Eigenvector analysis

The PLS algorithm is relatively fast because it only involves simple matrix multiplications. Eigenvalue/eigenvector analysis or matrix inversions are not needed. The determination of how many factors to take is a major decision. Just as for the other methods the right number of components can be determined by assessing the predictive ability of models of increasing dimensionality. This is more fully discussed in Section 36.5 on validation. 

Maeder, M. Gampp, H., Spectrophotometric data reduction by eigenvector analysis for equilibrium and kinetic studies and a new method of Btting exponentials, Anal. Chim. Act 122, 303-313 (1980). 

The goal of the TTFA method is to estimate the number of sources, to identify them and to calculate their contribution from the ambient sample matrix C (chemical component concentrations 1 measured during sampling periods or at sampling sites k) using as little a priori information as possible. As a first step, an eigenvector analysis of matrix C is performed. 

Overparameterization and frequently its sources are revealed by an eigenvalue-eigenvector analysis. In the module 1445 the matrix JT( )WJ(jB) is investigated. We call it normalized cross product matrix, because the partial derivatives are computed with respect to the normalized parameters... 

For multidimensional data it is not possible to determine the direction of greatest variance by experimenting with all possible angles. An eigenvector analysis of the covariance matrix will then be applied. The mathematical background of this method will not be discussed in detail here a chemist will usually make use of commercially available software for this computation. 

The extraction of the eigenvectors from a symmetric data matrix forms the basis and starting point of many multivariate chemometric procedures. The way in which the data are preprocessed and scaled, and how the resulting vectors are treated, has produced a wide range of related and similar techniques. By far the most common is principal components analysis. As we have seen, PCA provides n eigenvectors derived from a. nx n dispersion matrix of variances and covariances, or correlations. If the data are standardized prior to eigenvector analysis, then the variance-covariance matrix becomes the correlation matrix [see Equation (25) in Chapter 1, with Ji = 52]. Another technique, strongly related to PCA, is factor analysis. ... 

We will proceed, therefore, with an eigenvector analysis of the 5x5 covariance matrix obtained from zero-centred object data. TUs is referred to as Q-mode factor analysis and is complementary to the scheme illustrated pre-... 

We will proceed, therefore, with an eigenvector analysis of the 5x5 covariance matrix obtained from zero-centred object data. This is referred to as Q-mode factor analysis and is complementary to the scheme illustrated previously with principal components analysis. In the earlier examples the dispersion matrix was formed between the measured variables, and the technique is sometimes referred to as R-mode analysis. For the current MS data, processing by R-mode analysis would involve the data being scaled along each mjz row (as displayed in Table 3.8) and information about relative peak sizes in any single spectrum would be destroyed. In Q-mode analysis, any scaling is performed within a spectrum and the mass fragmentation pattern for each sample is preserved. 

At present, data projection is performed mainly by methods called PC A, FA, singular value decomposition (SVD), eigenvector projection, or rank annihilation. The different methods are linked to different science areas. They also differ mathematically in the way the projection is computed, that is, which dispersion matrix is the basis for data decomposition, which assumptions are valid, and whether the method is based on eigenvector analysis, SVD, or other iterative schemes. 

Eigenvector Analysis For a symmetric, real matrix, / , an eigenvector V is obtained by... 

Eigenvector analysis is necessary, for example, in Factorial Methods Section for projection of multidimensional data. 

The main point of this approach is to apply the method of eigenvalues and eigenvectors analysis to obtain the kinetic pattern through the sensitivity parameters. Information extracted in such a manner for different reaction times enables to identify effectively the unimportant steps in the reaction kinetic model. 

Following the determination of the hydrolytic rate constants of an epimeric pair of tricyclic nitriles (78) by a non-linear, least-squares fitting method, a novel eigenvalue-eigenvector analysis of the sensitivity coefficients permitted maximization of the kinetic dataJ ... 

The analysis of the transient motion of the previous simulation runs is performed based on an eigenvalue/eigenvector analysis of the matrices of the linearised system, determined at certain operating conditions. As described earlier, motion quantities are divided according to which modes they are associated with and to which they contribute more significantly. In this sense, some quantities of motion... 

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