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Machine Learning in Montpellier, Theory & Practice

Depth measures have gained popularity in the statistical literature for defining level sets in complex data structures such as multivariate data, functional data, and random graphs. Despite their versatility, integrating depth measures into regression modeling for establishing prediction regions remains a largely underexplored area. To address this gap, we propose novel model-free uncertainty quantification algorithms based on conditional depth measures and the notion of conditional kernel mean embeddings. The new algorithms can be used to define prediction and tolerance regions when the predictors and responses are defined in separable Hilbert spaces. To enhance the practical utility of the algorithms, we also introduce a conformal prediction version, providing non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results and fast convergence rates in some special homoscedastic cases. We evaluate the model’s finite-sample performance in extensive simulation studies involving different functional data objects and probability distributions. Finally, we apply the approach to a digital health application related to physical activity, aiming to offer personalized recommendations to the U.S. population based on individuals’ characteristics.

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