projection two-dimensional correlation analysis

Journal of Molecular Structure 974 (2010) 116–126

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Journal of Molecular Structure

journal homepage: www.elsevier .com/locate /molstruc

Projection two-dimensional correlation analysis

Isao Noda *

The Procter & Gamble Company, 8611 Beckett Road, West Chester, Ohio 45069, USA

a r t i c l e i n f o

Article history:Received 4 November 2009Accepted 18 November 2009Available online 22 November 2009

Keywords:Two-dimensional correlation spectroscopyProjection 2D correlationProjection matrixIR spectroscopy

0022-2860/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.molstruc.2009.11.047

* Tel.: +1 513 634 8949; fax: +1 513 277 6877.E-mail address: [email protected].

a b s t r a c t

Projection 2D correlation analysis, which may potentially become a useful tool in the simplification ofhighly congested 2D correlation spectra often encountered in practice, is introduced. By taking advantageof the fact that 2D correlation spectra can be expressed in terms of matrix multiplications of spectral data,one can apply matrix-based projection and null-space projection operations as an effective filtering toolto transform spectral data to generate new 2D correlation spectra with much more simplified features.Projection matrices used in this approach can be generated from various sources, including a part of ori-ginal spectral data, other spectra of different origin, perturbation variables, or even arbitrarily chosenmathematical functions. Furthermore, by linearly combining the projected and null-space projected spec-tra, application of a series of oblique projections to gradually attenuate or augment select features is alsopossible. Projection 2D correlation analysis applied to the ATR IR study of time-dependent compositionalchanges of a solution mixture demonstrated its ability to obtain pure component 2D spectra, 2D spectrabased on contributions of a reduced number of species, and various 2D spectra with specific featuresselectively accentuated, to assist the unambiguous interpretation of complex and highly overlapped spec-tral data.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Two-dimensional (2D) correlation spectroscopy has seen wide-spread applications in various branches of analytical science [1–4].By spreading peaks along the second dimension, apparent spectralresolution and selectivity are improved. One can often simplify over-lapped spectral information found in traditional one-dimensionalspectra by taking advantage of this useful feature of 2D spectra.However, for very complicated systems, such as multi componentmixtures, multiphase composite materials, and various complicatedbiomolecules, even 2D spectra become too congested for analysis.The appearance of many overlapped cross peaks makes the interpre-tation of 2D correlation spectra sometimes difficult. Thus, the devel-opment of a new and effective method is explored to make theinterpretation of congested 2D correlation spectra much simplerand easier.

In a separate article [5], the double 2D correlation technique,or the 2D correlation analysis of 2D spectra, was studied. Thistechnique has shown some promise in enhancing the selectivityand spectral resolution of 2D correlation spectra by accentuatingonly the doubly correlated portion of spectral information perti-nent to the analysis. During the course of the development of thistechnique, it was recognized that such double 2D correlationcould be understood in terms of a form of matrix projection oper-

ll rights reserved.

ation to convert a dataset to a different and sometimes more use-ful form.

In this communication, a technique called projection 2D correla-tion analysis is introduced to simplify and streamline the informa-tion contained in 2D correlation spectra. This method is based onthe use of mathematical matrix projection to selectively filter outthe unwanted portion of the information of spectral data. The com-bination of the projection and null-space projection operationsmakes it possible to attenuate or augment select features withincongested 2D correlation spectra for easier interpretation. An illus-trative example is provided to demonstrate the striking simplifica-tion of 2D correlation spectra having highly overlapped crosspeaks.

2. Background

2.1. Matrix representation of 2D correlation spectra

The basic concept of projection 2D correlation can be best under-stood in terms of a series of matrix manipulations. The response ofa given system, which is under the observation by a chosen spectro-scopic probe (e.g., IR, Raman, or UV) and stimulated by an externalperturbation (e.g., temperature, composition, or time-dependenteffects) can be captured by a series of spectra observed along the per-turbation variable. The spectral data matrix A consisting of m rows ofspectra with n columns of spectral variables, like wavenumber, rep-resents the system response. Generalized 2D correlation spectra can

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I. Noda / Journal of Molecular Structure 974 (2010) 116–126 117

be obtained by a simple matrix multiplication applied toward thespectral data matrix. For simplicity, it is assumed that each row ofthe spectral data matrix A has already being subtracted by a selectedreference spectrum, typically the average spectrum, which leads tothe simple mean-centering of the original spectra. Furthermore,the data is scaled by the square root of the degree of freedom term1/(m � 1)1/2, which only changes the intensity of spectra by a con-stant value.

The synchronous and asynchronous correlation spectra, U andW, are then obtained as

U ¼ ATA ð1ÞW ¼ ATNA: ð2Þ

Fig. 1. (a) A series of time-dependent ATR IR spectra observed for the solutionmixture of MEK, d-toluene and PS during the spontaneous evaporation of solventsand (b) profiles of select spectral intensities observed at different wavenumbers.

The m-by-m matrix N is the so-called Hilbert–Noda transformationmatrix, with the diagonal elements Njj being all zero and off-diago-nal elements given by Njk = 1/{p (k – j)}.

2.2. Projection matrix

Let us now consider an arbitrary matrix Y consisting of m rowsand k columns, which is different from the spectral data matrix A.We then define the projection matrix RY of Y as

RY ¼ YðYTYÞ�1YT: ð3Þ

The superscripts T and –1, respectively, stand for the transpose andinverse operation of the matrix. The above form of definition actu-ally is somewhat limited, because it assumes that the matrix prod-uct YTY is non-singular, such that the inverse (YTY)�1 exists. More

Fig. 2. (a) Synchronous and (b) asynchronous 2D IR correlation spectra constructedfrom the ATR IR data shown in Fig. 1.

118 I. Noda / Journal of Molecular Structure 974 (2010) 116–126

generally applicable definition of the projection matrix is obtainedby applying the singular value decomposition (SVD) to the matrix Y:

Y ¼ UYSYVTY þ EY: ð4Þ

The columns of the matrices UY and VY, respectively, correspond tothe principal eigenvectors of the matrix products YYT and YTY. Thediagonal matrix SY contains the singular values of Y, i.e., the positivesquare roots of the significant eigenvalues of either Y YT or YT Y, andEY is the residual matrix or the portion of data matrix not modeledby the SVD operation. For simplicity, we assume the residual matrixEY to be small enough to be negligible. The projection matrix thenbecomes

RY ¼ UYUTY: ð5Þ

This m-by-m matrix RY acts as a projector for the space spanned bythe columns of Y. We also define the accompanying null-space pro-jection matrix (I – RY),

Fig. 3. (a) Dynamic (mean-centered) spectra of the solvent evaporation process obtainedFig. 1; (b) spectra obtained by projecting the dynamic spectra onto the space spanned byprojection onto the same space.

ðI� RYÞ ¼ I� UYUTY ð6Þ

where I is the m-by-m identity matrix. The null-space projectionmatrix is a projector for the space spanned by the vectors orthogo-nal to the columns of Y. Both the projection matrix RY and null-space projection matrix (I – RY) are symmetric, i.e., RT

Y ¼ RY and(I – RY)T = (I – RY), as well as idempotent, i.e., R2

Y ¼ RY and (I –RY)2 = (I – RY).

2.3. Data transformation by projection

The spectral data matrix A can be transformed to a new form ofdata matrix by the projection operation. The projected data matrixAP is obtained by the simple multiplication of RY with A,

AP ¼ RYA: ð7Þ

The newly obtained projected data matrix AP represents the ma-trix projection of A onto the abstract mathematical space spannedby the columns of Y. In other words, AP is the closest possible

by subtracting the time-averaged spectrum as the reference from the raw spectra inthe intensity variations of PS at 1495 cm�1; and (c) spectra obtained by the positive


reconstruction of A by using only the linear combinations of allthe columns of Y. To make this operation possible, matrices Aand Y must have the same number of rows m. It is actually com-mon to select Y from several select columns of A.

The corresponding null-space projection is carried out as

AN ¼ ðI� RYÞA ¼ A� AP: ð8Þ

The null-space projected data matrix AN represents the projection ofA onto the space spanned by vectors which are orthogonal to thecolumns of Y. In other words, AN is the residual after the removalof AP from A. Thus, the projection operations separate the originaldata into two orthogonal parts, A = AP + AN, by using the informa-tion contained within the other chosen matrix Y.

It is now possible to combine two orthogonal matrices, AP andAN, in the form of a weighted sum to obtain a new data matrix with

Fig. 4. Synchronous spectra constructed (a) from the (fully) projected spectra ofFig. 3a and (b) from the positively projected spectra of Fig. 3b. In addition to thetime-averaged spectrum, power spectrum (dashed line) is provided for referencepurpose.

the projected matrix portion augmented or attenuated by a factora as

Aa ¼ aAP þ AN ¼ ðI� RY þ aRYÞA ð9Þ

where the constant a can be arbitrarily set to a value above zero toaccentuate or reduce some features of A. For example, if the value ofa is set to be one, the original data A is recovered. In contrast, if a iszero, the operation becomes the null-space projection. When atakes the value between one and zero, Aa becomes a data matrixwith the projected portion attenuated by the fraction (1 � a).

Fig. 5. Spectra obtained by positively projecting the dynamic spectra onto the spacespanned by the intensity variations at (a) 1390 cm�1 (d-toluene) and (b) 1360 cm�1

(MEK).


Finally, if a is greater than one, the operation leads to the augmen-tation or amplification of the projected portion. One may view(I � RY + a RY) as a form of oblique projection matrix, such that Ais projected onto an oblique space spanned by vectors which arenot necessarily parallel or perpendicular to the columns of Y.

2.4. 2D correlation of projected data

Data matrices created by various projection-based transforma-tion operations discussed above can be readily converted to 2Dcorrelation spectra. For example, by using Eq. (7), it is possible toobtain the 2D correlation spectra for the projected data matrix AP

UP ¼ ATPAP ¼ ATRYA ð10Þ

WP ¼ ATPNAP ¼ ATRYNRYA: ð11Þ

The term RY appears in Eq. (10) only once because this matrix isidempotent. 2D correlation spectra UP and WP for the projecteddata provide the correlation information among select signals ofA, which are in turn correlated with the projector matrix Y. In otherwords, all other signals not correlated Y will be filtered out prior tothe 2D correlation analysis.

Correlation analysis of the null-space projected data, AN = A -AP, results in the following set of 2D spectra.

UN ¼ ATNAN ¼ ATðI� RYÞA ð12Þ

WN ¼ ATNNAN ¼ ATðI� RYNRY � NRY � RYNÞA ð13Þ

Fig. 6. Synchronous spectra constructed from the spectra obtained by positively proje1390 cm�1 (d-toluene), (b) 1360 cm�1 (MEK), and (c) 1420 cm�1 (MEK).

The new set of 2D correlation spectra, UN and WN, for the null-space projected data provide the correlation information amongthe specific signals maintained within A, which are not correlatedwith Y. Eqs. (12) and (13) can be rewritten as

UN ¼ U�UP ð14ÞWN ¼ WþWP �WQ ð15Þ

The terms UP and WP are already described in Eqs. (10) and (11),while the new term WQ defined as

WQ ¼ ATðNRY þ RYNÞA ð16Þ

represents the portion of W generated by signals which are quadra-ture to (i.e., 90 degrees out of phase with) columns of Y.

Finally, 2D correlation spectra generated from the proportion-ally weighted sum of projected and null-space projected data ma-trix, Aa = A � AP + a AP, are given by

Ua ¼ ATaAa ¼ U� ð1� a2ÞUP ð17Þ

Wa ¼ ATaNAa ¼ Wþ ð1� aÞ2WP � ð1� aÞWQ ð18Þ

2.5. Projection onto a vector

The projection matrix can be built from a single column vectorinstead of a matrix. In fact, this type of projection matrix may bethe most useful one for many applications in 2D correlation

cting dynamic spectra onto the space spanned by the intensity variations at (a)

Fig. 7. (a) The positive portion of the synchronous 2D ATR IR correlation spectrumof the solution mixture. (b) Power spectra of the mixture and projected data.


analysis. We will, therefore, focus our attention to the use of vec-tor-based projection matrix. However, any development from thispoint on can be generalized to the situation involving multiple pro-jection vectors or a matrix.

For a given vector y, we define the vector-based projection ma-trix Ry as

Ry ¼ yðyTyÞ�1yT ð19Þ

or alternatively, it can also be written as

Ry ¼ uyuTy: ð20Þ

The normalized vector uy for y is given by

uy ¼ y=kyk ð21Þ

where ||y|| is the Euclidian norm of y, which is a scalar quantity.

kyk ¼ ðyTyÞ1=2 ð22Þ

By using the vector-based projection matrix Ry, it is again pos-sible to obtain the projected data

AP ¼ RyA ð23Þ

The vector projected data AP in this case represents the projec-tion of A onto the space spanned by a single vector y. In otherwords, each column of AP is built by the closest reproduction ofthe corresponding column of A by using only the vector y scaledwith an appropriate constant. It is straightforward to obtain the2D correlation spectra for the vector projected data in a mannersimilar to the steps described in the previous section.

2.6. Positive projection trick

If we define the loading vector vA of AP as

vA ¼ Auy ð24Þ

the projected data AP can be expressed as

AP ¼ uyvTA: ð25Þ

We now introduce an additional procedure for the vector projectionoperation to further manipulate data matrix. Given the loading vec-tor vA defined in Eq. (24), we can also define the corresponding po-sitive loading vector v+A as a vector similar to vA except that allnegative elements of vA are now replaced by zero. We then havea new data matrix created by the positive projection operation

AþP ¼ uYvTþA: ð26Þ

The positive projection of the data matrix retains only the col-umn of A which is oriented in the same direction as the vector yfor the projection operation. Thus, other columns of A oriented inthe opposite directions are no longer projected onto the spacespanned by y. The corresponding positive null-space projection pro-duces the data matrix.

AþN ¼ A� AþP ð27Þ

Likewise, the attenuated or augmented data for positive projectionbecomes

Aþa ¼ A� ð1� aÞAþP ð28Þ

We can obtain sets of 2D correlation spectra as before by usingthese data matrix newly created by the positive projectionoperations.

2.7. Source of the projector

Up to this point, the specific nature of the projecting matrix Y orvector y used to construct the projection matrices has not been

explicitly defined. There are several options to choose the sourceof the projector. The obvious choice of the projector is to select acolumn or several columns of the original data matrix A. In mostcases, a single column is selected as a projector vector y from thedata matrix. This vector corresponds to the spectral intensityy(tj) = a(mi, tj), which is varying along the perturbation variable tat a particular wavenumber mi. The projection operation appliedto A by using this vector will yield a new dataset consisting exclu-sively of the spectral intensity variations which are proportional tothe behavior of the chosen signal intensity at mi. Alternatively, onecan choose multiple columns of A to construct a projector matrix Y.The matrix projection operation can be applied in a straightfor-ward manner similar to the vector projection.

One can also select a column or columns from a matrix of an-other dataset B, as long as it has the same number of rows as A,to construct the projector y or Y, which in turn can be made intoan appropriate projection matrix. This type of projector selection


is essentially a variant form of hetero-spectral correlation. Thus,one may for example use a column of Raman data as a projectorfor transforming IR data, and vice versa. After the projection of IRdata onto the space spanned by the select column(s) of Ramandata, only the portion of IR data which shares the same trend asthe selected part of Raman data will be retained.

One can also use the perturbation variables, such as time, tem-perature, or concentration, as a source of the projector. In this case,the outcome of the projection or null-space projection operationresembles that of the partial least squares (PLS) or orthogonal sig-nal correction (OSC) operation. The specific projection operationwhich utilizes two projector columns comprising the perturbationvariable and its quadrature counterpart (i.e., the Hilbert transform)is known as the quadrature orthogonal signal correction (QOSC).This particular technique was explored in the past by Wu [6,7].

Finally, it is also possible to use even an arbitrarily chosenmathematical function as the projector. Thus, some convenientfunctional forms reflecting the specific features of interest, suchas exponential decay, sinusoid, and the like, may be applied to

Fig. 8. Dynamic spectra positively projected onto the null-space of the signals at (a) 149(exclusion of MEK).

transform the data matrix. The idea behind the projection opera-tion based on a mathematical function is somewhat similar tothe so-called model-based correlation analysis, such as km correla-tion developed by Dluhy [8,9].

3. Experimental

Time-dependent attenuated total reflectance (ATR) IR spectrawere collected for a solution of polystyrene (PS) dissolved in a mix-ture of methyl ethyl ketone (MEK) and perdeuterated toluene (d-toluene), which was undergoing the spontaneous evaporation ofvolatile solvent components [3]. The initial composition of the sol-vent mixture was set as a 1:1 blend by weight, and the PS concen-tration was about 1.0 wt.%. The transient ATR IR data werecollected with a Bio-Rad Model 165 FT-IR spectrometer. The sam-ple solution mixture was analyzed by depositing it on a horizontalZnSe ATR plate (Bio-Rad HATR accessory). The sample was exposedto open atmosphere, and IR spectra were collected as the volatile

5 cm�1 (exclusion of PS), (b) 1390 cm�1 (exclusion of d-toluene), and (c) 1360 cm�1


solvents evaporated, eventually leaving a PS film behind on theATR plate. Sets of IR spectra were collected at intervals of 9 s witheach set consisting of 8 coadded scans at 4 cm�1 resolution. A totalof 25 sets of spectra were collected in this manner. After the datacollection, a total of 12 consecutive scan sets were chosen to rep-resent the period of time when solvent evaporation was occurring.The transient IR spectra were converted to GRAMS (Glactic Indus-tries Corp.) format, then analyzed with the homebuilt 2D correla-tion software.

4. Applications

4.1. Conventional 2D correlation spectra

Fig. 1a shows a series of time-dependent IR spectra observed forthe solution mixture of MEK, d-toluene and PS during the sponta-neous evaporation of solvents. The gradual loss of MEK and d-tol-uene and accumulation of PS result in the continuous change in thespectral features. The profiles of the intensity evolution of four se-lect peaks at 1360, 1390, 1420 and 1495 cm�1 are shown in Fig. 1b.While these four profiles are, respectively, dominated by the con-tribution of MEK, d-toluene, MEK again, and PS, they do not repre-sent the pure component profiles due to the substantial overlap ofpeaks.

Fig. 2 shows the corresponding 2D IR correlation spectra, con-structed from the ATR IR data shown in Fig. 1. It is possible to iden-tify individual bands assignable to different components of thesolution mixture. Bands belonging to the same species are syn-chronously correlated with positive cross peaks (Fig. 2a), whilebands arising from different species can be more clearly identifiedin the asynchronous spectrum (Fig. 2b). The d-toluene band at1390 cm�1 is obscured by the contributions from the neighboringMEK bands in the synchronous spectrum. The appearance of asyn-chronous cross peaks at (1390, 1360), for example, makes it possi-ble to separate the two species.

Fig. 9. (a) Synchronous and (b) asynchronous 2D correlation spectra of constructedfrom dynamic spectra positively projected onto the null-space of the MEK signal at1360 cm�1.

4.2. Vector projection

Fig. 3 shows the transformation of data by the vector projectionwith the spectral intensity variation signal at 1495 cm�1, which isdominated by the contribution from the semi-circle stretchingvibration of PS phenyl ring. Fig. 3a is the original dynamic(mean-centered) spectra of the solvent evaporation process ob-tained by subtracting the time-averaged spectrum as the referencefrom the raw spectra (Fig. 1a). The entire dynamic spectra are thenprojected onto the space spanned by the intensity variation at1495 cm�1 to yield the projected data (Fig. 3b).

While the contributions from PS are much more enhanced inthe newly obtained projected data, features assignable to MEKand d-toluene are still clearly visible. This retention of the partialsignals from MEK and d-toluene, even after the projection opera-tion, arises from the fact that the portions of the solvent signalsare anti-correlated with the PS signal. In other words, there is somelevel of similarity between the profile of PS signal at 1495 cm�1

and those of solvents, even though the directions of change areopposite to each other.

To eliminate the contribution from this anti-correlation effect,we apply the positive correlation trick to restrict the projectionoperation only to those signals changing parallel to the1495 cm�1 signal of PS in the same direction. Fig. 3c shows the re-sult of such positive projection at 1495 cm�1 applied to the dy-namic spectra. This time, the projected spectra contain onlydynamic signals of PS, which are changing in the same directionas the 1495 cm�1 signal.

Fig. 4a is the synchronous 2D correlation spectrum constructedfrom the projected data (Fig. 3b). The asynchronous spectrum isnot shown here, because signals retained in the single vector pro-jected dynamic spectra are all synchronized, so there will be noasynchronous correlation peaks. The simple projection operationcan accentuate the spectral features of PS in the 2D spectrum,but correlation peaks associated with solvents are still present.Fig. 4b shows the synchronous spectrum constructed from the pos-itively projected data (Fig. 3c). We observe that the 2D correlationspectrum is now substantially simplified, such that peaks appearonly at the band positions of PS without any additional contribu-tions from MEK or d-toluene.

Fig. 5 shows the positively projected dynamic spectra by usingthe intensity variations at 1390 and 1360 cm�1, which respectivelycorrespond to the decrease in d-toluene and MEK, as the projectors.


These newly constructed spectra essentially represent the spectralintensity variations of pure d-toluene and MEK. Fig. 6a and b showsthe corresponding synchronous spectra of the projected data. Corre-lation peaks in Fig. 6a are assignable exclusively to d-toluene, whilethose in Fig. 6b are for MEK. Fig. 6c is the synchronous spectrumbased on the dynamic spectra obtained by the positive projectionat 1420 cm�1, which is another MEK-dominated region. It is interest-ing to note that Fig. 6b and c are essentially indistinguishable,although the intensity profiles at 1360 and 1420 cm�1 are not iden-tical to each other, due to the different contributions from the over-lapped adjacent peaks of other components (see Fig. 1b).

The synchronous spectra constructed from the positively pro-jected dynamic spectra at 1495, 1390 and 1360 cm�1 (Figs. 4b,6a and b) correspond to the pure component synchronous 2D cor-relation spectra of individual species in the solution mixture. Bysuperposing these three synchronous spectra, one can actuallyreconstruct the positive portions of the synchronous spectrum

Fig. 10. Power spectra obtained from the original dynamic spectra of the mixture, after tand (c) PS.

generated from the original data (Fig. 7a). The auto-power spectra,obtained as the correlation intensities along the diagonal of syn-chronous correlation spectra (Figs. 4b, 6a and b, and 7a) are com-pared in Fig. 7b. The additive nature of the pure component 2Dspectra is clearly illustrated here.

4.3. Null-space projection

Projected spectral data generated so far by the single vectorprojection operation provide highly simplified pure componentsynchronous spectra which are useful for the identification ofbands arising from the specific species. On the other hand, nomeaningful asynchronous spectrum could be constructed, as allsignals after the projection operation are fully synchronized.Therefore, some of the key advantages of the 2D correlation spec-troscopy will be lost. Null-space projection operation offers verydifferent and more promising opportunities.

he positive null-space projection, and positive projection for (a) MEK, (b) d-toluene,

Fig. 11. (a) Synchronous spectrum constructed from data attenuated for the 50% ofMEK signals by the projection at 1360 cm�1. (b) Synchronous spectrum constructedfrom data augmented by doubling the d-toluene signals by the projection at1390 cm�1.


Fig. 8 shows the dynamic spectra obtained by the positive null-space projection (Eq. (27)) by using the intensity profiles at 1495,1390, and 1360 cm�1. They represent the spectral intensity varia-tions with the contribution of a specific species selectively re-moved from the rest. Thus, null-space projected spectra shown inFig. 8a have no spectral contribution from PS. Likewise, Fig. 8bhas no d-toluene effect, and Fig. 8c does not show any MEK signalcontributions.

Fig. 9 shows the 2D correlation spectra constructed from thenull-space projected data with MEK signals removed (Fig. 8c) byusing the profile at 1360 cm�1 as the masking projector. Both syn-chronous and asynchronous spectra are free from the interferingcontributions from strong MEK signals. Indeed, the signals fromd-toluene, which were somewhat obscured by the overlappingMEK signals in Fig. 2, are now much more clearly visible. It is alsopossible to construct 2D correlation spectra without the interfer-ence from PS or d-toluene from the other null-space projected data.

It is important to point out that the signs of correlation peaksretained after the null-space projection do not change, so the con-ventional sign rules used in the interpretation of 2D correlationspectra are still fully applicable to make the proper judgmentabout the relative directions and sequential order of intensitychanges. The obvious advantage of the null-space projection seemsto be the reduction of interferences from dominant species to pro-duce much simplified correlation spectra for the specific species ofinterest.

4.4. Attenuation and augmentation by projection

Unlike the simple projection operation, which extracts the sig-nals of only one species at a time, null-space projection retains sig-nals from multiple species except one. Fig. 10 shows the set ofpower spectra extracted from synchronous spectra constructedfrom the original data (Fig. 2a), which are compared to the powerspectra generated from projected and null-space projected databuilt upon profiles of PS, d-toluene, and MEK. It is apparent thatprojection and null-space projection operations play complemen-tary roles by distributing the spectral data information into twoseparate spaces, which are orthogonal to each other.

It is straightforward to obtain a weighted sum of the projectedspectra and null-space projected spectra to create a new set ofspectral data, as formally expressed in Eq. (9). Such an oblique pro-jection, accomplished by the linear combination of the two orthog-onal components of spectral data, can selectively attenuate oraugment the particular aspect of information contained in the ori-ginal data in a flexible and gradual manner. The advantage of theoblique projection is that none of the contributions from individualspecies will be completely lost, as in the case of positive projectionor null-space projection. Certain information sought in 2D correla-tion analysis, such as the relative direction of signal changes andthe sequential order of intensity variations, is thus fully preserved.Their retrieval is effectively assisted by attenuating the dominantand overwhelming contributions and augmenting the weak andobscured signals.

Fig. 11 shows two examples of the results from oblique projec-tion operations. In Fig. 11a, a projection-attenuated spectrum isshown, where 50% of the MEK signals were reduced by utilizingthe profile at 1360 cm�1 as the projector and setting the value ofthe attenuation parameter a to be 0.5. Compared to the originalsynchronous spectrum (Fig. 2a) without any projection treatment,Fig. 11a is substantially simplified by reducing the interfering con-tribution of MEK. The overlapped peaks of MEK and d-toluene areclearly separated. On the other hand, Fig. 11b is obtained by aug-menting the signal intensities of d-toluene by doubling their con-tribution by using the profile at 1390 cm�1 as the projector andsetting the value of the parameter a to be 2.0. This time, d-toluene

peaks, which were barely observable before, now become defi-nitely visible.

5. Conclusions

We introduced the technique called projection 2D correlationanalysis, which may potentially become a very useful tool in thesimplification of highly congested 2D correlation spectra oftenencountered in practice. This technique takes advantage of the factthat 2D correlation spectra can be expressed in terms of matrixmultiplications of spectral data. Therefore, one can apply standardmatrix-based mathematical operations, such as projection andnull-space projection, as an effective filtering tool. Spectral data


can then be transformed to generate new 2D correlation spectrawith much more simplified features.

Projection matrices used in this study can be generated fromvarious sources. Some of the potential sources include a part of ori-ginal spectral data, other spectra of different origin, perturbationvariables, or even arbitrarily chosen mathematical functions. Fur-thermore, it is possible to linearly combine the projected andnull-space projected spectra to achieve oblique projection. A num-ber of such oblique projections become available to graduallyattenuate or augment select features.

Projection 2D correlation analysis applied to the ATR IR study oftime-dependent compositional changes of a solution mixture dem-onstrated its ability to obtain pure component 2D spectra, 2D spec-tra based on contributions of a reduced number of species, andvarious 2D spectra with specific features selectively accentuated,

to assist the unambiguous interpretation of complex and highlyoverlapped spectral data.

References

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236A.[4] I. Noda, Y. Ozaki, Two-dimensional Correlation Spectroscopy—Applications in

Vibrational and Optical Spectroscopy, Wiley, Chichester, 2004.[5] I. Noda, J. Molec. Struct., this issue, doi:10.1016/j.molstruc.2009.11.048.[6] Y. Wu, I. Noda, Appl. Spectrosc. 61 (2007) 1040.[7] Y. Wu, I. Noda, J. Mol. Struct. 883–884 (2008) 149.[8] S. Shanmukh, R.A. Dluhy, J. Phys. Chem. A 108 (2004) 5625.[9] S. Shanmukh, R.A. Dluhy, Vib. Spectrosc. 36 (2004) 167.

projection two-dimensional correlation analysis

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