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 presents a new combinatorial optimisation task, the Subset Sum Matching Problem (SSMP), which is an abstraction of common financial applications such as trades reconciliation. We present three algorithms, two suboptimal and one optimal, to solve this problem. We also generate a benchmark to cover different instances of SSMP varying in complexity, and carry out an experimental evaluation to assess the performance of the approaches.

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.

In human-robot interactions, human and robot agents maintain internal mental models of their environment, their shared task, and each other. The accuracy of these representations depends on each agent's ability to perform theory of mind, i.e. to understand the knowledge, preferences, and intentions of their teammate. When mental models diverge to the extent that it affects task execution, reconciliation becomes necessary to prevent the degradation of interaction. We propose a framework for bi-directional mental model reconciliation, leveraging large language models to facilitate alignment through semi-structured natural language dialogue. Our framework relaxes the assumption of prior model reconciliation work that either the human or robot agent begins with a correct model for the other agent to align to. Through our framework, both humans and robots are able to identify and communicate missing task-relevant context during interaction, iteratively progressing toward a shared mental model.

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.

Reconciliation has become an essential tool in multivariate point forecasting for hierarchical time series. However, there is still a lack of understanding of the theoretical properties of probabilistic Forecast Reconciliation techniques. Meanwhile, Conformal Prediction is a general framework with growing appeal that provides prediction sets with probabilistic guarantees in finite sample. In this paper, we propose a first step towards combining Conformal Prediction and Forecast Reconciliation by analyzing how including a reconciliation step in the Split Conformal Prediction (SCP) procedure enhances the resulting prediction sets. In particular, we show that the validity granted by SCP remains while improving the efficiency of the prediction sets. We also advocate a variation of the theoretical procedure for practical use. Finally, we illustrate these results with simulations.

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.