 Sensitivity of a tie-line fit to its weight matrix

Sensitivity of a tie-line fit to its weight matrix describes how changing the metric changes an inferred phase fraction, even when the measured compositions are held fixed. For distinct endpoints $\mathbf{a},\mathbf{b}$, define $\mathbf{d}=\mathbf{b}-\mathbf{a}$, $q={\mathbf{d}}^{T}W\mathbf{d}$ and the interior unconstrained tie-line phase-fraction fitDifferentiating the normal equation ${\mathbf{d}}^{T}W\mathbf{r}=0$, with $\mathbf{a},\mathbf{b},\mathbf{y}$ fixed, givesThis term supplements the composition and endpoint derivatives in tie-line phase-fraction sensitivity; it is not included in their fixed-metric formulas.

For an original example, use independent mole fraction coordinates $\mathbf{a}=(0.70,0.10)$, $\mathbf{b}=(0.30,0.50)$ and $\mathbf{y}=(0.45,0.38)$. With $W=I$, the fitted fraction is $0.6625$, $q=0.32$ and $\mathbf{r}=(0.015,0.015)$. Increasing the second diagonal weight from $1$ to $1.02$ predictsDirect refitting gives $0.21424/0.3232\approx 0.6628713$, a change of $0.0003713$. The small difference is the neglected higher-order term.

Two invariances clarify the result. An exactly balanced composition has $\mathbf{r}=0$ and the same fraction under every positive-definite metric. Also, replacing $W$ by $cW$ for positive $c$ leaves the fraction unchanged: the scalar rescaling invariance of a tie-line weight matrix follows from cancellation in its ratio. The differential agrees because ${\mathbf{d}}^{T}W\mathbf{r}=0$.

If $W={\Sigma }^{-1}$ for a composition covariance matrix, its perturbation is $\delta W=-W(\delta \Sigma )W$ to first order. Uncertain fitted weights can therefore add sensitivity, with correlations to the composition measurements retained in the measurement model. This is an application of the guide's general sensitivity-coefficient framework, not a tie-line formula quoted from it.^[ref] Boundary-constrained fits and coincident endpoints need separate treatment.