Separation of Additive Mixture Spectra by a Self-Modeling Method

Separation of Additive Mixture Spectra by a Self-Modeling Method

V L A D I M I R V. V O L K O V Institute of Crystallography, Russian Academy of Sciences, Leninsky pr. 59, Moscow 117333, Russia

A method for the separat ion of addit ive spectra of complex mixtures is deve loped on the basis of a l inear a lgebra technique and nonlinear opt imizat ion algori thms. It is shown to be possible, under certain condit ions, to uniquely separate a set of complex spectral curves consist ing of the same components , but with different pro- port ions, into the u n k n o w n spectra of the pure const i tuents and to give their respect ive relative concentrat ions . The method proposed is a variant of the se l f -model ing curve-reso lut ion approach based on the s ingular value decompos i t ion of the data matr ix formed by the set of digit ized spectra of mixtures . The spectra of component s are calculated as l inear combinat ions of left-side s ingular vectors of the data matr ix provided that both individual spectra and their concentrat ions are nonnegat ive and the shapes of the spectra are as diss imilar as possible . The technique provides a unique decomposition if each fundamenta l spec trum has at least one wave length with zero intensity and the other pure spectra are nonzero at this wavelength. The a lgor i thm is evaluated on an artificial data set to clearly demonstrate the method. The approach descr ibed in this paper m a y be applied to any exper iment whose outcome is a con- t inuous curve y(x) that is a sum of u n k n o w n , nonnegat ive , l inearly independent functions. Index Headings: S p e c t r a reso lu t ion; S e l f - m o d e l i n g ; Re l a t ed mixtures; Factor analysis .

I N T R O D U C T I O N

The numerical deconvolution of complex spectra is sometimes the only way to resolve spectra of pure constituents when mixtures cannot be completely separated by the use of analytical techniques or when separation is in principle not possible.

The resolution of any single mixture spectrum into un- known component spectra is not unique. Somet imes it is possible to obtain a set of spectra of related mixtures consisting of the same components with varying propor- tions. Spectra of related mixtures arise in some experimental circumstances, e.g., during the elution of poorly resolved chromatographic peaks, when a mixture is sam- pled at various times during the course of a reaction, or when mixture spectra are measured under different experimental conditions. These spectroscopic data allow the experimenter to develop and use efficient methods for spectral decomposit ion.

All approaches to the resolution of these complex curves published during the last two decades 1-11 require Beer 's law to hold; i.e., each mixed spectrum should be a sum of the constituent spectra multiplied by the corresponding concentration coefficients.

The numerical methods proposed can be classified into two groups: those in which assumptions about the system are made (e.g., identity of components or spectral contour shapes) and those in which no assumptions are made,

Received 8 May 1995; accepted 24 October 1995.

other than linear behavior (that is, Beer 's law holds) and nonnegativity of the pure spectra and their respective concentrations. The latter is, in practice, the most useful group of methods. Among these procedures the "self- model ing" technique, 4-6 which is closely related to the factor analysis method, 3,7-~3 seems to be most powerful.

Lawton and Sylwestre developed the self-modeling technique for the separation of two-component mixtures of optical spectra. 4 Later, this approach was extended to the resolution of three 5,6 and, potentially, morC components. These algorithms are, however, relatively sophis- ticated and their realization even for the 4-component case meets with significant difficulties in practice. 6

Additivity of mixed spectra coupled with nonnegativity, both of spectra and concentrations, cannot itself provide the uniqueness of the decomposit ion, so that extra constraints on the individual spectra are required. These constraints will be considered late~: The main limitations to the number of components depend in practice on Beer 's law violations.

T H E O R Y

Throughout, we will use upper case bold letters to denote matrices and lower case bold letters for vectors. The corresponding lower case letters, subscripted, will denote the array elements.

Consider the case of noiseless data. A spectrum measured at P wavelengths can be formally represented as a P-dimensional vector

d = {di} i = 1, P

where the ith element gives the signal intensity at the ith wavelength. It is conventional in linear algebra to consider column vectors, so the transposed vectors are written in row form. The totality of the measured spectra of mixtures can be represented by a matrix D (data matrix). The columns contain the spectra dj, where j is the ordinal number of the spectrum, j = 1 . . . . . M, and M is the number of mixtures involved in the analysis. Hence the data matrix is of dimensions P X M. A single matrix element is written as dij, i being the ordinal number of the wavelength and j the ordinal number of the mixture. It should be noted here that the number of mixture spectra involved in the analysis must exceed the number of components. This condition is checked in practice at the first step of the analysis procedure, namely, during the esti- mation of the number of components from the data matrix D (see below). I f the number of components is equal to K, a set of component spectra can be expressed with a P X K matrix E, where each column represents a pure spectrum. With mixture spectra adhering to Beer 's law, the following summation can be written:

320 Volume 50, Number 3, 1996 0003-7028/96/5003-032052.00/0 APPLIED SPECTROSCOPY © 1996 Society for Applied Spectrocopy

K

d O = ~ eik.ckj. (1) k=l

Here, eik is the intensity of the spectrum of the kth component at the ith wavelength, and ckj is the concentration of the kth component in the j th mixture. In matrix nota- tion this equation can be written as

D = E . C . ( 2 )

The goal is to find the number of components K, their spectral curves E, and the corresponding relative concentrations C, using the data matrix D. In this definition the problem has no unique solution in spite of the usual conditions

K --< M --< P, (3)

because the system of equations (Eq. 2) is not linear with respect to unknowns eik and ck/. The restrictions that spectra and their concentrations must be nonnegative also do not provide the uniqueness of the decomposit ion (Eq. 2). Additional constraints on the component spectra must be used either to provide the unique solution or to reduce the space of possible decomposit ions.

The main aim of this paper is to develop a robust algori thm that will enable the user to involve a rich variety of extra requirements to the calculated individual spectra.

Assuming the data to be noiseless, consider the geo- metrical vector representation of the problem (Fig. 1). Here, the number of spectral points P = 3 (spectral intensities are plotted on the axes I1,/2, and 13), the number of mixtures M = 4 (vector-spectra dj, d2, d3, and d4), and the number of components K = 2 (vectors e, and e2). All mixed spectra are linear combinations of the same component vectors; therefore, all possible vector spectra of mixtures must lie in the subspace spanned by the component vectors el and e2. This subspace is shown in Fig. 1 by the two-dimensional plane passing through OAD. Two important inferences should be made here. First, all possible component spectra obtained by any method must lie in the subspace spanned by the original vector data (columns of the matrix D). Second, for the relative concentrations of the components to be nonnegative, these component vector spectra must lie outside the simplex of max imum volume spanned by the data vectors dj. As shown in Fig. 1, this simplex is formed by vectors d, and d 4. The condition that the spectra must be nonnegative leads to the conclusion that component spectra must lie inside the positive ortant, i.e., inside the simplex spanned by vector intersections of the spectral subspace and coordinate planes. In Fig. 1 these intersections are shown as v e c t o r s e~l F and e AF. On this basis, it is easy to under- stand various self-modeling methods for decomposi t ion of complex spectra. For the two-component system, one of the pure spectra must lie in the region between ~ll v and d, and the second spectrum must lie between d 4 and ~F.

The key feature of self-modeling methods is to construct the trial component spectra as linear combinations of linearly independent base vectors lying in the spectral subspace (OAD in Fig. 1). Evidently, this procedure will generate pure spectra which will lie in this subspace and, therefore, precisely satisfy the experimental data vectors, as no experimental noise is considered here. The method proposed is also based on this approach.

D ^F

11 A

FIG. I. Vector representation of the two-component spectral data collected from four samples at three wavelengths. Spectral intensities are plotted on the axes I~, 12, and 13. Vectors d~, dz, d3, and d 4 are the mixture spectra; e~ and e z, the spectra of pure components ; ~1 F and ~F, the nonoverlapped individual spectra (corresponding vectors OA and OD are the intersections of the spectral plane passing through e~, e2, d~, dz, d3, and d 4 with the coordinate planes, here 1~12, and 1213). t is the spectrum which is not a component; p is its projection on the spectra subspace.

An analysis of the mathematical aspects of the problem (Eq. 2) allows one to put additional constraints on the solution. The first constraint is that the shapes of component spectra must be linearly independent, i.e., no one spectrum can be represented as a linear combination of others. This means that the dimension of spectral subspace must equal the number of components (matrix E must have the full rank K).

Another restriction is that the concentration of components must vary independently. In other words, none of the mixture spectra can be a linear combination of K - 1 other input spectra, and at least as many wavelengths and mixture spectra as there are components must be involved in the computations (Eq. 3). As a result, matrix D must also have the rank K. This requires a certain knowledge of the system to be analyzed, so that an appropriate number of starting mixture spectra is used: M should be greater than K. This condition is checked during the evaluation of the number of components f rom the matrix D (see later).

The central additional constraint on the solution we use is the max imum distinction between component spectra. Although it may seem somewhat artificial, there are some reasons to include it in the decomposit ion algorithm. The first is that distinct spectra should correspond to distinct species. Another reason results from the uniqueness con-

APPLIED SPECTROSCOPY 321

dition of the solution? 4 if each pair of pure component spectra contains at least two nonoverlapping po in t s - -one per spec t rum-- then a set of K nonnegative vectors lying in the column subspace of the matrix D and spanning the max imum angle in the positive ortant coincides with the true individual spectra (vectors e f t and e2 AF in Fig. 1). When a set of component spectra nonoverlapping at some wavelengths is obtained by some procedure, one cannot construct (using linear combinations of these spectra) a new set of spectra nonoverlapping at different wavelengths and, at the same time, stay inside the spectral subspace in the positive ortant (OAD in Fig. 1). In the case of three and more components, only two nonoverlapping points may be present in the mixture spectra, and one cannot find the decomposi t ion by simple subtraction of the spectral contours. The superscript AF is used here because Alentzev and Fok ~5 were the first to successfully use the nonoverlapped regions to resolve mixture spectra. Later, the same approach known as the "rat io method" was proposed independently by Hirschfeld 16 and then extended by Fogarty and Warner. ~7 In practice, the ratio method is very sensitive to experimental noise, restricting the max imum number of components to 2--4, and cannot be implemented for the resolution of even slightly overlapping spectra. Although the condition of pure wavelengths is almost never satisfied in practice, the spectral range can often be chosen so that at some wavelength a one-component spectrum has a small intensity whereas another is relatively large at the same spectral point. These spectral shapes may be considered to be significantly different.

Several numerical criteria can be used as a measure of the distinction, e.g., the average over angles (or cosine values) between vector-spectra. The cosine is computed as

P 2 ai'bi i=1 COSa,b = [ ]1/2 (4)

i=l i=1 where a and b are two vector-spectra to be compared. Another criterion is the sum of correlation coefficients over all pairs of individual spectra. Each of the terms of this sum is calculated as:

P ' ~ a i ' b i - (~a~) ' (~=~ ~=,

P . i _~l a i2 - i=l~ai "

• ( 211,,2 e'i=~l b] - ~b i ) ] = ,

(5)

In the present work the condition number of the matrix E is used as an integrated criterion for spectral similarity. This number correlates with the volume of the polyhe- dron spanned by the normalized column vectors of the matrix, i.e., by vector-spectra of pure components. The larger this volume, the better one can distinguish shapes of the spectra. The condition number is calculated as the ratio of the max i m um and min imum singular values of the matrix. The singular value decomposit ion (SVD) of

an arbitrary rectangular M × N matrix A is defined by the equation? 8

A = U.S.V T. (6)

Here U is the rectangular M × N matrix whose columns are the eigenvectors of the matrix A.A T, V is the N × N square matrix spanned by the eigenvectors of the matrix AT.A, and S is the diagonal matrix of the singular values si, i = 1 . . . . . N, that are nonnegative square roots of the nonzero eigenvalues of A.A T or of AT.A arranged in de- scending order. By this means the condition number of the component spectra matrix is defined by

cond(E) = Smax if E = U. S. V T. (7) Smin

Consider now the successive steps of the decomposition procedure. Self-modeling procedures call for the true number of pure components , which the user must either get independently or estimate from the spectral data set. It is best to obtain K in different ways and compare the results. All estimates being equal, the problem is nearly linear and suitable for numerical decomposit ion.

The min imum number of independent components can be determined from the data matrix D by a variety of statistical and empirical methods. 339-22 Most statistical methods require experimental errors to be normally dis- tributed. If this is not the case (as is usual in practice), the calculated estimate may significantly differ f rom the true value. Our experience shows that two procedures based on the singular value decomposi t ion of the spectra matrix:

D = UD'SD'V~ (8)

are easy to use in determining the number of components K.

In the noiseless case, the number of pure components is given by the number of nonzero singular values of the data matrix D. Assuming K < M < P, it must equal K.

In the presence of random noise, the rank of D be- comes equal to the min imum dimension involved, M. We use the first method to estimate the number of non-noise components on the basis of an examination of the mag- nitudes of singular values plotted against their ordinal numbers in the logarithmic scale. K may be determined by finding where the two lines passing through the largest and smallest values intersect 22 (if this plot is possible). The number of components is given by the ordinal number of the rightmost singular value in the left-side region of the steepest slope (e.g., region A in Fig. 3) not in- cluded in the right-side region of the low slope (region B in Fig. 3). In practice, this method is appropriate if the noise level is not too large, i.e., on the order of 0.1-0.3%.

The second method for estimating the number of components is based on analysis of the shape of the columns of Uo plotted against wavelengths as "abstract spectra" .23 Assuming the component spectra curves have Fourier spectra weighted toward the lower frequency coefficients (spectra are smooth) and the noise curve toward the high- er Fourier harmonics (the noise curve oscillates relatively rapidly), columns of Up representing spectral data will be as smooth as are the mixture spectra. In addition, their corresponding singular values are relatively large in magnitude. High-frequency fluctuations will be observed only

322 Volume 50, Number 3, 1996

in those singular vectors in which noise dominates. These vectors usually associate with the smallest singular values. A simple statistic for detecting such "nois iness" is the nonparametric criterion for randomness of vector data 24 or the autocorrelation coefficient. 2~ All nonrandom systematic errors should be considered as extra components in the mixture spectra.

The second stage of the decomposi t ion is a self-modeling procedure for searching a set of K linearly independent vectors lying in the subspace spanned by K "non-noise" singular vectors (uD)j and satisfying the criteria for nonnegativity and min imum similarity.

It is worth noting that every set of K independent vectors lying in the "spect ra l" subspace (plane OAD in Fig. 1) taken as the matrix E satisfies the Eq. 2 within the experimental tolerance. The true component spectra can be computed as linear combinations of K linearly independent vectors taken from this subspace. The problem then is to obtain the appropriate coefficients of this combination which yield the spectra of the pure components. With a set of K fixed independent base vectors, the coefficients can be found by minimizing the target function described below.

For the minimization procedure to be numerically robust, each trial matrix E is computed as a linear combination of K "non-noise" orthogonal singular vectors of the data matrix (uo)j, j = I . . . . . K, corresponding to the largest singular values:

E = U D x ' X (9)

where X is the K X K square matrix of coefficients varying by the search program (initially unity matrix), and UDx consists of K leading columns of U D. Each of the orthogonal singular vectors obtained from the SVD of the data matrix is a linear combination of the mixture spectra and, therefore, a linear combination of pure spectra. Therefore there must be some combination X of these base vectors giving the spectra of the components. The matrix of coefficients X is adjusted by the computer program so as to minimize the nonlinear penalty function: 25

W(X) = [ c o n d ( E ) - 1] 2 + c~. ~ e,2k + [3. ~'~ c~j Veik<O •Ckj<O

K K

+ ~ . ~ [len(ek)] 2 + 8 . ~ [cur(ek)] 2 + { . . . }. k-I k-1

(10)

Each pure trial spectrum (column of matrix E) is normalized to Euclidean unit vector norm, so that the term [cond(E) - 1 ]2 represents the criterion for minimum similarity of trial component spectra vectors. I f all spectra vectors are perpendicular to each other (minimum similarity), then cond(E) = 1 and this term attains its mini- m u m value. These orthogonal " spec t ra" must contain large negative intensities. The e~ and [3 terms impose penalties for the negativity of the spectra and of their concentrations, C, respectively. The latter is calculated as

C = E I.D. (11)

Concentrations may be computed through the already- existing SVD of the matrix E:

i f E : U . S . V T

t h e n E i : V • S - I . U T

a n d C = V . S I ' U T . D .

The summation in the c~ and [3 terms is done over all negative elements of matrices E and C, correspondingly.

The main advantage of this approach may be represented by the term " { . . . } " . In other words, any extra constraints on the solution properties can be easily in- stalled in the target function W(X) as penalty terms. For example, the ~ and 8 terms in Eq. 10 are penalties for the complexi ty of spectral shapes. Since any mixture spectrum should be more complex than the spectra of its constituents, a measure of the complexi ty of the spectral shape may be used to estimate the appropriate matrix of coefficients X. For example, the less the complexity, the less the approximated spectral curve length:

len(ek) = ~ h 2 + (ek,;,-i - e~,i) 2 i=l

or its overall curvature:

P-I _ 2 . e k , i + cur(ek) = Z e~'i ' ek'i+' i=2 h 2

where ek is the spectrum of the kth component, and h is the wavelength increment. All trial component spectra must be normalized either to unity vector length or to unity max imum intensity. These constraints allow the user to get the solution very close to the true individual spectra even in the case of significant overlap.

c~, [3, y, and 8 are user-defined coefficients whose values depend on the spectra features. Recommended values are c~ = 50, [3 = 1, "y = 1, and 8 = 1 if the experimenter has no experience in solving decomposi t ion problems.

At the start of the first iteration, the matrix of coefficients X is set to unity (i.e., we start f rom the orthogonal matrix E, which still contains large negative values in its elements). One can also include spectra of known components in UDx and prohibit the components f rom chang- ing in matrix E during the minimization of the target function (Eq. 10). This is done by fixing the corresponding column elements in the matrix X. If the known spectrum is obtained independently, it may not lie in the spectral subspace (OAD in Fig. 1) due to, for example, experimental errors (vector t in Fig. 1). Thus we should project it onto the subspace by the following:

if D = Uo. SD.V~ then projection p = UD-U T ' t

and install in the matrix Uox the vector p rather than t. The minimization program varies elements of X so as

to minimize the target function (Eq. 10). The minimization procedure is the well-known iterative Bro y d en - F le tcher -Goldfarb-Shanno variable metric method. 26 The min imum value of the penalty function (Eq. 10) corre- sponds to the set of component spectra E that may still have small negative intensities at some wavelengths. If it is desirable to reduce the spectral magnitude in negative regions, the search must be restarted with a larger value of c~. Before the procedure is restarted, it is better to re- place the matrix Uox in Eq. 9 by the calculated E to speed up the minimization. In this case the coefficient matrix


12

11

10

9

8

7

6

5

4

3

2

1

No. of mixture

I ' I , I . I . I . I . I , I

1 21 41 61 81 101 121 141

No. of spectral point Inflared absorbance spectra obtained during the elution of the Fie. 2.

mixture of six components: iso-octane, toluene, benzene, diethyl ethel; acetone, and amyl acetate. Zero absorbance levels are indicated. All spectra were normalized to unity intensity; wavelengths were selected to represent the most important spectra features.

X must be reset to unity. Experience shows that mixtures 1.0 of 4 -8 noninteracting components can be reasonably resolved into spectra of constituents suitable for their iden- tification. 0.5

EXPERIMENTAL

measured are shown in Fig. 2. The results of the analysis are shown in Figs. 2-6.

A P P L I C A T I O N S

To illustrate the power of the algorithm, we collected the infrared spectra of 10 mixtures as described above. A computer program REMIX was written with Microsoft FORTRAN and run on an IBM 486DX 33-MHz PC/AT.

Logari thms of singular values of the data matrix D are shown in Fig. 3. Following a technique previously described, 22 one can make the inference that the noisy part consists of four tailed singular values. It is difficult to decide whether the sixth value is noisy or its magnitude is determined by the spectral data. It is reasonable to suggest that the first six singular vectors plotted in Fig. 4 as " spec t ra" contain mainly spectroscopic information, whereas four others are noisy, because these four tailed contours contain too many large-amplitude peaks to represent spectral shapes of mixtures. In this way we make conclusions about six valuable components present in data set D.

If the component spectra obtained contain significant noise, then the number of components estimated is prob- ably greater than the true value. In this case the corresponding concentration matrix C contains large negative elements if one substitutes 13 -- 0 in Eq. 10. The number of components found being fewer than the true number, the corresponding component spectra will be more complex than those obtained with the true number of components. These situations are easily recognizable and may

0.0 Spectra of related mixtures were obtained with a Per- k in -E lmer 580B IR spectrometer coupled with a very low resolution liquid chromatograph. The chromatograph was a 200× 3.5-ram glass tube packed with 30-1xm Si- -0 .5 lasorb 600 (LC) and eluted with CC14 at pressures of 2 - 3 atm. The output of the chromatograph was connected to a 0 .15-mm KBr cell of the spectrometer. Spectra were -1 .0 measured every 30 s in the range 1750-1110 cm -i with a resolution of 5 cm J and digitized in steps of 2.5 cm -1, the solvent being immobile. Spectra were corrected by -1 .5 subtracting the spectrum of pure CC14 recorded at the same conditions. Some 141 wavelengths were then selected manually for the noninformative low-intensity -2 .0 spectral regions to be eliminated. For smooth graphical representation, the sampling distance was selected so that most narrow spectral peaks were represented by at least -2 .5

0 5 points lying above the half-intensity level. The spectral noise was --0.2% T. The test sample was prepared by mixing six compounds: iso-octane, toluene, benzene, diethyl ether, acetone, and amyl acetate. Ten sampling points FIG. 3. were selected during the elution time, and the data matrix D was built f rom l0 IR spectra. The normalized spectra

lg(s i)

B

I I I I I I I I I I I

1 2 3 4 5 6 7 8 9 10 ! l

i Log(singular values) of the data matrix plotted vs. their ordinal

numbers. Six components are present in the mixtures. A is the region of the steepest slope spanned by significant singular values s , i = 1, . . . . K; B is the region of noisy singular values.


No. of singular vector

10

9

8

7

6

5

4

3

2

1 I , I , I , I I I , I • I • I

1 21 41 61 81 101 121 141

No. of spec t ra l po in t FIG. 4. The singular vectors obtained from a set of six-component mixtures. The singular vectors are plotted as spectra.

be considered as additional practical criteria for the number of components.

Left-side singular vectors, Up, corresponding to the non-noisy singular values are supposed to contain all of the spectra features necessary to reconstruct the mixture of spectra. These six vectors were put together to form the matrix UDx the columns of which can be combined to form the spectra of the pure components (Eq. 9). The transformation matrix X, which consists of the coefficients by which the singular vectors were multiplied, was obtained by the program REMIX. This 6 × 6 matrix required a search time of 20-25 rain on the PC/AT. The default values of the weight coefficients used were tx = 50, 1~ = 1, 5/ = 1, and g = 0. The six component spectra that were calculated are shown in Fig. 5. The corresponding relative contributions of these pure spectra in the mixture spectra are shown in Fig. 6. Small negative values in the individual spectra and in the concentration matrix should be considered zero. The first term in the target function W(X) has an influence opposite to that of the tx and J3 terms, which are always obtained experimentally. The extra positive peaks in some calculated spectra arise from molecular interactions in the mixture solutions. These interactions lead to the dependence of the spectral shape on the relative concentration of the component and, therefore, to Beer 's law violations. Any violations of spectral additivity lead to extra spectral "componen t s" one obtains during numerical decomposit ion.

Although the algorithm presented cannot calculate component spectra that coincide with the true spectra in the case of completely overlapping pure spectra, it may

A

--...-.....J'X.,.A_

e

f

- - v w

I , I I I I I a I I I

1 21 41 61 81 101 121 141

No. of spectral point FIG. 5. Results of the decomposition of six-component mixtures of organic compounds. The known (upper curve in each pair) and calculated (lower trace) spectra of components: (a) amyl acetate; (b) acetone; (c) diethyl ether; (d) benzene; (e) iso-octane, and (f) toluene.

2.5 C, rel

2.0 a d

1.5 e

b f 1.0

0.5 C

0.0

3 4 5 6 7 8 9 10 11

t, min F]G. 6. Concentration profiles for the calculated component spectra: (a) iso-octane; (b) toluene; (c) benzene; (d) amyl acetate; (e) acetone; and (f) diethyl ether. The data point positions correspond to the sampling times the mixture spectra were recorded during the elution.


help obtain a useful solution. The solution is obtained without prior knowledge of the components and without making assumptions about curve shapes.

ACKNOWLEDGMENTS The author is grateful to Dr. B. N. Grechushnikov for extremely fruit-

ful and constructive discussions and to Dr. M. H. J. Koch for valuable comments on the manuscript. This work was supported by EMBO Fel- lowship ASTF 7796, by DAAD Fellowship 1995, and by NATO Lin- cage Grant LG 921231.

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Separation of Additive Mixture Spectra by a Self-Modeling Method

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