https://doi.org/10.1351/goldbook.10152
Multivariate calibration in which the coefficients of the regression are calculated directly from the indications and values of standards.
Notes:
- In \(\boldsymbol{X}\boldsymbol{b} = \boldsymbol{c} + \boldsymbol{e}\) (where \(\boldsymbol{X}\) is a vector of indications, \(\boldsymbol{b}\) are coefficients of the model, \(\boldsymbol{c}\) is a vector of values of the quantity of interest and \(\boldsymbol{e}\) a vector of errors), \(\boldsymbol{b} = \boldsymbol{X^{+}}\boldsymbol{c}\), where \(\boldsymbol{X^{+}}\) is the pseudo-inverse of \(\boldsymbol{X}\), calculated as \(\boldsymbol{X^{+}} = (\boldsymbol{X}^{\rm{T}}\boldsymbol{X})^{-1} \boldsymbol{X}^{\rm{T}}\)
- It is assumed that \(\boldsymbol{X}^{\rm{T}}\boldsymbol{X}\) has full rank, i.e. there are more objects than variables, and the variables are independent.
- If \(\boldsymbol{X}^{\rm{T}}\boldsymbol{X}\) has full rank only because of noise, the solution can become unstable.