Epistemic Error Decomposition for Multi-step Time Series Forecasting: Rethinking Bias-Variance in Recursive and Direct Strategies

Multi-step forecasting is often described through a simple rule of thumb: recursive strategies are said to have high bias and low variance, while direct strategies are said to have low bias and high variance. We revisit this belief by decomposing the expected multi-step forecast error into three parts: irreducible noise, a structural approximation gap, and an estimation-variance term. For linear predictors we show that the structural gap is identically zero for any dataset. For nonlinear predictors, however, the repeated composition used in recursion can increase model expressivity, making the structural gap depend on both the model and the data. We further show that the estimation variance of the recursive strategy at any horizon can be written as the one-step variance multiplied by a Jacobian-based amplification factor that measures how sensitive the composed predictor is to parameter error. This perspective explains when recursive forecasting may simultaneously have lower bias and higher variance than direct forecasting. Experiments with multilayer perceptrons on the ETTm1 dataset confirm these findings. The results offer practical guidance for choosing between recursive and direct strategies based on model nonlinearity and noise characteristics, rather than relying on traditional bias-variance intuition.

Stratify: Unifying Multi-Step Forecasting Strategies

A key aspect of temporal domains is the ability to make predictions multiple time steps into the future, a process known as multi-step forecasting (MSF). At the core of this process is selecting a forecasting strategy, however, with no existing frameworks to map out the space of strategies, practitioners are left with ad-hoc methods for strategy selection. In this work, we propose Stratify, a parameterised framework that addresses multi-step forecasting, unifying existing strategies and introducing novel, improved strategies. We evaluate Stratify on 18 benchmark datasets, five function classes, and short to long forecast horizons (10, 20, 40, 80). In over 84% of 1080 experiments, novel strategies in Stratify improved performance compared to all existing ones. Importantly, we find that no single strategy consistently outperforms others in all task settings, highlighting the need for practitioners explore the Stratify space to carefully search and select forecasting strategies based on task-specific requirements. Our results are the most comprehensive benchmarking of known and novel forecasting strategies. We make code available to reproduce our results.

Time-Series Classification for Dynamic Strategies in Multi-Step Forecasting

Multi-step forecasting (MSF) in time-series, the ability to make predictions multiple time steps into the future, is fundamental to almost all temporal domains. To make such forecasts, one must assume the recursive complexity of the temporal dynamics. Such assumptions are referred to as the forecasting strategy used to train a predictive model. Previous work shows that it is not clear which forecasting strategy is optimal a priori to evaluating on unseen data. Furthermore, current approaches to MSF use a single (fixed) forecasting strategy. In this paper, we characterise the instance-level variance of optimal forecasting strategies and propose Dynamic Strategies (DyStrat) for MSF. We experiment using 10 datasets from different scales, domains, and lengths of multi-step horizons. When using a random-forest-based classifier, DyStrat outperforms the best fixed strategy, which is not knowable a priori, 94% of the time, with an average reduction in mean-squared error of 11%. Our approach typically triples the top-1 accuracy compared to current approaches. Notably, we show DyStrat generalises well for any MSF task.