Parametric classes

In contrast to a class type (see Classes on page 28), at least one of the type elements of the class type is a generic parameter. 

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The classes specified in a robust Bayesian analysis can be defined in a variety of ways, depending on the nature of the analyst s uncertainty. For instance, one could specify parametric classes of distributions in one of the conjugate families (e.g., all the beta distributions having parameters in certain ranges). Alternatively, one could specify parametric classes of distributions but not take advantage of the conjugacies. 

FIG U RE 6.3 Two parametric classes of prior distributions having constant variance (left) or constant mean (right) shown as cumulative distribution functions (cdfs). The horizontal axis is some value for a random variable and the vertical axis is (cumulative) probability. 

ASSIST turned out to be very useful to restructure the code at this level and some attempts have been made by using some parametric classes of ASSISTlib (the library of ASSIST). To this end, in the AV routine the Scattering collective (SCATTER) is used on the producer side to distribute the matrices of Eq. 1. More in detail. SCATTER P carries out the distribution of blocks of rows and... 

Recently, Bacic et al.28 have studied the bending-stretching energy levels of HCN by SCF approximation with modes optimized within the class of elliptical (spheriodal) coordinates. This example will be examined here in some detail since (1) the improvement obtained by coordinate optimization in this case is very large and (2) this case illustrated how intuitive considerations based on molecular geometry, the shape of the potential surface, and so forth, can be useful in choosing a suitable parametrized class of coordinates. Whenever the trial set of parametrized coordinates is physically well motivated, the OC-SCF procedure of Eqs. (19)—(21) can be expected to yield excellent results. 

In (Haavardsson et al., 2008) a parametric class of admissible production strategies is introduced. This class has the property that it always contains an optimal production strategy. Finding an optimal strategy within this class is much simpler than searching within the entire class of admissible strategies B. In the next section we present this class. 

Haavardsson, N. F., Huseby, A. B. Holden, L. 2008, A Parametric Class of Production Strategies for Multi-Reservoir Production Optimization. Statistical Research Report, University of Oslo, 8. 

Derde MP, Kaufman L, Massart DL. A non-parametric class-modelling technique. J Chemometr 1989 3 375. 

In case of a generic class type, PARAMETRIC TYPE is a parametric class type or a variant class type. All the parameter types have to be class types. 

Each force field achieves good results only for a limited class of molecules, related to those for which it was parametrized. No force field can be generally used for all molecular systems of interest. 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

Now we want to compute the class of the complement D (X)<, = D (X) D (X)0. It parametrizes in a suitable sense the second order data of singular m-dimensional subvarieties of X. We will use a tool that will play a major role in the enumerative applications of higher order data in section 3.2, the Porteous formula. We will not quote the result in full generality but in the formulation in which we are going to use it. 

The models used can be either fixed or adaptive and parametric or non-parametric models. These methods have different performances depending on the kind of fault to be treated i.e., additive or multiplicative faults). Analytical model-based approaches require knowledge to be expressed in terms of input-output models or first principles quantitative models based on mass and energy balance equations. These methodologies give a consistent base to perform fault detection and isolation. The cost of these advantages relies on the modeling and computational efforts and on the restriction that one places on the class of acceptable models. 

Quadratic discriminant analysis (QDA) is a probabilistic parametric classification technique which represents an evolution of EDA for nonlinear class separations. Also QDA, like EDA, is based on the hypothesis that the probability density distributions are multivariate normal but, in this case, the dispersion is not the same for all of the categories. It follows that the categories differ for the position of their centroid and also for the variance-covariance matrix (different location and dispersion), as it is represented in Fig. 2.16A. Consequently, the ellipses of different categories differ not only for their position in the plane but also for eccentricity and axis orientation (Geisser, 1964). By coimecting the intersection points of each couple of corresponding ellipses (at the same Mahalanobis distance from the respective centroids), a parabolic delimiter is identified (see Fig. 2.16B). The name quadratic discriminant analysis is derived from this feature. 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

An improved version of the MTD approach would be of real interest as a mono-parametric Free-Wilson-type method (due to their meaning, the MTD and Free-Wilson parameters belong to the same class). The topological description of the molecular structure assures the easy to use character of the MTD method, and the hypermolecule concept allows to study widely differing structures within the data basis. 

A class of operations has been devised in which the process fluid is pumped through a particular kind of packed bed in one direction for a while, then in the reverse direction. Each flow direction is at a different level of an operating condition such as temperature, pressure, or pH to which the transfer process is sensitive. Such a periodic and synchronized variation of the flow direction and some operating parameter was given the name of parametric pumping by Wilhelm (1966). A difference in concentrations of an adsorbable-desorbable component, for instance, may develop at the two ends of the equipment as the number of cycles progresses. 

It should be noted that the above-mentioned LFER and parametric equations have been derived for a relatively small series of ligands and metals. Therefore, their application for prediction is limited to the class of metals or ligands used for fitting the empirical parameters of those equations. Nevertheless, they have contributed to the fundamental understanding of complexation phenomena, especially for the classes of ligands and metals studied. 

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