Model-Free Kernel Conformal Depth Measures Algorithm for Uncertainty Quantification in Regression Models in Separable Hilbert Spaces

Depth measures are powerful tools for defining level sets in emerging, non--standard, and complex random objects such as high-dimensional multivariate data, functional data, and random graphs. Despite their favorable theoretical properties, the integration of depth measures into regression modeling to provide prediction regions remains a largely underexplored area of research. To address this gap, we propose a novel, model-free uncertainty quantification algorithm based on conditional depth measures--specifically, conditional kernel mean embeddings and an integrated depth measure. These new algorithms can be used to define prediction and tolerance regions when predictors and responses are defined in separable Hilbert spaces. The use of kernel mean embeddings ensures faster convergence rates in prediction region estimation. To enhance the practical utility of the algorithms with finite samples, we also introduce a conformal prediction variant that provides marginal, non-asymptotic guarantees for the derived prediction regions. Additionally, we establish both conditional and unconditional consistency results, as well as fast convergence rates in certain homoscedastic settings. We evaluate the finite--sample performance of our model in extensive simulation studies involving various types of functional data and traditional Euclidean scenarios. Finally, we demonstrate the practical relevance of our approach through a digital health application related to physical activity, aiming to provide personalized recommendations

Deep Learning Framework with Uncertainty Quantification for Survey Data: Assessing and Predicting Diabetes Mellitus Risk in the American Population

Complex survey designs are commonly employed in many medical cohorts. In such scenarios, developing case-specific predictive risk score models that reflect the unique characteristics of the study design is essential. This approach is key to minimizing potential selective biases in results. The objectives of this paper are: (i) To propose a general predictive framework for regression and classification using neural network (NN) modeling, which incorporates survey weights into the estimation process; (ii) To introduce an uncertainty quantification algorithm for model prediction, tailored for data from complex survey designs; (iii) To apply this method in developing robust risk score models to assess the risk of Diabetes Mellitus in the US population, utilizing data from the NHANES 2011-2014 cohort. The theoretical properties of our estimators are designed to ensure minimal bias and the statistical consistency, thereby ensuring that our models yield reliable predictions and contribute novel scientific insights in diabetes research. While focused on diabetes, this NN predictive framework is adaptable to create clinical models for a diverse range of diseases and medical cohorts. The software and the data used in this paper is publicly available on GitHub.