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If we decrease the number of PCs below three, more changes are observed for the resulting 2D correlation spectra. Eventually, the 2D correlation spectra display the feature as if the system is behaving more or less synchronously. We can clearly create less noisy 2D spectra by simply using a smaller number of PCs. However, since we are discarding some real information from the raw data,
Chemometrics and 2D Correlation Spectroscopy
Synchronous 2D IR Correlation Spectrum Asynchronous 2D IR Correlation Spectrum
Figure 6.7 2D correlation spectra derived from the PCA-reconstructed data shown in Figure 6.5(b): (A) synchronous spectrum; (B) asynchronous spectrum
we may experience a certain loss of subtle feature from 2D correlation spectra. It is noted that the interpretation of such simplified 2D spectra is often much easier. In the extreme, the asynchronicity of dynamic spectra is preserved even for a data matrix comprising only two principal components. This result is consistent with the expectation from the basic 2D correlation theory, which requires only the existence of nominal nonlinearity in the sample dimension to exhibit detectable asynchronicity.
6.3.6 EIGENVALUE MANIPULATING TRANSFORMATION (EMT)
We introduced the idea of using a reconstructed data matrix from scores and loading vectors obtained by applying PCA to raw data before 2D correlation analysis. The 2D correlation analysis of such a reconstructed data matrix should more effectively accentuate the most important features of synchronicity and asyn-chronicity without being hampered by noise or insignificant minor components. We now explore another PCA-based chemometrics technique useful for 2D correlation analysis, known as eigenvalue manipulation transformation (EMT).24,25
It is well known that the matrix product L = WT W, obtained from the PCA score W, is a diagonal matrix where each diagonal element corresponds to the eigenvalue of a principal component. The eigenvalues are all positive real numbers and arranged in descending order of magnitude. It is often useful to express the score matrix in the form W = US. Here the square matrix S = L1/2 is another diagonal matrix, where the diagonal elements are now the positive square roots of eigenvalues, or singular values. The orthonormal matrix U can be obtained from W by normalizing each column of W. Alternatively, the matrix U can be obtained
Additional Developments in Two-dimensional Correlation Spectroscopy
directly from the original data matrix A, since columns of U correspond to the eigenvectors of the so-called association matrix AAT. It is noted here that the association matrix is proportional to the sample-sample correlation synchronous 2D correlation spectrum.
By combining the above results, the reconstructed data matrix A* can now be expressed in the familiar form known as the singular value decomposition (SVD)12-14,
A* = USVT (6.23)
As we discussed earlier, the separation of the original data A into a significant portion of the data A* and residual noise E is somewhat arbitrary, and determined by choosing exactly how many eigenvectors one would like to use to reconstruct A* to represent the original data A. One can, for example, employ a very large number of eigenvectors to reconstruct A*, but such additional eigenvectors often tend to contain no useful information but noise, and the corresponding eigenvalues are very small.
The singular values are arranged in the order of their size, so the first singular value, located at the upper left corner of S, is the largest. As we move down the rows and columns of S, the diagonal elements become smaller. If we replace smaller diagonal elements of S beyond a select point with zero, the contribution of the corresponding eigenvectors from U and V to the reconstruction of A* disappears. In other words, replacement of minor eigenvalues or singular values with zero is equivalent to the truncation of minor factors consisting mainly of noise contribution.
The above realization that the commonly practiced PCA-based noise truncation can be regarded as an intentional manipulation or replacement of certain (in this case, smaller magnitude) eigenvalues, leads to the idea designed to further transform a data set to extract useful information. The concept of eigenvalue manipulation transformation (EMT) of spectral data involves the systematic substitution of individual eigenvalues associated with the original data set. This process will generate a new set of transformed data with a very different emphasis placed on specific information content.
For our first attempt of producing a new reconstructed data matrix by EMT, we calculate a modified diagonal matrix Sq where the diagonal elements are now given by raising the corresponding singular values from S by the power of q, where q is a real number. Thus, the new data matrix will be obtained by
A** = USq VT (6.24)
We can then construct a new 2D correlation spectra based on this new transformed data matrix A** obtained by replacing eigenvalues according to Equation (6.24) instead of the usual PCA-reconstructed data matrix A*.