Modeling power

Stokes CL. Biological systems modeling powerful medicine for biomedical eR D. Am Inst Chem g/2000 46 430. 

Feature selection, i.e. the selection of variables that are meaningful for the classification and elimination of those that have no discriminating (or, for certain techniques, no modelling power). This step is discussed further in Section 33.3. 

In SIMCA, we can determine the modelling power of the variables, i.e. we measure the importance of the variables in modelling the elass. Moreover, it is possible to determine the discriminating power, i.e. which variables are important to discriminate two classes. The variables with both low discriminating and modelling power are deleted. This is more a variable elimination procedure than a selection procedure we do not try to select the minimum number of features that will lead to the best classification (or prediction rate), but rather eliminate those that carry no information at all. 

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

Another feature of SIMCA that is of considerable utility lies in the assistance the technique provides in selecting relevant variables. Information contained in the residuals, ei -, can be used to select variables relevant to the classification objective. If the residuals for a variable are not well predicted by the model, the standard deviation is large. An expression defined as modeling power has been defined to quantitatively express this relationship. The modeling power (MPOW) is defined as ... 

In SIHCA-3B, modeling power is defined to be a measure of the importance of each variable in a principal component term of the class model (18). The modeling power has a maximum value of one (1.0) if the variable is well described by the principal components model. Variables with modeling power of less than 0.2 can be eliminated from the data without a major loss of information (18). 

For the PCB mixtures we analyzed (Table i), the modeling power was determined on the basis of a three component model (A=3). These data revealed that most of the 105 GC-peaks play an important role in the class model for the four Aroclors and their mixtures. The modeling power of each variable is plotted (Figure 4) for each component term along with a plot of the concentration profile of sample 9 (Table i) this sample contains 1242 1248 1254 1260 in a 1 1 1 1 ratio and the plot represents its fractional composition. 

Figure 4. Plots of the Fractional Composition of an Aroclor Mixture (A, Sample 9, Table 1) and the Modeling Power for a Three Component Model of the Samples in Table 1 PC-1 (B) PC-2 (C) and PC-3 (D).

In SIMCA, a class modeling method, a parameter called modeling power is used as the basis of feature selection. This variable is defined in Equation 4, where is the standard deviation of a vari-... 

Furthermore, SIMCA computes a number of useful parameters, such as the modeling power of the variables (their contribution to the inner space), the discriminating power of the variables (when more than one... 

In SIMCA, feature selection is carried out by the deletion of variables with both low modelling power and low discriminating power for each category. Modelling power measures the contribution of a variable to the model of a category, and discriminating power measures the contribution of a variable to the separation of categories. 

PLS falls in the category of multivariate data analysis whereby the X-matrix containing the independent variables is related to the Y-matrix, containing the dependent variables, through a process where the variance in the Y-matrix influences the calculation of the components (latent variables) of the X-block and vice versa. It is important that the number of latent variables is correct so that overfitting of the model is avoided this can be achieved by cross-validation. The relevance of each variable in the PLS-metfiod is judged by the modelling power, which indicates how much the variable participates in the model. A value close to zero indicates an irrelevant variable which may be deleted. 

Four models (Power Law, Herschel-Bulkley, Casson, and Bingham) were used to fit the experimental data and to determine the yield stress of... 

The modelling power of each valuable for each separate class is defined by... 

Finally, using the standard deviations obtained in steps 1 and 4, the modelling power of the variables for this class is obtained. 

The modelling power varies between 1 (excellent) and 0 (no discrimination). Variables with M below 0.5 are of little use. In this example, it can be seen that variables 4 and 6 are not very helpful, whereas variables 2 and 5 are extremely useful. This information can be used to reduce the number of measurements. 

Another measure is how well a variable discriminates between two classes. This is distinct from modelling power - being able to model one class well does not necessarily imply being able to discriminate two groups effectively. In order to determine this, it is necessary to fit each sample to both class models. For example, fit sample 1 to the... 

To determine which properties of the reaction systems contribute to the observed selectivity, plots of the PLS weights of the x variables were used in combination with a statistical criterion, modelling power [43]. We will omit these details, full accounts are given in Refs. [1,80]. 

A prognostic model or score must always be independently validated. Simply because a model seems to include appropriately modeled powerful predictors does not mean that it will necessarily validate well, because associations might just occur by chance, and predictors may not be as powerful as they appear. 

Table 8-2 contains expressions for the velocity profiles and the volumetric flow rates of the three rheological models power law, Herschel-Bulkley, and the Bingham plastic models. 

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