Distribution of Test Statistic for Euclidean Distance Matrices

TL;DR

The paper tackles the practical problem of determining the distribution of the EDM-based test statistic $q = \frac{\lambda_4 + \lambda_5}{2 \lambda_1}$ under nominal GNSS noise. It develops a first-order eigenvalue perturbation framework to characterize fluctuations of the Gram-matrix eigenvalues and shows that, under nominal noise, the dominant eigenvalues are approximately Gaussian, while the zero-valued $\lambda_4$ and $\lambda_5$ require clock-bias stabilization to enable perturbation analysis. Because $q$ is a ratio of Gaussian-like quantities, the work adopts a Gaussian-ratio approximation to model its distribution and validates this approach with a Monte Carlo simulation (10{,}000 trials on $m=12$ satellites). The results provide a practical method to set fault-detection thresholds for EDM-based GNSS, though the authors caution that the approximation has conditions for validity and call for further validation, especially under larger noise or fault scenarios. Practically, this work offers a fast, geometry-dependent way to quantify the nominal distribution of the test statistic and supports EDM-based fault detection with a computable threshold criterion.

Abstract

Methods for global navigation satellite system fault detection using Euclidean Distance Matrices have been presented recently in the literature. Published methods define a test statistic in terms of eigenvalues of a certain matrix, but the distribution of the test statistic was not known, which presented a barrier to practical implementation. This document was a personal correspondence from Beatty to Derek Knowles. It includes a brief derivation of the distribution of the test statistic and a representative case showing that the theoretical distribution closely matches a simulated empirical distribution.

Figures (1)

• Figure 1: Comparison of empirical distribution of test statistic (blue bins) against theoretical normal distribution (orange line)