Reduced eigenvalue

Figure 52. Eigenvalues (...) and reduced eigenvalues ( ) for the spectra consisting of pure, normally distributed noise.

Table 8 contains the eigenvalues (EV s) and reduced eigenvalues (REV s) that we get for the data in our training set absorbance matrices, Al, and A2. These are also plotted in Figure 53. 

Table 8. Eigenvalues (EV) and reduced eigenvalues (REV) for the two training sets, Al and A2.

This lack of sharpness of the 1-way F-test on REV s is sometimes seen when there is information spanned by some eigenvectors that is at or below the level of the noise spanned by those eigenvectors. Our data sets are a good example of such data. Here we have a 4 component system that contains some nonlinearities. This means that, to span the information in our data, we should expect to need at least 4 eigenvectors — one for each of the components, plus at least one additional eigenvector to span the additional variance in the data caused by the non-linearity. But the F-test on the reduced eigenvalues only... 

From the A -electron Hilbert-space eigenvalue equation, Eq. (2), follows a hierarchy of p-electron reduced eigenvalue equations [13, 17, 18, 47] for 1 < p < N — 2. The pth equation of this hierarchy couples Dp,Dp+, and and can be expressed as... 

Even given a hypothetical set of necessary and sufficient Al-representability constraints, however, the solution of CSE(2) is only unique provided that the eigenvalue w is specified and fixed. Because w does not appear in the ICSEs, a unique solution of ICSE(l) and ICSE(2) is obtained only by simultaneous solution of these equations subject not only to Al-representability constraints but also subject to the constraint that w = tr(lT2 >2) remains fixed. For auxiliary constraint equations, such as the reduced eigenvalue equation for the operator... 

A simple hypothesis test can be devised using the reduced eigenvalues to test for the significance of a factor, j [17]... 

Diagonalizing matrix A7 (or we obtain the reduced eigenvalues which are the approximations... 

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