https://doi.org/10.1351/goldbook.10153
Calibration in which the indications are multivariate data.
Notes:
- In multivariate regression and calibration the problem is usually posed as a relation between the indications \(\boldsymbol{X}\) (which are multivariate) and the property to be measured \(\boldsymbol{y}\). Note that this is a reversal of the traditional form, \(x\) (concentration)/\(y\) (indication) of linear calibration. \(\boldsymbol{X}\) is a vector of observations (indications), and \(\boldsymbol{c}\) a vector of concentrations. In matrix form: \(\boldsymbol{X}\boldsymbol{b} = \boldsymbol{c} + \boldsymbol{e}\) where \(\boldsymbol{b}\) are the coefficients of the model and \(\boldsymbol{e}\) is a vector of errors.
- Regression establishes the coefficients \(\boldsymbol{b}\) (\(\boldsymbol{X}\boldsymbol{b} = \boldsymbol{c} + \boldsymbol{e}\)) from given indications \(\boldsymbol{X}\), or some factorization of \(\boldsymbol{X}\), and values \(\boldsymbol{c}\), or some factorization of \(\boldsymbol{c}\). Given indications \(\boldsymbol{X}_{\rm{u}}\) from an unknown sample, the quantity value \(c_{\rm{u}}\) can be calculated.