Multi-Step Prediction Horizons

• Multi-step prediction horizons are forecasting models that predict a vector of future values from past observations, capturing inter-step dependencies.
• Techniques include recursive, direct, and multi-output strategies, with adaptive methods mitigating error accumulation and enhancing calibration.
• Applications span resource allocation, risk management, autonomous systems, and reinforcement learning, with uncertainty quantification methods ensuring robust predictions.

Multi-step prediction horizons refer to the simultaneous forecasting of multiple future time points in temporal modeling, extending beyond the immediate next-step prediction. Instead of producing a single future estimate, models designed for multi-step horizons generate a vector of predictions $[y_{t+1}, y_{t+2}, \ldots, y_{t+H}]$, where $H$ is the desired lookahead. This paradigm is fundamental in practical applications such as resource allocation, risk management, control systems, and sequential decision-making, where understanding future trajectories and associated uncertainties is crucial.

1. Mathematical Formulation of Multi-Step Horizons

Let $\{y_t\}_{t=1}^T$ be a univariate time series. For a given horizon $H$, multi-step prediction aims to learn a mapping

$$\hat{\mathbf{y}}_{t+1:t+H} = \mathcal{F}(y_{t-w+1:t})$$

from a window of $w$ past observations to $H$ future values. The objective function is typically the mean squared error (MSE) over all horizon steps: $$L = \frac{1}{H} \sum_{h=1}^H \left(y_{t+h} - \hat{y}_{t+h}\right)^2$$ Alternative formulations enable point estimation, quantile regression for interval predictions, and control-specific loss functions.

Multi-step methods differ from recursive (iterated) forecasting, where a single-step model is repeatedly applied using its own predictions as inputs, potentially inducing compounding error and loss of dependency across steps. Multi-output approaches treat all future points jointly, mitigating this effect but increasing model complexity with horizon length.

2. Prediction Strategies and Methodologies

A. Canonical Strategies:

• Iterated (Recursive): Train a one-step model, apply recursively. Prone to error accumulation for large $H$.
• Direct: Train $H$ separate models, each predicting $y_{t+h}$. Reliable but ignores inter-horizon dependence and scales linearly in cost with $H$.
• Multi-output (MIMO): Train a single model predicting $[y_{t+1},...,y_{t+H}]$ in one shot. Preserves dependency structure between steps, better accuracy for longer horizons.
• Hybrid: Combine MIMO or direct models with iterated feedback, e.g., PSO-MISMO (Bao et al., 2013), DirRec, and Rectify methods (Green et al., 29 Dec 2024).

B. Adaptive Partitioning:

Advanced model selection such as PSO-MISMO (Bao et al., 2013) partitions $H$ into variable-length sub-horizons dynamically, tuning segment lengths with swarm optimization and assigning each sub-horizon to a dedicated neural network. This adapts the model architecture to local nonstationarities in step dependencies, outperforming static, equal-sized block methods.

C. Strategy Selection:

Meta-models and dynamic strategy selection (DyStrat) (Green et al., 13 Feb 2024) use time-series classification to pick the optimal forecasting strategy instance by instance, exploiting local bias-variance properties.

D. Unified Parameterization:

The Stratify framework (Green et al., 29 Dec 2024) unifies existing and hybrid strategies by parameterizing the horizon segmentation ("chunks" $\sigma$) and strategy type, recommending dynamic search across the $(\sigma_\text{base}, \sigma_\text{rectifier})$ plane for task-specific optimality.

3. Uncertainty Quantification and Conformal Prediction

Traditional uncertainty quantification (UQ) methods struggle to capture multi-step dependencies and temporal variation. Conformal Prediction (CP) for multi-step horizons augments model-agnostic statistical interval construction:

A. Dual-Splitting Conformal Prediction (DSCP):

DSCP (Yu et al., 27 Mar 2025) simultaneously clusters forecast vectors (vertical split) and merges adjacent horizon steps with similar residual distributions (horizontal split, via Kolmogorov–Smirnov test), yielding cluster-window cells of residuals. This enables quantile-based interval construction per horizon block, preserving coverage and producing interval widths that increase only mildly with horizon length.

• For calibration, $K_t$ is grouped using k-means; windows $w$ are determined by KS-tests.
• For test samples, intervals $$[L_{t,h}, U_{t,h}] = [\hat{y}_{t+h} + Q^{-}_{c,w(h)}, \hat{y}_{t+h} + Q^{+}_{c,w(h)}]$$ are produced per horizon step $h$.
• DSCP yields up to 23.59% improvement in Winkler Score versus other CP variants for $b>1$; coverage remains nominal as $H$ grows.

B. Autocorrelated Multi-step CP (AcMCP):

AcMCP (Wang et al., 17 Oct 2024) specifically models serial correlation in forecast errors up to lag $h-1$ for each horizon, fitting AR/MA residual models on calibration sets. Online interval quantiles are adjusted via PID-like rules and residual predictions, providing asymptotic coverage guarantees and narrower intervals than independent CP methods.

4. Deep, Graph, and Reinforcement-Learning Models

A. Graph and Spatiotemporal Models:

Hierarchical GCN architectures (STG2Seq (Bai et al., 2019)) segment long and short-term encoding, attenuating error propagation and leveraging separate feature streams. Attention is applied temporally and per-channel, further enhancing horizon-specific predictions while maintaining long-range dependencies.

B. Predictive Coding and Latent Abstraction:

Extended-horizon predictive coding (Gaudet et al., 2019, Ratzon et al., 12 Nov 2025) demonstrates that multi-step objectives and open-loop training (rather than K-step loss) induce learning of low-dimensional, global manifold representations. Sufficient horizon length results in a collapse onto structured latent representations, even in deep nonlinear networks. Participation ratio and principal component metrics confirm the emergent ordering effect of multi-step objectives.

C. Model-Based RL Multi-step Losses:

Multi-step weighted loss functions (Benechehab et al., 5 Feb 2024) for one-step dynamics stabilize long-horizon prediction, especially under observation noise. By minimizing weighted MSE across $H$ steps,

$$L(\theta) = \sum_{h=1}^H w_h \mathbb{E}_{(s_t,a_t)\sim D}\left[ \lVert s_{t+h} - \hat{p}_\theta^h(s_t, a_{t:t+h-1}) \rVert^2 \right]$$

multi-step objectives regularize optimization and reduce compounding errors, yielding up to 60% $\overline{R}^2$ improvement under nontrivial noise levels.

5. Practical Recommendations for Model Selection

A. Strategy Selection by Horizon Length:

• For short horizons ($H \lesssim 5$), recursive or direct methods suffice.
• For intermediate ($6 \leq H \leq 12$), multi-output models (MIMO, M-SVR (Bao et al., 2014)) and adaptive hybrid strategies deliver best trade-offs.
• For long horizons ($H>12$), chunked multi-output or dual-splitting conformal approaches retain accuracy and calibrated statistical coverage.

B. Ensemble Construction:

Dynamic weighting (arbitrating, windowing (Cerqueira et al., 2023)) yields benefits for $h \leq 3$; static equal-weighting becomes preferable as feedback weakens for $h > 10$.

C. Smoothing and Regularization:

Smooth multi-period regression (Tuzhilina et al., 2022) employs low-degree basis expansions for horizon-dependent coefficients, reducing variance and avoiding "wiggle" artifacts in longer-term forecasts.

D. Initialization and State Representation:

Efficient state initialization (NN-based) for sequence models (Mohajerin et al., 2018) is essential for stable multi-step rollouts, particularly for RNN and LSTM architectures in control systems.

6. Quantitative Trends, Evaluation Protocols, and Limitations

Across benchmarks:

• All methods exhibit error growth with increasing horizon, but multi-output, DSCP, and AcMCP approaches moderate this growth versus naive recursive extension.
• Allocation of model complexity and segment granularity is critical; PSO-MISMO dynamically tunes proper segmentation for improved stability.
• Ensemble dynamic weights lose efficacy as feedback becomes sparse at $h>10$.
• Deep networks trained with multi-step horizons show marked simplicity bias leading to better latent recovery (Ratzon et al., 12 Nov 2025).

Limitations include:

• Increased calibration demands for conformal methods at large $H$ (Wang et al., 17 Oct 2024).
• Model optimization cost for very high-dimensional joint multi-step predictors.
• Need to tune chunk sizes, merge thresholds ($\Theta$), cluster counts, or smoothing degrees by cross-validation in practice.

7. Application Domains and Impact

Multi-step prediction is integral to:

• Energy and IT resource trajectory planning (DSCP (Yu et al., 27 Mar 2025) yields 11.25% carbon emission reduction via predictive optimization).
• Financial and commodity forecasting—e.g., WTI crude oil (MIMO strategy achieves lowest SMAPE and computational load (Xiong et al., 2014, Bao et al., 2014)).
• Autonomous vehicles and robotics—e.g., TCN/RNN-based horizon predictions for quadrotors and driving cost maps (Looper et al., 2021, Amirloo et al., 2021).
• Reinforcement learning—Dyna-style policy training and cascaded latent planning (Gaudet et al., 2019, Fang et al., 2019, Wagner et al., 2021).

The choice of methodology, tuning strategy, and uncertainty quantification is dependent on both statistical properties of the series, desired coverage, and computational constraints. Multi-step prediction horizons remain an active area of research, with state-of-the-art performance determined by adaptive, model-agnostic, and horizon-aware innovations across the forecast, representation, and calibration stack.