Gaussian-Pareto Overbounding: A Method for Managing Risk in Safety-Critical Navigation Systems

Abstract

An innovative method for managing the integrity risk of safety-critical navigation systems is presented. The method builds upon the current statistical technique used in the field of navigation known as overbounding which uses conservative probability distributions to bound risk probabilities. In particular, this work replaces the Gaussian distribution currently in use with a hybrid Gaussian-Pareto distribution. This change is motivated by results from Extreme Value Theory (EVT) and a simulation study assessing the Pareto distribution's potential as a model for the tails of distributions. Both of which are discussed thoroughly. By utilizing the hybrid Gaussian-Pareto overbound, the extreme error probabilities for distributions that display heavier-than-Gaussian tails can be truly overbounded which is necessary for constructing system output overbounds from input overbounds. The model also uses observed data more efficiently than current methods because it separates the extreme portion of the distribution from the core portion. Furthermore, the model can be less conservative than the Gaussian distribution across the entire domain which would lead to greater availability. To demonstrate this, the Gaussian-Pareto overbound is shown to be an appropriate model for double-differenced pseudoranges from two Continuously Operating Reference Stations (CORS) of the Global Navigation Satellite System (GNSS), an important measurement in many safety-critical navigation systems. Lastly, a novel approach for overbounding multivariate probability risks called norm overbounding is developed so that the Gaussian-Pareto model can be applied in the multivariate domain. This approach utilizes hyperspherical coordinates as well as domain partitioning in order to construct a robust model for bounding norms of random vectors which can be considered as multivariate errors. It is demonstrated for a simple 2-dimensional navigation application.