7 NIR Data Exploration and Regression by Chemometrics—A Primer
147
7.4 Spectral Exploration by Principal Component Analysis
(PCA)
In practice, we can resolve the NIR spectral ensembles into a low
number of orthogonal latent variables by Principal Component Analysis:
X = T · P
T
+ E
Principal component analysis (PCA) is the workhorse of chemometrics. The
method is now more than 100 years old [29–31], is used in many research disciplines
for different purposes and is therefore unfortunately known under many different
names. Spectroscopic data are characterized by high colinearity, i.e., that two neighboring wavelengths are positively correlated. PCA is tailored to handle this type of
data, and it is in the analysis of spectroscopic data that PCA really shows its worth.
However, before we go to the analysis of NIR spectroscopic data, we will briefly
outline the principle of PCA. There is a striking similarity between the PCA and the
MCR models, as they both attempt to approximate the variation in the data with a
bilinear model. The difference between the two models lies in how the system of
equations is solved. For PCA, an algorithm is used that successively finds orthogonal components in a multivariate dataset X. This principle is, in contrast to the
MCR algorithms, extremely efficient and robust, but the solution has the interpretative disadvantage that the extracted components are forced to be orthogonal while
spectra in a mixture are not.
7.4.1 The PCA Method
A key concept in chemometric analysis is the reduction of variance in data into
a lower-dimensional space of principal components or latent variables. In PCA,
the multivariate dataset is decomposed into orthogonal components, whose linear
combinations approximate the original dataset in a least squares sense.
If we have an experiment of n observations (samples) of m independent variables
(wavelengths), a line describing the maximum observed variance in the variable space
can be defined as the least squares solution of minimizing each of the orthogonal
projected distances l n from the nth sample point onto to the principal line:
min
n
l
2
n
(7.15)
147
7.4 Spectral Exploration by Principal Component Analysis
(PCA)
In practice, we can resolve the NIR spectral ensembles into a low
number of orthogonal latent variables by Principal Component Analysis:
X = T · P
T
+ E
Principal component analysis (PCA) is the workhorse of chemometrics. The
method is now more than 100 years old [29–31], is used in many research disciplines
for different purposes and is therefore unfortunately known under many different
names. Spectroscopic data are characterized by high colinearity, i.e., that two neighboring wavelengths are positively correlated. PCA is tailored to handle this type of
data, and it is in the analysis of spectroscopic data that PCA really shows its worth.
However, before we go to the analysis of NIR spectroscopic data, we will briefly
outline the principle of PCA. There is a striking similarity between the PCA and the
MCR models, as they both attempt to approximate the variation in the data with a
bilinear model. The difference between the two models lies in how the system of
equations is solved. For PCA, an algorithm is used that successively finds orthogonal components in a multivariate dataset X. This principle is, in contrast to the
MCR algorithms, extremely efficient and robust, but the solution has the interpretative disadvantage that the extracted components are forced to be orthogonal while
spectra in a mixture are not.
7.4.1 The PCA Method
A key concept in chemometric analysis is the reduction of variance in data into
a lower-dimensional space of principal components or latent variables. In PCA,
the multivariate dataset is decomposed into orthogonal components, whose linear
combinations approximate the original dataset in a least squares sense.
If we have an experiment of n observations (samples) of m independent variables
(wavelengths), a line describing the maximum observed variance in the variable space
can be defined as the least squares solution of minimizing each of the orthogonal
projected distances l n from the nth sample point onto to the principal line:
min
n
l
2
n
(7.15)