Elementary Statistical Methods

In addition to the graphical techniques that have been illustrated in previous sections, some basic statistical tools should be brought to bear in the analysis of kinetic data. In fact, in most cases, graphical analyses merely set the stage for the efficient use of statistical analysis. Some of the most useful statistical tools are illustrated in the following example. 

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We shall start out with elementary general topological considerations of flow by studying network flow. We shall follow this by a variety of models from operations research that illustrate analytical methods and problems. No illustrations of statistical methods will be given here because statistics, a fundamental tool of science, is abundantly discussed in the literature of science. 

Statistics establish quality limits for the answers derived from a given method. A given laboratory result, or a sample giving rise to a given result, is considered good if it is within these limits. In order to understand how these limits are established, and therefore how it is known if a given result is unacceptable, some basic knowledge of statistics is needed. We now present a limited treatment of elementary statistics. 

In Moffat s review [1] an excellent account is given of the relation between the elementary approach and modern statistical methods in turbulence theory. In particular, by analogy with kinetic theory we easily derive for the diffusion coefficient of a scalar admixture the expression... 

Generally, the statistical methods assume that in the course of an elementary process a special state of the reacting system is reached where one of the internal degrees of freedom is changed into the translation along the reaction path. Hence, the statistical methods have two features in common. 

Having made the point that improvement in all business activities is of necessity data-driven, it is hopefully obvious that the emphases and methods of the subject of statistics are useful beyond the lab and even production. Of course, for broad implementation, it is the most elementary of statistical methods that are relevant. 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

Chapter 1 is an overview of statistical methods and elementary concepts for statistical model building. 

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. 

An essential component of calculations is to calibrate new methods, and to use the results of calculations to predict or rationalize the outcome of experiments. Both of these types of investigation compare two types of data and the interest is in characterizing how well one set of data can represent or predict the other. Unfortunately, one or both sets of data usually contain noise , and it may be difficult to decide whether a poor correlation is due to noisy data or to a fundamental lack of connection. Statistics is a tool for quantifying such relationships. We will start with some philosophical considerations and move into elementary statistical measures, before embarking on more advanced tools. 

With the statistical method, rate constants of the elementary reactions are generally not obtained alone, but in combination (see Table 15-7). 

A collection of several thousand such chains may be thought of as constituting a numerical model of the chain reaction it is analyzed, also by the computing machine, by statistical methods identical to the ones used for analyzing experimental observations of physical processes. In the language of probability, each chain is a point in a sample space. The probability measure in this space is necessarily very complicated, but from our knowledge of the elementary processes in a chain, it follows that such a probability measure exists and that our simulation procedure actually samples the distribution described by that measure. 

Thompson, M., Ellison, S. R. and Hardcastle, W. A. 2001. VAMSTAT II, LGC, Teddington. (A CD-ROM based program covering elementary statistics, with on-screen examples and tests. Very suitable for self-study, but multivariate methods not covered.)... 

H, Ohemoff and L. E. Moses, Elementary Decision Theory, John Wiley and Sons, Inc., New York, 1959 H. Cramer, Mathematical Method of Statistics, Princeton University Press, 1954. 

In order to compare various reacting-flow models, it is necessary to present them all in the same conceptual framework. In this book, a statistical approach based on the one-point, one-time joint probability density function (PDF) has been chosen as the common theoretical framework. A similar approach can be taken to describe turbulent flows (Pope 2000). This choice was made due to the fact that nearly all CFD models currently in use for turbulent reacting flows can be expressed in terms of quantities derived from a joint PDF (e.g., low-order moments, conditional moments, conditional PDF, etc.). Ample introductory material on PDF methods is provided for readers unfamiliar with the subject area. Additional discussion on the application of PDF methods in turbulence can be found in Pope (2000). Some previous exposure to engineering statistics or elementary probability theory should suffice for understanding most of the material presented in this book. 

Let us consider the compounds which show a small deviation from the stoichiometric composition and whose non-stoichiometry is derived from metal vacancies. The free energy of these compounds, which take the composition MX in the ideal or non-defect state, can be calculated by the method proposed by Libowitz. To readers who are well acquainted with the Fowler-Guggenheim style of statistical thermodynamics, the method here adopted may not be quite satisfactory however, the Libowitz method is understandable even to beginners who know only elementary thermodynamics and statistical mechanics. It goes without saying that the result calculated by the Libowitz method is essentially coincident with that calculated by the Fowler-Guggenheim method. 

Polymer identification starts with a series of preliminary tests. In contrast to low molecular weight organic compounds, which are frequently satisfactorily identified simply by their melting or boiling point, molecular weight and elementary composition, precise identification of polymers is difficult by the presence of copolymers, the statistical character of the composition, macromolecular properties and, by potential polymeric-analogous reactions. Exact classification of polymers is not usually possible from a few preliminary tests. Further physical data must be measured and specific reactions must be carried out in order to make a reliable classification. The efficiency of physical methods such as IR spectroscopy and NMR spectroscopy as well as pyrolysis gas chromatography makes them particularly important. 

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