Conformal Prediction for Hierarchical Data

Abstract: We consider conformal prediction of multivariate data series, which consists of outputting prediction regions based on empirical quantiles of point-estimate errors. We actually consider hierarchical multivariate data series, for which some components are linear combinations of others. The intuition is that the hierarchical structure may be leveraged to improve the prediction regions in terms of their sizes for given coverage levels. We implement this intuition by including a projection step (also called reconciliation step) in the split conformal prediction [SCP] procedure and prove that the resulting prediction regions are indeed globally smaller than without the projection step. The associated strategies and their analyses rely on the literatures of both SCP and forecast reconciliation. We also illustrate the theoretical findings, both on artificial and on real data.