ATEMPO: Amortized Transfer Entropy Model Per-lag Over-Time

Abstract

When one time series informs another, the delay at which that information arrives determines if models can utilize this relationship effectively. A source may be strongly predictive at one delay but uninformative at neighboring lags. Therefore, fixed input windows may obscure this structure by bundling informative and irrelevant history together. Transfer entropy (TE) can recover these delays by measuring how much a source's past tells us about a target beyond the target's own history. However, existing TE estimators require separate fits across source and conditioning depths, with further estimation on local windows to track delays over time. We introduce ATEMPO, a diffusion-based transfer entropy estimator amortized over lags and time. ATEMPO samples source and conditioning depths during training and conditions explicitly on temporal position, allowing a single score network to estimate the full TE surface without refitting. Differencing this surface recovers when information arrives. Across synthetic datasets with ground-truth TE, ATEMPO achieves the lowest mean relative error among baselines, and on datasets with known lags, ATEMPO achieves $0.98\pm0.04$ lag-detection accuracy. ATEMPO requires a single fit across source depths $K=6$ and target depths $L=3$, whereas existing estimators require 18 separate fits to evaluate the same $K \times L$ grid. On three systems with time-varying lag, ATEMPO is the only estimator to maintain perfect lag-detection accuracy across all evaluated window lengths. These recovered delays improve downstream forecasting: ATEMPO-selected inputs reduce MSE on WEATHER-5K by 7.25%, while 120 selected inputs outperform the benchmark's 240-feature default by 4.6%. Across three brain-computer interface tasks, using ATEMPO-selected lags achieves the highest or equal-best $R^2$, improving mean $R^2$ by 4.1% relative to the strongest competing baseline.