Significance of parameters

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

Specific adsorption on well defined materials has been the subject of many reviews [8-13]. Specific adsorption plays a key role in transport of nutrients and contaminants in the natural environment, and many studies with natural, complex, and ill defined materials have been carried out. Specific adsorption of ions by soils and other materials was reviewed by Barrow [14,15]. The components of complex mineral assemblies can differ in specific surface area and in affinity to certain solutes by many orders of magnitude. For example, in soils and rocks, (hydr)oxides of Fe(IH) and Mn(IV) are the main scavengers of metal cations and certain anions, even when their concentration expressed as mass fraction is very low. Traces of Ti02 present as impurities are responsible for the enhanced uptake of U by some natural kaolinites. In general, complex materials whose chemical composition seems very similar can substantially differ in their sorption properties due to different nature and concentration of impurities , which are dispersed in a relatively inert matrix, and which play a crucial role in the sorption process. In this respect the significance of parameters characterizing overall sorption properties of complex materials is limited. On the other hand the assessment of the contributions of particular components of a complex material to the overall sorption properties would be very tedious. 

The quality of data regression should be checked systematically by inspecting the deviation of individual points with respect to model prediction, as the statistical significance of parameters. As Prausnitz noted (1980), the choice of a model is not as important as the procedure used to obtain the parameters from limited and Inaccurate experimental data . 

FIGURE 2.2 Schematic representation of a Tafel plot showing curvature at low overvoltage and indicating significance of parameters a and b. 

It is always difficult to evaluate the health significance of parameter estimates obtained from regression analyses because they reflect only average effects. There may be sensitive individuals or defined groups within the population for whom the effects of exposure are far more serious. For example, our analyses of the 6- and 12-month Bayley data show consistent evidence of racial differences in the PbB-behaviour dose-effect relationship in this instance, each log increment in prenatal PbB was associated with an average decrement of 14 points in 12-month Bayley PDI among white infants (0.87 SD). 

As early as 1972 it was recognized that an explicit inclusion of terms associated with the charge-flux effects accompanying molecular vibrations may offer an opportunity for better understanding of the physical significance of parameters in infi-ared intensity models based on the bond moment concept [42]. 

The geometrical significance of parameter is not quite as clear-cut for clays as it is for ion-exchange resin-solution systems (Table I). There is a general increasing trend for d in the series kaolinite illitemontmorillonite, but in a plot of W vs. the ratio between computed and the specific values in the experiments reported... 

Figure 3 Feature relevance. The weight parameters for every component in the input vector multiplied with the standard deviation for that component are plotted. This is a measure of the significance of this feature (in this case, the logarithm of the power in a small frequency region.)...

Equation (2.14) has the advantage of simplicity its drawback is that we learn nothing about either the nature of viscosity or the nature of the sample from the result. In the next few sections we shall propose and develop a molecular model for the flow process. The goals of that development will be not only to describe the data, but also to do so in terms of parameters which have some significance at the molecular level. Before turning to this, it will be helpful if we consider a bit further the form of Eq. (2.14). 

We shall presently examine the physical significance of the shift factors, since they quantitatively embody the time-temperature equivalence principle. For the present, however, we shall regard these as purely empirical parameters. The following Ust enumerates some pertinent properties of a ... 

The significance of G G tan 5, Tj, and Tj is that they can be determined experimentally and used to characterize real materials. These parameters depend on frequency and temperature, and this dependence can be used to define behavior. For example, viscoelastic fluids are often characterized by log—log plots of one or more of these quantities vs the angular frequency CO, as shown in Figure 21, which illustrates the behavior of a polymer melt (149). 

Sample quantity to estimate moisture for specific material is influenced to various levels of significance by properties such as particle-size range as well as relative amounts or moisture distributed among denoted forms of retention. Practical sample size estimates require background knowledge of parameters derived from experience for specific materials. More detailed examination of moisture-sampling aspects is provided in reference texts (Pitard). 

The purpose is to develop estimates of significant model parameters that provide the best estimate of unit operation. The unit operation is embodied in the measurements. 

The pumose of fault detection is to interpret the set of measurements to determine whether the operation of the unit has changed. This interpretation is done by monitoring the set of the measurements or by monitoring values for the significant unit parameters. It is done automatically as part of the computer control of the unit or periodically as when comparing one unit test to a subsequent one. 

Numerous attempts have been made to develop hybrid methodologies along these lines. An obvious advantage of the method is its handiness, while its disadvantage is an artifact introduced at the boundary between the solute and solvent. You may obtain agreement between experiments and theory as close as you desire by introducing many adjustable parameters associated with the boundary conditions. However, the more adjustable parameters are introduced, the more the physical significance of the parameter is obscured. 

Table 3-2 Estimating the Significant Minimum Parameters of the Power Semiconductors...

---