= Multivariate calibration
<Multivariate calibration> relates several measured signals jointly to one or more chemical quantities, using reference samples whose relevant properties are known. Multiple <wavelengths>, detector channels or other signal coordinates can help distinguish overlapping component responses. The IUPAC multicomponent-calibration report treats both forward response models and inverse prediction, together with uncertainty and model validation.<Guidelines for calibration in analytical chemistry. Part 2: Multicomponent calibration by Klaus Danzer and others>[Sections 2–5; official project description, regression and validation scope]{ref}
In a forward linear <measurement model>, write
$$
\mathbf y=\mathbf b+M\mathbf c+\boldsymbol\varepsilon,
$$
where $\mathbf c$ contains <analyte> <concentrations>, $\mathbf b$ is a background response, and $M$ contains channel sensitivities determined from calibration mixtures. If $M$ has full column <matrix rank>, an unweighted <least-squares estimator> is
$$
\widehat{\mathbf c}=(M^TM)^{-1}M^T(\mathbf y-\mathbf b).
$$
The number of channels alone does not guarantee identification: their response vectors must provide independent information.
For an original example with fixed chosen signal/concentration units, set
$$
M=\begin{pmatrix}1&0\\0&1\\1&1\end{pmatrix},\qquad
\mathbf b=0,\qquad\mathbf y=\begin{pmatrix}1.0\\0.5\\1.8\end{pmatrix}.
$$
The estimated concentrations are $(1.1,0.6)^T$. Predicted signals are $(1.1,0.6,1.7)^T$, so the <calibration residuals> are $(-0.1,-0.1,0.1)^T$. They satisfy $M^T\mathbf r=0$: <least squares> balances the three channels rather than satisfying the first two exactly. This example treats the calibrated sensitivities as known; their uncertainty would add another contribution.
If the channel errors instead have known <covariance matrix> $\Sigma$, <generalized least squares> replaces the estimate by
$$
\widehat{\mathbf c}=(M^T\Sigma^{-1}M)^{-1}M^T\Sigma^{-1}(\mathbf y-\mathbf b).
$$
Correlated channels do not provide the same information as independent repetitions. Nearly dependent sensitivity columns also amplify noise, connecting wavelength selection with the <condition number> of the inverse problem.
An inverse model can instead fit concentrations directly from measured signal vectors. Methods such as <principal component regression> reduce the signal representation, but the retained variation must still predict the intended chemical quantity. <Cross-validation> and separately prepared mixtures test prediction; a high calibration fit alone cannot establish transfer to another sample matrix. Preparing standards in which component concentrations vary independently helps expose confounding between analytes, rather than teaching the model only one fixed mixture ratio.
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