the Koopman operator.
3. Process the resulting matrix to obtain a finite-dimensional Koopman matrix $K$.

D. Evaluation Protocols

To ensure rigorous validation, two strictly time-ordered protocols are used:

• LFO (Leave-Future-Out): For forecasting skill. The model is trained on data strictly prior to an anchor year $t_0$ and tested on $t_0+s$.
• LSO (Leave-s-Out): For spectral diagnostics. Used for hyperparameter selection and to ensure the extracted modes are robust, excluding specific forward blocks during training.

3. Key Contributions

1. Trajectory-Based State Space: The paper introduces a novel representation of SST dynamics using annual path segments rather than instantaneous fields, explicitly encoding finite-time history to address non-Markovianity.
2. Signature-Kernel kEDMD: It proposes a scalable, kernelized approach to Koopman learning on high-dimensional paths. This allows for the extraction of linear operators from nonlinear, history-dependent data without explicit feature truncation.
3. Unified Forecasting and Diagnostics: The framework provides a single estimator that simultaneously delivers:
   • Out-of-sample multi-year forecasts (superior to climatology and SPK baselines).
   • Spectral diagnostics (eigenvalues, eigenfunctions, and Koopman modes) that reveal coherent climate variability patterns.
4. Rigorous Time-Ordered Validation: The study strictly adheres to "no future data" constraints in both training and hyperparameter selection, ensuring the results are not artifacts of data leakage.

4. Results

The method was tested on the NOAA ERSSTv5 dataset (1854–2022).

• Forecast Skill (LFO):

  • The Signature-Kernel method (SigK-EDMD) outperformed both the Climatology baseline and the SPK-kEDMD baseline across all lead times (1 to 12 years).
  • The improvement was most significant at multi-year to decadal scales.
  • While SPK improved pattern correlation over climatology, it often failed to reduce RMSE. SigK-EDMD consistently reduced RMSE, indicating that respecting the temporal order of the path is crucial for accurate error minimization.
  • Spatial analysis showed improvements were heterogeneous, with significant gains in the extratropical North Pacific, Atlantic, and Indian Ocean.

• Spectral Diagnostics (LSO):

  • The estimated Koopman eigenvalues were concentrated near the unit circle, indicating weakly damped, persistent oscillatory modes.
  • Three dominant modes were identified that qualitatively match known climate phenomena:
    • ~20-year period: Is suggestive of the Kuroshio–Oyashio Extension (KOE).
    • ~9.1-year period: Is suggestive of the Pacific Decadal Oscillation (PDO).
    • ~2.9-year period: Is suggestive of Central-Pacific ENSO (CP-ENSO).
  • These modes were extracted directly from the operator without post-hoc clustering, demonstrating the method's ability to uncover intrinsic climate variability.

5. Significance

• Bridging Data Science and Climate Physics: The paper demonstrates that advanced machine learning techniques (Koopman theory, signature kernels) can be applied to geophysical data to produce interpretable, physics-consistent results (spectral modes) rather than just black-box predictions.
• Handling Partial Observability: It offers a robust solution for modeling complex systems where only a subset of variables (SST) is available, effectively "learning" the memory of the unobserved system through path signatures.
• Scalability: By using kernel methods, the approach scales to high-dimensional spatiotemporal fields (global SST grids) where explicit tensor-based signature methods would fail.
• Future Applications: The framework is presented as a general tool for any high-dimensional time series where history dependence is critical, extending beyond climate science to other complex dynamical systems.

Note on Data: The authors explicitly acknowledge that ERSSTv5 is a statistically reconstructed product. Therefore, the extracted modes represent the dynamics of the reconstructed dataset, serving as a proof-of-concept for the methodology rather than a definitive physical discovery of new climate modes.