Multivariate Data Analysis for Advancing the Interpretation of Bioprocess Measurement and Monitoring Data - Measurement, Monitoring, Modelling, and Control of Bioprocesses - page 173

3.2 Principal Component Analysis

Principal component analysis (PCA) [ 7 ] is probably the most established MVDA
method of feature extraction. It works by generating a new group of uncorrelated
variables (principal components, PCs) from a high-dimensional data set. The
approach transforms a matrix [X], containing measurements from n measured
variables, into a matrix of mutually uncorrelated PCs, t k (where k = 1ton), which
are transforms of the original data into a new basis defined by a set of orthogonal
loading vectors, p k . The individual values of the principal components are called
scores. The transformation can be described by Eq. ( 1 ):

½¼ X

np\n

t k p k þ E ð 1 Þ

k ¼ 1

The loadings are the eigenvectors of the data covariance matrix, X T X. The t k
and p k pairs are ordered such that the first pair captures the largest amount of
variation in the data and the last pair captures the least. This means that fewer PCs
are required to describe the relationship than the original process variables. This
data compression allows for easier visualisation of the data for the purposes of
feature extraction, which can be used in a variety of applications within

bioprocessing.
Literature sources on PCA applications within bioprocess data analysis are
numerous and cover all aspects of bioprocessing from raw material, seed culti-
vation, production batch or downstream process quality monitoring [ 2 , 9 , 44 ].
Typically in these applications, high-dimensional process data are compressed into
a set of principal components which are used either as inputs for further MVDA
process models or as a monitoring tool within a multivariate statistical process
control (MSPC) scheme.

3.2.1 Variants of PCA

There are a number of reported variants of PCA, introduced due to specific
application requirements; for example, Alexandrakis [ 1 ] discusses the need to
increase the robustness of PCA models in the analysis of NIR spectral data in order
to deal with external factors influencing the spectra in the industrial setting. These
factors include not only instrument effects not observed at laboratory scale during
process development (e.g. temperature effects and stray light), but more signifi-
cantly sample- (e.g. differences in particle sizes) and process-related effects (e.g.
variability in unit operation conditions). Whilst a number of alternative methods
have been discussed to address these issues, the use of orthogonal methods, such as
external parameter orthogonalization (EPO), transfer by orthogonal projection
(TOP) and dynamic orthogonal projection (DOP) [ 22 , 46 ] are highlighted as useful
tools for such an application.