Singular values significant

The above relation holds within the limited precision of our calculations of the singular values, which in the present example is about four significant digits. 

There are many advantages in selecting only the significant ne eigenvectors and singular values for the representation of Y. In fact, from now on we only use this selection and introduce an appropriate nomenclature. 

It is interesting to observe that the significant singular values are hardly affected by increasing noise, while the noise singular values move up together. 

Previously, we have seen in Magnitude of the Singular Values (p.219) that the number of significant singular values in S equals the number of linearly... 

The basic principle of EFA is very simple. Instead of subjecting the complete matrix Y to the Singular Value Decomposition, specific sub-matrices of Y are analysed. In the original EFA, these sub-matrices are formed by the first i spectra of Y where i increases from 1 to the total number of spectra, ns. The appearance of a new compound during the acquisition of the data is indicated by the emergence of a new significant singular value. 

The procedure is best explained graphically in Figure 5-33. The sub-matrix, indicated in grey, is subject to the SVD and the resulting ne significant singular values are stored as a row vector in a matrix EFA of the same number of rows as Y. 

The rank is the number of significant singular values. The significance level can be estimated as the first non-significant singular value of the total matrix Y. 

The human eye is very good at detecting patterns - in this case the appearance of a new significant singular value. The appearance of a new component, as indicated by the point where a new significant singular value rises above the noise level, is delayed by increasing noise. 

S diagonal matrix of significant singular values (nexne)... 

In the SISO case we look at magnitudes. In the multivariable case we look at singular values. Thus plots of the maximum singular value of the matrix wiU show the fiequency region where the uncertainties become significant. Then the... 

Figure 4.11 A singular value decomposition of the preconditioned in situ spectroscopic data showing the 1st, 4th, 5th, 7th significant vectors and the 713th vector. The marked extrema are those which were used to recover the organometallic pure component spectra as well as alkene and aldehyde by STEM. Atmospheric moisture and CO2, hexane, and dissolved CO were removed from the experimental data during preconditioning. (C. Li, E. Widjaja, M. Garland,/ Am. Chem. Soc., 2003, 725, 5540-5548.)...

Application of SVD analysis to the time-resolved SAXS data of Russell et al. (2002) clearly indicated two significant singular values, and therefore enabled data analysis by projection of each independent scattering profile onto two states. The folded and unfolded states were selected because of... 

There has been some discussion as to whether CD can distinguish parallel from antiparallel p sheets. As stable, well-defined model compounds are lacking, the spectra available have been derived from secondary structure deconvolutions (see below). Overall, the ability of CD to provide adequate estimates of both parallel and antiparallel p sheet contents is still an ongoing question. Johnson and co-workers were the first to derive basis spectra which corresponded to both parallel and antiparallel p sheet structures in globular proteins using the singular value deconvolution method [11, 12, 51-53], However, the basis spectra were significantly different from spectra reported for model sleet structures. Recently, Perczel et al. [54] employed another approach, convex curve analysis, to obtain improved p sheet baas spectra. The major improvement was to include more p sheet proteins into the data base. 

By far, singular value decomposition (SVD) is the most popular algorithm to estimate the rank of the data matrix D. As a drawback of SVD, the threshold that separates significant contributions from noise is difficult to settle. Other eigenvalue-based and error functions can be utilized in a similar way, but the arbitrariness in the selection of the significant factors still persists. For this reason, additional assays may be required, especially in the case of complex data sets. 

Figure 6.8. Example of the use of rank annihilation factor analysis for determining the concentration of tryptophane using fluorescence excitation-emission spectroscopy. In the top left plot the unknown sample is shown. It contains three different analytes. The standard sample (only tryptophane) is shown in the top right plot. In the lower right plot, it is shown that the smallest significant (third from top) singular value of the analyte-corrected unknown sample matrix reaches a clear minimum at the value 0.6. In the lower left plot the unknown sample is shown with 0.6 times the standard sample subtracted. It is evident that the contribution from the analyte is practically absent.

Figure 10.54. The singular values for the 14 x 600 matrix. There are four significant components. The dashed line is used in the elbow method to find significant components.

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