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Journal of Molecular Structure 883–884 (2008) 216–227 www.elsevier.com/locate/molstruc
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 2007 Available online 28 December 2007
Abstract Scaling techniques to enhance two-dimensional (2D) correlation spectra are examined by using Raman spectra obtained during the process-monitoring of an emulsion polymerization. 2D correlation spectra without any scaling suffer from the effect of dominant peaks arising from a few bands with strong spectral intensity variations, which can obscure other small but significant spectral features. Unitvariance scaling, which generates a correlation coefficient and asynchronous disrelation coefficient spectrum, retains purely correlational information while eliminating the magnitude effect of intensity variations. This scheme tends to amplify the effect of noise, especially in the region with relatively small signals. No useful information is retained around the main diagonal of a synchronous spectrum. The plaid appearance of correlation peaks in the contour map makes it difficult to determine band positions, if the neighboring peaks with the same signs are merged. Pareto scaling, which uses the square root of standard deviation as the scaling factor, circumvents the amplification of noise by retaining a small portion of magnitude information. Generalized form of Pareto scaling utilizes the scaling factor raised to a power anywhere between 0 and 1, with the optimal point often found near 0.8. Correlation enhancement of 2D spectra, which is achieved by preferentially scaling the correlation coefficient or disrelation coefficient portion, tends to sharpen the demarcation between overlapped peaks. Minor features are also amplified by this technique in a manner similar to Pareto scaling. The Pareto scaling technique can be effectively combined with the correlation enhancement scaling to achieve the synergistic effect of the two techniques. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Two-dimensional correlation spectroscopy; Unit-variance scaling; Pareto scaling; Correlation enhancement; Process monitoring; Raman spectroscopy
1. Introduction Two-dimensional (2D) correlation spectroscopy is routinely used today as a technique of choice in many practical applications [1–3]. The progress in this field has been discussed in a series of comprehensive reviews [4– 7]. In 2D correlation spectroscopy, a system of interest is stimulated by an external perturbation, and variations of spectral intensities induced by the applied perturbation are systematically examined by a simple cross correlation analysis. The correlation intensities between a pair of signals, varying either in-phase or out-of-phase with each other, are then plotted as a function of two independent spectral axes, like wavenumbers, to generate 2D spectra. *
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By spreading peaks over the second dimension, the spectral resolution is often enhanced. Cross peaks appearing in 2D correlation spectra provide the information about similarities or differences among spectral intensity variations. From the signs of cross peaks, one can determine the relative directions of changes and the order of sequential events reflected by the intensities of individual spectral bands [1,2]. The extraordinary sensitivity of 2D correlation spectroscopy to a very subtle change in spectral intensities often reveals features not readily observable in the original set of (1D) spectra. While this advantage of detecting fine spectral features is intrinsic to 2D correlation spectroscopy, it sometimes becomes desirable to further enhance the qualitative and quantitative appearance of 2D correlation spectra by applying an additional mathematical treatment for the correlation analysis. By transforming the 2D correla-
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tion spectra into a new enhanced form, which is capable of selectively emphasizing subtle but important small features of spectral data without amplifying the noise contribution, one can analyze the data more effectively. Fig. 1 shows an example of such a transformation, where an ordinary 2D Raman correlation spectrum is dramatically enhanced by the mathematical treatment called generalized scaling technique. Small spectral features overshadowed by the intense central autopeak of the original 2D Raman spectrum (Fig. 1a) now become clearly visible in the new scalingenhanced spectrum (Fig. 1b). Scaling of data, if carried out judiciously, becomes a powerful and versatile technique to achieve the goal of enhancing the quality of 2D spectra. There are many different ways to scale data, aimed at specific pretreatment objectives [8]. Unit-variance scaling or auto-scaling of 2D correlation spectra is probably the best known scaling technique to date [9–12]. The technique, leading to the con-
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struction of 2D correlation coefficient spectra, was proposed in the past to eliminate the effect of the magnitude of intensity variations by preserving only the purely correlational information of 2D spectra. This particular approach turned out to be of somewhat limited utility because of several major shortcomings, including the amplification of noise contributions in the resulting scaled spectra. In this work, a useful and more robust scaling technique called Pareto scaling [13–15] and its generalized form, as well as the scaling technique designed for correlation enhancement [16], will be explored both theoretically and experimentally. Raman spectra generated during the realtime process monitoring of an emulsion polymerization reaction of styrene and 1,3-butadiene mixture [17] is used to compare the performance of various scaling techniques to demonstrate the strength and shortcomings of each method.
Fig. 1. Pseudo-three-dimensional fishnet plots of 2D Raman correlation spectra derived from Raman data collected during the reaction monitoring of an emulsion polymerization of styrene and 1,3-butadiene: (a) 2D Raman correlation spectrum without any scaling treatment and (b) 2D Raman spectrum enhanced with the generalized scaling technique with the scaling constants set to a = 0.5 (i.e., Pareto scaling) and b = 1.
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2. Scaling techniques 2.1. Covariance, disvariance, and 2D correlation spectra Consider a measurement of spectral intensity yt(m) for a system under the influence of an external perturbation characterized by a physical variable t. The variable t can be, for example, the monitoring time for a chemical reaction. For a series of m sequentially collected spectral data yi(m) with a fixed increment along the external variable tj, where j = 1, 2, ... m, we define dynamic spectra ~y j (m) as ~y j ðmÞ ¼ y j ðmÞ y ðmÞ:
ð1Þ
Dynamic spectra simply reduce to the mean-centered spectra, if we select the reference spectrum y ðmÞ to be the average spectrum given by m 1 X y ðmÞ ¼ y ðmÞ: ð2Þ m j¼1 j It is also possible to calculate the standard deviation of spectral intensity variations observed at a selected wavenumber m given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u m 2 ð3Þ rðmÞ ¼ t ½y j ðmÞ y ðmÞ =ðm 1Þ: j¼1
The synchronous and asynchronous correlation intensities, U(m1, m2) and W(m1, m2) between two spectral intensity variations observed at m1 and m2 are given by [2,3] Uðm1 ; m2 Þ ¼ Wðm1 ; m2 Þ ¼
1 m1
m X
~y j ðm1 Þ ~y j ðm2 Þ
ð4Þ
m m X 1 X ~y j ðm1 Þ N jk ~y k ðm2 Þ; m 1 j¼1 k¼1
ð5Þ
j¼1
where Njk is the the jth row and kth column element of the Hilbert–Noda transformation matrix [18], defined as: 0 for j ¼ k N jk ¼ : ð6Þ p=ðk jÞ otherwise The synchronous correlation spectrum U(m1, m2) characterizes the in-phase or coincidental variations of a pair of spectral intensities measured at wavenumbers m1, and m2, while the asynchronous spectrum W(m1, m2) characterizes the out-of-phase variations along the external perturbation variable t. Generalized 2D correlation spectra given by Eqs. (4) and (5) correspond to the real and imaginary part of complex cross-correlation function [2,3]. As they contain both information about the magnitude and relative phase of the spectral intensity variations, signal variations with large amplitudes tend to dominate the 2D correlation spectra, often obscuring the intricate details arising from relatively small signals [19]. This problem is especially serious for a synchronous spectrum, where the congestion of strong autopeaks near the main diagonal makes the distinction of overlapped peaks difficult.
It is possible to make a direct connection between the 2D correlation intensity at a given coordinate with a well-established statistical quantity. Given the mean-centered 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 covariance between the spectral intensity variations measured at m1 and m2 [16,20]. The autopower spectrum, i.e., the correlation intensity along the main diagonal of a synchronous spectrum, corresponds to the continuous distribution of variance values along m, as long as the average spectrum is selected as the reference. Likewise, the asynchronous correlation intensity W(m1, m2) is viewed as a special form of covariance between the spectral intensity variations at m1 and the quadrature (i.e., 90 degrees out of phase along the external variable t) component of the intensity variations measured at m2. This type of covariance is called asynchronous disvariance. 2.2. Unit-variance scaling The standard deviations of spectral intensities measured at m1 and m2 are related to theffi synchronous p correlation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensities by rðm1 Þ ¼ Uðm1 ; m1 Þ and rðm2 Þ ¼ Uðm2 ; m2 Þ. We use the product of the two standard deviations [r(m1)r(m2)], sometimes referred to as the total joint variance [21], as the scaling factor to obtain the expression for the unit-variance scaled form of 2D spectra. qðm1 ; m2 Þ ¼ Uðm1 ; m2 Þ=½rðm1 Þ rðm2 Þ
ð7Þ
fðm1 ; m2 Þ ¼ Wðm1 ; m2 Þ=½rðm1 Þ rðm2 Þ
ð8Þ
Here, q(m1, m2) is the unit-variance scaled synchronous 2D correlation spectrum, which is equivalent to the spectrum of (Pearson’s second moment) correlation coefficient between spectral intensity variations measured at m1 and m2. Similarly, f(m1, m2) is the corresponding 2D spectrum of asynchronous disrelation coefficient. The same results can be obtained, if the individual mean-centered dynamic spectrum is scaled first by the standard deviation, and then the standard 2D correlation analysis is applied to the scaled data. Such a scaling operation of data is usually called auto-scaling. It should be pointed out that the asynchronous disrelation coefficient f(m1, m2) defined by Eq. (8) is different from the (total) disrelation coefficient n(m1, m2), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 given by jnðm1 ; m2 Þj ¼ 1 qðm1 ; m2 Þ , which is often useful in the analysis of data collected without the knowledge of the order of sampling [16]. The unit-variance auto-scaling operation provides purely correlational information, and consequently useful relative phase relationship among signals. The idea of utilizing the statistical 2D correlation coefficient plot was first proposed by Barton et al. [9] and later popularized by Sˇasˇic´ et al. [11,12] In comparison with the more conventional covariance-based synchronous 2D correlation spectrum (Eq. (4)), 2D correlation coefficient spectrum scaled by the product of standard deviations (Eq. (7)) was initially
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believed to have an advantage that the magnitude effect of intensity variations, which vary from wavenumber to wavenumber, can be effectively suppressed. Thus, only the purely correlational portion of information not influenced by the intensity effect is obtained. Unfortunately, it has been found that auto-scaling operation tends to greatly exaggerate the noise, especially in the spectral region with low signal amplitude, where the noise is amplified by the division with a small standard deviation value. Furthermore, no useful information is retained around the main diagonal of the 2D correlation coefficient spectrum, as the correlation intensity values are all scaled to unity. Peaks appearing in the 2D correlation coefficient and disrelation coefficient spectra have the characteristic plaid or patched tile like appearance without distinct local maxima or minima. These problems clearly limit some of the utilities of unit-variance scaled 2D correlation spectra when used by themselves. Unit-variance scaled 2D spectra are typically constructed in conjunction with conventional 2D spectra without any scaling treatment to compensate the mutual shortcomings [22–24]. 2.3. Pareto scaling Vilfredo Pareto (1848–1923) is an influential Italian economist, well known for the famous Pareto principle, also known as the heuristic 80–20 rule. It was postulated by Pareto that roughly 80% of the consequence often stems from 20% of the cause. Thus, 80% of income in Italy was found to be received by about 20% of Italian population. Other examples include, 80% of the significant work is done by only 20% of workers in any group, 80% of pertinent information is found in the 20% of spectral features, and so on. Pareto’s other famous intellectual contributions include the concept of Pareto optimality, i.e, a move to make one individual in a group better off without making any other in the same group worse off, as well as Pareto chart and Pareto distribution used extensively in economics and statistics field. The concept of Pareto scaling was first introduced by Svante Wold in 1993, who also coined the term to describe this surprisingly useful data pretreatment technique, in honor of the famous Italian economist [13]. The Paretoscaling operation is characterized by the scaling or dividing of 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 scaled by the standard deviation itself [8]. Pareto scaling provides an interesting opportunity to manipulate a dataset for 2D correlation analysis in order to emphasize important but subtle features, often obscured by the dominant spectral variations, without penalties commonly associated with unit-variance scaling. The inspection of the previously derived relationship (Eqs. (7) and (8)) readily shows that 2D correlation coefficient and disrelation coefficient spectrum are nothing but a scaled form of the conventional 2D correlation spectrum
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U(m1, m2) and W(m1, m2) with the total joint variance [r(m1)r(m2)] as a scaling factor. Obviously other forms of scaling factors may also be used instead. We now define the Pareto-scaled 2D correlation spectra as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ Uðm1 ; m2 ÞPareto ¼ Uðm1 ; m2 Þ= rðm1 Þ rðm2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pareto Wðm1 ; m2 Þ ¼ Wðm1 ; m2 Þ= rðm1 Þ rðm2 Þ ð10Þ by choosing the square root of the total joint variance as the new scaling factor. The same result can also be obtained if the 2D correlation spectra are directly calculated from dynamic spectra scaled by the square root of the standard deviation. Scaling by the square root of standard deviation will not completely remove the effect of the amplitude of signal variation, but it will provide a reasonable balance of contributions from high and low amplitude signals [8]. Unlike the case of unit-variance scaling, Pareto scaling does not seem to appreciably amplify noise. It retains the presence of discernible autopeaks along the diagonal in the scaled synchronous 2D spectrum, and cross peaks in Pareto-scaled 2D spectra have distinct local maxima or minima. It has also been noted that details of minor peaks become much more visible in Pareto-scaled spectra. This unexpected increase of the apparent spectral peak visibility probably is one of the more interesting features of Pareto-scaled 2D correlation spectra. The signs of cross peaks in both synchronous and asynchronous spectrum do not change. Thus, the so-called Noda’s rules to interpret sign relations still remain applicable to determine the sequential order of events encoded within the set of spectral data [2,3]. The global phase angle, obtained from the arctangent of the ratio between the asynchronous and synchronous correlation intensity [16,19], is also unchanged by the scaling operation, so the quantitative phase relationship among signals can 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 the total joint variance [r(m1)r(m2)] and either correlation coefficient q(m1, m2) or asynchronous disrelation coefficient f(m1, m2). The total joint variance represents the magnitude of signal variations, while correlation coefficient and disrelation coefficient, respectively, represent the degree of synchronicity and asynchronicity between signals. Based on the above, it is straightforward to propose a generalized scaling form for 2D correlation spectra. Uðm1 ; m2 Þ
ðScaledÞ
¼ Uðm1 ; m2 Þ ½rðm1 Þ rðm2 Þ
ðScaledÞ
¼ Wðm1 ; m2 Þ ½rðm1 Þ rðm2 Þ
Wðm1 ; m2 Þ
a a
b
jqðm1 ; m2 Þj ð11Þ b
jfðm1 ; m2 Þj ð12Þ
U(m1, m2)(Scaled) and W(m1, m2)(Scaled) are the scaled forms of synchronous and asynchronous correlation spectrum, and
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a and b are the scaling constants for the generalized scaling operation. If the values of scaling constants are set to be a = 0 and b = 0, then the scaled 2D spectra become identical to the ordinary form of 2D correlation spectra (Eqs. (4) and (5)). If the values of the scaling constants are set to be a = 1 and b = 0, then the scaled 2D correlation spectra reduce to the correlation coefficient spectrum q(m1, m2) and asynchronous 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 = 0.5 and b = 0. Obviously, the value of a can be set to any arbitrary number other than 0.5, preferably between 0 and 1. Closer the chosen value of a is to 0, the stronger the influence of the dominant signals with large amplitude of variations becomes. On the other hand, as the value of a approaches to 1, the effect of signal amplitude is gradually diminished, but the unwanted amplification of noise contribution to 2D correlation spectra may become more prominent. Thus, the scaling constant a may be used as a convenient adjustable parameter in search of the optimal point, where the balance between the desired visualization of fine features of 2D correlation spectra and suppression of the noise amplification are achieved. This practice of selecting an arbitrary value of a at will, while keeping the condition b = 0, may be viewed as an extension of the classical Pareto scaling technique originally proposed by Svante Wold [13], where a is fixed to the value of 0.5, to a more flexible functional form. The basic motivation for choosing an alternative scaling factor other than the unit variance in the new generalized Pareto scaling is the same. However, it has been found for many cases that the value of a can be set much greater than 0.5 to effectively emphasize the fine features of 2D correlation spectra without risking undesirable effects, such as the amplification of noise. Although there is no scientific basis for the selection of a particular value for a, experience has shown that the optimal value of a in practical applications is often found around 0.8. In other words, the majority of desired advantages of the unit-variance scaling (say about 80%) can be delivered without noise amplification by maintaining only a relatively small amount (maybe about 20%) of the variance magnitude information. It is humorous that this empirically found range of a value happens to coincide with the Pareto principle of 80–20 rule! 2.5. Correlation enhancement factor The other scaling constant b is sometimes referred to as the correlation enhancement factor. The idea of scaling the correlation coefficient to amplify the correlational information in 2D correlation analysis was originally introduced by Noda during the First International Symposium on TwoDimensional Correlation Spectroscopy (2DCOS-1) held in Kobe-Sanda, Japan in August 1999 [16]. The concept
was later revisited by Isakson, who multiplied the covariance value with the absolute value of the correlation coefficient raised by a positive integer [25]. Scaling of the disrelation coefficient to enhance the intrinsically strong discriminating power of asynchronous spectrum has not been explored. The correlation enhancement factor b does not have to be an integer, but it will be definitely more useful if it is set to be greater than 1, more preferably a positive number. For example, by setting the value of this scaling constant as b = 1, we completely lose the important correlational aspect of the information, and both 2D correlation spectra simply reduce to the total joint variance [r(m1)r(m2)] representing only the magnitude of signal variations. 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 scaled 2D correlation spectra with enhanced level of correlational features by selecting the value of b to be greater than 0. By setting b > 0, synchronous 2D correlation spectrum will be dominated only by autopeaks along the main diagonal and a few select cross peaks representing almost completely synchronized signals, i.e., q(m1, m2) ±1. For a very large value of b, the absence of perfectly coincidental signal variations (i.e., |q(m1, m2)|
