Title: Random Objects: Distance Profiles and Conformal Prediction

Abstract: Random objects are complex random variables taking values in general metric spaces. Although such data are increasingly common in scientific research, current statistical methodology and theory remain limited. The primary challenge in analyzing such data lies in the absence of vector space operations, such as addition, subtraction, scalar multiplication, and inner products, which are fundamental tools in conventional statistical methodologies.

This talk explores object data with distance profiles and their application to conformal prediction. We introduce conditional profile average transport costs by comparing distance profiles through the optimal transport. A novel score function for random objects is proposed, enabling the construction of prediction sets using the split conformal algorithm. We develop a theoretical framework to establish uniform convergence rates for the local linear estimator involving function classes defined on metric spaces and the asymptotic conditional validity of the prediction sets. The practical utility of our proposed methodology is demonstrated through applications to network data from New York taxi trips and compositional data from brain imaging studies.