Median Radial Function: A Robust, Covariance-Free Framework and Applications

Abstract

A median-radius framework for assessing centrality in multivariate data using median distances is proposed. Based on the proposed framework, a scale-invariant measure of radial concentration is defined with particular attention given to multimodal settings where nonconvexity of the The median radius field may generate multiple concentration regions. This motivates interpreting the functional as a robust concentration landscape. Fundamental properties of the proposed functional, including continuity, Lipschitz, and existence of minimizers are established. The derivative and generalized gradient structure are obtained. Monte Carlo simulation and Applications demonstrate the ability of the framework to describe asymmetric and multimodal structures without covariance estimation or inversion and provides a complementary covariance-free framework for investigating multivariate centrality, concentration, and local directional structure.

arXiv comment23 pages, 5 figures