Forecast reconciliation with non-linear constraints

2025-10-24

AI Summary

Forecasting ratio-type variables (e.g., mortality or unemployment rates) introduces nonlinear structural constraints that challenge conventional time-series forecast reconciliation, which traditionally assumes linear aggregation relationships. Method: We propose Nonlinear Constraint Reconciliation (NLCR), a novel reconciliation framework that projects inconsistent base forecasts—those violating nonlinear structural constraints—onto the feasible constraint manifold via constrained numerical optimization, thereby ensuring coherent and calibrated predictions. Contribution/Results: This is the first work to systematically extend forecast reconciliation to nonlinear constraints. We derive theoretical sufficient conditions under which NLCR improves forecast accuracy—addressing a key gap left by classical linear reconciliation methods. The approach balances interpretability with computational tractability through efficient gradient-based optimization. Empirical evaluation across multiple real-world demographic and economic datasets demonstrates that NLCR consistently outperforms state-of-the-art baselines, validating its effectiveness and superiority in settings with complex, nonlinear structural dependencies.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Extends forecast reconciliation to non-linearly constrained time series
• Proposes method adjusting forecasts to satisfy non-linear constraints
• Improves forecast accuracy for demographic and economic datasets

Innovation

• Extends forecast reconciliation to non-linear constraints
• Projects forecasts onto non-linear surface via optimization
• Ensures adjusted forecasts meet non-linear constraints