Statistical methods direct approach

Sampling studies can be classified Into two types - enumeratlve, or descriptive, and analytic (j ). The classification Is Important because the applicable statistical methods and approaches are different for these two types. The objective of either type of study Is to provide a basis for action. In an enumeratlve study the action Is directed to the population from which the samples were taken. How or why the population was formed Is not of primary Interest. In an analytic study, the primary Interest Is the causal system or process which created the conditions observed In the study. Action taken Is directed toward this process rather than the population sampled. 

The proposed approach leads directly to practical results such as the prediction—based upon the chemical potential—of whether or not a reaction runs spontaneously. Moreover, the chemical potential is key in dealing with physicochemical problems. Based upon this central concept, it is possible to explore many other fields. The dependence of the chemical potential upon temperature, pressure, and concentration is the gateway to the deduction of the mass action law, the calculation of equilibrium constants, solubilities, and many other data, the construction of phase diagrams, and so on. It is simple to expand the concept to colligative phenomena, diffusion processes, surface effects, electrochemical processes, etc. Furthermore, the same tools allow us to solve problems even at the atomic and molecular level, which are usually treated by quantum statistical methods. This approach allows us to eliminate many thermodynamic quantities that are traditionally used such as enthalpy H, Gibbs energy G, activity a, etc. The usage of these quantities is not excluded but superfluous in most cases. An optimized calculus results in short calculations, which are intuitively predictable and can be easily verified. 

The abbreviation QSAR stands for quantitative structure-activity relationships. QSPR means quantitative structure-property relationships. As the properties of an organic compound usually cannot be predicted directly from its molecular structure, an indirect approach Is used to overcome this problem. In the first step numerical descriptors encoding information about the molecular structure are calculated for a set of compounds. Secondly, statistical methods and artificial neural network models are used to predict the property or activity of interest, based on these descriptors or a suitable subset. A typical QSAR/QSPR study comprises the following steps structure entry or start from an existing structure database), descriptor calculation, descriptor selection, model building, model validation. 

Any QSAR method can be generally defined as an application of mathematical and statistical methods to the problem of finding empirical relationships (QSAR models) of the form ,- = k(D, D2,..., D ), where ,- are biological activities (or other properties of interest) of molecules, D, P>2,- ,Dn are calculated (or, sometimes, experimentally measured) structural properties (molecular descriptors) of compounds, and k is some empirically established mathematical transformation that should be applied to descriptors to calculate the property values for all molecules (Fig. 6.1). The goal of QSAR modeling is to establish a trend in the descriptor values, which parallels the trend in biological activity. In essence, all QSAR approaches imply, directly or indi-... 

Contribution plots presented in Section 7.4 provide an indirect approach to fault diagnosis by first determining process variables that have inflated the detection statistics. These variables are then related to equipment and disturbances. A direct approach would associate the trends in process data to faults explicitly. HMMs discussed in the first three sections of this chapter is one way of implementing this approach. Use of statistical discriminant analysis and classification techniques discussed in this section and in Section 7.6 provides alternative methods for implementing direct fault diagnosis. 

The first catenanes and rotaxanes were constructed by either statistical [2] or directed [3] synthetic approaches. The statistical method relies [2] on the formation of small quantities of a species in which a cyclic molecule is threaded by an acyclic molecule. After experiencing appropriate covalent bond formation, these threaded species are con-... 

An alternative approach to using traditional parametric statistical methods to calculate the significance of fitted correlations, would be to directly assess the model based on its ability to predict, rather than merely to assess how well the model fits the training set. When the quality of the model is assessed by the prediction of a test set, rather than the fit of the model to its training set, a statistic related to r or can be defined, and denoted q or q, to indicate that the quality measure is assessed in prediction. A q may be calculated by internal cross-validation techniques, or by the quality of predictions of an independent test set, in which case an upper-case is used. The equation to calculate q (or Q ) is shown in equation 9.3. 

Quantum chemistry has estabhshed itself as a valuable tool in the studies of polymerization processes [25,26]. However, direct quantum chemical studies on the relationship between the catalyst structure and the topology of the resulting polymer, as well as on the influence of the reaction conditions, are not practical without the aid of statistical methods. We have to this end proposed a combined approach in which quantum chemical methods are used to provide information on the microscopic energetics of elementary reactions in the catalytic cycle, that is required for a mesoscopic stochastic simulations of polymer growth [25]. A stochastic approach makes it possible to discuss the effects of temperature and olefin pressure. 

---