Metric Subtraction for Non-Euclidean Random Objects with Applications to Residual Analysis

Residual analysis for non-Euclidean regression models is challenging due to the lack of a natural subtraction operation and a corresponding notion of conditionally centered residuals. We introduce a metric subtraction framework that yields residuals in the original response space, with their conditional Fréchet mean equal to a fixed reference object when the model is correctly specified. We establish sufficient conditions for this conditional centering property, together with a separate condition ensuring distance preservation, providing a construction principle for suitable metric subtraction operators. The proposed metric subtraction framework accommodates several important classes of metric spaces, including spheres, symmetric positive definite matrices, simplices, and Wasserstein spaces. Building on this framework, we develop a randomized test for detecting remaining conditional location effects and establish its asymptotic size and power against fixed alternatives. We further introduce two complementary diagnostic tools, signed distance plots and distance-profile-based boxplots, to visualize systematic residual patterns and identify atypical observations. For uncertainty quantification, we construct residual-based prediction sets and establish their asymptotic validity. Simulation studies and applications to New York City bicycle usage distributions and U.S. electricity generation compositions demonstrate the practical utility of the proposed framework for model assessment, residual diagnostics, and prediction in non-Euclidean regression.