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Consistent Long-Term Forecasting of Ergodic Dynamical Systems

ICML 2024 · Time series › Causal and generative time series · approach: Dynamical system reconstruction · 0 citations (OpenAlex)

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Proposes deflate-learn-inflate (DLI), which removes the invariant-measure eigenpair via feature centering so any kernel Koopman/transfer operator estimator yields mass-preserving, uniformly consistent MMD forecasts of distribution flows for ergodic systems.

Problem

Kernel Koopman/transfer operator estimators forecast state distributions poorly over long horizons because spurious leading eigenvalues make the estimated flow decay or diverge instead of converging to the invariant measure.

Approach

Deflate the known leading eigenpair (1, 1_pi) of the Koopman operator, learn the deflated operator with empirically centered features (any KRR/PCR/RRR estimator), and inflate by adding back the empirical invariant-measure embedding; this yields asymptotically stable dynamics, mass preservation, and MMD bounds uniform over all future times.

Key results

• On the Ornstein-Uhlenbeck experiment, DLI-enhanced RRR yields consistently lower relative MMD than uncentered RRR across the entire trajectory.
• On US mortgage rate forecasting, DLI-RRR achieves average CRPS 0.203 ± 0.106, vs 0.213 for Cox-Ingersoll-Ross and 0.228 for RRR.
• Theory gives MMD bounds uniform over all t >= 1 with rate n^{-alpha/(2(alpha+beta))}, and the method adds only O(n^2) centering cost.

Measured effects reported in the paper

Surprising findings

• A transient hump in MMD error appears for both RRR and DLI-RRR in short-term forecasting, attributed to non-normality of the Koopman operator induced by the Gaussian kernel RKHS.

Limitations

• Analysis assumes i.i.d. or stationary ergodic data and a known invariant-measure structure with L2 density assumptions.
• Long-term guarantees rely on bounded Kreiss/transient constants; non-normal operators can cause transient error growth.
• Experiments are limited to two settings (simulated OU process and one mortgage-rate series).

Experimental setup

Datasets: simulated Ornstein-Uhlenbeck process weekly US 30-year fixed mortgage rates

Benchmarks: Ornstein-Uhlenbeck relative MMD US mortgage rates CRPS (2017-2018)

Metrics: relative MMD continuous ranked probability score

Baselines: Reduced Rank Regression Kernel Ridge Regression Principal Component Regression Cox-Ingersoll-Ross model

Builds on: Koopman operator regression kernel ridge regression reduced rank regression kernel mean embedding

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