In this paper, we introduce the forecast reconciliation packages FoReco and FoRecoML for R (RCore Team 2026). Forecast reconciliation adjusts forecasts for linearly constrained multiple time series (such as hierarchical or grouped series, or series observed at different temporal frequencies) so that they are coherent with respect to the underlying constraints, improving both accuracy and consistency for informed decision making. The contributions of the packages are threefold. First, FoReco and FoRecoML are the first to offer functionality for forecast reconciliation methods across cross-sectional, temporal and cross-temporal frameworks. Second, the packages provide a comprehensive set of forecast reconciliation approaches, including classical (e.g., top-down, bottom-up and middle-out) and regression based reconciliation methods - in FoReco - as well as non-linear reconciliation methods using machine learning - in FoRecoML. A third key contribution is their unified design, which enables easy-to-use forecast reconciliation functions built on the same philosophy, regardless of the reconciliation framework or method.

Premium automotive manufacturers face increasingly complex forecasting challenges due to high product variety, sparse variant-level data, and volatile market dynamics. This study addresses monthly automobile demand forecasting across a multi-product, multi-market, and multi-level hierarchy using data from a German premium manufacturer. The methodology combines point and probabilistic forecasts across strategic and operational planning levels, leveraging ensembles of LightGBM models with pooled training sets, quantile regression, and a mixed-integer linear programming reconciliation approach. Results highlight that spatiotemporal dependencies, as well as rounding bias, significantly affect forecast accuracy, underscoring the importance of integer forecasts for operational feasibility. Shapley analysis shows that short-term demand is reactive, shaped by life cycle maturity, autoregressive momentum, and operational signals, whereas medium-term demand reflects anticipatory drivers such as online engagement, planning targets, and competitive indicators, with online behavioral data considerably improving accuracy at disaggregated levels.

Methods for forecasting time series adhering to linear constraints have seen notable development in recent years, especially with the advent of forecast reconciliation. This paper extends forecast reconciliation to the open question of non-linearly constrained time series. Non-linear constraints can emerge with variables that are formed as ratios such as mortality rates and unemployment rates. On the methodological side, Non-linearly Constrained Reconciliation (NLCR) is proposed. This algorithm adjusts forecasts that fail to meet non-linear constraints, in a way that ensures the new forecasts meet the constraints. The NLCR method is a projection onto a non-linear surface, formulated as a constrained optimisation problem. On the theoretical side, optimisation methods are again used, this time to derive sufficient conditions for when the NLCR methodology is guaranteed to improve forecast accuracy. Finally on the empirical side, NLCR is applied to two datasets from demography and economics and shown to significantly improve forecast accuracy relative to relevant benchmarks.

This paper focuses on forecasting hierarchical time-series data, where each higher-level observation equals the sum of its corresponding lower-level time series. In such contexts, the forecast values should be coherent, meaning that the forecast value of each parent series exactly matches the sum of the forecast values of its child series. Existing hierarchical forecasting methods typically generate base forecasts independently for each series and then apply a reconciliation procedure to adjust them so that the resulting forecast values are coherent across the hierarchy. These methods generally derive an optimal reconciliation, using a covariance matrix of the forecast error. In practice, however, the true covariance matrix is unknown and has to be estimated from finite samples in advance. This gap between the true and estimated covariance matrix may degrade forecast performance. To address this issue, we propose a robust optimization framework for hierarchical reconciliation that accounts for uncertainty in the estimated covariance matrix. We first introduce an uncertainty set for the estimated covariance matrix and formulate a reconciliation problem that minimizes the worst-case expected squared error over this uncertainty set. We show that our problem can be cast as a semidefinite optimization problem. Numerical experiments demonstrate that the proposed robust reconciliation method achieved better forecast performance than existing hierarchical forecasting methods, which indicates the effectiveness of integrating uncertainty into the reconciliation process.

Hierarchical forecasting with reconciliation requires forecasting values of a hierarchy (e.g.~customer demand in a state and district), such that forecast values are linked (e.g.~ district forecasts should add up to the state forecast). Basic forecasting provides no guarantee for these desired structural relationships. Reconciliation addresses this problem, which is crucial for organizations requiring coherent predictions across multiple aggregation levels. Current methods like minimum trace (MinT) are mostly limited to tree structures and are computationally expensive. We introduce FlowRec, which reformulates hierarchical forecast reconciliation as a network flow optimization, enabling forecasting on generalized network structures. While reconciliation under the $\ell_0$ norm is NP-hard, we prove polynomial-time solvability for all $\ell_{p > 0}$ norms and , for any strictly convex and continuously differentiable loss function. For sparse networks, FlowRec achieves $O(n^2\log n)$ complexity, significantly improving upon MinT's $O(n^3)$. Furthermore, we prove that FlowRec extends MinT to handle general networks, replacing MinT's error-covariance estimation step with direct network structural information. A key novelty of our approach is its handling of dynamic scenarios: while traditional methods recompute both base forecasts and reconciliation, FlowRec provides efficient localised updates with optimality guarantees. Monotonicity ensures that when forecasts improve incrementally, the initial reconciliation remains optimal. We also establish efficient, error-bounded approximate reconciliation, enabling fast updates in time-critical applications. Experiments on both simulated and real benchmarks demonstrate that FlowRec improves accuracy, runtime by 3-40x and memory usage by 5-7x. These results establish FlowRec as a powerful tool for large-scale hierarchical forecasting applications.

Abstract--In this paper, we investigate meta-learning for combining forecasts generated by models of different types . While typical approaches for combining forecasts involve s imple averaging, machine learning techniques enable more sophis ti-cated methods of combining through meta-learning, leading to improved forecasting accuracy. We use linear regression, k - nearest neighbors, multilayer perceptron, random forest, and long short-term memory as meta-learners. We define global and local meta-learning variants for time series with compl ex seasonality and compare meta-learners on multiple forecas ting problems, demonstrating their superior performance compa red to simple averaging. Ensemble methods are widely recognized as a cornerstone of modern machine learning (ML) [1], commonly used for regression and classification problems. In addition, ensem bling has proven to be a highly effective approach for increasing the predictive power of forecasting models. The ensemble approach in forecasting, which involves combining the predictions of multiple models, can be justified for several reasons. First of all, it usually leads to increased accurac y. Ensemble models often outperform individual models, as the y leverage the strengths of different models and minimize the ir weaknesses. By combining diverse models, the ensemble can produce more accurate predictions by capturing a broader range of patterns and insights from the data. Ensembling als o allows for the incorporation of multiple drivers into the da ta generating process, mitigating uncertainties regarding m odel form and parameter specification [2].

Forecast reconciliation is considered an effective method for achieving coherence and improving forecast accuracy. However, the value of reconciled forecasts in downstream decision-making tasks has been mostly overlooked. In a multi-agent setup with heterogeneous loss functions, this oversight may lead to unfair outcomes, hence resulting in conflicts during the reconciliation process. To address this, we propose a value-oriented forecast reconciliation approach that focuses on the forecast value for individual agents. Fairness is ensured through the use of a Nash bargaining framework. Specifically, we model this problem as a cooperative bargaining game, where each agent aims to optimize their own gain while contributing to the overall reconciliation process. We then present a primal-dual algorithm for parameter estimation based on empirical risk minimization. From an application perspective, we consider an aggregated wind energy trading problem, where profits are distributed using a weighted allocation rule. We demonstrate the effectiveness of our approach through several numerical experiments, showing that it consistently results in increased profits for all agents involved.

Hierarchical time series forecasting plays a crucial role in decision-making in various domains while presenting significant challenges for modelling as they involve multiple levels of aggregation, constraints, and availability of information. This study explores the influence of distinct information utilisation on the accuracy of hierarchical forecasts, proposing and evaluating locals and a range of Global Forecasting Models (GFMs). In contrast to local models, which forecast each series independently, we develop GFMs to exploit cross-series and cross-hierarchies information, improving both forecasting performance and computational efficiency. We employ reconciliation methods to ensure coherency in forecasts and use the Mean Absolute Scaled Error (MASE) and Multiple Comparisons with the Best (MCB) tests to assess statistical significance. The findings indicate that GFMs possess significant advantages for hierarchical forecasting, providing more accurate and computationally efficient solutions across different levels in a hierarchy. Two specific GFMs based on LightGBM are introduced, demonstrating superior accuracy and lower model complexity than their counterpart local models and conventional methods such as Exponential Smoothing (ES) and Autoregressive Integrated Moving Average (ARIMA).

Time series forecasts of different temporal granularity are widely used in real-world applications, e.g., sales prediction in days and weeks for making different inventory plans. However, these tasks are usually solved separately without ensuring coherence, which is crucial for aligning downstream decisions. Previous works mainly focus on ensuring coherence with some straightforward methods, e.g., aggregation from the forecasts of fine granularity to the coarse ones, and allocation from the coarse granularity to the fine ones. These methods merely take the temporal hierarchical structure to maintain coherence without improving the forecasting accuracy. In this paper, we propose a novel granularity message-passing mechanism (GMP) that leverages temporal hierarchy information to improve forecasting performance and also utilizes an adaptive reconciliation (AR) strategy to maintain coherence without performance loss. Furthermore, we introduce an optimization module to achieve task-based targets while adhering to more real-world constraints. Experiments on real-world datasets demonstrate that our framework (GMP-AR) achieves superior performances on temporal hierarchical forecasting tasks compared to state-of-the-art methods. In addition, our framework has been successfully applied to a real-world task of payment traffic management in Alipay by integrating with the task-based optimization module.

Platform businesses operate on a digital core and their decision making requires high-dimensional accurate forecast streams at different levels of cross-sectional (e.g., geographical regions) and temporal aggregation (e.g., minutes to days). It also necessitates coherent forecasts across all levels of the hierarchy to ensure aligned decision making across different planning units such as pricing, product, controlling and strategy. Given that platform data streams feature complex characteristics and interdependencies, we introduce a non-linear hierarchical forecast reconciliation method that produces cross-temporal reconciled forecasts in a direct and automated way through the use of popular machine learning methods. The method is sufficiently fast to allow forecast-based high-frequency decision making that platforms require. We empirically test our framework on a unique, large-scale streaming dataset from a leading on-demand delivery platform in Europe.