Singular value decomposition algorithm

Similar results have been achieved for other realistic test vehicles [31]. We would especially like to mention a complex updating singular value decomposition algorithm, needed, for example, in data acquisition [15]. 

Factor analysis [144], a variation of PCA, is based on the conversion of the experimental NMR spectra Yj into a set of orthonormal subspectra Sj using the singular value decomposition algorithm (Eq. 20). 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Excel does not provide functions for the factor analysis of matrices. Further, Excel does not support iterative processes. Consequently, there are no Excel examples in Chapter 5, Model-Free Analyses. There are vast numbers of free add-ins available on the internet, e.g. for the Singular Value Decomposition. Alternatively, it is possible to write Visual Basic programs for the task and link them to Excel. We strongly believe that such algorithms are much better written in Matlab and decided not to include such options in our Excel collection. 

The Singular Value Decomposition, SVD, has superseded earlier algorithms that perform Factor Analysis, e.g. the NIPALS or vector iteration algorithms. SVD is one of the most stable, robust and powerful algorithms existing in the world of numerical computing. It is clearly the only algorithm that should be used for any calculation in the realm of Factor Analysis. 

Calculation of eigenvectors requires an iterative procedure. The traditional method for the calculation of eigenvectors is Jacobi rotation (Section 3.6.2). Another method—easy to program—is the NIPALS algorithm (Section 3.6.4). In most software products, singular value decomposition (SVD), see Sections A.2.7 and 3.6.3, is applied. The example in Figure A.2.7 can be performed in R as follows ... 

A variety algorithms can be used to calculate the loadings and. scores for PCA. A comiEonly employed approach is the singular value decomposition CS T>) algoriiim (Golub and Van Loan, 1983, Chapter 2). A matrix of arbitrary size can be 5sftten as R = USV. The U matrix contains the coordinates of the... 

The decomposition in eqn (3.30) is general for PCR, PLS and other regression methods. These methods differ in the criterion (and the algorithm) used for calculating P and, hence, they characterise the samples by different scores T. In PCR, T and P are found from the PCA of the data matrix R. Both the NIPALS algorithm [3] and the singular-value decomposition (SVD) (much used, see Appendix) of R can be used to obtain the T and P used in PCA/PCR. In PLS, other algorithms are used to obtain T and P (see Chapter 4). 

This is acceptable provided that the assumption is stated up front. The melt curves in the lower part of Fig. 17.3 were fit this way. Alternatively, the full model can be used, with additional steps taken to ensure the validity of the results. An example is shown in the upper part of Fig. 17.3. Here, the experiments were repeated multiple times (minimizing the measurement error), the data were fit simultaneously (using a global fitting algorithm), and the results were corroborated using a separate singular value decomposition analysis. 

NIPALS is a common, iterative algorithm often used for PCA. Some authors use another method called SVD (singular value decomposition). The main difference is that NIPALS extracts components one at a time, and can be stopped after the desired number of PCs has been obtained. In the case of large datasets with, for example, 200 variables (e.g. in spectroscopy), this can be very useful and reduce the amount of effort required. The steps are as follows. 

PCA is simple in Matlab. The singular value decomposition (SVD) algorithm is employed, but this should normally give equivalent results to NIPALS except diat all the PCs are calculated at once. One difference is that die scores and loadings are bodi normalised, so that for SVD... 

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. 

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