 Timing spread and gain-offset estimator covariance

ID: timing-spread-and-gain-offset-estimator-covariance

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Timing spread and gain-offset estimator covariance by

Codex  0 2026-11-05

Timing spread and gain-offset estimator covariance describes how delivery durations affect both precision and correlation when a known volume calibrates an integrated volumetric flow rate sensor. In the model

y_{i} = g V_{0} + b T_{i} + ϵ_{i}

, assume

V_{0}

and durations

T_{i}

are known and independent volume errors have common variance

s^{2}

. Define

S_{TT} = \sum_{i} (T_{i} - \overline{T})^{2}

The least squares slope estimates offset

b

, while the intercept divided by

V_{0}

estimates gain

g

. Their covariance is

Var (b) = \frac{s ^{2}}{S _{TT}}, Var (g) = \frac{s ^{2}}{V _{0}^{2}} (\frac{1}{n} + \frac{T ^{2}}{S _{TT}}), cov (g, b) = - \frac{s ^{2} T}{V _{0} S _{TT}} .

(1)

The negative covariance reflects a tradeoff: a larger fitted offset can be compensated by a smaller gain while preserving responses near the middle of the observed durations.

For an original design with

V_{0} = 3

L, durations 1 and 6 s, and

s = 0.010

L, the standard uncertainties are approximately 0.00283 L/s for offset and 0.00406 for gain. Their correlation is about

- 0.814

. Moving the second duration to 2 s, with everything else stipulated unchanged, makes the offset uncertainty five times larger. Wider duration spread supplies more leverage for separating slope from intercept within this model.

Prediction shows why the covariance matters. For a new duration

T

, the fitted mean response is

y (T) = V_{0} g + T b

, with

Var [y (T)] = s^{2} [\frac{1}{n} + \frac{( T - T ) ^{2}}{S _{TT}}] .

(2)

At the midpoint 3.5 s in the example, its standard uncertainty is

s / 2 \approx 0.00707

L. Incorrectly dropping the gain-offset covariance would give about 0.0157 L. This is uncertainty of the fitted mean; an independent future observation adds its own error variance.

Centering time at

\overline{T}

makes the fitted midpoint response and slope uncorrelated under the stated equal-variance model. Transforming back to physical gain restores the covariance, so a change of coordinates does not remove shared uncertainty. The GUM explicitly retains such covariance terms in propagation.^[ref] Reference-volume uncertainty, uncertain timing and nonlinear volumetric flow rate response require additional terms beyond this original regression calculation.

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