Angle-Constrained Alternating Least Squares

Angle-Constrained Alternating Least Squares

WILLEM WINDIG* and MICHAEL R. KEENANEigenvector Research, Inc., 3905 West Eaglerock Drive, Wenatchee, Washington 98801 (W.W.); and 8346 Roney Rd., Wolcott, New York 14590(M.R.K.)

When resolving mixture data sets using self-modeling mixture analysis

techniques, there are generally a range of possible solutions. There are

cases, however, in which a unique solution is possible. For example,

variables may be present (e.g., m/z values in mass spectrometry) that are

characteristic for each of the components (pure variables), in which case

the pure variables are proportional to the actual concentrations of the

components. Similarly, the presence of pure spectra in a data set leads to a

unique solution. This paper will show that these solutions can be obtained

by applying angle constraints in combination with non-negativity to the

solution vectors (resolved spectra and resolved concentrations). As will be

shown, the technique goes beyond resolving data sets with pure variables

and pure spectra by enabling the analyst to selectively enhance contrast in

either the spectral or concentration domain. Examples will be given of

Fourier transform infrared (FT-IR) microscopy of a polymer laminate,

secondary ion mass spectrometry (SIMS) images of a two-component

mixture, and energy dispersive spectrometry (EDS) of alloys.

Index Headings: Multivariate curve resolution; Alternating least squares;

MCR-ALS; Constraints; Fourier transform infrared spectroscopy; FT-IR

microscopy; Secondary ion mass spectrometry; SIMS; Energy dispersive

spectrometry; EDS; Chemometrics.

INTRODUCTION

The task of self-modeling mixture analysis is to express adata set in terms of its mixture composition as follows:

D ¼ CST ð1Þ

where D is the data matrix with nspec (number of spectra) rowsand nvar (number of variables) columns. C is the matrix ofconcentrations, of size nspec 3 ncomp (number of compo-nents), and S contains the spectra of the pure components, ofsize nvar 3 ncomp. Because, without a calibration step, thereis an unknown factor relating the results of these techniques toquantitative concentrations, the term ‘‘contributions’’ ispreferred over ‘‘concentrations’’. The matrices C and S are,by nature, positive. It is generally possible to obtain an infinitenumber of other positive solutions by the following transfor-mation:

D ¼ ðCTÞðT�1STÞ ð2Þ

where T is a transformation matrix of size ncomp 3 ncomp.The range of solutions, the so-called feasible solutions, wasdemonstrated in the first paper on self-modeling mixtureanalysis.1 A rather complicated geometrical method todetermine feasible solutions was described by Borgen,2,3

which was simplified by Gemperline4 and, more recently, hasbeen extensively studied by Tauler et al.5–7

There are numerous ways to limit the range of feasiblesolutions or find a unique solution. When pure variables or

pure spectra are present in the data set, techniques such asSIMPLISMA, OPA, or the purity approach can be used todirectly determine the pure variables or pure spectra andresolve the data set.8 Multivariate curve resolution based onalternating least squares (MCR-ALS) is an approach thatiteratively narrows down the feasible solutions using con-straints such as unimodality, positivity, closure, knowledge ofcertain components, selectivity, etc.9 When pure variables orpure spectra are known to be present, the solutions obtained bydirect methods often show some negative intensities caused byviolations of the linear model due to nonlinear counting, shiftsin peaks, noise, etc. When MCR-ALS is used to refine such apure variable or pure spectrum solution, one can begin with thedirect solution as a starting estimate9 and use a positivityconstraint to eliminate the negative intensities. It often appears,however, that the solution deviates significantly from the purevariable/spectrum solution. This paper introduces a constraintthat will result in solutions that remain closer to the originalestimate of the variable/spectrum solution. In order to have aperformance measure with which to compare the differentmethods, the relative root sum of square (RRSSQ) differencesis calculated, which has a value of zero for perfect similarityand 1 for maximum dissimilarity:

RRSSQ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

nspec

i¼1

X

nvar

j¼1

ðdi;j � dreconstructedi;j Þ2

X

nspec

i¼1

X

nvar

j¼1

d2i;j

v

u

u

u

u

u

u

u

t

ð3Þ

where the reconstructed data set Dreconstructed is calculated fromthe estimates C and S obtained by the self-modeling mixtureanalysis technique. For the mathematical details see theliterature.8,9

Dreconstructed ¼ CST ð4Þ

The proposed technique will result in changing the anglesbetween the columns within C and the columns within S. As ameasure for this, the determinant can be used, which calculatesthe volume of the space enclosed by the vectors. For theresolution vector C the equation is as follows:

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jCTlength scaledClength scaledj

q

ð5Þ

where the two vertical lines denote the determinant, a is thevolume enclosed by the column vectors in C, and Clength_scaled

is the length-scaled version of C.

ðclength scaledÞi;j ¼ci;jffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

nspec

i¼1

c2i;j

s ð6ÞReceived 30 September 2010; accepted 7 December 2010.

* Author to whom correspondence should be sent. E-mail: [email protected].

DOI: 10.1366/10-06139

Volume 65, Number 3, 2011 APPLIED SPECTROSCOPY 3490003-7028/11/6503-0349$2.00/0

� 2011 Society for Applied Spectroscopy

Similarly, the determinant of S can be calculated. Themaximum value of a is one.

MATERIALS AND METHODS

Fourier Transform Infrared Spectroscopy of a PolymerLaminate. A description of this data set, its availability, andMatlab code to resolve the data set have been published.10 Asymbolic representation of the sample and its analysis is shownin Fig. 1. The spectral data set was obtained from a sample ofpolymer laminate 240 lm thick. The inner layer of isophthalicpolyester (IP), 2–3 lm thick, was beyond the normal 10 lmresolution of standard micro-infrared analysis. The other layersconsist of polyethylene (PE) and polyethylene terephthalate(PET). The aperture was 10 by 30 lm. The polymer wasscanned in 5 lm steps.

Secondary Ion Mass Spectroscopy of a Two-ComponentMixture. A description and the data analysis of this data sethave been described previously.11 This paper will show onlypart of this data set. A mixed organic acid solution wasprepared by dissolving approximately 10 mg each of stearicacid and palmitic acid (Aldrich Chemical Company Milwau-kee, WI) in a single vial containing 3 mL USP ethanol. A 10lL syringe was used to place a drop of the solution on an

approximately 1 cm2 piece of Reynolds Wrapt aluminum foil.Once the ethanol had fully evaporated, the sample wasmounted in the standard backside-mounting sample holdersupplied with the ION TOF IV time-of-flight secondary-ionmass spectrometer (ToF-SIMS), manufactured by ION-TOFGmbH. The mass scale was 1 to 600 m/z and the high-resolution data was compressed into 1 m/z bins centered oneach nominal mass.

Energy Dispersive Spectrometry of Wires of DifferentAlloys. The analysis of this data set has been describedbefore.12 The sample consists of a series of six types of wiresembedded in an epoxy matrix, with the wire alloys themselvesbeing composed of a pallet of six different elements. Thesample was imaged in a scanning electron microscope (SEM)with an attached energy dispersive X-ray spectrometer (EDS).Typical data acquisition conditions for this type of spectralimage are described in Kotula et al.13 Figure 2a shows astandard SEM image of the sample together with thecomposition key. The image dimensions are 128 3 128 pixels,and a complete 1024-channel spectrum was collected at eachpixel. The mean spectrum, obtained by averaging over all ofthe pixels in the image, is shown in Fig. 2b. While the meanspectrum has good signal-to-noise ratio, this is not the case forany individual spectrum. Figure 2c shows a typical single-pixel

FIG. 1. A symbolic representation of the polymer laminate and its analysis. The spectra in row (a) are representative of the three layers, from left to right,polyethylene, isophthalic polyester, and polyethylene terephthalate. Due to the size, the observation area the middle spectrum has contributions from the otherpolymers. The stars in the representation of the laminate in (b) indicate the sample steps. The expected contribution profiles of the three components are given in (c).

350 Volume 65, Number 3, 2011

spectrum for the Cu/Mn/Ni wire. The discrete nature of the datais clearly evident, and the signal-to-noise ratio is sufficientlylow that the presence of Ni cannot be detected.

Data Analysis. All calculations were performed usingMatlab Version 7.9 (Mathworks, Natick, MA 01760-2098)on Windows-based computers in combination with PLS_Tool-box (Eigenvector Research, Inc., Wenatchee, WA). The angleconstraint was introduced in PLS_Toolbox version 6.0. For theinitialization of MCR-ALS the pure variables or pure spectra asestimated by the purity program,14 with an offset of 10, wereused.

RESULTS AND DISCUSSION

Angle Constraint. A visualization of the feasible solutionsfor a simulated mass spectrometry data set is shown in Fig. 3.The first column shows three mixture spectra. The task of self-modeling mixture analysis is to express this data set as amixture of, in this case, two components. Due to the nature ofspectrometry, the spectra of the pure components and theircontributions should be positive. A valid solution is given inthe second column. The pure components are spectra one andthree of the mixtures. This is the so-called pure spectrumsolution, indicating that the spectra of the pure components arepresent in the data set. The contributions of the twocomponents in the mixtures are also listed in the secondcolumn.

Another valid solution is given in the third column. In thiscase we assumed pure variables. A pure variable is a variablethat is present in only one of the components. For purespectrum 2a the pure variable is variable 1 and for spectrum 2bthe pure variable is variable 3.

An important observation is that solution 1 (spectra 1a and1b) has high contrast in the contributions (zero versus oneconcentrations of the third mixture spectrum) but low contrastin the spectra (no one versus zero intensities). Solution 2(spectra 2a and 2b) shows high contrast on the resolvedspectra: variable 1 has an intensity of 1 in component a and anintensity of 0 in component b and the reverse relation invariable 3. This reverse relation in contrast in the spectra andcontributions is also reflected in Eq. 2. In order to decrease thecontrast in C the matrix T has positive linear combinations inits columns. As a consequence, T represents a space of columnvectors with sharp angles between them. The operation CTthus is a projection of the data in C into a space defined by thevectors with sharp angles, resulting in the decrease in anglesand thus in contrast. Because the row vectors in the matrix T�1

will be orthogonal to all vectors in T, except for thecorresponding column vectors in T, the row vectors in T�1

represent a space of obtuse vectors. As a result, the operationT�1 results in an increase in angles and, as a consequence, incontrast. A similar reasoning can be applied to a decrease incontrast in S and increase in contrast in C. This can besummarized as follows.

In self-modeling mixture-analysis solutions a high contrastin resolved contributions relates to a low contrast in theresolved spectra and vice versa.

The contrast is also often referred to as simplicity. Examplesare the pure variable solution, which results in a high contrastin the resolved spectra, and the pure spectrum solution, whichresults in a high contrast in the resolved contributions.

A linear rearrangement of the spectra in Fig. 4 shows thefeasible regions for the solutions. The angles shown between

the spectra are based on their inner product. The high contrastof the pure variable solution is expressed as a large anglebetween the two spectra.

Any positive linear combination of the two spectra withineach range is a valid pure component. Outside these ranges novalid pure spectra exist. For example, a spectrum abovefeasible range A would have a negative intensity for the thirdvariable. A spectrum below feasible range A would require anegative linear combination to reproduce the first mixturespectrum (spectrum 2 in Fig. 4).

If it is reasonable to expect that pure spectra are present in adata set, then the MCR-ALS solution does not necessarily findthe pure spectra as a solution, even if the iterative procedurestarted with the pure spectra. In this case the resolved spectrumwill lie within the feasible region, and in order to obtain asolution closer to the pure spectrum solution the resolvedspectra should move inwards in the feasible region, asindicated in Fig. 4 by the arrows. This can be achieved byadding a small part of the mean of the resolved spectra to theresolved spectra:

scorrectedi;j ¼ ð1� f Þsi;j þ f

1

ncomp

X

ncomp

j¼1

si;j ð7Þ

where Scorrected represents the angle-corrected S and frepresents the contrast weight, for which the value 0.05 isused. This value was chosen because it has a minimal effect onthe RRSSQ value as shown in Table I, which will be discussedbelow, while achieving the goal of changing the contrast. In theimplementation of this constraint in PLS_Toolbox the contrastweight can be changed. A refinement would be to decrease theangle during the iterative process. When more data sets areavailable, this option will also be explored.

An example Matlab code implementing this procedure,including the resolution step, is given in the alternating leastsquares (ALS) code in Fig. 5. It has to be stressed that the codejust symbolizes the procedure and only shows constraints

FIG. 2. (a) A SEM image of the sample consisting of metal wires embedded inan epoxy matrix, together with the composition key. (b) The mean EDSspectrum computed from the data set. A 1024-channel spectrum was acquiredfrom each pixel in the 1283128 pixel image. (c) A single-pixel spectrum fromthe Cu/Mn/Ni wire. [This figure was reproduced with permission of Wiley,Chichester, UK, 2007 from: M. R. Keenan, ‘‘Multivariate Analysis of SpectralImages Composed of Count Data’’, in Techniques and Applications ofHyperspectral Image Analysis, H. F. Grahn and P. Geladi, Eds. (Wiley,Chichester, UK, 2007), pp. 89–126.]

APPLIED SPECTROSCOPY 351

important for the angle constraint and not other commonly used

constraints. In this code the initial estimate is random. While

this works in many cases, it is safer to start with an estimate

based on pure spectra. As mentioned before, this is done with

the purity program. Also, the simple non-negativity constraint

shown should in practice preferably be a non-negative least

squares technique such as the one available in PLS_Toolbox15

or faster implementations.16 Furthermore, in practice the loop

would be stopped by a convergence criterion. The angle

constraint used to obtain pure spectra by decreasing the angles

between the rows in the matrix purspec results in increasedangles, that is, higher contrast, for the columns in purint, the

FIG. 3. The mixture spectra under the header ‘‘mixtures’’ can be resolved into an infinite number of solutions, of which the two extreme solutions are given in theform of the resolved pure spectra and their contributions. The solution with pure spectra 1a and 1b represents the pure spectrum solution and the solution with purespectra 2a and 2b represents the pure variable solution.

FIG. 4. A linear representation of the mixture spectra (in bold, spectra 2, 3, and 4) and the extremes of the feasible solution ranges indicated (spectra 1 and 2; spectra4 and 5). The lines at the left give the angles between the spectra, as symbolized by a. The spectra in the two solution ranges with the smallest angle between them,spectra 2 and 4, give the highest possible contrast in the resolved contributions, and the spectra in the two solution ranges with the largest angle between them,spectra 1 and 5, give the highest possible contrast in the resolved spectra.


resolved contributions. In order to obtain a pure variablesolution the identical code is applied to the transposed matrix,resulting in a higher contrast in the resolved spectra.

To summarize, the algorithm increases contrast in theresolved contributions by decreasing the angle between theresolved component spectra and increases contrast in theresolved spectra by decreasing the angle between the resolvedcomponent contributions.

The concept of contrast/simplicity in resolution methods isnot new. A solution based on minimal angles, resulting in aspace with minimal volume, was recently discussed byRajko.17 The use of rotations of principal components usingorthonormality in the spectral (loading) or contribution (scores)domain has been described.18,19 The method discussed heredoes not require orthogonality, though. The concept ofsimplicity, another word for contrast, is also used in numerousrotation methods in factor analysis.20 The classical factoranalysis based methods do not generally lead to physicallyrealistic chemical components. Another advantage of theproposed method is that it can be used in combination withother appropriate constraints.

Fourier Transform Infrared Spectroscopy of a PolymerLaminate. For the analysis of a data set, one must decidewhether the solutions should have high contrast in the resolved

in the spectra or high contrast in the resolved contributions.Because we have three separate layers in this case, asconfirmed microscopically and spectroscopically,10 one ex-pects high contrast in the resolved contributions, as shown inthe expected contribution profiles in Fig. 1. Although highcontrast in the contributions is related to pure spectra, themiddle component does not have a pure spectrum in the dataset. As a consequence, a pure spectrum solution would containnegative contributions. However, applying MCR-ALS withpositivity constraints would seem to be an appropriate solution.

In order to display the results in a two-dimensional display,the data were normalized to total intensity prior to the dataanalysis. The initial estimates for the MCR-ALS function werethe pure spectra (spectra 1, 7, and 17) as estimated by the purityprogram with its default settings. Principal component analysis(PCA) of this normalized data set, with the resolved spectraadded, enables us to view the results in a two-dimensional plot(Fig. 6). In this figure the resolved spectra, as determined byMCR-ALS, are indicated by stars and connected by a solidtriangle. The pure spectra determined by the purity algorithmfor the initialization of MCR-ALS are connected by a dashedtriangle. Spectra 1 and 17 are expected to be pure, since theyrepresent pure components, while spectrum 7 is not pure from achemical point of view: because of the size of the observationarea this spectrum will have contributions from the other twocomponents. The triangle formed by the original pure spectrashows that several of the scores lie outside the triangle, whichindicates that using these spectra will result in negativecomponent contributions. The solution provided by MCR-ALS shows that all the scores lie within the triangle, indicatingthat the component contributions will be positive. It appearshowever, that the first spectrum does not coincide with thissolution, while this spectrum represents a pure polymer. In Fig.7 the resolved results are shown, which show again a deviationfrom the expected results, particularly that the componentcontribution of the middle component deviates from theexpected profile in Fig. 1.

TABLE I. The RRSSQ values of regular MCR-ALS and MCR-ALS withthe angle constraint. The increase in the RRSSQ value for the angleconstraint is relatively small.

RRSSQ

MCR-ALSMCR-ALS

contrast

FT-IR spectroscopy of a polymer laminate 0.033 0.041SIMS of a two-component mixture 0.092 0.099EDS of wires of different alloys, spectral contrast 0.228 0.249EDS of wires of different alloys, image contrast 0.228 0.230

FIG. 5. ALS code to decrease angle between the purspec vectors, resulting in higher contrast in the resolved contributions. The code in lines 10 and 11 representsthe code that needs to be added to regular MCR-ALS code.


Applying the angle constraint on the pure spectra of this data

set during the MCR-ALS procedure will nudge the contribu-

tions towards higher contrast. The constraint assumes that the

concentration profiles are as given in Fig. 1, which is known to

be appropriate for this model system. The assumption of these

profiles implies that the contributions from the first and third

layer in the middle spectrum are due to the limited resolution of

the microscope and not due to actual contributions of the

polymers constituting the first and third layer in the middle

component. This knowledge is crucial, since without it the

middle spectrum cannot be (more or less) uniquely resolved.21

The effect of narrowing the angle between the pure

component spectra in combination with a positivity constraint

is that the smallest possible triangle will be calculated

enclosing all the spectral points, as represented by the PCA

scores in Fig. 6. The smallest triangle will have the maximum

FIG. 6. The PCA scores of the original data (round markers) and the resolved component spectra (stars). The solid triangle connects the resolved component spectraand the dashed triangle connects the purest spectra in the data set. The arrow indicates the analysis sequence. (a) The results of regular MCR-ALS; (b) the results ofMCR-ALS with the angle constraint.

FIG. 7. The resolved contribution profiles and spectra of regular MCR-ALS and of MCR-ALS with the angle constraint.


number of spectral points on its sides, resulting in a highcontrast in the scores. The results as presented in Fig. 7 alsoshow a clear improvement. The value of 0.05 for the contrastweight (see Eq. 7) appears to be a good value: it results in thesolution expected for this data set and results in a similarRSSQ value (see Table I). The higher contrast in thecontributions is also confirmed in the determinant value inTable II. The value for the higher contrast results is close tothe maximum value of 1.

This example also shows that this method does not just workwith pure spectra: the middle component, IP, is not present in apure form in the data set, but it is clear that the angle constraintin combination with the positivity constraint results in theproper resolved spectrum for this component.

One has to realize that other MCR-ALS constraints are alsoavailable when (parts of) the data set are selective for singlecomponents.6 However, the determination of the local rank andthe extent of unique areas are more complicated to determinethan the angle constraint.

Secondary Ion Mass Spectrometry of a Two-ComponentMixture. This data set is from a sample that contains a singledrop of a mixture of two components. Although one would notexpect to be able to resolve this data set since the rank of the

mixture should be one, a complex evaporation behavior, relatedto the dense ring-like deposit of a coffee stain, results in aconcentration gradient within the drop.22

For mass spectrometry the contrast is to be expected in purevariables: generally, there will be m/z values that are unique foreach component. The pure variable approach was applied to thisdata set before and the presence of reference spectra for theresolved components confirmed the validity of the pure variableconcept for this data set.11 One certainly would not expectcontrast in the images (contribution profiles) since each pixel(spectrum) will contain the two components. From previousstudies we know that in addition to the two acid components,two additional background components need to be extracted.11

The data was preprocessed in order to account for the Poissonstatistics of the variables with an offset of 3 as described in theearlier work.23 MCR-ALS with positivity constraint was appliedto resolve four components. The results were corrected to undothe effects of the Poisson scaling. The results of regular MCR-ALS and MCR-ALS with the angle constraint, ignoring thebackground components, are shown in Fig. 8. The most strikingenhancement resulting from the angle constraint is the relativelyhigher intensities of the ions related to the molecular ion Mtypical for palmitic acid (MþH: 257 and MþH-H2O: 239) andstearic acid (MþH: 285 and MþH-H2O: 267) compared to thelower mass range. The lower mass range is dominated by peaksof the background spectra. Also, a slight contribution frompalmitic acid can be observed in the stearic acid spectrum of theMCR-ALS solution, which is not present after applying theangle constraint. This is a clear demonstration of the highercontrast, or smaller amount of similarity, in the spectra resultingfrom the application of the angle constraint. The images(contributions) clearly show ring-shaped concentration profiles.The angle-constrained solution had minimal effect on the RSSQ.The determinant values for the complete four-componentsolution and for just the two components shown in Table IIshow a clear increase in contrast.

Energy Dispersive Spectrometry of Wires of DifferentAlloys. The previous two examples showed that there are cases

TABLE II. Where two numbers are given, the first number is for thewhole data set and the second number is only for the spectra shown.

Determinant

MCR-ALSMCR-ALS

contrast

FT-IR spectroscopy of a polymer laminate 0.865 0.961SIMS of a two-component mixture 0.0534 0.3569

0.2955 0.8952EDS of wires of different alloys, spectral contrast 0.2260 0.7355

0.2275 0.2487EDS of wires of different alloys, image contrast 0.3587 0.6793

0.7109 0.9923

FIG. 8. The MCR-ALS solution and the angle constraint solution used to obtain high contrast in the resolved spectra.


in which spectral contrast is desirable in the solution and inwhich contribution contrast is desirable. The EDS data set is acase in which both solutions are worthwhile:

(1) Spectral contrast will lead to separating all the replicate setsof different elements.

(2) Image (contribution) contrast will lead to separating all thedifferent alloys.

Both these solutions are solid assumptions with this model dataset, based on the well-defined samples, as confirmed byprevious analysis.12

As shown in Fig. 2 the data set is extremely noisy. In orderto reduce the noise and speed up calculations, a reduced imageof the data set was calculated by averaging 333 blocks ofpixels. Poisson scaling was applied to the variables with anoffset of 3 prior to data analysis.23 The results of the MCR-ALS analysis were inversely scaled prior to display. Figure 9shows the effect of increasing the spectral contrast, which leadsto the elemental composition. The MCR-ALS solution does notclosely resemble this solution, which consists largely of thepure elements. Comparing the results with the knowncomposition as listed in Fig. 2 shows that the spectral contrastsolution is correct. The solution for the image (contribution)contrast shows the alloys correctly, while the regular MCR-ALS solution, again, is not close to this representation. Thesetwo examples show the significant improvements resultingfrom the angle constraint. The RSSQ values of this data set (seeTable I) are relatively high, which is due to the noisy character

of this data set. The determinant values again show theimprovement in the contrast, with the value for the imagecontribution of the spectra shown in Fig. 10 close to themaximum value of 1.

CONCLUSION

The results shown above are cases in which contrast wasdesirable in either the resolved contributions (for the polymerlaminate), the resolved spectra (SIMS data of the mixture), orboth (EDS of alloys). The term ‘‘desired solutions’’ is useddeliberately. Although finding the ‘‘true’’ solution, i.e., reflectingpure chemical components, seems the ultimate goal, the EDSdata set of alloys shows that there can be two equally valuablesolutions, and the ‘‘true’’ solution depends upon the intent of theresearcher. There are cases in which one may even deliberatelydeviate from trying to obtain a ‘‘true’’ solution. For example, ifone has a SIMS data set of a sample with a minor contaminant insome of the pixels, then the choice for contrast enhancementcould well be for the images instead of for the chemically morelogical ‘‘true’’ choice of contrast enhancement in the spectrasimply because the image contrast will help to determine thelocation of the problematic pixels. The resolved spectrum willthen be of limited use because of the limited spectral contrast,but the known location of the pixels will enable the researcher tostudy the spectra of the pixels by classical means.

Although for the examples shown in this paper of well-defined data sets with known composition the contrast requiredfor their solutions is appropriate, the solution is in general as

FIG. 9. The resolved images and spectra of regular MCR-ALS and of MCR-ALS with the angle constraint for spectral contrast.


valid as the hypothesis that the contrast is present in the data.This may seem to be a limitation of MCR-ALS techniques, butas long as the researcher understands the implication, aconstraint such as contrasts can be used to enhance differencesin the data that would be difficult to see in the ‘‘true’’ solution.

Although these are certainly not exceptional examples, thereare also many cases in which the desired solution is not as clearcut as for the data sets discussed above. For more complicateddata sets, contrast may not be sufficient as the only constraint,and other commonly available MCR-ALS constraints can beused, possibly in combination with the angle constraint.

Another option is to resolve the data sets both ways in orderto judge which solution makes the most sense from a chemicalpoint of view. There are also cases in which one might want tohave a stronger angle constraint for one component than for theothers. However, before making more complex optionsavailable, the application of the angle constraint to more datasets needs to be studied.

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(1986).4. P. J. Gemperline, Anal. Chem. 71, 5398 (1999).5. R. Tauler, J. Chemom. 15, 627 (2001).6. R. Tauler, M. Maeder, and A. de Juan, ‘‘Multiset Data Analysis: Extended

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analysis with SIMPLISMA’’, in Techniques and Applications of Hyper-spectral Image Analysis, H. F. Grahn and P. Geladi, Eds. (Wiley,Chichester, UK, 2007), pp. 69–87.

12. M. R. Keenan, ‘‘Multivariate Analysis of Spectral Images Composed ofCount Data’’, in Techniques and Applications of Hyperspectral ImageAnalysis, H. F. Grahn and P. Geladi, Eds. (Wiley, Chichester, UK, 2007),pp. 89–126.

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Anal. 41, 88 (2009).19. M. R. Keenan, Surf. Interface Anal. 41, 79 (2009).20. E. R. Malinowski, Factor Analysis in Chemistry (Wiley, New York, 2002),

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FIG. 10. The resolved images and spectra of regular MCR-ALS and of MCR-ALS with the angle constraint for contribution contrast.


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Angle-Constrained Alternating Least Squares