Eigenvalues, principal component analysis

Malinowski, E.R., Theory of die Distribution of Error Eigenvalues Resulting from Principal Component Analysis with Applications to Spectroscopic Data",... 

Then using these 91 peaks only, the original data set was reexamined by principal components analysis. Eigenvalues greater than one were plotted to determine how many factors should be retained. After variraax rotation, the factor scores were plotted and interpreted. 

Some of the linear combinations will be well defined and others poorly defined. The latter may be eliminated in a filtering procedure, referred to in the literature under the names characteristic value filtering, eigenvalue filtering, and principal component analysis. If the parameter set is not homogeneous, but includes different types, relative scaling is important. Watkin (1994) recommends that the unit be scaled such that similar shifts in all parameters lead to similar changes in the error function S. 

Principal component analysis is based on the eigenvalue-eigenvector decomposition of the n h empirical covariance matrix Cy = X X (ref. 22-24). The eigenvalues are denoted by > 2 — Vi > where the last inequality follows from the presence of same random error in the data. Using the eigenvectors u, U2,. . ., un, define the new variables... 

The mathematical procedure for finding such discriminant functions is to solve the eigenvalue problem of the quotient B i.e. to find the characteristic roots and eigenvectors as known from principal components analysis ... 

The multivariate autocorrelation function should contain the total variance of these autocorrelation matrices in dependence on the lag x. Principal components analysis (see Section 5.4) is one possibility of extracting the total variance from a correlation matrix. The total variance is equal to the sum of positive eigenvalues of the correlation matrices. This function of matrices is, therefore, reduced into a univariate function of multivariate relationships by the following instruction ... 

An anisometry descriptor defined as a function of the eigenvalues, obtained by - Principal Component Analysis applied to the correlation matrix calculated from the -> molecular matrix M ... 

Faber NM, Buydens LMC, Kateman G, Aspects of pseudorank estimation methods based on the eigenvalues of principal component analysis of random matrices, Chemometrics and Intelligent Laboratory Systems, 1994, 25, 203-225. 

Malinowski ER, Theory of the distribution of error eigenvalues resulting from principal component analysis with applications to spectroscopic data, Journal of Chemometrics, 1987, 1, 33-40. 

Eigenvalues, which are also sometimes called latent roots or characteristic roots, are important in determining the stability of a matrix to inversion and eigenvalues/ eigenvectors play an important role in many aspects of multivariate statistical analysis like principal component analysis. If X is a square symmetrical matrix then X can be decomposed into... 

Other whole molecule descriptors that do not require alignment include the Weighted Holistic Invariant Molecular (WHIM) indices developed by Todeschini et al. [70]. These indices are calculated from the 3D coordinates, which are weighted and centered to make them invariant to translation principal component analysis (PCA) is applied to obtain three principal components. These are used to produce new coordinates, which can be analyzed to obtain a series of 10 descriptors based on eigenvalues and the third-order and fourth-order moments of the three score column vectors. These descriptors are related to molecular size, shape, symmetry, and atom distribution and density. 

Fig. 6-15 Principal component analysis of multidimensional, chemical-genetic data, (a) Eigenvalues and associated variance, and eigenvectors and associated factor scores computed from the data in Fig. 6-14(a). The matrix of eigenvectors...

The most efficient method of data set compression in the joint basis is Principal Component Analysis (PCA). Principal Components (PCs) are constructed as a linear combination of original variables to maximize the description of data variance. They are eigenvectors of the auto-covariance matrix of data set. Each eigenvector is associated with the corresponding eigenvalue, which describes its importance in data variance description. For the studied IR library, 57 eigenvectors (principal components) are necessary to describe 95% of data variance, whereas as much as 109 eigenvectors are needed to describe 99% of data variance (see Fig.5). The mean value of RMS... 

One way to try to alleviate the problem of correlated descriptors is to perform a principal components analysis (see Section 9.13). Those principal components which explain (say) 90% of the variance may be retained for the subsequent calculations Alternatively, those principal components for which the associated eigenvalue exceeds unity may be chosen, or the principal components may be selected using more complex approaches based on cross-validation (see Section 12.12.3). It may be important to scale the descriptors (e.g. using autoscaling) prior to calculating the principal components. However, unless each principal component is largely associated with any particular descriptor it can be difficult to interpret the physical meaning of any subsequent results. ... 

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