Blend Weights and the Independence Assumption

Blend Weights and the Independence Assumption

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

foundations
math-foundations
calculation-audit
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Ada ADA · CALCULATION AUDIT

Blend Weights and the Independence Assumption

The chapter makes two claims: the gyro owns a fast tilt update, its fused 10.352 deg barely leaving the pure-gyro 10.4 deg, and averaging two sensors reading 10.0 and 10.4 shrinks uncertainty from 0.5 toward 0.35. The second claim hides a condition. This audit checks the blend weights and the independence assumption to show fusion shrinks random noise but cannot remove a shared bias.

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

Try

The chapter makes two claims: the gyro owns a fast tilt update, its fused 10.352 deg barely leaving the pure-gyro 10.4 deg , and averaging two sensors reading 10.0 and 10.4 shrinks uncertainty from 0.5 toward 0.35 . Calculate this case.

Observe

This audit checks the blend weights and the independence assumption to show fusion shrinks random noise but cannot remove a shared bias. Check shows this.

Explain

The audit conclusion is narrow: the smaller 0.35 uncertainty band is real only under independence. Fusion arithmetic can shrink random noise, but it cannot remove a shared bias — so the honest fused record publishes the common-calibration risk right next to the tighter-looking number. Check confirms it.

See the relationship before changing it

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

Independent Sensors changes average uncertainty A three-part teaching diagram connects independent sensors, the rule sigma = 0.5 / sqrt(sensor count), and average uncertainty. INPUT Independent sensors APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrow. Independent random noise shrinks; a shared bias does not.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2 sensors.

  2. 2

    Name the relationship. sigma = 0.5 / sqrt(sensor count)

  3. 3

    Substitute with units. 0.5 / sqrt(2) = 0.354

  4. 4

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

Predict, then change independent sensors

Try Predict how average uncertainty responds when independent sensors moves. Calculate independent sensors; compare average uncertainty with that prediction.

2 sensors
Chapter baseline
Average uncertainty

Observe Return to 2 sensors. Recheck average uncertainty with independent sensors at its chapter value.

Explain Independent random noise shrinks; a shared bias does not.

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 independent sensors moves here. The average uncertainty calculation excludes field effects listed below.

Honesty note. The average uncertainty result holds independent sensors as the only changing input. It is not a field prediction.

Technical boundaries

Beyond the scope of “Blend Weights and the Independence Assumption” sit correlated errors, bias, non-Gaussian noise, time alignment, sensor failure, or uncertainty in the assumed variances; “Blend Weights and the Independence Assumption” therefore reports only its named fixtures.

Ada: This chapter makes two numeric claims — that the gyro carries most of one update, and that averaging can shrink uncertainty. Both are true, but the second one hides a condition. Let me check both with the chapter’s own values.

For the tilt update, the previous fused angle is 10.0 deg, the gyro rate is 20.0 deg/s, the time step is 0.020 s, the accelerometer estimate is 8.0 deg, and alpha is 0.98:

  • Gyro prediction: 10.0 + 20.0 x 0.020 = 10.0 + 0.4 = 10.400000 deg
  • Fused angle: 0.98 x 10.400000 + 0.02 x 8.0 = 10.192000 + 0.160000 = 10.352000 deg

The fused value sits 10.400 - 10.352 = 0.048 deg below the pure gyro prediction, so the accelerometer moved it only a hair toward gravity — the gyro owns this fast update.

Now the shared-bias example. Sensor A reads 10.0 and sensor B reads 10.4, each with random uncertainty of about 0.5:

  • Simple average: (10.0 + 10.4) / 2 = 10.200000
  • If the two errors are independent, the average’s uncertainty shrinks by 1 / sqrt(2): 0.5 / sqrt(2) = 0.5 / 1.414214 = 0.353553, about 0.35 — smaller than either sensor’s 0.5.
  • If instead both sensors share a +0.4 calibration offset, the average still carries the full +0.4, because averaging two copies of the same bias returns that bias unchanged.

The audit conclusion is narrow: the smaller 0.35 uncertainty band is real only under independence. Fusion arithmetic can shrink random noise, but it cannot remove a shared bias — so the honest fused record publishes the common-calibration risk right next to the tighter-looking number.

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