Theoretical Results on the Optimal Detection Statistics for Autonomous Integrity Monitoring

Abstract

One of the most commonly used detection statistics in autonomous integrity monitoring is the solution separation statistic. In this paper, we show that the solution separation statistic is closely linked to the optimal detection statistics. More precisely, we show that in the case of one threat, even multi-dimensional, the optimal detection statistic is the solution separation, and that in the case of many one-dimensional threats, the optimal statistics can be expressed as a function of the solution separations corresponding to each threat. This is achieved by casting the search of the optimal detection region as a mini-max problem and by using the Neyman-Pearson lemma to limit the search of the detection regions to a class of regions parameterized by a bias. These results allow us to establish a lower bound on the minimum achievable integrity risk.

Integrity Risk Minimisation in RAIM Part 1: Optimal Detector Design

“…As mentioned earlier, this multiple integration process is computationally expensive, and in this paper, is only used for analytical purposes in two illustrative examples. Equation (37) describes the detection region as a union of half-planes, and is consistent with the assertion proved in Blanch et al (2013a), which states that the optimal detection boundary defines a closed, convex curve. Equation (37) is also written as a 'mini-max' problem.…”

“…This result is confirmed using the Neyman-Pearson (NP)-based approach (Blanch et al, 2013a), which, by sampling the parity space point by point, provides a precise depiction of the optimal no-detection region (grey area in Figure 3). In Figure 3, the optimal no-detection region closely matches the χ 2 RAIM circular boundary.…”

“…Blanch et al (2013a) demonstrates the tight relationship between the optimal detection region and the solution separation statistics for a given set of navigation requirements. In contrast, in this work, the proposed algorithm approximates the optimal detection region…”

“…Optimal Piecewise Linear Detection Boundary. Blanch et al (2013a) recently developed an approach, based on the Neyman-Pearson (NP) theorem, to visualize the optimal no-detection region by determining the set of points in parity space that contribute to reducing the integrity risk. This NP approach does not include a process to evaluate the integrity risk P HMI .…”

“…This method is computationally expensive, but enables integrity risk evaluation (unlike (Blanch et al, 2013a)). It also provides the means to show that the optimal detection region varies with navigation system parameters, and that assuming realistic requirements, the SS polytopic boundary can approach the optimal detection region.…”