https://doi.org/10.1351/goldbook.10111
Interactive method to obtain concentrations and pure spectra from spectra of mixtures using directly-measured variables.
Notes:
- The directly-measured variables are called "pure variables" in the method.
- A data matrix \(\boldsymbol{D} = \boldsymbol{C} \times \boldsymbol{P}^{\rm{T}} + \boldsymbol{E}\) where \(\boldsymbol{C}\) is a concentration matrix, \(\boldsymbol{P}\) pure spectra of mixture components and \(\boldsymbol{E}\) an error matrix. Pure spectra are estimated \(\boldsymbol{\hat Y}\) which allows projection of a concentration matrix \(\boldsymbol{C^{*}}\) from which the data matrix can be reconstructed and compared with the measured spectra. \[\left\{ {\matrix{ {\boldsymbol{\hat P} = \boldsymbol{D}^{\rm{T}} \boldsymbol{C}(\boldsymbol{C}^{\rm{T}}\boldsymbol{C})}^{-1} \cr {\boldsymbol{C^{*}} = \boldsymbol{C}\boldsymbol{\hat P}(\boldsymbol{\hat P}^{\rm{T}}\boldsymbol{\hat P})}^{-1} \cr }} \right. {\rm{then }}\;{\boldsymbol{D}_{{\rm{recon}}}} = \boldsymbol{C^{*}} \times \boldsymbol{\hat P}^{\rm{T}}\]
- Second derivatives of spectra can be used for modelling.