Scaling techniques to enhance two-dimensional correlation spectra

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Journal of Molecular Structure 883–884 (2008) 216–227

Scaling techniques to enhance two-dimensional correlation spectra

Isao Noda *

The Procter & Gamble Company, 8566 Beckett Road, West Chester, OH 45069, USA

Received 16 November 2007; received in revised form 19 December 2007; accepted 19 December 2007Available online 28 December 2007

Abstract

Scaling techniques to enhance two-dimensional (2D) correlation spectra are examined by using Raman spectra obtained during theprocess-monitoring of an emulsion polymerization. 2D correlation spectra without any scaling suffer from the effect of dominant peaksarising from a few bands with strong spectral intensity variations, which can obscure other small but significant spectral features. Unit-variance scaling, which generates a correlation coefficient and asynchronous disrelation coefficient spectrum, retains purely correlationalinformation while eliminating the magnitude effect of intensity variations. This scheme tends to amplify the effect of noise, especially inthe region with relatively small signals. No useful information is retained around the main diagonal of a synchronous spectrum. Theplaid appearance of correlation peaks in the contour map makes it difficult to determine band positions, if the neighboring peaks withthe same signs are merged. Pareto scaling, which uses the square root of standard deviation as the scaling factor, circumvents the ampli-fication of noise by retaining a small portion of magnitude information. Generalized form of Pareto scaling utilizes the scaling factorraised to a power anywhere between 0 and 1, with the optimal point often found near 0.8. Correlation enhancement of 2D spectra, whichis achieved by preferentially scaling the correlation coefficient or disrelation coefficient portion, tends to sharpen the demarcationbetween overlapped peaks. Minor features are also amplified by this technique in a manner similar to Pareto scaling. The Pareto scalingtechnique can be effectively combined with the correlation enhancement scaling to achieve the synergistic effect of the two techniques.� 2008 Elsevier B.V. All rights reserved.

Keywords: Two-dimensional correlation spectroscopy; Unit-variance scaling; Pareto scaling; Correlation enhancement; Process monitoring; Ramanspectroscopy

1. Introduction

Two-dimensional (2D) correlation spectroscopy is rou-tinely used today as a technique of choice in many prac-tical applications [1–3]. The progress in this field hasbeen discussed in a series of comprehensive reviews [4–7]. In 2D correlation spectroscopy, a system of interestis stimulated by an external perturbation, and variationsof spectral intensities induced by the applied perturbationare systematically examined by a simple cross correlationanalysis. The correlation intensities between a pair of sig-nals, varying either in-phase or out-of-phase with eachother, are then plotted as a function of two independentspectral axes, like wavenumbers, to generate 2D spectra.

0022-2860/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.molstruc.2007.12.026

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

By spreading peaks over the second dimension, the spec-tral resolution is often enhanced. Cross peaks appearingin 2D correlation spectra provide the information aboutsimilarities or differences among spectral intensity varia-tions. From the signs of cross peaks, one can determinethe relative directions of changes and the order ofsequential events reflected by the intensities of individualspectral bands [1,2].

The extraordinary sensitivity of 2D correlation spectros-copy to a very subtle change in spectral intensities oftenreveals features not readily observable in the original setof (1D) spectra. While this advantage of detecting fine spec-tral features is intrinsic to 2D correlation spectroscopy, itsometimes becomes desirable to further enhance the quali-tative and quantitative appearance of 2D correlation spec-tra by applying an additional mathematical treatment forthe correlation analysis. By transforming the 2D correla-

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I. Noda / Journal of Molecular Structure 883–884 (2008) 216–227 217

tion spectra into a new enhanced form, which is capable ofselectively emphasizing subtle but important small featuresof spectral data without amplifying the noise contribution,one can analyze the data more effectively. Fig. 1 shows anexample of such a transformation, where an ordinary 2DRaman correlation spectrum is dramatically enhanced bythe mathematical treatment called generalized scaling tech-nique. Small spectral features overshadowed by the intensecentral autopeak of the original 2D Raman spectrum(Fig. 1a) now become clearly visible in the new scaling-enhanced spectrum (Fig. 1b).

Scaling of data, if carried out judiciously, becomes apowerful and versatile technique to achieve the goal ofenhancing the quality of 2D spectra. There are many differ-ent ways to scale data, aimed at specific pretreatmentobjectives [8]. Unit-variance scaling or auto-scaling of 2Dcorrelation spectra is probably the best known scaling tech-nique to date [9–12]. The technique, leading to the con-

Fig. 1. Pseudo-three-dimensional fishnet plots of 2D Raman correlation spectremulsion polymerization of styrene and 1,3-butadiene: (a) 2D Raman correlatenhanced with the generalized scaling technique with the scaling constants set

struction of 2D correlation coefficient spectra, wasproposed in the past to eliminate the effect of the magni-tude of intensity variations by preserving only the purelycorrelational information of 2D spectra. This particularapproach turned out to be of somewhat limited utilitybecause of several major shortcomings, including theamplification of noise contributions in the resulting scaledspectra.

In this work, a useful and more robust scaling techniquecalled Pareto scaling [13–15] and its generalized form, aswell as the scaling technique designed for correlationenhancement [16], will be explored both theoretically andexperimentally. Raman spectra generated during the real-time process monitoring of an emulsion polymerizationreaction of styrene and 1,3-butadiene mixture [17] is usedto compare the performance of various scaling techniquesto demonstrate the strength and shortcomings of eachmethod.

a derived from Raman data collected during the reaction monitoring of anion spectrum without any scaling treatment and (b) 2D Raman spectrumto a = 0.5 (i.e., Pareto scaling) and b = 1.

218 I. Noda / Journal of Molecular Structure 883–884 (2008) 216–227

2. Scaling techniques

2.1. Covariance, disvariance, and 2D correlation spectra

Consider a measurement of spectral intensity yt(m) for asystem under the influence of an external perturbationcharacterized by a physical variable t. The variable t canbe, for example, the monitoring time for a chemical reac-tion. For a series of m sequentially collected spectral datayi(m) with a fixed increment along the external variable tj,where j = 1, 2, ...m, we define dynamic spectra ~yj(m) as

~yjðmÞ ¼ yjðmÞ � �yðmÞ: ð1Þ

Dynamic spectra simply reduce to the mean-centered spec-tra, if we select the reference spectrum �yðmÞ to be the aver-age spectrum given by

�yðmÞ ¼ 1

m

Xm

j¼1

yjðmÞ: ð2Þ

It is also possible to calculate the standard deviation ofspectral intensity variations observed at a selected wave-number m given by

rðmÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXm

j¼1

½yjðmÞ � �yðmÞ�2=ðm� 1Þ

vuut : ð3Þ

The synchronous and asynchronous correlation intensities,U(m1,m2) and W(m1,m2) between two spectral intensity varia-tions observed at m1 and m2 are given by [2,3]

Uðm1; m2Þ ¼1

m� 1

Xm

j¼1

~yjðm1Þ � ~yjðm2Þ ð4Þ

Wðm1; m2Þ ¼1

m� 1

Xm

j¼1

~yjðm1Þ �Xm

k¼1

N jk � ~ykðm2Þ; ð5Þ

where Njk is the the jth row and kth column element of theHilbert–Noda transformation matrix [18], defined as:

N jk ¼0 for j ¼ k

p=ðk � jÞ otherwise

�: ð6Þ

The synchronous correlation spectrum U(m1,m2) character-izes the in-phase or coincidental variations of a pair ofspectral intensities measured at wavenumbers m1, and m2,while the asynchronous spectrum W(m1,m2) characterizesthe out-of-phase variations along the external perturbationvariable t. Generalized 2D correlation spectra given byEqs. (4) and (5) correspond to the real and imaginary partof complex cross-correlation function [2,3]. As they containboth information about the magnitude and relative phaseof the spectral intensity variations, signal variations withlarge amplitudes tend to dominate the 2D correlation spec-tra, often obscuring the intricate details arising from rela-tively small signals [19]. This problem is especially seriousfor a synchronous spectrum, where the congestion ofstrong autopeaks near the main diagonal makes the distinc-tion of overlapped peaks difficult.

It is possible to make a direct connection between the2D correlation intensity at a given coordinate with awell-established statistical quantity. Given the mean-cen-tered nature of dynamic spectra, specifically set by Eqs.(1) and (2), the synchronous correlation intensity U(m1,m2)in Eq. (4) turns out to be equivalent to the statistical covari-ance between the spectral intensity variations measured atm1 and m2 [16,20]. The autopower spectrum, i.e., the corre-lation intensity along the main diagonal of a synchronousspectrum, corresponds to the continuous distribution ofvariance values along m, as long as the average spectrumis selected as the reference. Likewise, the asynchronous cor-relation intensity W(m1,m2) is viewed as a special form ofcovariance between the spectral intensity variations at m1

and the quadrature (i.e., 90 degrees out of phase alongthe external variable t) component of the intensity varia-tions measured at m2. This type of covariance is called asyn-

chronous disvariance.

2.2. Unit-variance scaling

The standard deviations of spectral intensities measuredat m1 and m2 are related to the synchronous correlationintensities by rðm1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUðm1; m1Þ

pand rðm2Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUðm2; m2Þ

p.

We use the product of the two standard deviations[r(m1)�r(m2)], sometimes referred to as the total joint vari-

ance [21], as the scaling factor to obtain the expressionfor the unit-variance scaled form of 2D spectra.

qðm1; m2Þ ¼ Uðm1; m2Þ=½rðm1Þ � rðm2Þ� ð7Þfðm1; m2Þ ¼ Wðm1; m2Þ=½rðm1Þ � rðm2Þ� ð8Þ

Here, q(m1,m2) is the unit-variance scaled synchronous 2Dcorrelation spectrum, which is equivalent to the spectrumof (Pearson’s second moment) correlation coefficient be-tween spectral intensity variations measured at m1 and m2.Similarly, f(m1,m2) is the corresponding 2D spectrum ofasynchronous disrelation coefficient. The same results canbe obtained, if the individual mean-centered dynamic spec-trum is scaled first by the standard deviation, and then thestandard 2D correlation analysis is applied to the scaleddata. Such a scaling operation of data is usually calledauto-scaling. It should be pointed out that the asynchro-nous disrelation coefficient f(m1,m2) defined by Eq. (8) isdifferent from the (total) disrelation coefficient n(m1,m2),

given by jnðm1; m2Þj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� qðm1; m2Þ2

q, which is often useful

in the analysis of data collected without the knowledge ofthe order of sampling [16].

The unit-variance auto-scaling operation providespurely correlational information, and consequently usefulrelative phase relationship among signals. The idea of utiliz-ing the statistical 2D correlation coefficient plot was firstproposed by Barton et al. [9] and later popularized by Sasicet al. [11,12] In comparison with the more conventionalcovariance-based synchronous 2D correlation spectrum(Eq. (4)), 2D correlation coefficient spectrum scaled bythe product of standard deviations (Eq. (7)) was initially


believed to have an advantage that the magnitude effect ofintensity variations, which vary from wavenumber to wave-number, can be effectively suppressed. Thus, only thepurely correlational portion of information not influencedby the intensity effect is obtained.

Unfortunately, it has been found that auto-scaling oper-ation tends to greatly exaggerate the noise, especially in thespectral region with low signal amplitude, where the noiseis amplified by the division with a small standard deviationvalue. Furthermore, no useful information is retainedaround the main diagonal of the 2D correlation coefficientspectrum, as the correlation intensity values are all scaledto unity. Peaks appearing in the 2D correlation coefficientand disrelation coefficient spectra have the characteristicplaid or patched tile like appearance without distinct localmaxima or minima. These problems clearly limit some ofthe utilities of unit-variance scaled 2D correlation spectrawhen used by themselves. Unit-variance scaled 2D spectraare typically constructed in conjunction with conventional2D spectra without any scaling treatment to compensatethe mutual shortcomings [22–24].

2.3. Pareto scaling

Vilfredo Pareto (1848–1923) is an influential Italianeconomist, well known for the famous Pareto principle,also known as the heuristic 80–20 rule. It was postulatedby Pareto that roughly 80% of the consequence often stemsfrom 20% of the cause. Thus, 80% of income in Italy wasfound to be received by about 20% of Italian population.Other examples include, 80% of the significant work isdone by only 20% of workers in any group, 80% of perti-nent information is found in the 20% of spectral features,and so on. Pareto’s other famous intellectual contributionsinclude the concept of Pareto optimality, i.e, a move tomake one individual in a group better off without makingany other in the same group worse off, as well as Pareto

chart and Pareto distribution used extensively in economicsand statistics field.

The concept of Pareto scaling was first introduced bySvante Wold in 1993, who also coined the term to describethis surprisingly useful data pretreatment technique, inhonor of the famous Italian economist [13]. The Pareto-scaling operation is characterized by the scaling or dividingof dataset by the square root of its standard deviation [13–15]. It is contrasted to the more conventional unit-variance(Pearson) scaling operation, where dataset is simply scaledby the standard deviation itself [8]. Pareto scaling providesan interesting opportunity to manipulate a dataset for 2Dcorrelation analysis in order to emphasize important butsubtle features, often obscured by the dominant spectralvariations, without penalties commonly associated withunit-variance scaling.

The inspection of the previously derived relationship(Eqs. (7) and (8)) readily shows that 2D correlation coeffi-cient and disrelation coefficient spectrum are nothing but ascaled form of the conventional 2D correlation spectrum

U(m1,m2) and W(m1,m2) with the total joint variance[r(m1)�r(m2)] as a scaling factor. Obviously other forms ofscaling factors may also be used instead. We now definethe Pareto-scaled 2D correlation spectra as:

Uðm1; m2ÞPareto ¼ Uðm1; m2Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðm1Þ � rðm2Þ

pð9Þ

Wðm1; m2ÞPareto ¼ Wðm1; m2Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðm1Þ � rðm2Þ

pð10Þ

by choosing the square root of the total joint variance asthe new scaling factor. The same result can also be ob-tained if the 2D correlation spectra are directly calculatedfrom dynamic spectra scaled by the square root of the stan-dard deviation.

Scaling by the square root of standard deviation will notcompletely remove the effect of the amplitude of signal var-iation, but it will provide a reasonable balance of contribu-tions from high and low amplitude signals [8]. Unlike thecase of unit-variance scaling, Pareto scaling does not seemto appreciably amplify noise. It retains the presence of dis-cernible autopeaks along the diagonal in the scaled syn-chronous 2D spectrum, and cross peaks in Pareto-scaled2D spectra have distinct local maxima or minima. It hasalso been noted that details of minor peaks become muchmore visible in Pareto-scaled spectra. This unexpectedincrease of the apparent spectral peak visibility probablyis one of the more interesting features of Pareto-scaled2D correlation spectra. The signs of cross peaks in bothsynchronous and asynchronous spectrum do not change.Thus, the so-called Noda’s rules to interpret sign relationsstill remain applicable to determine the sequential order ofevents encoded within the set of spectral data [2,3]. The glo-bal phase angle, obtained from the arctangent of the ratiobetween the asynchronous and synchronous correlationintensity [16,19], is also unchanged by the scaling opera-tion, so the quantitative phase relationship among signalscan be deduced.

2.4. Generalized scaling

The simple rearrangement of terms in Eq. (7) and (8)reveals that 2D correlation spectra, U(m1,m2) and W(m1,m2),may be viewed as the mathematical product between thetotal joint variance [r(m1)�r(m2)] and either correlation coef-ficient q(m1,m2) or asynchronous disrelation coefficientf(m1,m2). The total joint variance represents the magnitudeof signal variations, while correlation coefficient and disre-lation coefficient, respectively, represent the degree of syn-chronicity and asynchronicity between signals. Based onthe above, it is straightforward to propose a generalizedscaling form for 2D correlation spectra.

Uðm1; m2ÞðScaledÞ ¼ Uðm1; m2Þ � ½rðm1Þ � rðm2Þ��a � jqðm1; m2Þjb ð11ÞWðm1; m2ÞðScaledÞ ¼ Wðm1; m2Þ � ½rðm1Þ � rðm2Þ��a � jfðm1; m2Þjb ð12Þ

U(m1,m2)(Scaled) and W(m1,m2)(Scaled) are the scaled forms ofsynchronous and asynchronous correlation spectrum, and


a and b are the scaling constants for the generalized scalingoperation.

If the values of scaling constants are set to be a = 0 andb = 0, then the scaled 2D spectra become identical to theordinary form of 2D correlation spectra (Eqs. (4) and(5)). If the values of the scaling constants are set to bea = 1 and b = 0, then the scaled 2D correlation spectrareduce to the correlation coefficient spectrum q(m1,m2) andasynchronous disrelation coefficient spectrum f(m1,m2)obtained by the unit-variance scaling (Eqs. (7) and (8)).Pareto-scaled 2D correlation spectra (Eqs. (9) and (10))are obtained by setting the scaling constants as a = 0.5and b = 0.

Obviously, the value of a can be set to any arbitrarynumber other than 0.5, preferably between 0 and 1. Closerthe chosen value of a is to 0, the stronger the influence ofthe dominant signals with large amplitude of variationsbecomes. On the other hand, as the value of a approachesto 1, the effect of signal amplitude is gradually diminished,but the unwanted amplification of noise contribution to 2Dcorrelation spectra may become more prominent. Thus, thescaling constant a may be used as a convenient adjustableparameter in search of the optimal point, where the balancebetween the desired visualization of fine features of 2D cor-relation spectra and suppression of the noise amplificationare achieved.

This practice of selecting an arbitrary value of a at will,while keeping the condition b = 0, may be viewed as anextension of the classical Pareto scaling technique origi-nally proposed by Svante Wold [13], where a is fixed tothe value of 0.5, to a more flexible functional form. Thebasic motivation for choosing an alternative scaling factorother than the unit variance in the new generalized Paretoscaling is the same. However, it has been found for manycases that the value of a can be set much greater than 0.5to effectively emphasize the fine features of 2D correlationspectra without risking undesirable effects, such as theamplification of noise. Although there is no scientific basisfor the selection of a particular value for a, experience hasshown that the optimal value of a in practical applicationsis often found around 0.8. In other words, the majority ofdesired advantages of the unit-variance scaling (say about80%) can be delivered without noise amplification by main-taining only a relatively small amount (maybe about 20%)of the variance magnitude information. It is humorous thatthis empirically found range of a value happens to coincidewith the Pareto principle of 80–20 rule!

2.5. Correlation enhancement factor

The other scaling constant b is sometimes referred to asthe correlation enhancement factor. The idea of scaling thecorrelation coefficient to amplify the correlational informa-tion in 2D correlation analysis was originally introduced byNoda during the First International Symposium on Two-Dimensional Correlation Spectroscopy (2DCOS-1) heldin Kobe-Sanda, Japan in August 1999 [16]. The concept

was later revisited by Isakson, who multiplied the covari-ance value with the absolute value of the correlation coef-ficient raised by a positive integer [25]. Scaling of thedisrelation coefficient to enhance the intrinsically strongdiscriminating power of asynchronous spectrum has notbeen explored.

The correlation enhancement factor b does not have tobe an integer, but it will be definitely more useful if it isset to be greater than �1, more preferably a positive num-ber. For example, by setting the value of this scaling con-stant as b = �1, we completely lose the importantcorrelational aspect of the information, and both 2D corre-lation spectra simply reduce to the total joint variance[r(m1)�r(m2)] representing only the magnitude of signal vari-ations. Ordinary 2D correlation spectra (Eqs. (4) and (5))are obtained under the condition of b = 0. It is, however,much more interesting to generate a new class of scaled2D correlation spectra with enhanced level of correlationalfeatures by selecting the value of b to be greater than 0.

By setting b > 0, synchronous 2D correlation spectrumwill be dominated only by autopeaks along the main diag-onal and a few select cross peaks representing almost com-pletely synchronized signals, i.e., q(m1,m2) � ±1. For a verylarge value of b, the absence of perfectly coincidental signalvariations (i.e., |q(m1,m2)|<1) will make the term |q(m1, m2)|b

in Eq. (11) so small that cross peak intensity will be sub-stantially diminished even for slightly asynchronous sig-nals. Thus, a synchronous 2D correlation spectrum whichis scaled with a high b value becomes much more selective,emphasizing only the correlations among truly synchro-nized spectral variations.

The similar effect is observed for the b-scaled asynchro-nous 2D correlation spectrum. However, in this case crosspeaks corresponding to signal pairs with higher levels ofasynchronicity are selectively preserved instead, while thosechanging more or less coincidentally are eliminated. Thiseffect often results in highly enhanced spectral resolutionof overlapped bands. Fine spectral features not observableeven in the conventional asynchronous 2D correlationspectrum start to emerge, as the value for correlationenhancement factor b is increased. In practice, even a mod-erate value of b (e.g., 1 or 2) can still profoundly enhancethe correlational features. Excessively high value of b actu-ally is not recommended, as it will exaggerate the contribu-tions of noise. We now apply these different scalingtechniques discussed so far to a set of experimentallyobtained Raman spectra generated during the processmonitoring of a polymerization reaction to explore thepractical merit and shortcomings of each method.

3. Experimental

3.1. Materials

Styrene and 1,3-butadiene monomers were, respectively,received from Sigma–Aldrich, inc. and Matheson Tri-Gas,Inc. A free radical initiator 2,20-azobis(2-amidinopropane)


dihydrochloride (V-50) was supplied by Wako USA, Inc.An oligomeric nonionic surfactant, oleyl ethoxylate withabout 20 ethoxylate unit (Volpo-20), was received fromCroda, Inc. They were all used as received without furtherpurification. The composition of the reaction mixture is ini-tially set such that 700 parts water, 40 parts styrene, 60parts 1,3-butaduene, 5 parts oleyl ethoxylate, and 1 partinitiator are all mixed together.

3.2. Method

The reaction mixture was placed in a sealed glass reactorquipped with a heater and agitator. A series of Ramanspectra were collected during the emulsion polymerizationof styrene and butadiene charged within a laboratory-scalereactor (Reaction Calorimeter RC1 Classic, MettlerToledo) coupled with a Raman probe (Raman ReactionMonitor, Kaiser Optical Systems). The reaction tempera-ture was set to 50 �C, and the polymerization reactionwas continuously monitored for 2 h.

4. Results

4.1. 2D Raman correlation spectra

Raman spectra obtained during the monitoring of anemulsion polymerization reaction of styrene and 1,3-buta-diene are used to demonstrate the performance of differentscaling techniques [17]. Fig. 2 shows the set of Raman spec-

Fig. 2. Raman spectra collected during the reaction monitoring of anemulsion polymerization of styrene and 1,3-butadiene.

tra collected every 20 min for the first 120 min. The contin-uous decrease of Raman intensity around 1638 cm�1,which corresponds to the vinyl C@C stretching mode ofstyrene and butadiene monomers, and the correspondinggradual build-up of the intensities around 1582 and1668 cm�1 reflecting the formation of styrene-butadienerubber (SBR) are clearly observed. Additional fine featuresarising from the contributions of monomers and polymerare also present.

Fig. 3 shows the counter map representation of the syn-chronous and asynchronous 2D correlation spectra directlycalculated from the raw data presented in Fig. 2 by usingEqs. (4) and (5). The average spectrum (Eq. (2)) is placedas a reference at the top and side of 2D spectra. The

Fig. 3. Contour map representation of ordinary 2D Raman correlationspectra without any scaling operation: (a) synchronous spectrum and (b)asynchronous spectrum.

Fig. 4. Ordinary 2D Raman correlation spectra with a very low contourline threshold: (a) synchronous spectrum and (b) asynchronous spectrum.


pseudo-three-dimensional fishnet representation of the syn-chronous 2D Raman spectrum corresponding to the coun-ter map in Fig. 3a has already been shown in Fig. 1a. Theminimum counter line threshold is set to be one-tenth ofthe maximum correlation intensity of the 2D map. The syn-chronous spectrum (Fig. 3a) is dominated by the intenseautopeak around 1638 cm�1, obscuring the rest of fine fea-tures in this region of the correlation spectrum. Crosspeaks at 1600 and 1668 cm�1 are barely visible, and otherautopeaks and cross peaks do not even show up in the2D map at this level of contour line threshold. Having aband with a large magnitude of intensity variations createsthe problem in simultaneously displaying the fine featuresof bands exhibiting only small amount of intensityvariations.

The corresponding asynchronous spectrum (Fig. 3b)shows cross peaks indicating the clear split of the1638 cm�1 peak into two separate bands at 1632 and1640 cm�1. These two bands represent, respectively, thecontribution from styrene and 1,3-butadiene. The intensi-ties of the two cross peaks are very strong, such that otherasynchronous cross peaks are somewhat obscured,although the styrene peak around 1600 cm�1 and theSBR peak around 1668 cm�1 are still visible. Some of thesmall but important features clearly observable in the origi-nal Raman spectra shown in Fig. 2 do not appear in the 2Dspectra.

By substantially lowering the minimum threshold levelof contour lines, it is possible to see much more details of2D correlation spectra. Fig. 4 shows the same 2D correla-tion spectra now plotted with the contour line thresholdlevel set to be one-hundredth of the maximum correlationintensity value. Minor cross peaks, as well as some of thesmall autopeaks, now become visible in the synchronousspectrum (Fig. 4a). However, the correlation spectra arefilled with unwanted features arising from the contributionof noise to make the determination of true cross peaksmuch more difficult. This problem is especially true forthe asynchronous spectrum (Fig. 4b), where pertinent crosspeaks are surrounded by the noise-induced features. Theapparent spectral resolution of 2D correlation spectra can-not be easily improved much by merely adjusting the con-tour line threshold.

4.2. 2D Raman spectra with unit-variance scaling

Fig. 5 shows the unit-variance scaled 2D correlationspectra, i.e., the correlation coefficient spectrum and asyn-chronous disrelation coefficient spectrum, obtained fromthe auto-scaled experimental Raman spectra by usingEqs. (7) and (8). Such fully scaled correlation coefficientspectra were suppose to provide the advantage of purelycorrelational information not obscured by the effect ofthe magnitude of signal variations. Consequently, thescaled spectra should be better suited for identifying signalcontributions from relatively small spectral intensity varia-tions. Unfortunately, it is often the case that the desired

advantage of auto-scaled 2D spectra is overshadowed bythe serious problems associated with this type of scalingtechnique, such as noise amplification and shape distortionof correlation peaks.

It is clear from Fig. 5 that spectral features in the 2Dcorrelation coefficient or asynchronous disrelation coeffi-cient map created from the experimental Raman data arevirtually impossible to interpret. The entire spectral planeof the unit-variance scaled synchronous spectrum(Fig. 5a) is essentially all covered with numerous correla-tion coefficient ‘‘peaks” with the intensities nearing either+1 or �1. Many of them actually arise from the fortuitoussynchronization of small noise contributions and baselinefluctuations, which are grossly amplified by the scalingoperation. There is no simple way of distinguishing para-

Fig. 5. Unit-variance scaled 2D Raman spectra: (a) correlation coefficientspectrum and (b) asynchronous disrelation coefficient spectrum.


sitic correlation coefficient peaks from actual correlationpeaks arising from the significant spectral intensity varia-tions, if the correlation peak intensities are all scaled tonearly the same value. Individual correlation peaks havethe patched-tile-like appearance of small squares with noapparent local maxima or minima, making the identifica-tion of the center point of peaks difficult. Neighboringpeaks with the same signs also tend to merge into a contin-uous rectangle without clear demarcation, thus compro-mising the resolution of the 2D spectrum.

The situation becomes even worse for the unit-variancescaled 2D asynchronous spectrum, or disrelation coefficientplot (Fig. 5b). Noise tends to have much more asynchro-nous elements, which is apparent in the development of

numerous asynchronous cross peaks all scaled to a rela-tively large value. The hopeless congestion of the unit-var-iance scaled asynchronous spectrum is especially acute inthe region with little spectral signals. Again, the task of dif-ferentiating the real correlation peak from the noise-induced artifacts becomes a major challenge, if the scaledpeak intensities all become very similar.

So far, the unit-variance or auto-scaled 2D correlationspectra may seem useless in the practical analysis of real-world spectra with inevitable inclusion of noise effect.However, it should be pointed out that the constructionof unit-variance scaled 2D correlation spectra actuallyhas some significant merit when used in conjunction withthe conventional 2D correlation spectra. The feature ofhighlighting the correlation intensity signs everywherewithin the scaled 2D spectra becomes especially useful,when some of minor synchronous correlation peaks areobscured by a few intense peaks of high magnitude withouta proper scaling operation.

For example, we have identified several correlationpeaks carrying pertinent information in Fig. 3. With thespecific knowledge of spectral coordinates for select corre-lation peaks, one can determine the corresponding intensi-ties and signs of the correlation and disrelation coefficientby examining the same coordinate position in Fig. 5. Theso-called Noda’s rules for determining the sequentialorders of events still apply with the scaled 2D spectra.Thus, one can definitively conclude from Fig. 5 that theRaman intensity change around 1600 cm�1 attributed tothe consumption of styrene monomer occurs after thataround 1640 cm�1 assigned to the reaction of 1,3-butadiene.

4.3. 2D Raman spectra with Pareto scaling

Fig. 6 shows the Pareto-scaled 2D Raman spectraobtained by using Eqs. (9) and (10), or alternatively fromEqs. (11) and (12) with the scaling constants set asa = 0.5 and b = 0. It is apparent that much more detailedfeatures of the correlation spectra are now visible. The Par-eto-scaled synchronous spectrum (Fig. 6a) shows crosspeaks at 1578 and 1584 cm�1 attributable to styrene mono-mer and SBR polymer. They were not visible in Fig. 3a,because of the intense dominant autopeak around1638 cm�1 obscuring the fine features of the spectrum.The same is true for the asynchronous spectrum(Fig. 6b). By scaling down the dominant asynchronouscross peaks at 1632 and 1640 cm�1, other minor crosspeaks become much more noticeable.

The comparison between the ordinary 2D Raman spec-tra with a low contour line threshold (Fig. 4) and Pareto-scaled 2D Raman spectra (Fig. 6) is instructive. It is possi-ble to highlight fine features of 2D Raman correlation spec-tra by simply lowering the contour line level to a very lowvalue. This operation, however, will also pick up the contri-bution of noise which produces unwanted parasitic featuresin the 2D map. The selection of a proper contour line

Fig. 6. Pareto-scaled 2D Raman spectra: (a) synchronous spectrum and(b) asynchronous spectrum.


threshold seems to be a balancing act between two compet-ing objectives of the detection of pertinent peaks and therejection of unwanted peaks. On the other hand, the Paretoscaling approach seems to accomplish the objective ofsmall peak detection without significantly sacrificing theother important need to keep the noise effect from becom-ing a problem. This interesting feature of Pareto-scalingtechnique is actually in accordance with the objective ofso-called Pareto optimality principle.

Fig. 7 shows the result of the generalized form of Paretoscaling applied to the construction of a synchronous 2DRaman correlation spectrum. In the generalized Paretoscaling, the scaling factor a in Eqs. (11) and (12) is nolonger fixed to the value of 0.5 as in Fig. 6a, but insteadallowed to take an arbitrary value between 0 and 1. In

Fig. 7a–c, the a value is chosen, respectively, to be 0.7,0.8 and 0.9. We actually already have two more additionalexamples of generalized Pareto scaling. One is the 2D cor-relation coefficient spectrum (Fig. 5a) where a is set to 1,and the other is the ordinary 2D Raman correlation spec-trum (Fig. 3a) where a is 0.

It is clear by comparing these Pareto-scaled synchronous2D correlation spectra that, as the scaling factor a isincreased gradually from 0, much more detailed featuresof 2D synchronous spectrum become visible. The influenceof the overwhelming predominance of the intense autopeakat 1638 cm�1 is reduced, and other smaller peaks becomenoticeable. In addition to the cross peaks, which emergedat 1578 and 1584 cm�1 for styrene monomer and SBRpolymer in the classical Pareto scaled 2D spectrum(Fig. 6a), many more cross peaks attributable to eithermonomers or polymer became identifiable, as the valueof the scaling factor a is increased above 0.5 (Fig. 7).

The optimal range of the useful a value to detect asmany correlation peaks as possible in this example proba-bly is around 0.8 (Fig. 7b). Indeed, a dozen or so distinctRaman bands arising from different constituents of thepolymerization reaction system can be identified. By thetime the a is increased to 0.9 (Fig. 7c), the appearance ofthe generalized Pareto-scaled 2D spectrum starts resem-bling that of the unit-variance scaled correlation coefficientplot (Fig. 5a). Characteristic problems, such as the exagger-ation of noise contributions in the spectral region of lowsignals and merging of the neighboring correlation peakshaving the same sign, makes it difficult to identify pertinentRaman bands, as the a value becomes higher than 0.8.

4.4. 2D Raman spectra with correlation enhancement

Fig. 8 shows the correlation enhanced 2D Raman spec-tra obtained by using Eqs. (11) and (12). The correlationenhancement factor b is now set to be 1, while the general-ized Pareto scaling factor a is kept at 0. This particularform of correlation enhanced 2D spectra may be viewedas the point-by-point or pair-wise product (i.e., equivalentto the so-called Hadamard products in matrix multiplica-tion) between the ordinary 2D Raman correlation spectra(Fig. 3) and their unit-variance scaled counterparts(Fig. 4). Thus, the individual synchronous correlationintensity is multiplied by the correlation coefficient at thesame coordinate, while asynchronous correlation intensityis multiplied by the disrelation coefficient.

The correlation enhancement operation shows an inter-esting outcome. The demarcation between overlapped syn-chronous correlation peaks tends to become much sharper(Fig. 8a) compared to the original synchronous spectrum(Fig. 3a). Thus, a cluster of peaks connected by ridgesare now fragmented into distinct peaks separated by clearboundaries. This sharpening of correlation peak bound-aries occurs because the region of a spectrum having mixedcontributions from different band signals tends to have alower absolute correlation coefficient value compared to

Fig. 7. Generalized Pareto scaling applied to synchronous 2D Raman spectrum with varying scaling constant: (a) a = 0.7, (b) a = 0.8, and (c) a = 0.9.


the pure signal region. Thus, within the overlapped portionof Raman peaks, the section with about the same amountof contributions from two different bands is actually pref-erentially eliminated first, when the correlation coefficientis raised by a positive power b value.

A similar effect is observed for the asynchronous spec-trum (Fig. 8b). Correlation peak clusters connected bylocalized ridges in Fig. 3b are again separated into individ-ual peaks with distinct boundaries. Interestingly, the corre-lation enhanced asynchronous spectrum shows moredetailed features arising from smaller signal intensity vari-ations than the original Raman spectra. The influence ofthe disrelation coefficient term in Eq. (12) is amplified bythe scaling and playing much greater role than that ofthe total joint variance term in determining the appearanceof the scaled asynchronous spectrum. The consequence of

this effect is somewhat analogous to the Pareto scaling,where the influence of the total joint variance is decreasedby the scaling operation.

4.5. Combination of pareto scaling and correlation

enhancement

We have so far examined two different types of scalingtechniques to enhance the appearance of 2D Raman corre-lation spectra. Pareto scaling brings out the detailed fea-tures of 2D correlation spectra by reducing theoverwhelming effect of a few dominant intense correlationpeaks in the display. Correlation-enhancement scaling, onthe other hand, separates the overlapped peaks by prefer-entially emphasizing spectral intensity variation signalswhich are more synchronously or asynchronously corre-

Fig. 8. Correlation enhancement applied to 2D Raman spectra withb = 1: (a) synchronous spectrum and (b) asynchronous spectrum.

Fig. 9. 2D Raman spectra subjected to generalized scaling treatment witha = 0.5 (i.e., Pareto scaling) and b = 1: (a) synchronous spectrum and (b)asynchronous spectrum.


lated. It is also possible to combine these two scaling tech-niques to further enhance the appearance and utility of 2Dcorrelation spectra by taking advantage of the synergisticeffect of the two scaling techniques.

Fig. 9 shows the scaled 2D Raman spectra obtained byusing Eqs. (11) and (12) with values of the two scaling con-stants set to a = 0.5 and b = 1. The scaled 2D Raman cor-relation spectra thus obtained are equivalent to theHadamard (i.e., coordinate-by-coordinate) productsbetween the unit-variance scaled 2D Raman correlationcoefficient or disrelation coefficient spectrum (Fig. 4) andthe corresponding Pareto-scaled spectra (Fig. 5). The emer-gence of finer features in 2D spectra from bands with smal-ler signals and isolation of previously connected correlation

peak clusters into individual peaks are both clearlyobserved.

By revealing surprisingly detailed spectral features with-out noise amplification, the advantage of scaling operationson the improvement of 2D correlation spectra becomes mostapparent when the appropriately scaled 2D Raman spectra(Fig. 9) are directly compared with the original 2D Ramancorrelation spectra obtained without any scaling treatment(Fig. 3). The side-by-side comparison of the synchronous2D Raman spectra in the pseudo-three-dimensional fishnetrepresentation (Fig. 1) probably is even more dramatic inshowcasing the difference. The difference in the detailed fea-tures of the two 2D Raman correlation spectra is quite obvi-ous. Without the scaling operation, the synchronous 2DRaman correlation spectrum is dominated by the most


intense autopeak around 1638 cm�1. Other small but impor-tant features arising from bands with weaker spectral inten-sity variation signals are difficult to detect. However, oncethe scaling operation is applied, fine correlational featuresbecome visible all over the spectral region. Numerous corre-lation peaks emerge in the spectral region away from the1638 cm�1 autopeak, while the neighboring overlapped peakclusters now become clearly distinguishable as separatepeaks with boundaries. Thus, the power and utility of gener-alized scaling technique to enhance two-dimensional corre-lation spectra is demonstrated.

5. Conclusions

Various scaling techniques to enhance two-dimensionalcorrelation spectra have been explored, both theoreticallyand with actual experimental data based on the emulsionpolymerization reaction of styrene and 1,3-butadiene, mon-itored with Raman spectroscopy. 2D correlation spectrawithout any scaling treatment often suffer from the effectof intense dominant correlation peaks arising from a fewbands with strong spectral intensity variations, which canobscure relatively small but important fine features of 2Dcorrelation spectra. Neighboring correlation peaks areoften agglutinated to clusters by ridges and saddles to makethe differentiation of individual peaks difficult.

Unit-variance scaling or auto-scaling operation, leadingto the generation of 2D correlation coefficient and disrela-tion coefficient spectrum, successfully eliminates the effectof magnitude of intensity variations completely by retain-ing only the purely correlational (i.e., phase) information.Unfortunately, this scheme also tends to grossly amplifythe effect of noise contributions, especially in the spectralregion with relatively small signals. No useful informationis retained around the main diagonal of the synchronousspectrum, since the correlation intensity is all normalizedto unity. The characteristic plaid or patched-tile likeappearance of correlation peaks in the contour maps ofauto-scaled 2D correlation coefficient or disrelation coeffi-cient spectra makes the determination of the band positiondifficult, especially if the neighboring peaks with the samesigns are merged into one for such spectra.

Pareto-scaled 2D correlation spectra, on the other hand,provide an excellent alternative to the unit-variance scaledspectra, by scaling the dataset by the square roots of the stan-dard deviations. The Pareto scaling approach seems to elim-inate the most obvious shortcomings of the unit-variancescaling by retaining a small portion of the magnitude infor-mation of the spectral intensity variations. The unwantedamplification of noise contributions by unit-variance scalingis effectively circumvented. By relaxing the constraint of theclassical Pareto scaling, where the scaling factor is set to 0.5,the generalized form of Pareto scaling can utilize the scalingfactor raised to a power anywhere between 0 and 1, with theoptimal point often found near 0.8.

Correlation enhancement of 2D spectra is achieved byscaling the correlation or disrelation coefficient of 2D corre-

lation spectra. The operation tends to sharpen the demarca-tion between overlapped correlation peaks. Individual peaksare clearly separated with distinct boundaries, instead offorming a cluster of peaks connected by ridges. Minor fea-tures are also amplified by this technique in a manner similarto the Pareto scaling procedure. Finally, the generalized Par-eto scaling technique can be simultaneously combined withthe correlation enhancement scaling to achieve the synergis-tic effect of the two scaling techniques.

Acknowledgement

The author thanks Professor Svante Wold of UmeaUniversity for providing the pertinent literature and guid-ance on the Pareto scaling concept.

References

[1] I. Noda, Appl. Spectrosc. 47 (1993) 1329.[2] I. Noda, A.E. Dowrey, C. Marcott, G.M. Story, Y. Ozaki, Appl.

Spectrosc. 54 (2000) 236A.[3] I. Noda, Y. Ozaki, Two-dimensional Correlation Spectroscopy—

Applications in Vibrational and Optical Spectroscopy, Wiley, Chich-ester, 2004.

[4] I. Noda, J. Molec. Struct. 883-884 (2008) 26.[5] I. Noda, J. Mol. Struct. 799 (2006) 2.[6] I. Noda, Vib. Spectrosc. 36 (2004) 143.[7] I. Noda, in: Y. Ozaki, I. Noda (Eds.), Two-Dimensional Correlation

Spectroscopy, AIP Press, Melville, 2000, pp. 3–17.[8] R.A. van den Berg, H.C.J. Hoefsloot, J.A. Westerhuis, A.K. Smilde,

M.J. van den Werf, BMC Genomics 7 (2006) 1.[9] F.E. Barton II, D.S. Himmelsbach, J.H. Duckworth, M.J. Smith,

Appl. Spectrosc. 46 (1992) 420.[10] P. Malkavaara, R. Alen, E. Kolehmainen, J. Chem. Inf. Comput. Sci.

40 (2000) 438.[11] S. Sasic, Y. Ozaki, Anal. Chem. 73 (2001) 2294.[12] S. Sasic, J.-H. Jiang, Y. Ozaki, Hemom. Intel. Lab. Syst. 65 (2003) 1.[13] S. Wold, E. Johansson, M. Cocchi, in: H. Kubinyi (Ed.), 3D QSAR

in Drug Design; Theory, Methods and Applications, ESCOM SciencePublishers, Leiden, 1993, pp. 523–550.

[14] L. Eriksson, J. Jaworska, A.P. Worth, M.T.D. Cronin, R.M.McDowell, P. Gramatica, Environ. Health Perspect. 111 (2003)1361.

[15] M.R. Pears, J.D. Cooper, H.M. Mitchison, R.J. Mortishire-Smith,D.A. Pearce, J.L. Griffin, J. Biol. Chem. 280 (2005) 42508.

[16] I. Noda, in: Y. Ozaki, I. Noda (Eds.), Two-Dimensional CorrelationSpectroscopy, AIP Press, Melville, 2000, pp. 201–204.

[17] I. Noda, W.M. Allen, S.E. Lindberg, Presented at the Federation ofAnalytical Chemistry and Spectroscopy Societies (FACSS) andSociety for Applied Spectroscopy (SAS) National Meeting, LakeBuena Vista, FL, 25 September 2006.

[18] I. Noda, Appl. Spectrosc. 54 (2000) 994.[19] S. Morita, Y. Ozaki, I. Noda, Appl. Spectrosc. 55 (2001) 1618.[20] C. Marcott, G.M. Story, A.E. Dowrey, I. Noda, in: C.L. Wilkins

(Ed.), Computer-Enhanced Analytical Spectroscopy, vol.4, Plenum,New York, 1993, pp. 237–255,.

[21] I. Noda, Vib. Spectrosc. 36 (2004) 261.[22] F.E. Barton II, D.S. Himmelsbach, A.M. McClung, E.L. Cham-

pagne, Cereal Chem. 79 (2002) 143.[23] Z.W. Yu, L. Chen, S.Q. Sun, I. Noda, J. Phys. Chem. A 106 (2002) 6683.[24] A. Jirasek, G. Schulze, M.W. Blades, R.F.B. Turner, Appl. Spectrosc.

57 (2003) 1551.[25] T. Isaksson, Vib. Spectrosc. 36 (2004) 251.

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Scaling techniques to enhance two-dimensional correlation spectra