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THEORY-1092: Read over the sensor domains, the discrepancies between each observation and the restrictions of others can single out a faulty sensor where the consistency radius alone does not: in Robinson's simulated search, a 2° bearing bias was attributed to the right sensor, without prior knowledge of it, because every large discrepancy involved that sensor's domains, while the radius was lower than in the unbiased case

3 min read

Properties3
StatusProposed
Tagsmathematics, information-theory
SourceLIT-1658 — Sheaves are the canonical data structure for sensor integration

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Source

Robinson (2017), LIT-1658, Example 24 (pp. 25–29 of arXiv 1603.01446v3), Tables 1–2 and Figures 11–13; read in NOTE-1221.

What was actually shown

The setup. A simulated search for a downed airliner, with a sheaf whose restrictions are geometric maps between positions, velocities, crash time and RDF bearings (Example 11, p. 16). Three cases, each one realization of simulated noise. The true crash site was withheld from the model (p. 25).

Case 2. As Case 1, with RDF sensor 2 “biased southward by 2°”. “This error is immediately detectable in Figure 12(a) as an anomalously high error between the Field office (X) and the RDF sensor 2 (U4). Since the only high errors occur between sensor domains involving RDF sensor 2 (U4 and V2), the sheaf model makes it clear that the errors are due exclusively to RDF sensor 2. It should be noted that Figure 12(a) does not include any a priori knowledge of the sensor bias” (p. 28).

The radius did not do it. The consistency radius was 11.6 km in Case 2 and 15.7 km in Case 1, which had no bias (Table 1, p. 26). The bias showed in where the discrepancies sat, not in their maximum. In Case 1 too, “nearly all the error is concentrated in two places” (p. 28). In Case 3, with a wrong velocity added, the radius rose to 152 km, “an indication that something is amiss” (p. 29), but the text does not localize the velocity error.

What could have come out otherwise: the large discrepancies could have been spread over domains that share no sensor, or have pointed at the wrong one. In this one case they did not.

What this does not say

  • That localization works in general. One case, one realization, read by eye. No procedure, threshold or error rate is given.
  • That a measurement fault can be told from a model fault. The restriction maps are fixed and trusted (Axiom 5, p. 15). A 2° bias in one bearing gives the same discrepancy as an error in that station’s coordinates inside maps B and D (p. 16). The paper’s attribution to the sensor rests on trusting the map.
  • That the residuals find inadequate models. The paper gives that role to higher cohomology, which uses no data (pp. 3, 34).
  • That the system recalibrates. Nothing is revised from the residuals: fusion replaces the data by the nearest global section and leaves the maps alone. QUESTION-285’s description of the paper as sensor networks that “recalibrate from consistency residuals” goes beyond it.

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