Singular value decomposition evaluation

To improve reproducibility in MRSI in human brain, simultaneous acquisition of the internal water reference and metabolite signals was evaluated. Use of singular value decomposition techniques and finite impulse response filters proved effective in separating water and metabolite signals and providing estimations of the metabolite concentrations. 

In principle both the classical and the inverse approach use a multivariate data set. But in the classical approach the variance is minimised, whereas in the inverse approach one tries to find an equilibrium between bias and variance. Therefore the bias is reduced and by the procedure of predictive receivable error sum of squares either via a singular value decomposition or the bidiagonalisation method estimated values, either according to principle component regression or partial least squares, are found. The multilinear regression on the other hand will find the best linear unbiased estimation as an approach to a true concentration. Besides applications in absorption spectroscopy, fluorescence spectra can also be evaluated [74]. 

The anal3Aical fitting procedure involves the explicit evaluation of all the matrix elements of G and its subsequent inversion, which is achieved by diagonalization. We have implemented several methods for the diagonalization step. Initially we employed singular value decomposition (SVD) (Press et al., 1992) by setting the inverse of the eigenvalue to zero if it is below a certain cutoff. However, this method produces undesirable numerical instabilities (noise) when... 

Once the singular value decomposition has been performed on A, we can obtain a set of stoichiometric coefficients from (11.2.15), as shown above. This differs from a traditional balancing of reactions. In a traditional balancing, we do not select values for the scale factors X. Instead, we select a value for one stoichiometric coefficient in each reaction, compute (in effect) the corresponding scale factors, and then use (11.2.15) to obtain all remaining stoichiometric coefficients Vy for the reaction j. We refer to this as an explicit evaluation of stoichiometric coefficients. We prefer this procedure when performing calculations by hand, for it can ensure that the stoichiometric coefficients are integers, or at least simple fractions. 

Singular value decomposition is now applied to all the Level 1 matrices. The product of the absolute value of the first column of the left singular vector of each of the SVD evaluations... 

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