Gate and Exclusion Thresholds

Gate and Exclusion Thresholds

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

Gate and Exclusion Thresholds

The chapter gates a 25.0 C sample against a 22.0 C prediction and rejects it — its normalized innovation of 18.0 sits far past the 3.841 chi-square limit — then excludes a sensor whose 0.72 m residual breaks a 0.30 m range gate. Every one of these is a single number compared to a threshold. This audit reproduces each gate and exclusion threshold to show one comparison decides accept, reject, exclude, and readmit.

Companion to the chapter Sensor Fusion Best Practices — every number here comes from that chapter.

Try

The chapter gates a 25.0 C sample against a 22.0 C prediction and rejects it — its normalized innovation of 18.0 sits far past the 3.841 chi-square limit — then excludes a sensor whose 0.72 m residual breaks a 0.30 m range gate. Calculate this case.

Observe

This audit reproduces each gate and exclusion threshold to show one comparison decides accept, reject, exclude, and readmit. Check shows this.

Explain

The audit conclusion is narrow: one one-line comparison decides accept, reject, exclude, and readmit. Recording the gate value beside the residual is what lets a later reviewer reproduce every one of those decisions instead of trusting a smooth output. Check confirms it.

See the relationship before changing it

The figure reads from left to right. The blue input is measurement. The middle card names the page’s rule. The green output is normalised innovation squared. The arrow matters: change the input, apply the rule once, then read the result with its unit.

Walk the arrow. The gate measures distance from the prediction in uncertainty units.

Derive the baseline in four named moves

1. 1

   Name the input. The chapter baseline is 25 C.

2. 2

   Name the relationship. NIS = (measurement - 22)^2 / (0.25 + 0.25)

3. 3

   Substitute with units. (25 - 22)^2 / 0.50 = 18.00

4. 4

   Read the result. Keep the unit beside the value, then use the result only inside the technical boundary below.

Predict, then change measurement

Try Predict how normalised innovation squared responds when measurement moves. Calculate measurement; compare normalised innovation squared with that prediction.

Measurement

25 C

Chapter baseline

Normalised innovation squared

Observe Return to 25 C. Recheck normalised innovation squared with measurement at its chapter value.

Explain The gate measures distance from the prediction in uncertainty units.

Check yourself

What should you do before trusting a moved-slider result?

Answer: Predict its direction, apply the displayed relationship, keep the units, and compare the reset value with the chapter’s worked baseline.

What does this small model leave out?

Answer: Only measurement moves here. The normalised innovation squared calculation excludes field effects listed below.

Honesty note. The normalised innovation squared result holds measurement as the only changing input. It is not a field prediction.

Technical boundaries

The “Gate and Exclusion Thresholds” model leaves out non-Gaussian residuals, correlated dimensions, model drift, missing observations, or the cost of false exclusion and false acceptance; “Gate and Exclusion Thresholds” therefore reports only its named fixtures.

Ada: Every gate in this chapter is a comparison between a number and a threshold, so let me confirm each number lands on the side the text claims, using only the stated values.

For the scalar innovation gate, the prediction is 22.0 C with variance P = 0.25, and the measurement is 25.0 C with variance R = 0.25:

• Innovation: y = 25.0 - 22.0 = 3.000000
• Innovation variance: S = P + R = 0.25 + 0.25 = 0.500000
• Normalized innovation squared: NIS = y^2 / S = 9.000000 / 0.500000 = 18.000000

The one-dimensional 95 percent chi-square gate is 3.841 (equivalently 1.96^2 = 3.8416). Since 18.0 > 3.841, the 25.0 C sample is rejected — not because 25 C is hot, but because it sits sqrt(18) = 4.242641, about 4.24 standard deviations, from the prediction.

The fault-exclusion example uses a fixed range gate of 0.30 m against three residuals:

• Sensor A residual 0.08 <= 0.30: keep.
• Sensor B residual 0.11 <= 0.30: keep.
• Sensor C residual 0.72 > 0.30: exclude — it is 0.72 / 0.30 = 2.4x the gate.

Recovery reads the same threshold forward. If C returns at 0.09, 0.07, 0.06 m, all three are <= 0.30, so the recovery rule can readmit it. If C instead returns at 0.09 m and then jumps to 0.41 m, that 0.41 > 0.30 keeps it quarantined on the second sample.

The audit conclusion is narrow: one one-line comparison decides accept, reject, exclude, and readmit. Recording the gate value beside the residual is what lets a later reviewer reproduce every one of those decisions instead of trusting a smooth output.

Every number above is taken from the chapter’s own material and re-derived step by step.