Instrumentation & Measurement Magazine 24-4 - 42

Chemometrics for Data
Interpretation: Application of
Principal Components Analysis
(PCA) to Multivariate
Spectroscopic Measurements
Leonardo Iannucci
E
xtracting relevant and useful information from measurements
is an issue of paramount importance and
it can be considered as complementary to the process
of data acquisition. This is a crucial point especially in the
field of chemical measurements, where data sets can consist
of hundreds or even thousands of variables so their interpretation
can require long time. Chemometrics try to tackle this
issue by applying mathematical and statistical tools to data
coming from chemical, biological or medical analyses. Among
possible methods, Principal Components Analysis (PCA) has
found wide application in the I&M field thanks to its ability to
identify patterns in acquired measurements and classify data
in different groups. Possible applications span from chemicals
detection [1] to concentration estimation of compounds in
a given system [2]. Actually, many studies demonstrated the
possibility to use PCA to process different kinds of data [3], in
some cases coupling PCA to other tools such as artificial neural
networks to improve the processing performance [4].
Many books addressed PCA in an exhaustive and complete
way, so readers can refer to them in order to obtain a thorough
discussion of this topic [5], [6]. The aim of this article is rather
to provide only a simple dissertation for a beginner in this field
to show how powerful the PCA is. Specifically, detailed indications
for the analysis of spectroscopic measurements are
provided, as this kind of data is often left out in many reviews
concerning PCA. Moreover, tips and instructions are given to
let the reader write his/her own code and implement such data
processing using the Python programming language not only
for spectroscopic data, but more in general for any kind of data.
PCA Basics
Modern instruments allow researchers to collect huge amount
of data in an easy and often automated way. Obviously, this is
true also for the field of chemistry, where the possibility of performing
fast and inexpensive analyses has generally moved
the critical point in the design of experiments from phenomenon
measurement to data interpretation.
When more than one quantity is measured for each sample,
it is possible to define this data set as multivariate, i.e., each
42
measurement is composed of many variables. This is the case,
for example, when measuring length, width, and weight of
different objects. It is possible to describe multivariate measurements
as a matrix composed of m rows, representing
m analyzed samples, and n columns, indicating each of the
analyzed variables. In chemistry, common examples for multivariate
data sets are spectroscopic measurements, i.e., those
analyses in which the interaction between the sample and an
electromagnetic radiation is studied. Actually, techniques like
Infra-Red spectroscopy or UV-Visible spectroscopy probe the
sample using not just a single wavelength, but rather using
a range of wavelengths in order to identify and study different
chemical bonds. So, when dealing with spectroscopic
measurements, it is possible to talk about multivariate measurements,
and each of the analyzed wavelengths is a variable
composing the acquired data set.
In multivariate measurements there is real possibility to
have correlation between different variables, so a great benefit
can derive from removal of redundant information. Principal
Components Analysis (PCA) is, in its simplest definition,
a method to perform variables reduction in an acquired data
set. The original data matrix is transposed into a new space
having lower dimension but where the new variables constituting
the model (named principal components) account
for most of the variability contained in the original data set.
In matrix notation, it is possible to describe this operation as
follows:
X TP E
 
T
(1)
where X is the original data matrix (having dimension m x n),
P is the loading matrix that is the eigenvectors representing
the new space (with dimension n x k, where k is the number of
variables in the PCA model usually much lower than n), T is
the score matrix, composed of the eigenvalues derived from X
matrix decomposition and having dimension m x k and eventually
E is the residual matrix, sometimes referred to as error
matrix, which contains the variance burden not explained by
the PCA model.
IEEE Instrumentation & Measurement Magazine
1094-6969/21/$25.00©2021IEEE
June 2021

Instrumentation & Measurement Magazine 24-4

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