Discriminant surface

Fig. 12.37 Discriminant analysis defines a discriminant function (dotted line) and a discriminant surface (solid) line.

Feature mapping (i.e., numeric-symbolic mapping) requires decision mechanisms that can distinguish between possible label classes. As shown in Fig. 5, widely used decision mechanisms include linear discriminant surfaces, local data cluster criteria, and simple decision limits. Depending on the nature of the features and the feature extraction approaches, one or more of these decision mechanisms can be selected to assign labels. 

Figure 7.4 2D representation of discriminant analysis. The dotted line represents the discriminant function and the solid line represents a discriminant surface that separates the two classes of samples. (From Livingstone, D.J., Data Analysis for Chemists Applications to QSAR and Chemical Product Design, Oxford University Press, Oxford, 1995. Reproduced with permission of Oxford University Press.)... 

State of the art composite membranes for reverse osmosis consist of three layers 1) the discriminating surface layer (commonly a polyamide produced by interfacial polymerization of m-phenylenediamine - trimesoyl chloride), 2) a supporting ultrafiltration layer (commonly polysulfone), and 3) a non-woven fabric that provides the majority of the mechanical strength [28, 30]. This trilayer composite reduces the resistance to permeation in the supporting layers without compromising mechanical integrity. 

Fig. 7.1. Two-dimensional representation of discriminant analysis. The dotted line represents the discriminant function and the solid line a discriminant surface which separates the two...

The accuracy of Lindemann s criterion depends also on the steepness of its variation with temperature at phase transition and on the specification of its percentage increase which should be adequate to discriminate surface (partial) from all-cluster melting. In the present case, we assign to the temperature at which Lindemann s index starts increasing. This allows us... 

Takmg advantage of the synunetry changes induced by the presence of a surface. Many nonlinear teclmiques rely on the fact that the surface breaks the centrosynuuetrical nature of the bulk. The use of polarized light can also discriminate among dipole moments in different orientations. 

On metals in particular, the dependence of the radiation absorption by surface species on the orientation of the electrical vector can be fiilly exploited by using one of the several polarization techniques developed over the past few decades [27, 28, 29 and 30], The idea behind all those approaches is to acquire the p-to-s polarized light intensity ratio during each single IR interferometer scan since the adsorbate only absorbs the p-polarized component, that spectral ratio provides absorbance infonnation for the surface species exclusively. Polarization-modulation mediods provide the added advantage of being able to discriminate between the signals due to adsorbates and those from gas or liquid molecules. Thanks to this, RAIRS data on species chemisorbed on metals have been successfidly acquired in situ under catalytic conditions [31], and even in electrochemical cells [32]. 

The surface that actually separates the classes is orthogonal to this discriminant function, as shown in Figure 12.37, and is chosen to maximise the number of compounds correctly classified. To use the results of a discriminant analysis, one simply calculates the appropriate value of the discriminant function, from which the class can be determined. 

To obtain the monolayer capacity from the isotherm, it is necessary to interpret the (Type II) isotherm in quantitative terms. A number of theories have been advanced for this purpose from time to time, none with complete success. The best known of them, and perhaps the most useful in relation to surface area determination, is that of Brunauer, Emmett and Teller. Though based on a model which is admittedly over-simplified and open to criticism on a number of grounds, the theory leads to an expression—the BET equation —which, when applied with discrimination, has proved remarkably successful in evaluating the specific surface from a Type II isotherm. 

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