1 INTRODUCTION: GENERAL CONCEPT, AMBIGUITIES, RESOLUTION THEOREMS
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In the resolution of any multicomponent system, the main goal is to transform
the raw experimental measurements into useful information. By doing so, we aim
to obtain a clear description of the contribution of each of the components present
in the mixture or the process from the overall measured variation in our chemical
data. Despite the diverse nature of multicomponent systems, the variation in their
related experimental measurements can, in many cases, be expressed as a simple
composition-weighted linear additive model of pure responses, with a single term
per component contribution. Although such a model is often known to be followed
because of the nature of the instrumental responses measured (e.g., in the case of
spectroscopic measurements), the information related to the individual contributions
involved cannot be derived in a straightforward way from the raw measurements.
The common purpose of all multivariate resolution methods is to ﬁll in this gap and
provide a linear model of individual component contributions using solely the raw
experimental measurements. Resolution methods are powerful approaches that do
not require a lot of prior information because neither the number nor the nature of
the pure components in a system need to be known beforehand. Any information
available about the system may be used, but it is not required. Actually, the only
mandatory prerequisite is the inner linear structure of the data set. The mild requirements needed have promoted the use of resolution methods to tackle many chemical
problems that could not be solved otherwise.
All resolution methods mathematically decompose a global instrumental
response of mixtures into the contributions linked to each of the pure components
in the system [1–10]. This global response is organized into a matrix D containing
raw measurements about all of the components present in the data set. Resolution
methods allow for the decomposition of the initial mixture data matrix D into the
product of two data matrices C and ST, each of them containing the pure response
proﬁles of the n mixture or process components associated with the row and the
column directions of the initial data matrix, respectively (see Figure 11.2). In matrix
notation, the expression for all resolution methods is:
D = CST + E
(11.1)
where D (r × c) is the original data matrix, C (r × n) and ST (n × c) are the matrices
containing the pure-component proﬁles related to the data variation in the row
direction and in the column direction, respectively, and E (r × c) is the error matrix,
i.e., the residual variation of the data set that is not related to any chemical contribution. The variables r and c represent the number of rows and the number of
columns of the original data matrix, respectively, and n is the number of chemical
components in the mixture or process. C and ST often refer to concentration proﬁles
and spectra (hence their abbreviations and the denomination we will adopt often in
this chapter), although resolution methods are proven to work in many other diverse
problems [13–20].
From the early days in resolution research, the mathematical decomposition of
a single data matrix, no matter the method used, has been known to be subject to
ambiguities [1, 2]. This means that many pairs of C- and ST-type matrices can be
found that reproduce the original data set with the same ﬁt quality. In plain words,
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Mixed information
Pure component information
s1
sn
=
c1
D
ST
cn
Absorbance
C
Rete
ntion
time
s
hs
ngt
vele
a
W
Retention times
Pure concentration profiles
Wavelengths
Pure signals
FIGURE 11.2 Resolution of a multicomponent chromatographic HPLC-DAD run (D matrix)
into their pure concentration proﬁles (C matrix, chromatograms) and responses (ST matrix,
spectra) [10].
the correct reproduction of the original data matrix can be achieved by using component proﬁles differing in shape (rotational ambiguity) or in magnitude (intensity
ambiguity) from the sought (true) ones [21].
These two kinds of ambiguities can be easily explained. The basic equation
associated with resolution methods, D = CST, can be transformed as follows:
D = C (T T−1) ST
(11.2)
D = (CT) (T−1 ST)
(11.3)
D = C′ S′T
(11.4)
where C¢ = CT and S¢T = (T−1 ST) describe the D matrix as correctly as the true
C and ST matrices do, though C¢ and S¢T are not the sought solutions. As a result
of the rotational ambiguity problem, a resolution method can potentially provide as
many solutions as T matrices can exist. Often this may represent an inﬁnite set of
solutions, unless C and S are forced to obey certain conditions. In a hypothetical
case with no rotational ambiguity, that is, in the case where the shapes of the proﬁles
in C and S are correctly recovered, the basic resolution model could still be subject
to intensity ambiguity, as shown in Equation 11.5
n
D=
i
i =1
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1
∑ k c ( k s )
i
i
T
i
(11.5)
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where ki are scalars and n refers to the number of components. Each concentration
proﬁle of the new C¢ matrix (Equation 11.4) would have the same shape as the
real one, but it would be ki times smaller, whereas the related spectra of the new
S¢T matrix (Equation 11.4) would be equal in shape to the real spectra, though ki
times more intense.
The correct performance of any curve-resolution (CR) method depends strongly
on the complexity of the multicomponent system. In particular, the ability to correctly
recover dyads of pure proﬁles and spectra for each of the components in the system
depends on the degree of overlap among the pure proﬁles of the different components
and the speciﬁc way in which the regions of existence of these proﬁles (the so-called
concentration or spectral windows) are distributed along the row and column directions of the data set. Manne stated the necessary conditions for correct resolution
of the concentration proﬁle and spectrum of a component in the 2 following
theorems [22]:
1. The true concentration proﬁle of a compound can be recovered when all
of the compounds inside its concentration window are also present outside.
2. The true spectrum of a compound can be recovered if its concentration
window is not completely embedded inside the concentration window of
a different compound.
According to Figure 11.3, the pure concentration proﬁle of component B can
be recovered because A is inside and outside B's concentration window; however,
B's pure spectrum cannot be recovered because its concentration proﬁle is totally
embedded under the major compound, A. Analogously, the pure spectrum of A can
be obtained, but not the pure concentration proﬁle because B is present inside its
concentration window, but not outside.
The same formulation of these two theorems holds when, instead of looking at
the concentration windows in rows, the “spectral” windows in columns are considered. In this context, the theorems show that the goodness of the resolution results
depends more strongly on the features of the data set than on the mathematical
background of the CR method selected. Therefore, a good knowledge of the properties of the data sets before carrying out a resolution calculation provides a clear
idea about the quality of the results that can be expected.
A
B
FIGURE 11.3 Concentration proﬁles for a two-component system (see comments in text
related to resolution).
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11.2 HISTORICAL BACKGROUND
The ﬁeld of curve resolution was born in response to the need for a tool to analyze
multivariate experimental data from multicomponent dynamic systems. The common
goal of all curve-resolution methods is to mathematically decompose the global
instrumental response into the pure-component proﬁles of each of the components
in the system. The use of these methods has become a valuable aid for resolving
complex systems, especially when obtaining selective signals for individual species
is not experimentally possible, too complex, or too time consuming.
Two pioneering papers on curve resolution were published by Lawton and Sylvestre early in the 1970s [1, 2]. In particular, a mixture analysis resolution problem was
described in mathematical terms for the case of a simple two-component spectral
mixture. Interestingly, several concepts introduced in these early papers were the
precursors of the ideas underlying most of the curve-resolution methods developed
afterward. For instance, the concept of pure-component solutions as a linear combination of the measured spectra and vice versa was presented; the concept of a
subspace spanned by “true” solutions in relation to the subspace spanned by PCA
(principal component analysis) solutions was presented; and the concept of a range
or band of feasible solutions, and how to reduce the width of this band by means
of constraints, such as nonnegativity and closure (mass balance) equations, was
presented. Later on, these ideas were reformulated more precisely using the concepts
of rotational and intensity ambiguities [23], which are found ubiquitously in all
factor-analysis matrix bilinear decomposition methods.
The extension of Lawton and Sylvestre’s curve resolution from two- to threecomponent systems was presented by Borgen et al., [3, 4] focusing on the optimization of ranges of feasible solutions. At the same time, the ﬁrst edition of
Malinowski's book [24] Factor Analysis in Chemistry appeared [25], which presented
a review of updated concepts and applications. In a way, Malinowski’s book could be
considered for many researchers in this ﬁeld as the consolidation of the incipient subject
of chemometrics, at a time when this term was still not widely accepted.
The main goal of factor analysis, i.e., the recovery of the underlying “true”
factors causing the observed variance in the data, is identical to the main goal of
curve-resolution methods. In factor analysis, “abstract” factors are clearly distinguished from “true” factors, and the key operation is to ﬁnd a transformation from
abstract factors to the true factors using rotation methods. Two types of rotations
are usually used, orthogonal rotations and oblique rotations. Principal component
analysis, PCA, (or principal factor analysis, PFA) produces an orthogonal bilinear
matrix decomposition, where components or factors are obtained in a sequential
way to explain maximum variance (see Chapter 4, Section 4.3, for more details).
Using these constraints plus normalization during the bilinear matrix decomposition,
PCA produces unique solutions. These “abstract” unique and orthogonal (independent) solutions are very helpful in deducing the number of different sources of
variation present in the data. However, these solutions are “abstract” solutions in
the sense that they are not the “true” underlying factors causing the data variation,
but orthogonal linear combinations of them. On the other hand, in curve-resolution
methods, the goal is to unravel the “true” underlying sources of data variation. It is
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not only a question of how many different sources are present and how they can be
interpreted, but to ﬁnd out how they are in reality. The price to pay is that unique
solutions are not usually obtained by means of curve-resolution methods unless
external information is provided during the matrix decomposition.
Different approaches have been proposed during recent years to improve the
solutions obtained by curve-resolution methods, and some of them are summarized
in the next sections. The ﬁeld is already mature and, as it has been recently pointed
out [26], multivariate curve resolution can be considered as a “sleeping giant of
chemometrics,” with a slow but persistent growth.
Whenever the goals of curve resolution are achieved, the understanding of a
chemical system is dramatically increased and facilitated, avoiding the use of
enhanced and much more costly experimental techniques. Through multivariateresolution methods, the ubiquitous mixture analysis problem in chemistry (and other
scientiﬁc ﬁelds) is solved directly by mathematical and software tools instead of
using costly analytical chemistry and instrumental tools, for example, as in sophisticated “hyphenated” mass spectrometry-chromatographic methods.
11.3 LOCAL RANK AND RESOLUTION: EVOLVING
FACTOR ANALYSIS AND RELATED TECHNIQUES
Manne’s resolution theorems clearly stated how the distribution of the concentration
and spectral windows of the different components in a data set could affect the
quality of the pure proﬁles recovered after data analysis [22]. The correct knowledge
of these windows is the cornerstone of some resolution methods, and in others where
it is not essential, information derived from this knowledge can be introduced to
generally improve the results obtained.
Setting the boundaries of windows of the different components can only be done
if we are able to know how the number and nature of the components change in the
data set. Obtaining this information is the main goal of local-rank analysis methods,
which are used to locate and describe the evolution of each component in a system.
This is accomplished by combining the information obtained from multiple rank
analyses performed locally on limited zones (row or column windows) of the data set.
Some of the local-rank analysis methods, such as evolving-factor analysis (EFA)
[27–29], are more process oriented and rely on the sequential evolution of the
components as a function of time or any other variable in the data set, while others,
such as ﬁxed-size moving-window–evolving-factor analysis (FSMW-EFA) [30, 31],
can be applied to processes and mixtures. EFA and FSMW-EFA are the two
pioneering local-rank analysis methods and can still be considered the most representative and widely used.
Evolving-factor analysis was born as the chemometric way to monitor chemicalevolving processes, such as HPLC diode-array data, batch reactions, or titration data
[27–28]. The evolution of a chemical system is gradually measured by recording a
new response vector at each stage of the process under study. Mimicking the experimental protocol, EFA performs principal component analyses on submatrices of
gradually increasing size in the process direction, enlarged by adding a row
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(response), one at a time. This procedure is performed from top to bottom of the
data set (forward EFA) and from bottom to top (backward EFA) to investigate the
emergence and the decay of the process contributions, respectively. Figure 11.4b
displays the information provided by EFA for an HPLC-DAD example and how to
interpret the results.
Each time a new row is added to the expanding submatrix (Figure 11.4b), a
PCA model is computed and the corresponding singular values or eigenvalues are
saved. The forward EFA curves (thin solid lines) are produced by plotting the saved
singular values or log (eigenvalues) obtained from PCA analyses of the submatrix
expanding in the forward direction. The backward EFA curves (thin dashed lines)
are produced by plotting the singular values or log (eigenvalues) obtained from the
PCA analysis of the submatrix expanding in the backward direction. The lines
connecting corresponding singular values (s.v.), i.e., all of the ﬁrst s.v., the second
s.v., the ith s.v., indicate the evolution of the singular values along the process and,
as a consequence, the variation of the process components. Emergence of a new
singular value above the noise level delineated by the pool of nonsigniﬁcant singular
values indicates the emergence of a new component (forward EFA) or the disappearance of a component (backward EFA) in the process.
Figure 11.4b also shows how to build initial estimates of concentration proﬁles
from the overlapped forward and backward EFA curves as long as the process evolves
in a sequential way (see the thick lines in Figure 11.4b). For a system with n
signiﬁcant components, the proﬁle of the ﬁrst component is obtained combining the
curve representing the ﬁrst s.v. of the forward EFA plot and the curve representing
the nth s.v. of the backward EFA plot. Note that the nth s.v. in the backward EFA
plot is related to the disappearance of the ﬁrst component in the forward EFA plot.
The proﬁle of the second component is obtained by splicing the curve representing the
second s.v. in the forward EFA plot to the curve representing (n − 1)th s.v. from the
backward EFA plot, and so forth. Combining the two proﬁles into one proﬁle is easily
accomplished in a computer program by selecting the minimum value from the two
s.v. lines to be combined. It can be seen that the resulting four elution proﬁles
obtained by EFA are good approximations of the real proﬁles shown in Figure 11.4a.
The information provided by the EFA plots can be used for the detection and
location of the emergence and decay of the compounds in an evolving process.
As a consequence, the concentration window and the zero-concentration region
for each component in the system are easily determined for any process that evolves
such that the emergence and decay of each component occurs sequentially. For
example, the concentration window of the ﬁrst component to elute is shown as a
shadowed zone in Figure 11.4b. Uses of this type of information have given rise
to most of the noniterative resolution methods, explained in Section 11.4 [32–39].
Iterative resolution methods, explained in Section 11.5, use the EFA-derived estimates of the concentration proﬁles as a starting point in an iterative optimization
[40, 41]. The location of selective zones and zones with a number of compounds
smaller than the total rank can also be introduced as additional information to
minimize the ambiguity in the resolved proﬁles [21, 41, 42].
As mentioned earlier, FSMW-EFA is not restricted in its applicability to evolving
processes, although the interpretation of the ﬁnal results is richer for this kind of problem.
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