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R(i, j) =
'T.xjyi - (E j (Ey;)
([«Exj2 - (Exj)2] [nEyf2 - (E y;)2])
The subscripts i and j are the column indices of the X and Y matrices ranging from 1 to nx and 1 to ny, respectively. They also indicate the corresponding element in the R matrix of correlation coefficients. Therefore, the calculated matrix element R(i, j) represents the correlation coefficient of the IR responses
Statistical 2D Correlation Spectroscopy
at wavenumber i with the NIR responses at wavelength j. One can construct the correlation coefficient spectrum of all IR wavenumbers with the individual NIR wavelength at index i by extracting a single row i from the R matrix. Likewise, by extracting a single column j from the R matrix one can obtain the correlation spectrum of all NIR wavelengths with the individual IR wavenumber at index j.
Figure 7.1 shows an example of a 2D contour map plot of the NIR versus mid-IR against the coefficient of determination (r2), generated from the spectra of complex agricultural samples that differ in wax (cuticle), carbohydrate, protein, and lignin content.21 One-dimensional IR and NIR spectra of the samples are placed on the top and right side of the contour map, respectively. A high r2 value at a point of the map indicates the presence of a strong correlation between mid-IR and NIR band intensities. It is noted that r2 at 2130 nm (0.5) increases across the OH and CH stretching band regions of the IR spectra from about 3700 cm-1 to a maximum at 2915-2850 cm-1; then it decreases down to the minimum (0.1) around 1850cm-1. Although the OH stretching bands are broader and stronger in the spectra (see the top spectrum in Figure 7.1), the C-H stretching bands are correlated more intensely. The fingerprint region of the IR spectra yields similar results, but the overall correlations are smaller below
Wavenumber (cm 1)
Figure 7.1 Contour map plot of the NIR versus mid-IR against the coefficient of determination (R2). The numerals on the contours are R2 rounded to the nearest tenth. The map in this figure is a broad-range map which depicts the general shape of the correlation over the entire regions (NIR and IR). The number contours in this case is 5, which shows the effects without appearing overly ‘busy’. The R2 values for the contours are 0.1, 0.3, 0.5, 0.7 and 0.9, respectively (Reproduced with permission from F.E. Barton II et al, Appl. Spectrosc., 46, 420 (1992) (Ref. 21). Copyright (1992) Society for Applied Spectroscopy.)
Other Types of Two-dimensional Spectroscopy
1500 cm-1. The NIR regions where most correlation activity is seen are 1385, 1730, and 2200-2450 nm. The IR regions correlating with the above regions are the 2900-2700 cm-1, 1700-1500 cm-1, and 1100-800 cm-1 regions. These IR regions correspond to those for CH stretching band region, C=O and C=C stretching, and C-C, C-O, and C-N stretching band region, respectively.
In 2D correlation spectroscopy, slice spectra play important roles in interpreting the correlations between the two spectral regions.21 The slices, obtained by holding a wavelength in one region constant, and letting the wavelengths in the other vary, can give a lot of information about the relationship between an absorber in one region and many absorbers in another. Barton II et at.21 employed the CH stretching modes of the ‘waxy’ material in the samples as the most prominent example to show how the correlation patterns from one region can be used to interpret the other region of the spectrum. Figure 7.2(A) and (B) shows NIR and mid IR spectra of beeswax, respectively.21 The three bands at 2307, 1726, and 1396 nm in the NIR spectrum are assigned to the CH combination mode, the first overtone of the CH stretching mode, and its second overtone. These bands have the major correlations to the IR bands at 2919 and 2854 cm-1 due to the CH stretching modes, as shown in the correlation slices in Figure 7.3(A) and (B).21 The correlation pattern is virtually identical for the two IR slices, as it is for the three NIR slices (Figure 7.4(A), (B), and (C)). It is clear from Figure 7.4 that the patterns at all three wavelengths, 1390, 1729, and 2312 nm, respectively, are identical to the major correlations at 2919 and 2850 cm-1.
This type of 2D correlation spectroscopy has two advantages. First, based on an NIR correlation slice of a spectrum in another region (for example, a mid-IR spectrum), one can examine which component in a sample contributes to a particular NIR band. Second, 2D correlation spectroscopy assists one to develop or interpret a chemometrics model. With the aid of an IR correlation slice of an NIR spectrum, it may be possible to predict useful wavelengths for a chemometrics model. This 2D approach has been used to construct NIR and Raman heterocorrelation 2D maps as well as NIR and IR ones. This type of 2D correlation spectroscopy does not have an asynchronous spectrum, so that one cannot investigate the dynamic behavior of spectral changes.
7.2.2 STATISTICAL 2D CORRELATION BY SaSiC AND OZAKI