Development statistical models

Developing statistical models for image transform domains other than the DCT, e.g. DWT, FFT, sub-images DCT etc. The goal is to find the transform that provides robust statistical features and is less affected by common image processing operations. 

The mean free path of molecules in air at atmospheric pressure is /free — 1 /(Niiyg), where Nl 2.69 10 cm is the number density of gas molecules and cTg 10 " cm is the cross section for elastic collisions of molecules. These numbers result in /free — 3.7 10 cm, or 37 nm. The mean pore radius of the GDL is in the order of 10 pm, which means that the flow in the GDL pores occurs in a continuum regime. Thus, pressure-driven oxygen transport in a dry porous GDL can be modeled as a viscous Hagen-Poiseuille flow in an equivalent duct. However, determination of the equivalent duct radius and the dependence of this radius on the GDL porosity is a nontrivial task (Tamayol et al., 2012). Much workhas recently been done to develop statistical models of porous GDLs and to calculate viscous gas flows in these systems using Navier-Stokes equations (Thiedmann et al., 2012). 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

A general purpose program has been developed for the analysis of NMR spectra of polymers. A database contains the peak assignments, stereosequence names for homopolymers or monomer sequence names for copolymers, and intensities are analyzed automatically in terms of Bernoullian or Markov statistical propagation models. A calculated spectrum is compared with the experimental spectrum until optimized probabilities, for addition of the next polymer unit, that are associated with the statistical model are produced. 

In the past three decades, industrial polymerization research and development aimed at controlling average polymer properties such as molecular weight averages, melt flow index and copolymer composition. These properties were modeled using either first principle models or empirical models represented by differential equations or statistical model equations. However, recent advances in polymerization chemistry, polymerization catalysis, polymer characterization techniques, and computational tools are making the molecular level design and control of polymer microstructure a reality. 

Pelander et al. [81] developed a computer program for optimization of the mobile phase composition in TLC. They used the desirability function technique combined with the PRISMA model to enhance the quahty of TLC separation. They apphed the statistical models for prediction of retardation and band broadening at different mobile phase compositions they obtained using the PRISMA method the optimum mobile phase mixtures and a good separation for cyanobacterial hepatotoxins on a normal phase TLC plate and for phenolic compound on reversed-phase layers. 

Though statistical models are important, they may not provide a complete picture of the microscopic reaction dynamics. There are several basic questions associated with the microscopic dynamics of gas-phase SN2 nucleophilic substitution that are important to the development of accurate theoretical models for bimolecular and unimolecular reactions.1 Collisional association of X" with RY to form the X-—RY... 

Finally, accurate theoretical kinetic and dynamical models are needed for calculating Sn2 rate constants and product energy distributions. The comparisons described here, between experimental measurements and statistical theory predictions for Cl"+CHjBr, show that statistical theories may be incomplete theoretical models for Sn2 nucleophilic substitution. Accurate kinetic and dynamical models for SN2 nucleophilic substitution might be formulated by introducing dynamical attributes into the statistical models or developing models based on only dynamical assumptions. 

This chapter has outlined specifically how quantitative data on somewhat idealized reaction systems can be used as a basis for demonstrating the validity of our empirical electronic models in the field of reactivity. The multiparameter statistical models derived for the systems studied (PA, acidity, etc.) have limited direct application in EROS themselves. The next section develops the theme of applying the models in a much more general way, leading up to general reactivity prediction in EROS itself. 

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