Exploration of Random Objects Through Increasing Ball Statistics

We introduce a novel method for exploring complex data where the atoms of a sample correspond to metric space-valued random objects. The method is based on constructing Fréchet means of such random elements within a ball centered at a specific point in the space as the radius of the ball varies. Specifically, we investigate the behavior of these ball Fréchet means as the ball radius increases. We obtain real-valued functions by taking the distance of the ball Fréchet means from the global Fréchet mean, which leads to functional data objects with the radius and the ball center as arguments, enabling the application of traditional functional data analysis techniques. The proposed methodology is supported by theory and is illustrated with simulations and applications to human mortality data and U.S. electricity generation data, demonstrating its potential for visually representing object data as random functions, as well as for exploratory clustering and outlier detection.