We consider conformal prediction for multivariate data and focus on hierarchical data, where some components are linear combinations of others. Intuitively, the hierarchical structure can be leveraged to reduce the size of prediction regions for the same coverage level. We implement this intuition by including a projection step (also called a reconciliation step) in the split conformal prediction [SCP] procedure, and prove that the resulting prediction regions are indeed globally smaller. We do so both under the classic objective of joint coverage and under a new and challenging task: component-wise coverage, for which efficiency results are more difficult to obtain. The associated strategies and their analyses are based both on the literature of SCP and of forecast reconciliation, which we connect. We also illustrate the theoretical findings, for different scales of hierarchies on simulated data.

翻译：我们考虑多元数据的共形预测问题，并重点关注分层数据，其中部分分量是其他分量的线性组合。直观而言，分层结构可用于在相同覆盖水平下缩小预测区域的规模。我们通过在分割共形预测（SCP）流程中引入投影步骤（亦称协调步骤）来实现这一直觉，并证明由此得到的预测区域确实全局更小。我们在经典联合覆盖目标以及具有挑战性的新任务——分量级覆盖（其效率结果更难获取）下均验证了这一点。相关策略及其分析基于SCP与预测协调两个领域的文献，我们在此建立了二者之间的关联。我们还通过不同规模分层结构的模拟数据，对理论发现进行了说明。