Exploring Multivariate Data Using Median Absolute Deviation Depth

TL;DR

This paper introduces the moving median absolute deviation (MMAD) as a robust, computationally efficient depth measure for multivariate data, providing geometric insights and aligning with classical depth notions.

Contribution

The paper develops MMAD as a novel robust depth measure with efficient computation and geometric interpretation for multivariate data analysis.

Findings

MMAD effectively identifies central observations similar to classical depth measures.

MMAD provides additional geometric and directional insights into data structure.

The method is computationally efficient, relying on distance calculations without complex optimization.

Abstract

We propose and analyze the moving median absolute deviation (MMAD) as a robust depth construction based on the median absolute distance functional with particular emphasis on its local geometry and probabilistic structure. In the univariate setting, we derive the derivative of the MMAD scale and interpret it through boundary mass imbalance, thereby establishing a direct connection to a robust skewness measure. This idea extends naturally to a multivariate setting that describes how observations are arranged along the 50% central region using a directional derivative, a gradient representation, and a spherical boundary distribution. From a computational perspective, MMAD can be estimated efficiently using distance calculations without needing complex optimization or projection schemes. Multivariate applications based on depth correlations, contour visualizations, and central region overlap demonstrate that MMAD identifies essentially the same central observations as classical depth notions while delivering additional information and geometric insight about directional structure. These features make MMAD a practical and informative approach for robust multivariate data analysis.