Heterogeneous data analysis

Section 10.4 Heterogeneous Data Analysis for Reactor Design... 

A good model is consistent with physical phenomena (i.e., 01 has a physically plausible form) and reduces crresidual to experimental error using as few adjustable parameters as possible. There is a philosophical principle known as Occam s razor that is particularly appropriate to statistical data analysis when two theories can explain the data, the simpler theory is preferred. In complex reactions, particularly heterogeneous reactions, several models may fit the data equally well. As seen in Section 5.1 on the various forms of Arrhenius temperature dependence, it is usually impossible to distinguish between mechanisms based on goodness of fit. The choice of the simplest form of Arrhenius behavior (m = 0) is based on Occam s razor. 

To extract this knowledge from the various heterogeneous data sources made accessible to the KSP and the UltraLink, we combine several steps of normalization and analysis. These procedures are applied at loading time whenever the documents are displayed in the browser. 

Accommodate access to heterogeneous data sources and analysis tools. 

The ideal situation in a multi-centre trial is to have a small number of large centres (or pre-defined pseudo-centres). This gives the necessary consistency and control yet still allows the evaluation of heterogeneity. In practice, however, we do not always end up in this situation and combining centres at the data analysis stage inevitably needs to be considered. From a statistical perspective adjusting for small centres in the analysis is problematic and leads to unreliable estimates of treatment effect so we generally have to combine. 

Different data interpretation models have been applied simple dissociation constants (Langford and Khan, 1975), discrete multi-component models (Lavigne et al., 1987 Plankey and Patterson, 1987 Sojo and de Haan, 1991 Langford and Gutzman, 1992), discrete kinetic spectra (Cabaniss, 1990), continuous kinetic spectra (Olson and Shuman, 1983 Nederlof et al., 1994) and log normal distribution (Rate et al., 1992 1993). It should be noted that for heterogeneous systems, analysis of rate constant distributions is a mathematically ill-posed problem and slight perturbations in the input experimental data can yield artefactual information (Stanley et al., 1994). 

No precise information about the olefin polymerisation mechanism has been obtained from kinetic measurements in systems with heterogeneous catalysts analysis of kinetic data has not yet afforded consistent indications either concerning monomer adsorption on the catalyst surface or concerning the existence of two steps, i.e. monomer coordination and insertion of the coordinated monomer, in the polymerisation [scheme (2) in chapter 2], Note that, under suitable conditions, each step can be, in principle, the polymerisation rate determining step [241]. Furthermore, no % complexes have been directly identified in the polymerisation process. Indirect indications, however, may favour particular steps [242]. Actually, no general olefin polymerisation mechanism that may be operating in the presence of Ziegler-Natta catalysts exists, but rather the reaction pathway depends on the type of catalyst, the kind of monomer and the polymerisation conditions. 

For colloidal semiconductor systems, Albery et al. observed good agreement between the value of the radial dispersion obtained from dynamic light scattering and the value found from application of the above kinetic analysis to flash photolysis experiments [144], It should be remembered that this disperse kinetics model can only be applied to the decay of heterogeneous species under unimolecular or pseudo-first order conditions and that for colloidal semiconductors it may only be applied to dispersions whose particle radii conform to equation (37), i.e., a log normal distribution. However, other authors [145] have recently refined the model so that assumptions about the particle size distribution may be avoided in the kinetic data analysis. 

The molecular weight distribution of SAN copolymers can be determined by gel permeation chromatography [10]. SAN copolymers are soluble in common solvents such as tetrahydrofuran. The solubility characteristics of SAN copolymers, combined with the commercial availability of columns and data analysis software, make gel permeation chromatographic analysis a rapid and routine procedure. The choice of detectors can allow for the determination of absolute molecular weight [11]. Selection of multiple detectors enables characterization of compositional heterogeneity as a function of molecular weight [12], as illustrated in Figure 13.3. 

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