Treatment of Epistemic Uncertainty in Conjunction Analysis with Dempster-Shafer Theory

TL;DR

This paper tackles epistemic uncertainty in conjunction analysis by treating CDMs as draws from an unknown distribution family and employing the DKW inequality to construct p-box bounds on uncertain parameters. A Dempster–Shafer-based framework then yields Belief and Plausibility for the probability of collision, enabling a six-class classification of conjunction events that incorporates both evidence and uncertainty. A time-weighted CDM scheme and alpha-cut p-box construction provide a practical, scalable method to quantify uncertainty and guide decision-making for CAM planning. The approach aligns with ESA SDO decisions when CDMs are coherent, while offering more conservative, uncertainty-quantified guidance in inconsistent sequences and providing complementary insights relative to CNES sPoC. Overall, the method delivers a principled uncertainty-aware tool for Space Traffic Management and collision avoidance decision support.

Abstract

The paper presents an approach to the modelling of epistemic uncertainty in Conjunction Data Messages (CDM) and the classification of conjunction events according to the confidence in the probability of collision. The approach proposed in this paper is based on the Dempster-Shafer Theory (DSt) of evidence and starts from the assumption that the observed CDMs are drawn from a family of unknown distributions. The Dvoretzky-Kiefer-Wolfowitz (DKW) inequality is used to construct robust bounds on such a family of unknown distributions starting from a time series of CDMs. A DSt structure is then derived from the probability boxes constructed with DKW inequality. The DSt structure encapsulates the uncertainty in the CDMs at every point along the time series and allows the computation of the belief and plausibility in the realisation of a given probability of collision. The methodology proposed in this paper is tested on a number of real events and compared against existing practices in the European and French Space Agencies. We will show that the classification system proposed in this paper is more conservative than the approach taken by the European Space Agency but provides an added quantification of uncertainty in the probability of collision.

Figures (23)

• Figure 1: Support to the value of $PoC$ being greater than a given value: $Bel$ -black solid line; $Pl$ - black dashed line. The dotted purple line represents a possible $PoC_0$.
• Figure 2: Example of intervals derivation form the eCDF. (a) eCDF (solid blue), individual sample's Gaussian pdf distributions (solid grey), pdf of the sum of Gaussian distributions for the eCDF fit (solid orange) (b) eCDF (solid blue), DKW bands (dashed green), fitted eCDF with weighted sum of Gaussian distributions (dashed-pointed orange). (c) eCDF (solid blue), DKW bands (dashed green), p-box optimising the weighted sum of Gaussian distributions (dashed-pointed red), 1% and 99% percentiles (vertical pointed black lines). (d) eCDF (solid blue), p-box (dashed-pointed red), 1 $\alpha$-cut 2 intervals' Pl and Bel (dashed blue), 7 $\alpha$-cuts 8 intervals' Pl and Bel (dashed black). Dotted thin horizontal lines for the $\alpha$-cuts: light blue at 0.5 for the 2 intervals partition, grey lines spaced 0.125 for the 8 intervals partition.
• Figure 3: Fitting law: (a) $y'=e^{-3t'}$ (thick red line) and the dimensionless covariance determinant for a number of sequences of CDM (thinner lines), (b) Fitted law (dashed-pointed red) of a single CDM sequence (dashed-pointed black).
• Figure 4: eCDF for $\mu_\xi$ weighing the samples (blue) and with samples equally weighted (dashed red).
• Figure 5: Plausibility and Belief of $PoC\geq PoC_0$. Black: 1 $\alpha$-cut (two intervals) per variable, 32 FEs. Blue: 7 $\alpha$-cut (eight intervals) per variable, 32768 FEs. Solid lines: belief. Dashed lines: plausibility. Dotted purple vertical line: $PoC_0$.