Right States, Wrong Paths: Controlling Temporal Dependence in Chaotic Forecasting

In chaotic systems, small forecast errors grow over time, setting a predictability horizon beyond which forecasts lose track of the reference trajectory. Past this horizon, the goal shifts from tracking a realization to reproducing the correct statistics. Yet matching the distribution of visited states (state-space fidelity) is not enough: those states must also be organized correctly in time (path-space fidelity). We show that the Energy Score (ES), a standard probabilistic objective whose expectation over whole trajectories is uniquely minimized by the true joint distribution, becomes increasingly insensitive to errors in temporal dependence as the effective number of spatial dimensions grows, while remaining sensitive to marginal-variance errors. Similar ES values can therefore hide very different temporal dynamics. We propose a post-hoc calibration method that makes temporal dependence an explicit, adjustable target. For generators with per-time stochastic inputs, such as the coarse-to-fine JUMP–COMPLETE forecaster we use, correlating these inputs across forecast times changes temporal dependence while keeping the learned weights fixed and the one-time latent marginals unchanged. The coupling is selected with a temporal Variogram Score; we show that its standard finite-ensemble estimator is biased and can move temporal correlations away from the reference, and use an unbiased version instead. Across three chaotic systems, calibration substantially improves temporal correlations and, for two of them, temporal spectra, while joint ES remains nearly unchanged.