A robust distance concept for Random Surfaces with application to classification

Leopold Micheler${}^{a,b}$, Marcus Mayrhofer${}^a$, Una Radojicic${}^a$, Peter Filzmoser${}^a$

${}^a$ Institute of Statistics and Mathematical Methods in Economics, TU Wien
${}^b$ AC2T research GmbH (AC${}^{\mathrm{2}}$T), Austria
This contribution introduces a new distance measure and a robust method for covariance estimation of random surfaces with a separable covariance structure. The approach links separable random surfaces to the matrix-variate distribution of their basis function representations, which provides a convenient and principled framework for estimation.

Building on this connection, we develop a robust procedure based on the Matrix Minimum Covariance Determinant (MMCD) [1] estimator, combined with a truncated functional Mahalanobis semi-distance to estimate mean and covariance functions. This formulation is designed to remain stable under contamination and to provide reliable distance-based inference for functional data.

To improve interpretability, we extend the Shapley value [2] methodology to the functional setting. This allows us to decompose the proposed distance measure and thus functional outlyingness into contributions from specific spatial and temporal regions. We further introduce a novel formulation for computing these functional Shapley values while preserving their fundamental axiomatic properties.

The resulting framework integrates matrix-variate modeling, robust distance measures, and interpretable decomposition techniques into a unified approach for analyzing random surfaces. We provide theoretical justification alongside empirical results on real-world datasets, demonstrating strong performance in classification and robust outlier detection tasks.

Keywords: Functional data analysis, Robustness, Classification