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.

翻译：近年来，遵循线性约束的时间序列预测方法取得了显著发展，尤其是随着预测协调技术的出现。本文针对非线性约束时间序列这一开放性问题，将预测协调方法进行了扩展。非线性约束可能出现在以比率形式构成的变量中，例如死亡率和失业率。在方法论层面，本文提出了非线性约束协调算法。该算法对不满足非线性约束的预测进行调整，以确保新预测满足约束条件。NLCR方法本质上是向非线性曲面的投影，可表述为一个约束优化问题。在理论层面，本文再次运用优化方法，推导出NLCR方法能够保证提升预测精度的充分条件。最后在实证层面，将NLCR应用于人口学和经济学中的两个数据集，结果表明相较于相关基准方法，NLCR显著提升了预测精度。