BUCHI NIRCal User Manual
Page 43
Chemometrics
NIRCal 5.5 Manual, Version A
43
Scores: v
in
Weightings of each PC after the pretreated spectrum has been transformed into the cluster. A
score is the portion of a PC used for the reconstruction of the original spectrum. Each
spectrum has different and up to max. i scores.
Residuum: R n (k)
The difference between the pretreated spectrum and the reconstructed spectrum is the
residuum spectrum. When the residuum is summed across the wavelength, a number is
obtained, the Residual.
The scores and the residual are variables for each spectrum.
Leverage
The direct distance of the spectrum from the centre of the coordinate system in the score
place.
Reconstruction of a spectrum
Now, each spectrum can be reconstructed on the basis of a sum through multiplying of scores and
PCs.
Any desired spectrum I n (k) of the calibration set is developed:
Formula:
For the reconstruction it is now only necessary to save the scores v
in
and the residual R
n
of each
individual spectrum, since the mean spectrum and PCs are constant for the entire set of spectra
in a calibration.
The spectrum can also be described as the linear combination of the PCs -U i (k)- and their scores
(mean-centred data matrix).
The following figure shows the reconstruction of a spectrum:
Mathematically speaking, the Principal Component Analysis is then a breakdown of the spectrum
matrix into 2 smaller matrices. This matrix operation can be represented graphically for, e.g., 15 PC's
in the following way: