Reconstructed data matrix

Proving From the properties of wavelet packet transformation and wavelet packet reconstruction we will see that if the wavelet packet coefficients are not treated, the wavelet packet reconstructed signals are the same as the original signals. When all principle components are reserved, the reconstructed data matrix is the same with the original data matrix. Done. 

A reconstruction of the original data matrix A is computed by using the preselected number of principal components (i.e., columns in our T and V matrices) as... 

Spectra taken during the Auger sputter profile were transformed into a data matrix and treated by the PHI-MATLAB , version 4.0, software. Each element was treated separately, that is the number of matrices was the same as the number of significant elements, and each matrix was completely independent of the others. The number of factors used for reconstruction was the one predicted by the minimum of the indicator function (IND). Unless specified, no mathematical treatment was performed on the data (differentiation, smoothing). 

Malinowski introduced an indicator function (IND) as a criterion to define the minimum number of eigenvalues, and therefore the number of eigenvectors, which are necessary to reconstruct the original data matrix. 

Knowing R, it is a simple matter to reconstruct the concentration profiles by including the scores over die entire data matrix as above, and similarly the spectra. 

Principle components regression (PCR) is one of the supervised methods commonly employed to analyze NMR data. This method is typically used for developing a quantitative model. In simple terms, PCR can be thought of as PCA followed by a regression step. In PCR, the scores matrix (T) obtained in PCA (Section 3.1) is related to an external variable in a least squares sense. Recall that the data matrix can be reconstructed or estimated using a limited number of factors (/ffact), such that only the fc = Mfaet PCA loadings (l fc) are required to describe the data matrix. Eq. (15) can be reconstructed as... 

After creating the concentration profiles (Figure 6), an absorption profile was generated fi-om the pure component spectra using eq 9, and the spectral data matrix was reconstructed as a linear combination of the component spectra. The best fit was obtained using the simplest mechanism, A - B C, and was taken as the plausible mechanism. Using this mechanism and the extinction profiles, absorption spectra at different time points were calculated (Figure 7). TTie SVD derived kinetic parameters were compared with those obtained (7) from the... 

In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

Result of reconstruction is a 3D matrix of output data assigned with the values of the local density inside elementary volumes. The ways of obtaining the 3D matrix of output data can be various. They are determined by the structure of tomographic system and chosen way of collected data processing. 

The graphs in Figure 5-23 are convincing. The top panel displays the original data for a simple first order reaction A—>B. The next panel shows the same data after the addition of a substantial amount of noise. The third panel features the reconstructed matrix Y = USV with 2 eigenvectors. Clearly a substantial amount, but not all, of the noise, was removed. 

The A-matrix can be reconstructed from the PCA scores, T. Usually, only a few PCs are used (the maximum number is the minimum of n and m), corresponding to the main structure of the data. This results in an approximated A-matrix with reduced noise (Figure 3.3). If all possible PCs would be used, the error (residual) matrix E would be zero. 

The procedure was repeated for the density matrix matrix of unpolarized excited states. In Fig. 3 we display = O p (f) / ,0 = 0) 2 for j = 134, derived from the data of Fig. 1, at different times. As in the pure case of Figs. 2, our procedure is able to prefectly reconstruct the true density-matrix at all times considered. 

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