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<I>SS = -YYT (5.4)
n — 1
The form of the covariance matrix yields a new type of 2D synchronous correlation spectrum with samples on both axes. Each point in the 2D map represents a correlation between a given pair of sample traces, such as those measured at different concentration or temperature. The covariance matrix given by Equation (5.4) is well suited for analyzing features characteristic to samples, such as concentration dynamics of components, while the previous form given by Equation (5.2) is more convenient for elucidating detailed spectral features. The two types of covariance matrices, thus, represent different but complementary aspects of spectral data and their dependence on the external perturbation.
Further Expansion of Generalized 2D Correlation Spectroscopy
Closer examination of the structure of association product Y YT used in sample-sample correlation provides the insight into how this scheme actually works. Each element of the association product matrix is obtained by calculating the dot (inner) product of two spectral vectors, i.e., different rows of Y. In other words, the sample-sample correlation effectively carries out the detection of similarity or collinearity of two spectra. The synchronous sample-sample correlation intensity becomes significant only if the spectra of two samples are similar in shape. If, on the other hand, the spectral shapes of two samples are substantially
different, $ss becomes small. Thus, the underlying concept of synchronous sam-
ple-sample correlation is essentially identical to the classical spectral matching method based on the calculation of inner products of spectral vectors.8 Such a comparison method has been extensively used in spectral library searches.
It is obviously tempting to set the corresponding asynchronous sample-sample correlation spectrum in the form
*ss = rYNvYT (5.5)
n — 1
The n x n matrix Nv performs the orthogonal transformation of spectral traces in the wavenumber domain in a manner similar to the Hilbert-Noda transformation matrix N. The above equation, at least purely from a mathematical viewpoint, seems like a reasonable extension of the relationships already provided in Equations (5.2)-(5.4). However, unlike the well-established asynchronous wavenumber-wavenumber correlation case, the indiscriminate use of asynchronous sample-sample correlation can potentially be a problematic exercise.
The physical significance of asynchronous sample-sample correlation intensities, especially those generated from a set of optical spectra, is unfortunately not established. In asynchronous sample-sample correlation, one spectrum is converted to the orthogonal form by calculating the Hilbert transform along the wavenumber axis before being cross-correlated against another spectrum. The Hilbert transformation of an absorbance spectrum, which is also commonly known as the Kramers-Kronig transformation in optical spectroscopy, merely produces the corresponding refractive index spectrum. Simplistic comparison of absorbance and refractive index by direct cross-correlation obviously does not carry much interpretable information, other than an indication if two spectra compared are different. For example, Wu et al. reported that asynchronous sample-sample correlation spectra they generated for their data did not show any regular feature except fluctuations.9
The possible saving utility of asynchronous sample-sample correlation may lie in the fact that asynchronous correlation is often sensitive to the dissimilarity between two spectral traces belonging to different samples. Thus, one can expect the value of ^ss to become small if two spectral traces measured for different samples are nearly matched. The mismatch of spectral patterns usually generates
Sample-Sample Correlation Spectroscopy
some intensity for Ô^. Unlike the conventional variable-variable 2D correlation, however, one cannot analyze the sequential order of events directly from the sign of Ô^ and Ô^. The sign of Ô^ simply indicates which side of the spectral region the mismatch resides for different sample traces. Finally, the fundamental constraint imposed on the structure of dynamic spectra (Equation 2.1), which makes the link between the Hilbert transformation (Equation 2.48) and the generalized 2D correlation function (Equation 2.5) possible via the Wiener-Khintchine theorem, actually does not apply to the wavenumber domain, as implicitly assumed in Equation (5.5). Thus, the mathematical interpretation of Ô^, at least in the form given by Equation (5.5), remains ambiguous from the view point of classical correlation analysis.
In order to emphasize the true dissimilarity of spectral traces associated with individual samples, it is probably more fruitful to consider the construction of sample-sample correlation spectra based on disrelation analysis. The sample-sample disrelation matrix Ass is constructed such that the ith row and jth column element of the disrelation is given by