Correlation among localized

The density cumulant therefore measures the correlation among local density fluctuations p(r) — (p(r). It is these fluctuations which lead to scattering. 

Correlations among specific segments can be measured only in computer experiments. Normal scattering experiments measure the segment density correlation function, to which all segments contribute equally. To define this quantity we introduce the local segment, density... 

The correlation among these records is so high that local or regional causes—rather than global conditions—might be considered to explain such a... 

In the proposed C-JIT modeling, samples stored in the database are divided into several data sets. Although the method of generating data sets is arbitrary, each data set is generated so that it consists of successive samples included in a certain period of time in this work, because the correlation among variables in such a data set is expected to be very similar. To build a local model, the index J in Eq. (5) is calculated for each data set, and the data set that minimizes J is selected as the modeling data set. 

The estimation results are shown in Table 1. In this table, r denotes the correlation coefficient between measurements and estimates, and RMSE is the root mean square error. The results show that neither recursive PLS nor JIT modeling functions well. In general, recursive PLS is suitable only for slow changes in process characteristics. On the other hand, the reason for the poor performance of JIT modeling seems that JIT modeling does not take account of correlation among variables when a local model is built. 

The correlated methods discussed up to this point provide a delocalized description of the electronic system. The delocalized nature of these methods arises from their use of canonical orbitals (i.e., the eigenvectors of the Hartree-Fock equations) of Eq. (33). To treat large systems, it is better to express the theory in terms of orbitals that are localized in space, extending over only a few atoms. The virtual excitations then occur predominantly locally in the molecule (among localized occupied and virtual orbitals). As a result, the number of excitation amplitudes increases only linearly with system size. 

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