Parametric clustering methods

Q is parametrized exponentially in the coupled cluster method. A particularly compact form is obtained with the normal ordered form Q = exp(5). ... 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

However, due to the availability of numerous techniques, it is important to point out here the differences and equivalence between schemes. To summarize, two EDA families can be applied to force field parametrization. The first EDA type of approach is labelled SAPT (Symmetry Adapted Perturbation Theory). It uses non orthogonal orbitals and recomputes the total interaction upon perturbation theory. As computations can be performed up to the Coupled-Cluster Singles Doubles (CCSD) level, SAPT can be seen as a reference method. However, due to the cost of the use of non-orthogonal molecular orbitals, pure SAPT approaches remain limited... 

De Gunst, M. C., and Luebeck, E. G. (1998). A method for parametric estimation of the number and size distribution of cell clusters from observations in a section plane. Biometrics 54, 100-112. 

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

While the use of direct absorption methods has grown, indirect action spectroscopic methods continue to be widely and successfully used in the study of neutral molecular clusters. As mentioned earlier, there are two commonly used detection methods, mass spectrometers and bolometers. Because of the variety of mass-spectroscopic methods, there is an equally wide range of techniques used in neutral cluster spectroscopy. One of the oldest among these involves electron-impact mass spectrometry of a cw neutral beam combined with vibrational predissociation spectroscopy using a tunable cw or pulsed laser. The advent of continuously tunable infrared sources (such as color center lasers and LiNbOa optical parametric oscillators) allowed for detailed studies of size and composition variation in neutral clusters. However, fragmentation of the clusters within the ionizer of the mass spectrometer, severely limited the identification of particular clusters with specific masses. Isotopic methods were able to mitigate some of the limitations, but only in a few cases. 

COMPACT (computer-optimized molecular parametric analysis of chemical toxicity) [582, 583], a discriminant analysis approach, is described to predict carcinogenicity and other forms of toxicity involving the formation of reactive intermediates by determining the structural criteria for substrate specificity towards cytochrome P-450 enzymes it is claimed that the method is about 75% predictive for rodent carcinogenicity [583]. Recently, a discriminant-regression model was described [584]. It applies stepwise discriminant analysis to form clusters of compounds for which quantitative relationships are derived by multiple regression analysis. 

The different models compared and discussed in this paper include mDC The method described in the previous section and parametrized to the S22 [29], S66 [41], JSCH [28,29,43], SCAl [5] databases and to a database of sulfur containing molecules and water clusters. The description of the parametrization procedure and a detailed and... 

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