Frame theory in directional statistics

TL;DR

This paper integrates frame theory with directional statistics to identify non-uniform distributions that evade uniformity tests, and applies these findings to granular rod experiments.

Contribution

It introduces probabilistic tight frames into directional statistics, revealing distributions that challenge traditional uniformity tests and modeling experimental patterns.

Findings

Probabilistic tight frames produce non-uniform distributions that fail the Bingham test.

Application to granular rod experiments demonstrates real-world relevance.

Provides a new theoretical framework for analyzing directional data.

Abstract

Distinguishing between uniform and non-uniform sample distributions is a common problem in directional data analysis; however for many tests, non-uniform distributions exist that fail uniformity rejection. By merging directional statistics with frame theory, we find that probabilistic tight frames yield non-uniform distributions that minimize directional potentials, leading to failure of uniformity rejection for the Bingham test. Finally, we apply our results to model patterns found in granular rod experiments.