Classification potential functions

Fig. 33.15. Classification of an unknown object u. f(/0 and f(L) indicate the potential functions for classes K and L.

By dividing the cumulative potential function of a class by the number of samples contributing to it, one obtains the (mean) potential function of the class. In this way, the potential function assumes a probabilistic character and, therefore, the density method permits probabilistic classification. 

The classification of a new object u into one of the given classes is determined by the value of the potential function for that class in u. It is classified into the class which has the largest value. A one-dimensional example is given in Fig. 33.15. Object u is considered to belong to K, because at the location of u the potential value of K is larger than that of L. The boundary between two classes is given by those positions where the potentials caused by these two classes have the same value. The boundaries can assume irregular values as shown in Fig. 33.3. 

Ionic potential — Function defined by = zjr, where z and r are the valence and radius of an ion, respectively. This function was introduced by G.H. Cartledge [i,ii], who used it as a quantitative basis of the periodic classification of elements. The ionic potential is directly connected with the heat of hydration of ions (see - Born equation), and thus related to the heat of solution of salts, acidic properties of ions, and others. It is also known that the ionic potential is correlated with electrochemical redox potentials (e.g., for solid metal hexacyanomet-allates [iii]). 

The most popular classification methods are Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA), Regularized Discriminant Analysis (RDA), K th Nearest Neighbours (KNN), classification tree methods (such as CART), Soft-Independent Modeling of Class Analogy (SIMCA), potential function classifiers (PFC), Nearest Mean Classifier (NMC) and Weighted Nearest Mean Classifier (WNMC). Moreover, several classification methods can be found among the artificial neural networks. 

The above classification tends to explain the properties of a more complex fluid in terms of an excess over a less complex (simpler) fluid pointing to a perturbation treatment as a suitable tool for both theory and applications. The properties of fluids belonging to different classes seem thus to be determined by the different types of predominant interactions. Consequently, to be able to understand and thus to predict the macroscopic properties of fluids, it is natural (and important) to determine the effect of the individual terms contributing to u on the macroscopic behavior. However, this need not be the case when the origins of the potential functions are considered. With the advance of computer technology, quantum chemical computation methods have also made considerable progress in the development of reasonably accurate effective pair potentials, but in a form which differs from that of Eqs. (2)-(4). Consequently, the simple physical picture of intermolecular interactions is lost and decompositions (3)-(4) become of little use. 

The gradient dynamical system and the catastrophe theories are two very useful and complementary mathematical tools for the study of the energetic and mechanisms of chemical reactions. We propose a classification of the potential functions and of the control space parameters. It emerges that the structural stability is a central concept for the understanding of chemical reactions and of chemical reactivity. 

Low-Frequency Vibrations in Small Ring Molecules Table 3.1. Classification of asymmetric potential functions. (Z4 + BZ2 + CZ3) 36B > 9C2 > 0... 

The above classification of asymmetric potential functions is convenient for comparison of different molecules or as a systematic basis for making an initial fit to experimental data. However, when the Schrodinger equation is being solved by the linear variation method with harmonic-oscillator basis functions, it may not provide the best choice of origin for the basis function. For example, a better choice in the case of an asymmetric double-minimum oscillator, where accurate solutions are required in both wells, would be somewhere between the two wells. Systematic variation of the parameters may still be made as outlined above, but the origin should be translated before the Hamiltonian matrix is set up. The equations given earlier... 

Olivieri et al. (2011) worked out the exploration of three different class-modelling techniques to evaluate classification abilities based on geographical origin of two PDO food products olive oil from Liguria and honey from Corsica. Authors developed the best models for both Ligurian olive oil and Corsican honey by a potential function technique (POTFUN) with values of correctly classified around 83%. 

Coulomb potential functions and the standard Tripos CoMFA probes (the Csp probe was used for calculation of steric interactions and the probe for calculation of elec-bostatic interactions, respectively). A PCA (factor analysis without axes rotation) was done on the descriptor matrix and a classification of the heteroaromatic substiments into families was performed using the Sybyl hierarchical clustering procedure of the obtained principal component... 

A powerful yet laborious classification method uses potential functions to decide on the class membership of a point x in the pattern space. At each pattern point of the data set an "electric charge" is located. Each pattern is therefore surrounded by its own potential field. At a distance D from the pattern point the potential Z(D) may be given by equation (75) C3693 or equation (76) C763. 

In principle, the form of the potential function and other controlling parameters can be varied as a function of the class membership, region of pattern space and the probability of the presence of different classes to achieve optimum classification. However, for practical applications, this optimization seem to be impracticable C128T. 

In a simpler and faster version of the classification by potential functions, only a number of neighbours nearest to the unknown are used for the calculation of the overall potentials. 

FIGURE 32. Classification with potential functions. To each member of class 1 and 2 was assigned a unit charge (positive or negative, resp.). Lines of equal superposition potential Z were calculated by equation (75) for q = 0.1. The decision boundary between the two classes is equal to the line Z = 0. The absolute magnitude of Z can be used as a confidence measure. 

FIGURE 33- Classification by potential functions. Adjustment of "electric charges in the data set C3693. 

A first approach is the estimation of p(x m) for each pattern x to be classified by using known patterns in the neighbourhood of x. For this computations the KNN-technique or potential functions may be used (Chapter 3). The advantage of this approach is that no special assumptions have been made about the form of the probability density function the disadvantage is that the whole set of known patterns is necessary for each classification. 

D. Zhao, W. Chen, and S. Hu, Comput. Chem., 22, 385 (1998). Potential Function Based Neural Networks and Its Applications to the Classification of Complex Chemical Patterns. 

To apply the decision diagram proposed by RCM philosophy (MOUBRAY, 2000) it is necessary to carry out the classification of failures of components, as a function of the consequences of their failure on the operational performance. On the point of view of decision making in maintenance, failures can be functional or potential. Functional failures are non-fulfillment of desired... 

A classification of dispersed systems on this basis has been worked out by Pawlow (30) (1910), who introduces a new variable called the concentration of the dispersed phase, i.e., the ratio of the masses of the two constituents of an emulsion, etc. When the dispersed phase is finely divided the thermodynamic potential is a homogeneous function of zero degree in respect of this concentration. 

There is an extremely wide range of potentially useful chemical treatments available, and for any boiler system, proper selection, utilization, and control are vital considerations that may largely determine the ultimate success of the overall program. These chemicals usually are organized by type of compound, function, mode of action, or similar classification, but, because many chemicals are multifunctional in character, may be used in either a primary or supplementary (adjunct or conjunctional treatment) role, and additionally may be branded (especially many modem polymers) or otherwise disguised, such classifications may be quite arbitrary. 

Other ion channels are closed at rest, but may be opened by a change in membrane potential, by intracellular messengers such as Ca + ions, or by neurotransmitters. These are responsible for the active signalling properties of nerve cells and are discussed below (see Hille 1992, for a comprehensive account). A large number of ion channels have now been cloned. This chapter concerns function, rather than structure, and hence does not systematically follow the structural classification. 

The hazards of a rigid classification of substances which may modify the course of a free radical polymerization are well illustrated by the examples of inhibitors and retarders which have been cited. The distinction between an inhibitor or retarder, on the one hand, and a co-monomer or a transfer agent, on the other, is not sharply defined. Moreover, if the substance is a free radical, it is potentially either an initiator or an inhibitor, and it may perform both functions as in the case of triphenylmethyl. If the substance with which the chain radicals react is a molecule rather than a radical, three possibilities may arise (i) The adduct radicals may be completely unreactive toward monomer. They must then disappear ultimately through mutual interaction, and we have a clear-cut case of either inhibition or retarda-... 

A classification of electrodes has already been given in Section 1.3.1. The function of the indicator electrode is to indicate by means of its potential the concentration of an ion or the ratio of the concentrations of two ions belonging to the same redox system. Under non-faradaic conditions, the relationship between the potential and these concentrations is given by the Nemst or the more extended Nernst-Van t Hoff equation, as explained below. As a single potential between an electrode and a solution cannot be measured in the absolute sense but only in a relative manner, a reference electrode is needed its function is merely to possess preferably a constant potential or at any rate a known potential under the prevailing experimental conditions. Often both electrodes cannot be placed in the same solution, so that a second solution... 

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