3 LOCAL RANK AND RESOLUTION: EVOLVING FACTOR ANALYSIS AND RELATED TECHNIQUES
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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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(a)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
5
425
10
15
20
25
30
Retention times
35
40
45
50
Forward EFA
(b)
Log (eigenvalues)
PCA
11
10.5
10
9.5
9
8.5
8
7.5
PCA
5
10
PCA
15
PCA
20
25
30
Retention times
PCA
PCA
35
40
PCA
45
50
PCA
PCA
Backward EFA
(c)
PCA
PCA
PCA
PCA
PCA
Log (eigenvalues)
5.5
5
4.5
0
4
3.5
0
1
5
10
2
15
1
2
20
25
30
Retention times
2
1
35
40
0
45
50
FIGURE 11.4 (a) Concentration proﬁles of an HPLC-DAD data set. (b) Information derived from
the data set in Figure 11.4a by EFA: scheme of PCA runs performed. Combined plot of forward EFA
(solid black lines) and backward EFA (dashed black lines). The thick lines with different line styles
are the derived concentration proﬁles. The shaded zone marks the concentration window for the ﬁrst
eluting compound. The rest of the elution range is the zero-concentration window. (c) Information
derived from the data set in Figure 11.4a by FSMW-EFA: scheme of the PCA runs performed. The
straight lines and associated numbers mark the different windows along the data set as a function of
their local rank (number). The shaded zones mark the selective concentration windows (rank 1).
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FSMW-EFA does not focus on a description of the evolution of the different components in a system as EFA does; rather, it focuses on the local rank of windows in the
concentration domain (rows) or the local rank of windows in the spectral response
domain (columns).
FSMW-EFA is carried out by conducting a series of PCA analyses on submatrices obtained by moving a window of a ﬁxed size through the data set, starting at
the top of the matrix and moving downward, one row at a time. The singular values
or eigenvalues from the repeated analyses are saved, and a plot is constructed by
connecting the corresponding singular values as done in EFA. Visual examination
of these plots gives a local-rank map of the data set, i.e., a representation of how
many components are simultaneously present in the different zones of the data set
(Figure 11.4c). For each window analyzed, the number of singular values exceeding
the noise level threshold is used to determine the local rank. The local-rank map
helps to identify selective zones in the data set (e.g., zones where the local rank is 1)
and to know the degree of compound overlap in the data set. The unambiguous
determination of the number of compounds present and their identities is only
possible in processes where components evolve sequentially or when more external
information is available. The window size is a parameter that has an effect on the
information obtained (e.g., local-rank maps). Wider windows increase the sensitivity
for detecting minor components, including components completely embedded under
major compounds. Narrower windows can provide more accurate resolution of
boundaries between zones of different rank.
New algorithms based on EFA and FSMW-EFA have reﬁned the performance
of the parent methods [43, 44] and have widened their applicability to the study of
systems with concurrent processes [45] or complex spatial structure, such as spectroscopic images [46].
11.4 NONITERATIVE RESOLUTION METHODS
Resolution methods are often divided in iterative and noniterative methods. Most
noniterative methods are one-step calculation algorithms that focus on the one-ata-time recovery of either the concentration or the response proﬁle of each component.
Once all of the concentration (C) or response (S) proﬁles are recovered, the other
member of the matrix pair, C and S, is obtained by least-squares according to the
general CR model, D = CST [32–38].
Noniterative methods use information from local-rank maps or concentration
windows in a characteristic way. In mathematical terms, these windows deﬁne
subspaces where the different compounds are present or absent. The subspaces can
be combined in clever ways through projections or by extraction of common vectors
(proﬁles) to obtain the proﬁles sought.
As mentioned in Section 11.3, the cornerstone of these procedures is the correct
location of concentration windows of the compounds of interest. Limitations of these
methods are linked to this point. Thus, data sets where the compositional evolution
of the compounds does not follow any clear pattern, such as in a series of mixtures
or image pixels, cannot be resolved by these methods because it is practically impossible to determine the concentration windows of components. Evolving processes are
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the most suitable systems to be analyzed but, again, attention should be paid to
situations where the pattern by which components emerge and decay is not sequentially ordered. Some examples that violate this requirement are nonunimodal concentration proﬁles or small embedded peaks under major peaks. In cases such as these,
specialized EFA derivations [39] should be used to avoid incorrect assignment of
component windows. Other problems associated with locating window boundaries
are due to the presence of noise that can blur the extremes of the concentration
windows. Errors from this source can also affect the quality of the ﬁnal results.
Noniterative methods are fast, but they have clear limitations in their applicability
because of the difﬁculties associated with correct deﬁnition of concentration windows
and local rank. Their use is practically restricted to processes with sequentially
evolving components like chromatography, the components of which fulﬁll the conditions required by Manne’s theorems, to ensure a correct component resolution [22].
11.4.1 WINDOW FACTOR ANALYSIS (WFA)
Window factor analysis (WFA) was described by Malinowski and is likely the most
representative and widely used noniterative resolution method [34, 35]. WFA recovers the concentration proﬁles of all components in the data set one at a time. To do
so, WFA uses the information in the complete original data set and in the subspace
where the component to be resolved is absent, i.e., all rows outside of the concentration window. The original data set is projected into the subspace spanned by where
the component of interest is absent, thus producing a vector that represents the
spectral variation of the component of interest that is uncorrelated to all other
components. This speciﬁc spectral information, combined appropriately with the
original data set, yields the concentration proﬁle of the related component. To ensure
the speciﬁcity of this spectral information, all other components in the data set should
be present outside of the concentration window of the component to be resolved.
This means, in practice, that component peaks with embedded peak proﬁles under
them cannot be adequately resolved.
Figure 11.5 illustrates the scheme followed in the WFA resolution. The steps of
the WFA method are listed below, followed by a description clarifying their meaning.
1. A PCA model of the original data matrix, D, is computed.
2. The concentration windows of each component in the data set are
determined.
For each component:
3. A PCA model of a submatrix, Do, is computed where the rows related to
the concentration window of the nth component to be resolved have been
removed.
4. The vector, pnoT, is computed, which is the part of the spectrum of the
nth component orthogonal to the spectra of all other components in the
original matrix.
5. The true concentration proﬁle of the nth component is recovered using
pnoT and D.
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PT
Conc. window
nth component
D
=
Rank n
T
(a)
PTo
=
To
Rank (n −1)
Do
(b)
PTo
PT
∈
⊥
To
pn
(c)
=
D
pno
cn
(d)
FIGURE 11.5 Recovery of the concentration proﬁle of the nth compound by window factor
analysis. (a) PCA of the raw data matrix and determination of the concentration window, D
(steps 1 and 2); (b) PCA of the matrix formed by suppression of the concentration window of
the nth component, D (step 3); (c) recovery of the part of the spectrum of the nth component
orthogonal to all the spectra in D, pnTo (step 4); and (d) recovery of the concentration proﬁle
of the nth component (step 5).
As a general last step after obtaining the concentration proﬁles of all components:
6. The pure-spectrum data matrix ST is estimated by least squares using D
and C.
WFA starts with the PCA decomposition of the D matrix, giving the product of
scores and loadings, TPT. In general, the D matrix will have n components, i.e., rank
n. The determination of the location of concentration windows for each component is
carried out using EFA (see Figure 11.4b) or other methods. Steps 3 to 5 are the core
of the WFA method and should be performed as many times as compounds are present
in matrix D to recover the concentration proﬁles of the C matrix, one at a time.
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For each component, a Do submatrix is constructed by removing the rows related
to its concentration window. Then, a PCA model is computed and the product ToPoT
is obtained. Note that Do has rank n − 1 because the variation due to the component
of interest disappears when its corresponding window (rows) in the data matrix D
is deleted. The loading matrices, PT and PoT, describe the space of the n pure spectra
in D and the (n − 1) pure spectra in Do, respectively. The rows in these loading
matrices are actually “abstract spectra,” and the real spectra can be expressed as a
linear combination of them. Using these two loading matrices, PT and PoT, it is
possible to calculate a vector pnoT that is orthogonal to the (n − 1) pioT vectors and
that belongs to the space deﬁned by PT. This vector completes the set of vectors in
PoT and contains the part of the spectra of the removed component uncorrelated to
the spectra of the other (n − 1) components in the data matrix. Using this vector
with information exclusively related to the removed component, the true concentration proﬁle of this compound can be calculated as follows:
Dpno = cn
(11.6)
The complete C matrix is then formed by appending row-wise the column
concentration proﬁles found for each component in the D matrix. The matrix of
spectra, ST, is obtained by least squares using the D and C matrices and the basic
equation of CR methods, D = CST:
ST = (CTC)−1CTD
(11.7)
Recent modiﬁcations of the WFA method attempt to solve some of the problems
caused by poorly deﬁned boundaries for concentration windows [35].
11.4.2 OTHER TECHNIQUES: SUBWINDOW FACTOR ANALYSIS
(SFA) AND HEURISTIC EVOLVING LATENT PROJECTIONS
(HELP)
Following the idea of using concentration windows and the subspaces that can be
derived, other noniterative methods are focused on the recovery of the response
proﬁles (spectra). This is the case of subwindow factor analysis (SFA), proposed by
Manne [38], and other derivations of this method, like parallel vector analysis (PVA)
[39]. Unlike WFA, SFA recovers the pure response proﬁle of each component. The
individual row response proﬁles are appended in a columnwise fashion, until the
complete ST matrix is built. The C matrix is easily derived by least-squares according
to the CR model, D = CST, as follows:
C = DS(STS)−1
(11.8)
In SFA, the knowledge of the concentration windows is used in such a way
that each pure spectrum is calculated as the intersection of two subspaces that
have only the compound to be resolved in common. Figure 11.6 illustrates the
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Retention times
Wavelengths
A, B subspace
A
B
C
B, C subspace
(a)
sA
sB
B, C
sC
A, B
(b)
FIGURE 11.6 Application of subwindow factor analysis (SFA) for resolution. (a) Concentration proﬁles of A, B, and C and subwindows used for the resolution of component B (ﬁrst
containing A and B compounds and second containing B and C compounds). (b) The A,B
plane is deﬁned by the pure spectra of A and B (sA, sB) and the plane B,C by the pure spectra
of B and C (sB, sC). The intersection of both planes must be necessarily the pure spectrum of B.
idea behind SFA for a three-component HPLC-DAD system (A, B, and C). Once
the concentration windows of the three components are known, one subwindow
can be constructed with rows including only A and B and another one with rows
where only B and C are present. The intersection of the two planes derived from
these subspaces must necessarily give the pure spectrum of B as an answer. The
same strategy would be applied to recover the spectra of the rest of the compounds
in a general example.
To conduct SFA in practice, the singular-value decomposition (SVD, see Chapter 4)
of the two subwindows yields a basis of orthogonal vectors spanning the (A,B)
subspace, called {ei}, and another basis for the (B,C) subspace, called {fi}. The
spectrum of B, sB, can be obtained from these two sets of basis vectors as shown in
Equation 11.9,
sB =
∑a e = ∑b f
i i
i
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i i
i
(11.9)
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The SFA algorithm computes the ai and bi values that minimize Equation 11.10,
min
ai , bi
∑
i
aiei −
∑
2
bi fi
(11.10)
i
after which sB can be obtained by using any of the resulting linear combinations, Σi aiei or Σi aiei .
The spectra of C and A can be obtained in a straightforward fashion, since these
components have selective zones in their elution proﬁles.
The HELP method is another pioneering noniterative method using local-rank
information [36, 37] and based on the local-rank analysis of the data set and
focuses on ﬁnding selective concentration or response windows. When these
selective zones exist, the resolution of the system is clear. Thus, for an HPLCDAD data set, a row related to a selective elution time directly provides the shape
of the spectrum of the only component present at that stage of the chromatographic
elution. In a similar manner, a column related to a selective wavelength directly
provides the chromatographic peak of the only absorbing compound at that
wavelength.
HELP works by exploring both the concentration and spectral response spaces
with a powerful graphical tool (the so-called datascope) to visually detect potential
selective zones in the scores and then loading plots of the data matrix, which are
seen as points (representing rows or columns of the original data set) lying on straight
lines centered near the origin. A statistical method to conﬁrm the presence of
selectivity in the concentration or spectral windows is based on the use of an F-test
to compare the magnitude of eigenvalues related to potential selective zones of the
data set with eigenvalues related to noise zones of the data matrix, i.e., those regions
where no chemical components are supposed to be present. The conﬁrmation of a
selective zone in the data set, which is actually a rank-one window in the data matrix,
will then be obtained when no signiﬁcant differences are found between the ﬁrst
eigenvalue of a noise-related zone of the data matrix and the second eigenvalue of
the potential selective zone. Components with selective concentration and response
zones are straightforwardly resolved. Subtraction of the cisiT contribution of the
resolved components from the raw data set can facilitate the resolution from components originally lacking selectivity.
11.5 ITERATIVE METHODS
Iterative resolution methods obtain the resolved concentration and response matrices
through the one-at-a-time reﬁnement or simultaneous reﬁnement of the proﬁles in
C, in ST, or in both matrices at each cycle of the optimization process. The proﬁles
in C or ST are “tailored” according to the chemical properties and the mathematical
features of each particular data set. The iterative process stops when a convergence
criterion (e.g., a preset number of iterative cycles is exceeded or the lack of ﬁt goes
below a certain value) is fulﬁlled [21, 42, 47–50].
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Iterative resolution methods are in general more versatile than noniterative
methods. They can be applied to more diverse problems, e.g., data sets with partial
or incomplete selectivity in the concentration or spectral domains, and to data sets
with concentration proﬁles that evolve sequentially or nonsequentially. Prior knowledge about the data set (chemical or related to mathematical features) can be used
in the optimization process, but it is not strictly necessary. The main complaint about
iterative resolution methods has often been the longer calculation times required to
obtain optimal results; however, improved fast algorithms and more powerful PCs
have overcome this historical limitation.
The next subsection deals ﬁrst with aspects common to all resolution methods.
These include (1) issues related to the initial estimates, i.e., how to obtain the proﬁles
used as the starting point in the iterative optimization, and (2) issues related to the
use of mathematical and chemical information available about the data set in the
form of so-called constraints. The last part of this section describes two of the most
widely used iterative methods: iterative target transformation factor analysis (ITTFA)
and multivariate curve resolution–alternating least squares (MCR-ALS).
11.5.1 GENERATION
OF INITIAL
ESTIMATES
Starting the iterative optimization of the proﬁles in C or ST requires a matrix or a
set of proﬁles sized as C or as ST with rough approximations of the concentration
proﬁles or spectra that will be obtained as the ﬁnal results. This matrix contains the
initial estimates of the resolution process. In general, the use of nonrandom estimates
helps shorten the iterative optimization process and helps to avoid convergence to
local optima different from the desired solution. It is sensible to use chemically
meaningful estimates if we have a way of obtaining them or if the necessary
information is available. Whether the initial estimates are either a C-type or an STtype matrix can depend on which type of proﬁles are less overlapped, on which
direction of the matrix (rows or columns) has more information, or simply on the
will of the chemist.
There are many chemometric methods to build initial estimates: some are particularly suitable when the data consists of the evolutionary proﬁles of a process,
such as evolving factor analysis (see Figure 11.4b in Section 11.3) [27, 28, 51],
whereas some others mathematically select the purest rows or the purest columns
of the data matrix as initial proﬁles. Of the latter approach, key-set factor analysis
(KSFA) [52] works in the FA abstract domain, and other procedures, such as the
simple-to-use interactive self-modeling analysis (SIMPLISMA) [53] and the orthogonal projection approach (OPA) [54], work with the real variables in the data set to
select rows of “purest” variables or columns of “purest” spectra, that are most
dissimilar to each other. In these latter two methods, the proﬁles are selected sequentially so that any new proﬁle included in the estimate is the most uncorrelated to all
of the previously selected ones.
Apart from using chemometric methods, a matrix of initial estimates can always
be formed with the rows or columns of the data set that the researcher considers most
representative because of chemical reasons, and it can also include external information, such as some reference spectra or concentration proﬁles, when available.
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11.5.2 CONSTRAINTS, DEFINITION, CLASSIFICATION: EQUALITY
AND INEQUALITY CONSTRAINTS BASED ON CHEMICAL
OR MATHEMATICAL PROPERTIES
Although resolution does not require previous information about the chemical system
under study, additional knowledge, when it exists, can be used to tailor the sought
pure proﬁles according to certain known features and, as a consequence, to minimize
the ambiguity in the data decomposition and in the results obtained.
The introduction of this information is carried out through the implementation
of constraints. A constraint can be deﬁned as any mathematical or chemical property
systematically fulﬁlled by the whole system or by some of its pure contributions
[55]. Constraints are translated into mathematical language and force the iterative
optimization to model the proﬁles while respecting the conditions desired.
The application of constraints should always be prudent and soundly grounded,
and constraints should only be set when there is an absolute certainty about the
validity of the constraint. Even a potentially useful constraint can play a negative
role in the resolution process when factors like experimental noise or instrumental
problems distort the related proﬁle or when the proﬁle is modiﬁed so roughly that
the convergence of the optimization process is seriously damaged. When well implemented and fulﬁlled by the data set, constraints can be seen as the driving forces of
the iterative process to the right solution and, often, they are found not to be active
in the last part of the optimization process.
The efﬁcient and reliable use of constraints has improved signiﬁcantly with
the development of methods and software that allow them to be easily used in
ﬂexible ways. This increase in ﬂexibility allows complete freedom in the way
combinations of constraints can be used for proﬁles linked in the different concentration and spectral domains. This increase in ﬂexibility also makes it possible
to apply a certain constraint with variable degrees of tolerance to cope with noisy
real data. For example, the implementation of constraints often allows for small
deviations from ideal behavior before correcting a proﬁle [7, 21, 55]. Methods for
correcting the proﬁle to be constrained have evolved into smoother methodologies,
which modify the poorly behaving proﬁle so that the global shape is retained as
much as possible and the convergence of the iterative optimization is minimally
upset [56–61].
There are several ways to classify constraints: the main ones relate either to the
nature of the constraints or to the way they are implemented. In terms of their nature,
constraints can be based on either chemical or mathematical features of the data set.
In terms of implementation, we can distinguish between equality constraints or
inequality constraints [56]. An equality constraint sets the elements in a proﬁle to
be equal to a certain value, whereas an inequality constraint forces the elements in
a proﬁle to be unequal (higher or lower) than a certain value. The most widely used
types of constraints will be described using the classiﬁcation scheme based on the
constraint nature. In some of the descriptions that follow, comments on the implementation (as equality or inequality constraints) will be added to illustrate this
concept. Figure 11.7 shows the effects of some of these constraints on the correction
of a proﬁle.
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0
(a)
(b)
c11
.
ci1
.
cn1
cn2
ct
c12
.
ci2
.
Σ = ct
(c)
FIGURE 11.7 Effects of some constraints on the shape of resolved proﬁles. The thin and
the thick lines represent the proﬁles before and after being constrained, respectively. Constraints shown are (a) nonnegativity, (b) unimodality, and (c) closure.
11.5.2.1 Nonnegativity
The nonnegativity constraint is applied when it can be assumed that the measured
values in an experiment will always be positive. For example, it can be applied to
all concentration proﬁles and to many experimental responses, such as UV (ultraviolet) absorbances and ﬂuorescence intensities [42, 47, 48, 56, 59]. This constraint
forces the values in a proﬁle to be equal to or greater than zero. It is an example of
an inequality constraint (see Figure 11.7).
11.5.2.2 Unimodality
The unimodality constraint allows the presence of only one maximum per proﬁle
(see Figure 11.7) [42, 55, 60]. This condition is fulﬁlled by many peak-shaped
concentration proﬁles, like chromatograms or some types of reaction proﬁles, and
by some instrumental signals, like certain voltammetric responses. It is important
to note that this constraint does not only apply to peaks, but to proﬁles that have a
constant maximum (plateau) or a decreasing tendency. This is the case for many
monotonic reaction proﬁles that show only the decay or the emergence of a compound [47, 48, 51, 61], such as the most protonated and deprotonated species in an
acid-base titration, respectively.
11.5.2.3 Closure
The closure constraint is applied to closed reaction systems, where the principle of
mass balance is fulﬁlled. With this constraint, the sum of the concentrations of all of
the species involved in the reaction (the suitable elements in each row of the C matrix)
is forced to be equal to a constant value (the total concentration) at each stage in the
reaction [27, 41, 42]. The closure constraint is an example of an equality constraint.
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