Leaky integrators reduce error buildup in time series forecasting
Leaky-integrator reconstruction: taming error accumulation in recursive differenced time-series forecasting
Artificial Intelligence
Summary
Forecasting future values in time series often involves predicting small changes and summing them up, but this can cause errors to build up and get worse over time. The authors explain why this happens mathematically and suggest a simple fix: instead of adding all changes equally, slightly reduce the weight of past changes during integration. This adjustment prevents errors from exploding and improves prediction accuracy over long time horizons without needing to retrain models.
What this means in practice
- •For data scientists: Improve accuracy of existing time series forecasting models by applying leaky integration without retraining models.
- •For financial analysts: Enhance long-term forecasts in financial time series by reducing error growth using a stable integration method.
Authors
Zijiang Yang
Abstract
We introduce leaky-integrator reconstruction, a training-free method that cures the error accumulation of recursive differenced forecasting. Our first contribution is diagnostic: predicting one-step changes and integrating them by cumulative summation, the standard remedy for non-stationarity, is a discrete integrator with a pole on the unit circle, and we show this makes recursive rollout of a nonlinear model diverge, its 336-step error reaching several times that of a well-behaved forecaster (normalised MAE 1.6-3.8 versus about 0.8) across every neural architecture tested. Our second, central contribution is the fix: move the pole inside the unit circle with a leaky integrator H(z) = 1/(1 - gamma z^-1), gamma < 1, which provably bounds the accumulated error variance. Applied at reconstruction time with a single fixed gamma=0.9 (no retraining, a two-line change to any deployed one-step or foundation-model forecaster), it shrinks error at every horizon, the mean gain over seven diverging architectures and twenty datasets growing from ~3% at H=24 to 23% at H=96, 37% at H=192 and 51% (43-74% across those architectures) at H=336 (78% with an oracle pole). Crucially, it is provably inert where no pathology exists (stable or joint predictors already at the irreducible rate), making it a safe, general default.