Class-Modelling

A first distinction which is often made is that between methods focusing on discrimination and those that are directed towards modelling classes. Most methods explicitly or implicitly try to find a boundary between classes. Some methods such as linear discriminant analysis (LDA, Sections 33.2.2 and 33.2.3) are designed to find explicit boundaries between classes while the k-nearest neighbours (A -NN, Section 33.2.4) method does this implicitly. Methods such as SIMCA (Section 33.2.7) put the emphasis more on similarity within a class than on discrimination between classes. Such methods are sometimes called disjoint class modelling methods. While the discrimination oriented methods build models based on all the classes concerned in the discrimination, the disjoint class modelling methods model each class separately. 

M. Forina, C. Armanino, R. Leardi and G. Drava, A class-modelling technique based on potential functions. J. Chemom. 5 (1991) 435-453. 

Derde MP, Massart DL (1988) Comparison of the performance of the class modelling techniques UNEQ, SIMCA and PRIMA. Chemom Intell Lab Syst 4 65... 

Green, B. F. (1951). A general solution for the latent class model of latent structure analysis. Psychometrika, 16, 151—166. 

When the 575 remaining Coso spectra (from samples not used in the 5-class model) were tested against the model, 574 out of 575 were correctly classified with the other Coso samples in the model. In other words, the model remains valid even for unknown samples (Table 2). 

A class model is shown in Figure 11.5. The dynamics of the inheritance design can be shown on an enhanced interaction diagram, separating the inherited and locally defined parts of an object to show calls that go up or down the inheritance chain. 

We can now define the design s class model the classes that compose the system, the interfaces they implement and use, their attributes and operations, and the references between them. The initial class model is derived by reifying each model type to a class, adding the operations that the class must implement from the interaction diagram, and designing the attributes and directed associations that the class needs to implement those operations. 

Kemsley, E. K. Discriminant Analysis and Class Modelling of Spectroscopic Data. Wiley, Chichester, United Kingdom, 1998. 

The similarity of samples can be evaluated by using geometrical constructs based on the standard deviation of the objects modeled by SIMCA. By enclosing classes in volume elements in descriptor space, the SIMCA method provides information about the existence of similarities among the members of the defined classes. Relations among samples, when visualized in this way, increase one s ability to formulate questions or hypotheses about the data being examined. The selection of variables on the basis of MPOW also provides clues as to how samples within a class are similar, and the derived class model describes how the objects are similar, with regard to the internal variation of these variables. 

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. 

Such a measure of the separation between classes will work best when It can be assumed that the classes approximate multivariate normal distributions. That Is a reasonable assumption for the classes modeled by the output of the FCV algorithms. 

Up to this point the methods of classification operate in the same way. They differ considerably, however, in the way that rules for classification are derived. In this regard the various methods are of three types 1) class discrimination or hyperplane methods, 2) distance methods, and 3) class modeling methods. 

Only one class modeling method is conmonly applied to analytical data and this is the SIMCA method ( ) of pattern recognition. In this method the class structure (cluster) is approximated by a point, line, plane, or hyperplane. Distances around these geometric functions can be used to define volumes where the classes are located in variable space, and these volumes are the basis for the classification of unknowns. This method allows the development of information beyond class assignment ( ). 

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-... 

Since SIMCA is a class modeling method, class assignment is based on fit of the unknowns to the class models. This assignment allows the classification result that the unknown is none of the described classes, and has the advantage of providing the relative geometric portion of the newly classified object. This makes it possible to assess or quantitate the test sample in terms of external variables that are available for the training sets. 

The class belonging can be determined when the distance of an object to the class model is compared with the typical distance of the class objects to the same model. 

Caiculategaincipal components for the samples in the individual classes and determine theiiitial setting for the rank of each class model. 

Scores Plot The score plot displays the relationship of the samples to each other in row space. It does not show the residual information and only contains that fraction of the total variation that is described by the PCs that are examined. Thece arc a series of score plots (score 2 vs. score 1, score 3 vs. score 2, etc.) avaii e for each unknown and each class model. 

Instead, class-modeling techniques verify whether a sample is compatible or not with the characteristics of a given class of interest. In fact. 

Nevertheless, in most of the electronic tongue applications found in the literature, classification techniques like linear discriminant analysis (LDA) and partial least squares discriminant analysis (PLS-DA) have been used in place of more appropriate class-modeling methods. Moreover, in the few cases in which a class-modeling technique such as soft independent modeling of class analogy (SIMCA) is applied, attention is frequently focused only on its classification performance (e.g., correct classification rate). Use of such a restricted focus considerably underutilizes the significant characteristics of the class-modeling approach. 

Classification and class-modeling techniques belong to three main families ... 

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