EDMS: Ensembled Direct Multi-Step Forecasting

• The paper demonstrates that EDMS, by combining weighted ensembles with limited DMS retraining, achieves an average forecast improvement of 33.32% and reduces error propagation.
• EDMS employs a heterogeneous ensemble of models—including Linear, Polynomial, ETS, and LSTM—to mitigate model-specific weaknesses and adapt to varying data complexities.
• The methodology uses selective retraining at predefined intervals, balancing computational efficiency with robust multi-step forecast performance in volatile macroeconomic and financial series.

to=arxiv_search 开号网址json code Ensembled Direct Multi-Step (EDMS) is a time-series forecasting methodology that combines a performance-weighted ensemble of heterogeneous forecasting models with a limited Direct Multi-Step (DMS) retraining procedure. It was introduced for macroeconomic and financial forecasting, with the stated objective of leveraging the strengths of ensembling and DMS while exploiting their synergy: the ensemble reduces model-specific weakness, and DMS reduces the error accumulation typical of iterative forecasting. In the reported experiments on publicly available macroeconomic datasets and common financial indices data, EDMS was benchmarked against an ensemble applied to Iterative Multi-Step (IMS) forecasts and showed a significant performance improvement, averaging a 33.32% enhancement across the analysed datasets, with some cases above 60% (Łapiński et al., 17 Sep 2025).

1. Definition, motivation, and relation to iterative forecasting

The EDMS methodology was proposed in response to a central limitation of standard Iterative Multi-Step forecasting. In IMS, a model is trained once and then rolled forward recursively: the forecast from step $t+1$ is used as input for $t+2$, then the forecast for $t+2$ is used for $t+3$, and so on. This is computationally cheap because the model is not retrained at each horizon, but it is vulnerable to error propagation: a small error in the first forecast is fed into the next step, the error compounds across the horizon, and long-horizon forecasts can degrade sharply (Łapiński et al., 17 Sep 2025).

The paper identifies this problem as especially serious for non-stationary series, series with structural breaks, macroeconomic indicators such as GDP, CPI, and unemployment, and volatile financial data. Its DMS component addresses the issue by retraining the model on forecast-augmented data so that the forecasting system can adapt to the evolving series and reduce compounding errors. However, pure DMS is described as computationally costly because retraining at every step is expensive (Łapiński et al., 17 Sep 2025).

EDMS is designed as a compromise between these two regimes. It uses an ensemble at selected retraining points, does not retrain at every step, and uses IMS between retraining points. This suggests that the method should be understood not as a replacement for IMS everywhere, but as a hybrid schedule intended to preserve much of the computational efficiency of IMS while borrowing some of the robustness and adaptability of DMS (Łapiński et al., 17 Sep 2025).

2. Forecasting pipeline and retraining schedule

The EDMS pipeline has two key components: the construction of a weighted ensemble of several forecasting models and the use of that ensemble within a limited DMS schedule. The training and weighting procedure begins with multiple candidate models. Their out-of-sample performance is estimated using an 80/20 split, with 80% of the data used for training and 20% used for testing. Weights are then derived from inverse error, so that better-performing models receive higher weights (Łapiński et al., 17 Sep 2025).

The paper uses MAE as the primary weighting metric because it gave the best final results, although MSE, RMSE, and MAPE were also tested. The model-specific MAE is defined as

$$mae_i = \frac{1}{H}\sum_{h=1}^{H}\left|\widehat{y}_{i,t+h} - y_{t+h}\right|$$

and the weights are obtained by inversion and normalization: $$w_i = \frac{mae'_i}{\sum_i mae'_i}, \qquad mae'_i = 1 - \frac{mae_i}{\sum_i mae_i}.$$ The source notes that the typography is malformed, but the intended logic is explicit: lower MAE implies higher weight (Łapiński et al., 17 Sep 2025).

The ensemble forecast is a weighted average,

$$\widehat{y}_{t+h} = \sum_i w_i \widetilde{y}_{t+h,i},$$

where $\widehat{y}_{t+h}$ is the ensemble forecast, $\widetilde{y}_{t+h,i}$ is the forecast from model $i$, and $t+2$0 is the weight of model $t+2$1 (Łapiński et al., 17 Sep 2025).

The limited DMS schedule is the distinguishing operational feature. Rather than retraining at every horizon, EDMS retrains only at selected points. For economic data, retraining is performed at 1 year and 5 years; for financial data, the same logic is adapted to 1 month and 5 months. Between these retraining points, standard IMS is used (Łapiński et al., 17 Sep 2025).

The paper describes the method as a three-step process. First, the system forecasts the first horizon and ensembles the model outputs using weights derived from the validation split. Second, the training data are extended with the ensembled forecast from the first step, the models are retrained, and another ensembled forecast is generated further out. Third, the data are extended again with the next ensembled forecast, the models are retrained again, and the remainder of the horizon is forecast. The corresponding formulas are presented as

$t+2$2

$t+2$3

$t+2$4

with $t+2$5 denoting the sequence of forecasts up to that retraining point (Łapiński et al., 17 Sep 2025).

3. Ensemble composition and model heterogeneity

The ensemble used in EDMS is deliberately heterogeneous, spanning simple, medium-complexity, and complex model classes. The selection criterion is stated in terms of performance, complexity, and computational resource requirements, and the resulting set is intended to cover a full spectrum from simple Linear and Polynomial Regression to medium-complexity ETS and complex LSTM models (Łapiński et al., 17 Sep 2025).

Model	Characterization in the paper	Role in the ensemble
Average growth rate	Very simple baseline	Cheap, simple enrichment
Linear Regression	Low-complexity statistical model	Fits simple, regular trends
Polynomial Regression	More flexible than linear regression	Captures curvature
ETS / Holt’s exponential smoothing	Medium-complexity classical time-series model	Intermediate complexity
LSTM	Most complex model	Sequential dependence and long-range memory

For the average growth rate baseline, the paper gives

$t+2$6

and forecasts via repeated growth extrapolation: $t+2$7 Linear Regression and Polynomial Regression are included as low-complexity alternatives, with Polynomial Regression described as a slightly more flexible version of linear regression that can fit curvature (Łapiński et al., 17 Sep 2025).

For the ETS component, the paper reports that Holt’s additive trend model was the best ETS variant in the experiments, using

$t+2$8

$t+2$9

and

$t+2$0

This model is positioned between simple regression and deep learning in complexity (Łapiński et al., 17 Sep 2025).

The LSTM component is the most complex model in the ensemble. The paper briefly specifies the standard gating equations: $t+2$1

$t+2$2

$t+2$3

$t+2$4

The rationale for this mixture is that different series and different horizons may favor different model classes, and the ensemble can exploit this heterogeneity by assigning greater weight to the model that performs best on the validation set (Łapiński et al., 17 Sep 2025).

A further practical design choice is explicit exclusion. Prophet was excluded because it was computationally expensive and brought little or no accuracy gain. This is not a general claim about Prophet, but a paper-specific statement about model selection within the reported EDMS experiments (Łapiński et al., 17 Sep 2025).

4. Mathematical formulation and distinction from conventional ensemble forecasting

The mathematical core of EDMS is the combination of performance-weighted averaging with selective retraining. The ensemble forecast is

$t+2$5

and the weights are determined from model performance under the MAE-based construction already described. For cross-dataset evaluation, the paper uses average MAPE,

$t+2$6

while noting that the printed equation appears garbled in the source and that the intended meaning is standard average MAPE across all series and horizons. The reported improvement percentage is defined as

$t+2$7

again with the caveat that the source formatting is incomplete, but the intended measure is clear (Łapiński et al., 17 Sep 2025).

A crucial distinction in the paper is the contrast between EDMS and what it treats as the benchmark method, namely an ensemble conventionally applied to IMS-generated forecasts, denoted here as EIMS. In that benchmark, models are trained once, forecasts are recursively generated for the entire horizon, and the ensemble combines those IMS outputs: $t+2$8 Under that setup, ensemble weights are fixed from the original training/validation split (Łapiński et al., 17 Sep 2025).

EDMS differs in three linked ways: the ensemble is used at selected retraining points, forecasted values are fed back into training, and ensemble weights are recomputed after retraining. The paper therefore frames EDMS not as “ensemble + IMS” in the ordinary sense, but as an ensemble embedded within a limited DMS retraining framework. The authors’ interpretation is that the empirical gains arise from synergy: the ensemble improves forecast quality at each retraining step, DMS retrains on these better forecasts, the next-stage models therefore receive a more accurate training extension, and subsequent forecasts improve further (Łapiński et al., 17 Sep 2025).

A common misconception would be to equate EDMS with a simple weighted average of recursive forecasts. The methodological description in the source explicitly rejects that reading. The defining feature is not weighting alone, but weighting combined with forecast-augmented retraining and weight recomputation at selected horizons (Łapiński et al., 17 Sep 2025).

5. Experimental setup, datasets, and empirical findings

The empirical study evaluates EDMS on publicly available macroeconomic and financial data. The macroeconomic datasets include CPI, Employment, GDP in real terms, GDP in nominal terms, Population, Wholesale Producer Price Index, Unemployment rate, Interest rates, and HICP variants. The data are described as coming from national statistical offices, Eurostat, and S&P Global Market Intelligence. The financial data consist of daily stock indices from Yahoo Finance: Dow Jones Industrial Average, FTSE 100, Hang Seng, Nasdaq Composite, Nikkei 225, and S&P 500 (Łapiński et al., 17 Sep 2025).

The retraining schedule is adapted to data frequency. For annual macro data, retraining occurs at 1 year and 5 years. For quarterly data, the paper describes an analogous long-horizon retraining schedule. For monthly data, retraining occurs at 1 month and 5 months. For daily data, the ensemble is tested on index data with the same general framework, although the paper does not present an explicit week-based retraining schedule in the results tables (Łapiński et al., 17 Sep 2025).

The main evaluation metric is average MAPE, selected because it is scale-independent and suitable for comparing many heterogeneous time series. EDMS is benchmarked against EIMS. The headline result is an average improvement of 33.32% across all datasets. Broken down by frequency class, the reported averages are 28.26% for annual economic data, 36.84% for quarterly economic data, 48.53% for monthly economic data, and 20.49% for daily financial data (Łapiński et al., 17 Sep 2025).

Several individual cases exceed or approach the 60% range. The paper reports 64.21% for GDP Nominal (quarterly), 67.52% for Employment (monthly), 63.06% for CPI (monthly), 59.76% for Short-Term Interest Rates (monthly), 55.52% for Long-Term Interest Rates (monthly), and 50.73% for Long Interest Rates (annual). The abstract’s statement that performance sometimes reaches improvement above 60% is therefore supported by the reported quarterly and monthly macroeconomic results (Łapiński et al., 17 Sep 2025).

The improvements are not universal. Employment (annual) slightly favored EIMS, and Nikkei 225 also slightly favored EIMS. The paper interprets these as cases where the simpler iterative approach may be sufficient, or where the sample is less representative of EDMS’s strengths. This suggests that the method should not be treated as uniformly dominant over all datasets and all data-generating regimes, even within the paper’s own benchmark suite (Łapiński et al., 17 Sep 2025).

6. Interpretation, scope, and acronym ambiguity

The paper’s own explanation for EDMS centers on two effects. First, ensembling reduces model-specific weakness: better models get more weight, weaker models can still contribute useful information, and forecasts become more robust. Second, DMS reduces recursive error accumulation by retraining on forecast-augmented data, which the paper presents as particularly useful in non-stationary or structurally changing series. The authors state explicitly that DMS alone would not explain the full improvement and that the key mechanism is the synergy between DMS and performance-weighted ensembling (Łapiński et al., 17 Sep 2025).

Within time-series forecasting, EDMS therefore denotes a specific framework: a diverse ensemble of models, weighted by validation performance, operating inside a limited Direct Multi-Step retraining schedule, with IMS used only between retraining points to control computation. A plausible implication is that EDMS occupies an intermediate methodological position between fully recursive forecasting and full retraining at every step, with the ensemble serving as the stabilizing component during each retraining stage (Łapiński et al., 17 Sep 2025).

The acronym, however, is not unique across arXiv. In a separate control-theoretic context, a 2026 paper uses an EDMS-style formulation for direct multi-step Extended Dynamic Mode Decomposition in Koopman-based MPC. There, the method directly learns the condensed finite-horizon state-control map $t+2$9 rather than first identifying one-step lifted dynamics $t+3$0. Its stated advantages are convex least-squares learning, horizon-wise and coordinate-wise decomposition, optional $t+3$1-regularization and pruning, and the avoidance of recursive error compounding in long-horizon prediction (Wu et al., 17 Jan 2026).

That later usage does not define EDMS as “Ensembled Direct Multi-Step.” It instead refers to a direct multi-step EDMD framework for Koopman-MPC. The overlap is conceptual rather than terminological: both methods are motivated by the limitations of recursively applying one-step predictors over long horizons, and both seek to avoid error accumulation by aligning model fitting with multi-step prediction targets. The forecasting EDMS methodology and the Koopman-MPC direct multi-step EDMD method should therefore be distinguished carefully despite the shared emphasis on multi-step learning and the avoidance of recursive degradation (Wu et al., 17 Jan 2026).