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Cauchy-Gaussian Overbound for Heavy-tailed GNSS Measurement Errors

Zhengdao Li, Penggao Yan, Weisong Wen, Li-Ta Hsu

TL;DR

This work addresses the challenge of tightly bounding heavy-tailed GNSS measurement errors for integrity monitoring by introducing a Cauchy-Gaussian overbound that couples a Cauchy core with Gaussian tails. It develops explicit three-step procedures for both symmetric unimodal and non-symmetric unimodal error profiles, and proves that overbounding properties are preserved under convolution, enabling placement-domain bounds from range-domain distributions. The methods yield significantly tighter bounds than traditional Gaussian overbounds and NavDEN, with average vertical protection level reductions of about 15% for symmetric errors and up to 47.7% for non-symmetric errors in real urban data. The approach improves integrity risk management in GNSS by delivering tighter, broadcast-friendly bounds that better capture core and tail behavior, potentially benefiting fault detection and high-availability positioning systems.

Abstract

Overbounds of heavy-tailed measurement errors are essential to meet stringent navigation requirements in integrity monitoring applications. This paper proposes to leverage the bounding sharpness of the Cauchy distribution in the core and the Gaussian distribution in the tails to tightly bound heavy-tailed GNSS measurement errors. We develop a procedure to determine the overbounding parameters for both symmetric unimodal (s.u.) and not symmetric unimodal (n.s.u.) heavy-tailed errors and prove that the overbounding property is preserved through convolution. The experiment results on both simulated and real-world datasets reveal that our method can sharply bound heavy-tailed errors at both core and tail regions. In the position domain, the proposed method reduces the average vertical protection level by 15% for s.u. heavy-tailed errors compared to the single-CDF Gaussian overbound, and by 21% to 47% for n.s.u. heavy-tailed errors compared to the Navigation Discrete ENvelope and two-step Gaussian overbounds.

Cauchy-Gaussian Overbound for Heavy-tailed GNSS Measurement Errors

TL;DR

This work addresses the challenge of tightly bounding heavy-tailed GNSS measurement errors for integrity monitoring by introducing a Cauchy-Gaussian overbound that couples a Cauchy core with Gaussian tails. It develops explicit three-step procedures for both symmetric unimodal and non-symmetric unimodal error profiles, and proves that overbounding properties are preserved under convolution, enabling placement-domain bounds from range-domain distributions. The methods yield significantly tighter bounds than traditional Gaussian overbounds and NavDEN, with average vertical protection level reductions of about 15% for symmetric errors and up to 47.7% for non-symmetric errors in real urban data. The approach improves integrity risk management in GNSS by delivering tighter, broadcast-friendly bounds that better capture core and tail behavior, potentially benefiting fault detection and high-availability positioning systems.

Abstract

Overbounds of heavy-tailed measurement errors are essential to meet stringent navigation requirements in integrity monitoring applications. This paper proposes to leverage the bounding sharpness of the Cauchy distribution in the core and the Gaussian distribution in the tails to tightly bound heavy-tailed GNSS measurement errors. We develop a procedure to determine the overbounding parameters for both symmetric unimodal (s.u.) and not symmetric unimodal (n.s.u.) heavy-tailed errors and prove that the overbounding property is preserved through convolution. The experiment results on both simulated and real-world datasets reveal that our method can sharply bound heavy-tailed errors at both core and tail regions. In the position domain, the proposed method reduces the average vertical protection level by 15% for s.u. heavy-tailed errors compared to the single-CDF Gaussian overbound, and by 21% to 47% for n.s.u. heavy-tailed errors compared to the Navigation Discrete ENvelope and two-step Gaussian overbounds.
Paper Structure (21 sections, 2 theorems, 69 equations, 11 figures, 6 tables)

This paper contains 21 sections, 2 theorems, 69 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Review on Gaussian Overbounding Methods
  3. Single-CDF Gaussian overbound
  4. Paired Gaussian overbound
  5. Two-step Gaussian overbound
  6. Cauchy-Gaussian Overbound
  7. Cauchy distribution
  8. Procedures of bounding s.u. errors
  9. Procedures of bounding n.s.u. errors
  10. Position domain bounding
  11. Numerical Experiments
  12. Bounding performance for s.u. errors
  13. Bounding performance for n.s.u. errors
  14. Impact of heavy-tailedness
  15. Conclusion and Future Work
  16. ...and 6 more sections

Key Result

Theorem 1

For a Gaussian model $N(\mu,\sigma)$ and a Cauchy model $C(m, \lambda)$, if their medians are aligned (i.e., $\mu=m=M_0$, where $M_0$ denotes the known value of the median), then the sufficient and necessary condition of $C(M_0,\lambda)$ overbounding $N(M_0,\sigma)$ is $\lambda \geq \sqrt{\frac{2}{\

Figures (11)

  • Figure 1: Flowcharts of the Cauchy-Gaussian overbound for (a) s.u. and (b) n.s.u. error distributions. The black curve in each subfigure represents the empirical error distribution. For better visualization at tail regions, the subfigures in (b) show the CDF values transformed into their equivalent standard normal quantiles.
  • Figure 2: Comparison between the standard Cauchy and Gaussian distributions, through (a) PDF and (b) folded CDF on a logarithmic scale.
  • Figure 3: The tentative combined overbounding distribution for zero-mean BGMM error defined in Equation \ref{['eq:example_error_type_I']} (s.u. profile) in two views: (a) CDF; (b) PDF. The Gaussian and Cauchy overbounds are also plotted for reference in each subfigure.
  • Figure 4: The Cauchy-Gaussian overbounding results for zero-mean BGMM error defined in Equation \ref{['eq:example_error_type_I']} (s.u. profile). Three views are exhibited, including (a) CDF; (b) quantile-scale CDF; (c) PDF. The gray-shaded rectangles denote the two tangential transition regions on the left and right. The Gaussian and Cauchy overbounds are plotted for reference in all the subfigures.
  • Figure 5: The Cauchy-Gaussian overbounding results for biased BGMM errors defined in Equation \ref{['eq:example_error_type_II']} (n.s.u. profile): (a) CDF of the optimized paired CGCM overbound in Step 1; (b) CDF of the optimized paired Gaussian overbound in Step 2; quantile-scale (c) right bound (defined in Equation \ref{['equ: synthesized CG R']}) and (d) analog single-CDF (defined in Equation \ref{['equ: F_ob nsu single-cdf analog']}) of finalized Cauchy-Gaussian overbound.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Theorem
  • proof : Proof of sufficiency
  • Lemma
  • proof
  • proof : Proof of necessity
  • proof : Proof of sufficiency
  • proof : Proof of necessity