Minimum volume ellipsoid scaled to contain a tangent sphere, with application to integrity monitoring

Robust Chi-Square Monitor Performance with Noise Covariance of Unknown Aspect-Ratio

“…Early proposed solutions for (12) and (13) assumed Q was bounded using a positivedefinite matrix comparison [14,15]. Subsequent work simplified this criterion slightly, assuming the inner and outer bounding positive-definite matrices were spherical [6,16]. For spherical inner and outer bounds, the covariance matrices can be characterized simply in terms of their minimum and maximum eigenvalues, ̲ λ Q and λ Q , respectively.…”

“…The solutions a P md and a P fa given by (16) and (17) are useful but may be overly conservative in a particular situation, specifically the situation when the absolute maximum and the absolute minimum eigenvalues occur for different covariance matrices over the range of all possible Q . This distinction between the ratio of worst-case eigenvalues over the full threat space and the ratio for any particular covariance matrix can be quantified by introducing the notation ϕ Q to describe the aspect ratio for any individual matrix Q.…”

“…The SDM example discussed in this paper is one such application. For this application, a tighter bound on missed-detection probability is desired than can be delivered by (16).…”

Comparing Geometric Approximations of Heavy-Tail Effects for Chi-Square Integrity Monitors

“…As one example, prior work on overbounding has considered how to convolve multiple (not necessarily Gaussian) error PDFs of the same form . As another example, the authors have conducted recent research on overbounding an unknown covariance for Gaussian‐distributed SDM noise, an uncertainty that can cause the PDF of the monitor statistic to diverge from its expected chi‐square model .…”

Overbounding False-Alarm Probability for a Chi-Square Monitor with Natural Biases

“…Overbounding position-error distributions has been the subject of intense prior research [2][3][4][5][6][7][8][9][10]. In recent years, a particular emphasis has been placed on overbounding for integrity monitors, too, with the goal of obtaining rigorous bounds for false-alarm and missed-detection risks [11][12][13][14]. As experience has been gained in operating integrity augmentations for GPS, such as ground-based augmentation systems (GBAS) and satellite-based augmentation systems (SBAS) [15][16][17][18], it has become increasingly clear that error distributions for integrity monitoring algorithms are often poorly characterized and that monitor performance analyses must take this fact into account.…”