Direct Multi-Step Forecasting

• Direct Multi-Step Forecasting (DMS) is a methodology that builds distinct models for each forecast horizon to directly predict future time series values.
• It encompasses both single-output and multi-output techniques that effectively capture cross-horizon dependencies and address seasonal or noisy patterns.
• Practical enhancements like deseasonalization, automated input selection, and forecast combination significantly improve DMS performance while managing computational trade-offs.

Direct Multi-Step Forecasting (DMS) refers to a family of strategies for predicting the future values of a time series over multiple steps ahead by employing specialized models that directly target each forecast horizon, as opposed to cascading one-step predictions. DMS encompasses a range of single-output and multi-output techniques with differing abilities to address serial dependence, error accumulation, temporal structure, and operational constraints. The strategy is widely studied in the context of financial, economic, energy, epidemiological, and complex network time series.

1. Formulation and Core Strategies

The core principle of Direct Multi-Step Forecasting is to construct a function or model for each forecast horizon $h$:

$y_{t+h} = f_h(y_t, y_{t-1}, \dots, y_{t-d+1}) + w$

where $f_h$ may be any regression or machine learning model specifically trained to predict the value at time $t+h$ using a lagged window of $d$ past observations (Taieb et al., 2011, Xiong et al., 2014, Duarte et al., 26 Sep 2025). For horizon $H$, this approach requires the construction of $H$ distinct models.

An alternative is the multi-output variant, or Multi-Input Multi-Output (MIMO) approach:

$$\left[y_{t+1},\ldots, y_{t+H}\right] = F(y_t, y_{t-1}, \dots, y_{t-d+1}) + w$$

where a single model is trained to jointly estimate the entire sequence of the next $H$ future values. DIRMO strategies generalize between single-output and multi-output by operating on blocks of size $s \mid H$.

DMS is often contrasted with:

• Recursive (Iterative) methods, which use a single model for one-step prediction and repeatedly feed back predicted values as inputs—leading to error accumulation.
• DirRec (Direct-Recursive hybrid) methods, which combine some aspects of iterative feeding of past forecasts with horizon-specific modeling.

These strategies can be unified and parameterized under the Stratify framework as particular instances of a two-stage parameterized space (Green et al., 29 Dec 2024).

2. Comparative Performance: Single-output vs. Multi-output DMS

Extensive empirical studies demonstrate that multi-output DMS methods (MIMO, DIRMO) consistently yield superior results over single-output DMS (Direct) and Recursive methods, especially in the presence of serial dependence and complex seasonal patterns (Taieb et al., 2011, Cerqueira et al., 2023, Zhang et al., 2023). For instance, in the NN5 competition, MIMO-based models achieved test SMAPE* scores of approximately 18.8%, outperforming all single-output (Direct, Recursive, DirRec) approaches. This advantage is attributed to the preservation of cross-horizon dependencies, which is crucial for trend- and seasonality-rich data.

Direct multi-step methods, by design, avoid the error propagation seen in recursive strategies, since each forecast is computed from actual historical data rather than from previous model outputs. However, this comes at the cost of increased computational workload, as a distinct model must be trained for each horizon and horizon-specific dynamics might be underexploited if dependencies are strong (Xiong et al., 2014, Duarte et al., 26 Sep 2025).

Summary Table: Direct vs. Recursive vs. Multi-output

Strategy	Number of Models	Error Propagation	Cross-Horizon Dependency	Computational Cost
Recursive	1	High	Not preserved	Low
Direct	H	None	Not preserved	High
Multi-output	1 (joint)	None	Preserved	Intermediate/High

3. Practical Enhancements: Deseasonalization, Input Selection, and Combination

DMS forecasting accuracy, particularly in real-world applications, can be significantly improved by targeted data preprocessing and model selection strategies:

• Deseasonalization: Removing seasonality before model fitting leads to consistent improvements in all DMS strategies. Forecasts generated on deseasonalized data uniformly achieve lower error rates, with both single-output and multi-output models benefiting (Taieb et al., 2011).
• Input Variable Selection: Employing automated selection methods such as the Delta Test and forward–backward selection identifies informative lagged variables. The performance gain is most substantial when selection is applied to deseasonalized data, since this reduces redundancy and avoids the removal of seasonality signals inadvertantly.
• Forecast Combination: Rather than relying on a single best model, combining forecasts through simple means (equal weighting, COMB) or weighted averages (weights inversely proportional to test error, WCOMB) reduces overall SMAPE and mitigates overfitting to transient data structure. Combination is especially effective with multi-output strategies.

4. Computational and Model Selection Trade-offs

While DMS avoids the error accumulation of recursive strategies, it introduces computational and modeling challenges:

• Model Proliferation: The direct approach requires $\mathcal{O}(H)$ models for horizon $H$, increasing estimation and maintenance cost, particularly for high-dimensional or long-horizon forecasting (Xiong et al., 2014, Duarte et al., 26 Sep 2025).
• Variance and Dependency: Since each direct model is trained independently, inter-horizon stochastic dependencies may be omitted unless multi-output (MIMO/DIRMO) strategies are used (Taieb et al., 2011). In short or noisy series (e.g., mortality data), this can lead to higher variance and less reliable predictions (Duarte et al., 26 Sep 2025).
• Strategy Identification: No single DMS or hybrid method is universally optimal. Empirical evidence from the Stratify and DyStrat frameworks confirms large instance-level variance in optimal strategy, necessitating data-driven or classifier-based approaches for strategy selection (Green et al., 29 Dec 2024, Green et al., 13 Feb 2024).

5. Uncertainty Quantification and Interval Forecasting

Quantifying predictive uncertainty in DMS is addressed through several advanced conformal prediction methods tailored for multi-step settings:

• CopulaCPTS: Employs copula functions to capture cross-step dependencies in nonconformity scores, yielding more efficient, calibrated joint prediction regions compared to naive Bonferroni adjustments (Sun et al., 2022).
• AcMCP: Modifies online conformal prediction updates to account for the serial correlation structure of forecast errors in DMS, providing theoretically valid, adaptive prediction intervals across horizons, even under nonstationarity (Wang et al., 17 Oct 2024).
• DSCP: Introduces dual splitting (horizontal and vertical via trend clustering and KS-based error merging) to produce sharper, temporally consistent prediction intervals. This approach achieved up to 23.59% improvement in Winkler Score over previous conformal methods and delivered real-world benefits such as 11.25% CO₂ reduction in power management via predictive scheduling (Yu et al., 27 Mar 2025).

6. Domains of Application and Empirical Evidence

DMS frameworks are widely applied in:

• Economic and Financial Forecasting: Enhanced ensemble DMS (EDMS) techniques integrating regression, smoothing, and deep models yield on average a 33.32% improvement (and up to 60%) on macroeconomic indicators. This is particularly effective for data with regime shifts and long-memory effects (Łapiński et al., 17 Sep 2025).
• Volatile and Noisy Series: For short mortality series, DMS is less robust than recursive methods in hybrid systems, due to increased parameterization and data limitations (Duarte et al., 26 Sep 2025).
• Complex Network and Knowledge Graph Dynamics: DMS strategies based on multi-output and discriminatively trained models (e.g., SeDyT) achieve significant improvements in event link forecasting, as measured by MRR (Zhou et al., 2021).
• Energy, Demand, and Transportation: Hybrid DMS models with graph convolution, attention, or ensemble scenarios drive advances in passenger demand forecasting, wind power, and energy management with robust uncertainty quantification (Bai et al., 2019, Sarkar et al., 2023, Alruqimi et al., 15 Jul 2024).

7. Unified Strategy Spaces and the Need for Dynamic Selection

Recent advances (Green et al., 29 Dec 2024, Green et al., 13 Feb 2024) formalize the space of multi-step forecasting strategies, showing that:

• All common and novel DMS (direct, recursive, hybrid) strategies can be understood within a single parameterized “base-plus-rectifier” framework, such as Stratify.
• There is no single optimal DMS strategy across tasks: empirical benchmarking over hundreds of dataset-model-horizon combinations shows that practitioners must systematically search the parameterized space to optimize performance for context-specific requirements.
• Data-driven dynamic selection schemes (DyStrat) that map time series instances to optimal DMS strategies via classifiers (e.g., time series random forests) yield an average 11% reduction in MSE and improve top-1 strategy assignment accuracy threefold compared to static methods.

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In summary, Direct Multi-Step Forecasting encompasses a methodology for mapping lagged historical features to future time points by direct modeling, with extensions to multi-output and hybrid approaches. Real-world effectiveness hinges on careful model selection, input engineering, data preprocessing, uncertainty quantification, and, increasingly, dynamic or parameterized strategy selection. While multi-output DMS dominates in many settings, the space of optimal strategies is highly instance- and data-dependent. Frameworks that integrate dynamic model selection, ensemble weighting, and principled interval forecasting continue to advance the state of the art in this challenging domain.