Quadratic analysis, reasons

Friedman and Frank [75] have shown that SIMCA is similar in form to quadratic discriminant analysis. The maximum-likelihood estimate of the inverse of the covariance matrix, which conveys information about the size, shape, and orientation of the data cloud for each class, is replaced by a principal component estimate. Because of the success of SIMCA, statisticians have recently investigated methods other than maximum likelihood to estimate the inverse of the covariance matrix, e.g., regularized discriminant analysis [76], For this reason, SIMCA is often viewed as the first successful attempt by scientists to develop robust procedures for carrying out statistical discriminant analysis on data sets where maximum-likelihood estimates fail because there are more features than samples in the data set. 

The energy levels of the vibrational modes can be predicted with a reasonable accuracy on the basis of the standard Wilson vibrational analysis (241,244) (called GF analysis). The vibrational motion of atoms in the polyatomic system is approximated by harmonic oscillations in a quadratic force field. Computations of the force constants are the subject of quantum chemistry. 

An unbiased statistical analysis of the relationship between observed red-shifts and the rate of quasar expansion, when ascribed to a light beam or other relativistic motion in the source frame of reference, leads directly to the quadratic redshift-distance relationship and predicts an eminently reasonable radius of the universe. 

Based on this reasoning we have performed a similar analysis for the case of our quadratic extension. As a result we could prove the equality of the moments only up to second order [38]. One has to emphasize, that in the former case the results obtained by using the LVC Hamiltonian H were compared to those obtained from the LVC Hamiltonian H. Correspondingly, in the present case, we compare the cumulants to the QVC Hamiltonians H and H. It should be recognized, that generally the QVC Hamiltonian H is more accurate than the LVC Hamiltonian, as it provide a better description of the actual electronic potential energy surfaces. 

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