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Conformal Prediction for Hierarchical Data

Guillaume Principato, Gilles Stoltz, Yvenn Amara-Ouali, Yannig Goude, Bachir Hamrouche, Jean-Michel Poggi

TL;DR

This paper develops a framework that combines conformal prediction with forecast reconciliation to construct valid, efficient prediction regions for hierarchical multivariate data. It introduces joint-coverage ellipsoidal SCP and a hierarchical, component-wise SCP with a projection-based reconciliation that enforces coherence across levels. The authors prove that reconciliation can yield smaller prediction regions than non-reconciled methods under both joint and component-wise guarantees, and they provide oracle and practical implementations (including minimum-trace projections) with efficiency guarantees. Empirical results on synthetic hierarchies confirm substantial efficiency gains, particularly for weighted component-wise objectives, and highlight robustness considerations for large-scale hierarchies. The work lays a theoretical and practical foundation for hierarchy-aware, distribution-free uncertainty quantification in forecasting settings, with potential extensions to hierarchical time series and adaptive conformal inference.

Abstract

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.

Conformal Prediction for Hierarchical Data

TL;DR

This paper develops a framework that combines conformal prediction with forecast reconciliation to construct valid, efficient prediction regions for hierarchical multivariate data. It introduces joint-coverage ellipsoidal SCP and a hierarchical, component-wise SCP with a projection-based reconciliation that enforces coherence across levels. The authors prove that reconciliation can yield smaller prediction regions than non-reconciled methods under both joint and component-wise guarantees, and they provide oracle and practical implementations (including minimum-trace projections) with efficiency guarantees. Empirical results on synthetic hierarchies confirm substantial efficiency gains, particularly for weighted component-wise objectives, and highlight robustness considerations for large-scale hierarchies. The work lays a theoretical and practical foundation for hierarchy-aware, distribution-free uncertainty quantification in forecasting settings, with potential extensions to hierarchical time series and adaptive conformal inference.

Abstract

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.

Paper Structure

This paper contains 60 sections, 18 theorems, 119 equations, 6 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work
  4. Forecast reconciliation.
  5. Conformal prediction.
  6. Contributions and challenges
  7. Main contributions.
  8. Technical challenges.
  9. Outline
  10. Objectives and methodology
  11. Objectives, part 1: joint coverage vs. component-wise coverage guarantees
  12. Objectives, part 2: efficiency
  13. Methodology: split conformal prediction [SCP], performed jointly or component-wise
  14. Multivariate SCP for joint coverage based on ellipsoidal sets
  15. Component-wise SCP for component-wise coverage
  16. ...and 45 more sections

Key Result

Theorem 1

Algorithms eq:alg:messoudi and eq:alg:reconciled messoudi, used with any regression algorithm $\mathcal{A}$ and any estimation procedure $\mathcal{E}$, ensure that whenever data $(\boldsymbol{x}_t,\boldsymbol{y}_t)_{1 \leqslant t \leqslant T+1}$ is i.i.d., In addition, and under no assumption on the data, Algorithm eq:alg:reconciled messoudi outputs prediction ellipsoids $\mathring{E}$ that are u

Figures (6)

  • Figure 1: An example of a hierarchical structure with its associated structural matrix $H$.
  • Figure 2: How to report concisely the indicators defined in \ref{['eq:indMCcomponent1']} for a given element $i$ in the hierarchy (and two methods).
  • Figure 3: Component-wise coverages and efficiencies achieved for the methods considered on Configuration 1 of Appendix \ref{['app:datageneration']}. This figure should be read as indicated in Figure \ref{['fig:given-i']}.
  • Figure 4: Component-wise coverages and efficiencies achieved for the methods considered on Configuration 2 of Appendix \ref{['app:datageneration']}. This figure should be read as indicated in Figure \ref{['fig:given-i']}.
  • Figure : Plain component-wise SCP with signed scores
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1
  • Example 1
  • Remark 1
  • Theorem 1: informal statement
  • Theorem 2: Coverage
  • Definition 2
  • Definition 3
  • Theorem 3
  • Lemma 1: Minimum-trace projection, wickramasuriya2019optimal
  • Theorem 4
  • ...and 24 more