= Reference lineage in covariance models
{title2=$U_x=A U_z A^{\mathsf T}$}
\b[<Reference lineage in covariance models>] records which primitive observations and calibration quantities enter each derived measurement, with what coefficients. Two inputs can share the same reference yet respond to it differently. A shared name alone therefore does not establish either equal errors or exact cancellation.
For a linear measurement model, collect primitive inputs in $\boldsymbol z$, derived inputs in $\boldsymbol x=A\boldsymbol z$ and their <covariance matrices> in $U_z,U_x$. Then
$$
U_x=A U_z A^{\mathsf T},\qquad
Y=\boldsymbol g^{\mathsf T}\boldsymbol x
=(A^{\mathsf T}\boldsymbol g)^{\mathsf T}\boldsymbol z,
$$
so
$$
u^2(Y)=\boldsymbol g^{\mathsf T}U_x\boldsymbol g
=(A^{\mathsf T}\boldsymbol g)^{\mathsf T}U_z(A^{\mathsf T}\boldsymbol g).
$$
The matrix records the data lineage as well as the algebra. This implements the GUM approach of expressing correlated quantities in terms of their common inputs.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.2.2 and 5.2.4, correlated inputs and common-input formulations]{ref}
An original example uses independent inputs $T,B,C,\epsilon_1,\epsilon_2$ with variances $1,9,9,4,4\,\mathrm{mg^2}$, respectively. Let two reference readings be $A_1=C+\epsilon_1$ and $A_2=2C+\epsilon_2$. Correct a specimen by $Y=T-B+A_1-A_2$. Substitution gives
$$
Y=T-B-C+\epsilon_1-\epsilon_2,\qquad
u^2(Y)=1+9+9+4+4=27\,\mathrm{mg^2}.
$$
The shared $C$ does not cancel, because its sensitivities are one and two. Equivalently, $\operatorname{Var}(A_1)=13$, $\operatorname{Var}(A_2)=40$ and $\operatorname{Cov}(A_1,A_2)=18$ in squared milligrams. Their difference has variance $13+40-2(18)=17$, giving the same total 27. Treating the reference readings as independent would incorrectly give 63 for the total variance.
If both readings instead contained $C$ with coefficient one, that shared contribution would cancel and the total would be 18 in the same units. These are different acquisition models, not two uncertainty conventions for the same data.
A usable lineage record therefore includes acquisition identifiers, calibration versions, reuse of numerical values and sensitivity coefficients. For nonlinear data reduction, a local sensitivity matrix replaces the linear $A$ when first-order propagation is justified. Covariance must follow the actual reuse and dependence structure; assigning independence because derived columns have different labels can discard the information needed for correct cancellation.
Back to article page
