Instrumentation & Measurement Magazine 24-4 - 45

Fig. 2. (a) Cumulative explained variance as a function of the components number in the PCA model. Each marker is labelled with the percent variance captured
by the corresponding PC. (b) Loadings associated to the first three components in the PCA model.
eigenvectors = model_pca.components_
It is possible to use the loadings in order to check which features
(i.e., peaks) the explained variance is representing and
thus reach a more clear results interpretation. Actually, the
original spectrum can be expressed as a linear combination of
eigenvectors and eigenvalues, according to the following expression,
equivalent to (1):
X tp t p
   
11 2 2
TT T
kk
t p E
(3)
As it is not possible to have negative peaks in Raman spectroscopy,
features in negative domain will correspond to
negative eigenvalues or they will be compensated by another
component so as to obtain positive values in the linear combination
(3).
As an example, in Fig. 2 it is possible to see that the first
two components are characterized by a sharp peak before 1000
cm−1
, which is characteristic of sulphates, slightly shifted depending
on the specific mineral. Immediately close to it and
also in other regions of the spectrum there are minimum points
that thus can be explained by looking at the corresponding
eigenvalues.
The effect of the addition of an increasing number of components
can be observed in Fig. 3. The figure shows how the
model progressively approximates the acquired spectrum,
thanks to the use of a higher number of components, until
reaching a satisfactory result with three PC. Specifically, the
Fig. 3. Reconstruction of two spectra using an increasing number of PCs. Original spectrum is represented in black, colored lines are used for the model. It is
possible to see how the addition of PCs improves the goodness of fit. Sample names stand for the spectra Rruff ID. (a) R060090. (b) R100199.
June 2021
IEEE Instrumentation & Measurement Magazine
45

Instrumentation & Measurement Magazine 24-4

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