Class-modelling methods equivalent determinant

Method Based on the Equivalent Determinant The second approach for estimating the critical limits of the cumulative potential to be used for class modelling was proposed by Forina et al. and is based on the concept of equivalent determinant [59,60]. By the term equivalent determinant, the authors indicate the determinant of the variance/covariance matrix of the multivariate normal distribution having the same value of the mean probability density as the one estimated by the density-based method. Without entering in the details of the procedure, it is possible to demonstrate... 

This classification method can be considered a qualitative equivalent to the subset regression strategy for improving quantitative models, discussed earlier in Section 8.3.8.5. Instead of defining a common sample space for all classes and then determining class membership based on the location of the unknown sample in the common space, SIMCA actually defines a different space for each class. For this discussion, the parameter Z will be used to denote the number of known classes in the calibration data. 

It is important to note, however, that there are fundamental differences between FSCC and SRCC with respect to the nature of their excitation operators. For a given truncation of the cluster operators beyond simple double excitations, the determinantal expansion space available in an FSCC calculation is smaller than those of SRCC calculations for the various model space determinants. A class of excitations called spectator triple excitations must be added to the FSCCSD method to achieve an expansion space that is in some sense equivalent to that of the SRCC. But even then, the FSCC amplitudes are restricted by the necessity to represent several ionized states simultaneously. Thus, we should not expect the FSCCSD to produce results identical to a single reference CCSD, nor should we expect triple excitation corrections to behave in the same way. The differences between FSCC and SRCC shown in Table I and others, below, should be interpreted as a manifestation of these differences. 

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