Eigenvector reconstruction

There are other methods for noise reduction such as wavelets, eigenvector reconstruction, and artificial neural networks (ANN) (22,23). 

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. 

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. 

Rank annihilation methods employ eigenvalue-eigenvector analyses for direct determination of analyte concentration with or without intrinsic profile determination. With the exception of rank annihilation factor analysis, these methods obtain a direct, noniterative solution by solving various reconstructions of the generalized eigenvalue-eigenvector problem. 

Figure 3.22 shows the three abstract eigenvectors (the columns of Urc(l Sied) that were used to reproduce the spectra. Clearly they do not correspond to the spectra of the species Bj B3. Rather, the first eigenvector (solid line) represents an average of the whole spectral series and is positive throughout. The second (dotted line) has one node and allows for a first correction in reconstructing the individual spectra and the last (dashed line,... 

The appropriate SVD-derived spectral and temporal eigenvectors were selected and the temporal vectors were modeled. Ideally, the temporal vectors are the kinetic traces of individual components, each one being associated with a spectrum of a pure component Le., the spectral vector). Once the temporal vectors had been modeled the pure component spectra were reconstructed as a function of the pre-exponential multiplier obtained from the analysis, SVD determined spectral eigenvectors, and the corresponding eigenvalues. After the spectra of the component species were determined, the extinction profile was calculated and used along with the calculated decay times to construct a linear combination of the pure component species contributions to the observed... 

The variation spectra are often called eigenvectors (a.k.a., spectral loadings, loading vectors, principal components, or factors) for the methods used to calculate them. The scaling constants used to reconstruct the spectra are generally known as scores. 

Fig. 6. By multiplying PCI and PC2 (eigenvectors) by the set of representative scalar fractions (scores) and summing the results (along with the mean spectrum if the data were mean centered), the original calibration spectra can be re-created. The spectral residual is the difference between this reconstruction and the original.

However, since the Fock matrix is defined in terms of its own eigenvectors, the canonical spin orbitals and the orbital energies can only be obtained by means of an iterative procedure, where the Fock matrix is repeatedly reconstructed and rediagonalized until the spin orbitals generated by its diagonalization become identical to those from which the Fock matrix has been constructed. This iterative procedure is known as the self-consistent field (SCF) method and the resulting wave function - which satisfies (5.4.3) - is called the SCF wave function. 

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