Session B2b: Autonomous Navigation and Safety-Critical Applications

An Extension of Gaussian Overbounding to Multivariate Distributions with an Application to GNSS Integrity
Juan Blanch, Rebecca Wang, Todd Walter, Stanford University
Location: Beacon B

To provide reliable position error bounds, GNSS based systems require a reliable characterization of the statistical properties of the measurement errors. A large part, if not the main one, of any integrity analysis is the determination of error models for each type of error, like clock and ephemeris, tropospheric delay, or multipath. These error models are usually based on data collected in measurement campaigns. The measurement residuals (that is, the differences between observed and truth, or a best approximation of it) are then used to form sample error distributions. These sample distributions are then approximated, or bounded in some sense, by simpler continuous distributions, like Gaussian distributions. This is necessary because to compute the user position error bound, we must characterize the linear combination of all the error sources, and therefore compute the convolution of all error distributions corresponding to each source. For arbitrary distributions like the ones defined by the sample distribution, this would be very computationally demanding. For Gaussian distributions, it is straightforward, because the distribution is completely defined by the variance and the mean, which are easy to compute for a linear combination of random variables. Because we are replacing the actual distribution with a new one, we need to make sure that the overbounding property is conserved after the linear combination operation. That is, the overbounding property must be stable through convolution.
Previous work
Three very important sets of overbounding criteria are symmetric unimodal CDF bounding [1] paired overbounding [2], and excess mass bounding [3]. These three overbounding methods and their combined use [4] are very powerful and have been key in several safety-of-life systems. They are however limited to the combination of independent distributions. If the distributions of the errors being combined are not strictly independent, there is no guarantee that the errors of the linear combination of the errors will be bounded by the convolution of the Gaussian overbounding distributions. While the independence assumption is a reasonable one in many situations it is not valid for some important cases. One such case is when using estimators that exploit the temporal structure of the errors, like Kalman filters, factor graphs, or batch approaches, because most of the error sources are correlated in time. Determining an overbounding model that captures and bounds the correlation is not trivial. One approach that has been proposed is frequency domain overbounding [5]. This is a powerful method, but it only guarantees overbounding for Gaussian errors. For all other distributions, it only guarantees that the variance of the bounding distribution bounds the variance of the actual distribution. Another approach that has been proposed is symmetric overbounding [6], but it requires symmetry conditions on the sample distribution that are seldom met in practice.
The definition of a practical overbounding concept for correlated errors remains an open problem.
Contribution of the paper
In this work, we describe a “weak” overbounding concept for random vectors, in a precisely defined weak sense. This overbounding concept results in a practical criterion to determine whether a given multivariate Gaussian overbounds a random vector distribution (that is, a multivariate random variable). The criterion can be stated as follows: the distribution of the squared norm of the vector (for the norm defined by the multivariate Gaussian) must be bounded by a chi-square distribution with n degrees of freedom where n is equal to the size of the vector. The sufficiency of this criterion (for the weak sense overbounding property) is demonstrated mathematically.
As an example, we use this concept to determine temporal error models for the GNSS clock and ephemeris errors.
REFERENCES
[1] B. DeCleene, "Defining Pseudorange Integrity - Overbounding," Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000), Salt Lake City, UT, September 2000, pp. 1916-1924
[2] J. Rife, S. Pullen, B. Pervan, and P. Enge. Paired Overbounding for Nonideal LAAS and WAAS Error Distributions. IEEE Transactions on Aerospace and Electronic Systems, 2006, 42, 4, 1386 -1395
[3] J. Rife, T. Walter, and J. Blanch, “Overbounding SBAS and GBAS error distributions with excess-mass functions,” Proceedings of the 2004 International Symposium on GPS/GNSS, Sydney, Australia, 6-8 December, 2004.
[4] Blanch J., Walter, T., and Enge, P., “Gaussian Bounds of Sample Distributions for Integrity Analysis”. IEEE Transactions on Aerospace and Electronic Systems.” PP. 1-1. 10.1109/TAES.2018.2876583.
[5] Langel, Steven, Crespillo, Omar García, Joerger, Mathieu, "A New Approach for Modeling Correlated Gaussian Errors Using Frequency Domain Overbounding," 2020 IEEE/ION Position, Location and Navigation Symposium (PLANS), Portland, Oregon, April 2020, pp. 868-876. https://doi.org/10.1109/PLANS46316.2020.9110192
[6] Rife, J. and Gebre-Egziabher, D. (2007), Symmetric Overbounding of Correlated Errors. Navigation, 54: 109-124. https://doi.org/10.1002/j.2161-4296.2007.tb00398.x