 Stilbene composition consistency residual

A stilbene composition consistency residual is the difference between an additional optical observation and the value predicted from a fitted stilbene isomer mixture. It tests agreement with the measurement model rather than directly measuring an impurity or proving sample purity.

In a stilbene isomer composition measurement, two calibrated wavelengths can determine modeled concentrations $c_E,c_Z$ of trans-Stilbene and cis-Stilbene. Let $a_j$ be background-subtracted absorbance divided by path length, and let $E_{12}$ contain the first two wavelengths' molar absorption coefficients. For nonsingular $E_{12}$,The vector $e_3$ contains the third wavelength's coefficients. That third observation was not used to fit the two concentrations, so it provides an additional consistency condition.

An original example uses coefficient rows $(2,1)$, $(1,3)$ and $(1,1)$ in consistent scaled units. Observations $(a_1,a_2)=(0.70,0.60)$ give $({\hat{c}}_E,{\hat{c}}_Z)=(0.30,0.10)$ and third-channel prediction $0.40$. An observed $a_3=0.43$ gives $r_3=0.03$. These are constructed coefficients, not measured stilbene spectra; the primary identity records establish that the two geometric isomers are chemically distinct despite their shared formula.^[ref][ref]

The fitted concentrations also carry reading error. In this example,With $q=(-0.4,-0.2,1{)}^{\mathsf{T}}$ and observation covariance matrix $\Sigma$, linear uncertainty propagation gives $u_r^2=q^{\mathsf{T}}\Sigma q$ when the coefficients are treated as fixed. Independent readings with equal standard uncertainty $0.01$ give $u_r=0.01\sqrt{1.2}\simeq 0.01095$. The residual is then about $2.74$ such standard uncertainties; interpreting that as a probability statement needs an additional distribution model. Calibration-coefficient uncertainty must also be included when material.^[ref]

A common additive offset $b$ changes this residual by $0.4b$, so it does not cancel automatically. A nonzero residual can reflect background, changing illumination, coefficient errors or an unmodeled absorber. Conversely, an extra component whose responses mimic a combination of the two modeled spectra can remain hidden even when the residual is zero. Consistency is evidence about a stated model, not a unique identification of the sample's full composition.