Eigenanalysis analysis

When merging more than two signals, yAlatent is obtained either by the n-dimensional principle of Pythagoras or - better - by principal component analysis (PCA, eigenanalysis see Sect. 6.3 Danzer [1990]). 

The proceeding of common methods of data analysis can be traced back to a few fundamental principles the most essential of which are dimensionality reduction, transformation of coordinates, and eigenanalysis. 

We now have both the data matrix A and the concentration vector c required to calculate PLS S VD. Both A and c are necessary to calculate the special case of PLS singular value decomposition (PLSSVD). The operation performed in PLSSVD is sometimes referred to as the PLS form of eigenanalysis, or factor analysis. If we perform PLSSVD on the A matrix and the c vector, the result is three matrices, termed the left singular values (LSV) matrix or the V matrix the singular values matrix (SVM) or the S matrix and the right singular values matrix (RSV) or the V matrix. 

Tables I and II report the morphological indices and the anisotropy measures obtained by Falcone et al. (2004b), respectively, from the stereological analysis and the eigenanalysis of the MIL tensor in order to characterize the inner crumb micro structure. By using these indices, it is possible to compare with high sensitivity the spatial arrangement of the cell walls in bread samples.

Eigenanalysis An analysis to determine the characteristic vectors (eigenvectors) of a matrix. These are a measure of the principal axes of the matrix. 

We have seen that it is possible to place upper bounds on the continuity of a scheme by carrying out eigenanalysis around a mark point. In principle these upper bounds can be tightened by doing this analysis for powers of the scheme, which give additional markpoints. 

We have seen how a lower bound on the continuity of the limit curve can be determined by -transform analysis, and an upper bound by eigenanalysis. 

In the eigenanalysis chapter we used the support analysis to tell us how many control points influenced a neighbourhood of a point of the limit curve, and thus how large the matrix needed to be on which to carry out the eigenanalysis. 

Principle components analysis (PCA), a form of factor analysis (FA), is one of the most common unsupervised methods used in the analysis of NMR data. Also known as Eigenanalysis or principal factor analysis (PEA), this method involves the transformation of data matrix D into an orthogonal basis set which describes the variance within the data set. The data matrix D can be described as the product of a scores matrix T, and a loading matrix P,... 

Mathematically, the technique operates through an eigenanalysis of either the eovariance matrix Cov N, Np), or of the eorrelation matrix R(N, N ), see Sections 4.5.3 and 4.6.1. Full formal details of the derivation of PC s are given in most texts on multivariate analysis (see [53]). Mathematical treatments are also given by... 

MacLeod, N. (2005b). Principal components analysis (eigenanalysis regression 5). Palaeontological Association Newsletter, 59 42—54. 

If the origin ( 0 ) is chosen at the centroid of the atoms, then it can be shown that distances from this point can be computed from the interatomic distances alone. A fundamental theorem of distance geometry states that a set of distances can correspond to a three-dimensional object only if the metric matrix g is rank three, i.e., if it has toee positive and N — 3 zero eigenvalues. This is not a trivial theorem, but it may be made plausible by thinking of the eigenanalysis as a principal component analysis all of the distance properties of the molecule should be describable in terms of three components , which would be the x, y and z coordinates. If we denote the eigenvector matrix as w and the eigenvalues A., the metric matrix can be written in two ways ... 

A systematic investigation of explosions based on the eigenanalysis of the Jacobian is called chemical explosive mode analysis (CEMA) (Lu et al. 2010 Luo et al. 2012c). An explosion index is defined for the explosive modes, which is similar to the radical pointer of the CSP method discussed below. This indicates... 

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