Conformal uncertainty quantification using kernel depth measures in separable Hilbert spaces

TL;DR

This paper introduces a model-free conformal uncertainty quantification method using depth measures and kernel embeddings for complex data in Hilbert spaces, with applications in functional data and digital health.

Contribution

It proposes a novel, practical approach combining depth measures, kernel embeddings, and conformal prediction for uncertainty quantification in complex regression models.

Findings

Provides non-asymptotic guarantees for prediction regions

Demonstrates fast convergence rates in homoscedastic cases

Shows strong finite sample performance in simulations

Abstract

Depth measures have gained popularity in the statistical literature for defining level sets in complex data structures like multivariate data, functional data, and graphs. Despite their versatility, integrating depth measures into regression modeling for establishing prediction regions remains underexplored. To address this gap, we propose a novel method utilizing a model-free uncertainty quantification algorithm based on conditional depth measures and conditional kernel mean embeddings. This enables the creation of tailored prediction and tolerance regions in regression models handling complex statistical responses and predictors in separable Hilbert spaces. Our focus in this paper is exclusively on examples where the response is a functional data object. To enhance practicality, we introduce a conformal prediction algorithm, providing non-asymptotic guarantees in the derived prediction region. Additionally, we establish both conditional and unconditional consistency results and fast convergence rates in some special homoscedastic cases. We evaluate the model finite sample performance in extensive simulation studies with different function objects as probability distributions and functional data. Finally, we apply the approach in a digital health application related to physical activity, aiming to offer personalized recommendations in the US. population based on individuals' characteristics.