= Product-null spectral contrasts
A product-null spectral contrast is a weighted combination of spectral measurements that cancels a modeled product contribution while retaining sensitivity to the carrier of interest. It is useful when carrier consumption and product formation both change the <absorbance> recorded in an <inhibited autoxidation assay>.
After background correction and with path length included in the response coefficients, write a measurement vector as
$$
\mathbf A=\mathbf d\,c_D+\mathbf p\,c_P+\boldsymbol\epsilon.
$$
Here $\mathbf d$ and $\mathbf p$ are known carrier and product spectral response columns. A contrast $\widehat c_D=\mathbf w^{\mathsf T}\mathbf A$ removes the product and has unit carrier response when
$$
\mathbf w^{\mathsf T}\mathbf p=0,\qquad
\mathbf w^{\mathsf T}\mathbf d=1.
$$
If the columns are proportional, these constraints are incompatible. More measurements do not help if they add no independent spectral information.
An original three-channel example uses scaled coefficient columns $\mathbf d=(3,1,0)^{\mathsf T}$ and $\mathbf p=(1,1,1)^{\mathsf T}$. The weights $\mathbf w=(5,-1,-4)^{\mathsf T}/14$ satisfy both constraints. Stipulated concentrations $(c_D,c_P)=(0.2,0.1)$ give $\mathbf A=(0.7,0.3,0.1)^{\mathsf T}$ and the contrast returns $0.2$. The coefficient and concentration units have been absorbed into the scaling; these are not measured dye coefficients.
For independent equal-variance channel noise $\sigma^2$, the example has variance $\sigma^2\|\mathbf w\|^2=3\sigma^2/14$. These are the minimum-variance weights under the stated noise model: project $\mathbf d$ perpendicular to $\mathbf p$, then normalize that residual to unit carrier response. The <Cauchy-Schwarz inequality> shows that any other vector with the same constraints has at least as large a norm. A two-channel difference using just the first two measurements also isolates the carrier, but weights $(1/2,-1/2,0)$ give the larger variance $\sigma^2/2$.
For general <covariance matrix> $\Sigma$, the variance is $\mathbf w^{\mathsf T}\Sigma\mathbf w$, so correlated or unequal noise changes the optimal weighting.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, covariance propagation]{ref} The coupled chemical readout remains the one described by the primary signal-carrier method.<A Continuous Visible Light Spectrophotometric Approach To Accurately Determine the Reactivity of Radical-Trapping Antioxidants by Evan A. Haidasz and others>[Abstract]{ref} Additional products or changing protonation require additional response columns; a cancellation calibrated for one product does not remove every unknown spectral contribution.
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