Statistical analysis of mathematical models

The limitations of mathematical modeling described above increase the importance of statistical analysis of accidental explosions. However, gathering all needed data to perform a statistical analysis is often very complicated, so results are often incomplete. Wherever possible, both theoretical and statistical models should both be applied in estimating effects. 

While computers are a substantial aid in statistical analysis, it is also true that statistical methods have helped in certain computer applications. In Section V the subject of mathematical models will be discussed. These are in many cases based on empirical correlations. When these have been obtained by regression methods, not only is the significance of the results better understood, but also the correlation is expressed directly in a mathematical form suitable for programming. 

By assuming a reactor model, it is possible to determine reaction rates from experimental results. Then, various factors affecting yields, selectivities and reaction rates become evident. Experimental rate laws are deduced from results, e.g. in the classical form involving reaction orders and activation energies. At this stage, computers are used for solving numerically the mathematical models of reaction and reactor (Sect. 4) and for making a statistical analysis of experimental results (Sect. 5). 

Hugh Chipman is Associate Professor and Canada Research Chair in Mathematical Modelling in the Department of Mathematics and Statistics at Acadia University. His interests include the design and analysis of experiments, model selection, Bayesian methods, and data mining. 

The statistical techniques which have been discussed to this point were primarily concerned with the testing of hypotheses. A more important and useful area of statistical analysis in engineering design is the development of mathematical models to represent physical situations. This type of analysis, called regression analysis, is concerned with the development of a specific mathematical relationship including the mathematical model and its statistical significance and reliability. It can be shown to be closely related to the Analysis of Variance model. 

Fig. 2 The red blood cell has played a special role in the development of mathematical models of metabolism given its relative simplicity and the detailed knowledge about its molecular components. The model comprises 44 enzymatic reactions and membrane transport systems and 34 metabolites and ions. The model includes glycolysis, the Rapaport-Leubering shunt, the pentose phosphate pathway, nucleotide metabolism reactions, the sodium/potassium pump, and other membrane transport processes. Analysis of the dynamic model using phase planes, temporal decomposition, and statistical analysis shows that hRBC metabolism is characterized by the formation of pseudoequilibrium concentration states pools or aggregates of concentration variables. (From Ref...

To demonstrate this we use the simple example that was introduced in chapter 4, that of solubility in a mixed surfactant system. The treatment is in two stages, the first being a intuitive rather than mathematical demonstration of testing for lack of fit and curvature of a response surface. Then, in section Il.B, we will carry out a more detailed, statistical analysis of the same process, showing how prediction confidence limits are calculated and the use of ANOVA in validating a model. 

In the simplest case, a discriminant analysis is performed in order to check the affiliation (yes/no decision) of an unknown to a particular class, e.g. in case of a pur-ity/quality check or a substance identification. A sample may equally well be assigned between various classes (e.g., quahty levels) if a corresponding series of mathematical models has been estabhshed. Models are based on a series of test spectra, which has to completely cover the variations of particular substances in particular chemical classes. From this series of test spectra, classes of similar objects are formed by means of so-called discriminant functions. The model is optimized with respect to the separation among the classes. The evaluation of the assignment of objects to the classes of an established model is performed by statistically backed distance and scattering measures. 

Chapter 3 ean be used as the basis for CHEN 3650, Chemical Engineering Analysis (whieh eovers mathematical modeling and analytical, numerical, and statistical analysis of chemical processes). Statistical process eontrol (SPC) is not, of course, eovered in this book and the course reading should be supplemented by another book on SPC (e.g., Amitava Mitra, Fundamentals of Quality Control and Improvement, Prentice Hall, New York, 1998). 

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