  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.
