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First-order propagation of measurement uncertainty

Statement

Let $d \geq 1$ , let $\hat{x} \in U$ for an open set $U \subseteq R^{d}$ , and let $f : U \to R$ be differentiable at $\hat{x}$ . Let $X$ be an $R^{d}$ -valued random element on one probability space with square-integrable coordinate variables $X_{i}$ , means $E [X_{i}] = {\hat{x}}_{i}$ , and $X \in U$ almost surely. Let $Σ_{i j} = Cov (X_{i}, X_{j})$ , set $g = \nabla f (\hat{x})$ , and define $Q_{l i n} = f (\hat{x}) + g^{T} (X - \hat{x}), u_{c}^{2} = Var (Q_{l i n}) .$ Then $u_{c}^{2} = g^{T} Σ g = \sum_{i = 1}^{d} g_{i}^{2} Σ_{i i} + 2 \sum_{1 \leq i < j \leq d} g_{i} g_{j} Σ_{i j} .$ The equality is exact for the first-order linearized output. To interpret it as an approximation to uncertainty in the nonlinear output $f (X)$ , require $f (X)$ to be a measurable, square-integrable random output (including a measurable definition on any null event where $X \notin U$ ) and require the first-order model to be adequate. This interpretation is not an assertion of exact finite-sample variance or interval coverage.

Facts & Assumptions

Given: The objects and hypotheses of the Statement.

[F1]

Moments, variance, and covariance on a probability space: for square-integrable real random variables, variance and covariance are the expectations of the centered square and centered product.

[F2]

Covariance is symmetric and bilinear in finite linear combinations: covariance is symmetric and bilinear on finite linear combinations of square-integrable real random variables on one common probability space; constant summands are allowed.

[F3]

The real Frechet derivative on real and complex matrix spaces with the Frobenius norm: after identifying $R^{d}$ with $d \times 1$ real matrices, differentiability at $\hat{x}$ means a real-linear first-order map whose remainder divided by the Euclidean input norm tends to zero.

[F4]

Measurands, calibration, and measurement uncertainty: the input estimates are ${\hat{x}}_{i} = E [X_{i}]$ , $Σ_{i j} = Cov (X_{i}, X_{j})$ , and $u_{c}^{2} = Var (Q_{l i n})$ for the stated first-order linearized output.

Proof

1.1givenF3choosealgebra

Identify $R^{d}$ with real $d \times 1$ matrices and let $L = D f (\hat{x})$ . By [F3], $f (\hat{x} + h) = f (\hat{x}) + L [h] + r (h)$ with $r (h) / ∥ h ∥ \to 0$ . For the standard basis $e_{i}$ , put $g_{i} = L [e_{i}]$ ; linearity gives $L [h] = \sum_{i} g_{i} h_{i} = g^{T} h$ . Thus $Q_{l i n}$ is the first-order Taylor term.

1.2givenF1algebra

Put $c = f (\hat{x}) - g^{T} \hat{x}$ . Then $(\sum_{i} g_{i} X_{i})^{2} \leq d \sum_{i} g_{i}^{2} X_{i}^{2}$ , and $(c + \sum_{i} g_{i} X_{i})^{2} \leq 2 c^{2} + 2 d \sum_{i} g_{i}^{2} X_{i}^{2}$ . Since each $X_{i}$ is square-integrable and $d$ is finite, the right side has finite expectation, so $Q_{l i n}$ is square-integrable. Thus [F1] defines its variance and all covariances below.

2.1F1F2step 1.2algebra

By [F1], $Var (Q_{l i n}) = Cov (Q_{l i n}, Q_{l i n})$ . Apply the finite bilinearity in [F2] to step 1.2; the constant has zero covariance, so $Var (Q_{l i n}) = \sum_{i = 1}^{d} \sum_{j = 1}^{d} g_{i} g_{j} Cov (X_{i}, X_{j})$ .

3.1F1F2F4step 2.1algebra∎

By [F4], each covariance in step 2.1 is $Σ_{i j}$ , so the double sum is $g^{T} Σ g$ . Symmetry in [F2] pairs the $(i, j)$ and $(j, i)$ terms to give the stated diagonal and off-diagonal form. It is nonnegative because it equals the variance in [F1], and [F4] therefore gives the stated $u_{c}^{2}$ formula.

Remarks

No independence assumption is used. The formula includes covariance between calibration coefficients, reference standards, corrections, and any later reading included in $X$ . The first-order Taylor rule is appropriate only when the nonlinear remainder is negligible for the measurement at hand. JCGM 100:2008/Amd.1:2026 calls for higher-order terms when nonlinearity is significant; Monte Carlo methods are outside this result. Neither $u_{c}$ nor $u_{c}^{2}$ alone supplies a confidence interval or coverage probability.

Depends on

Used by

Calibration curve with uncertain reference standards and batch drift Example

Dependency tree · two levels

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Sources

JCGM 100:2008, Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement, clauses 4.1 and 5.1–5.2 (standard reference, not scraped)
JCGM 100:2008/Amd.1:2026, Amendment 1: Nonlinearity in Measurement Models (standard reference, not scraped)
NIST/SEMATECH Engineering Statistics Handbook, Chapter 2: Measurement Process Characterization (standard reference, not scraped)
Sai Ravela, MIT OpenCourseWare, Quantifying Uncertainty: Summary lecture notes, Fall 2012 (standard reference, not scraped)