Disjoint 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. 

Theory. SIMCA is a parametric classification method introduced by Wold (29), which supposes that the objects of a given class are normally distributed. The particularity of this PCA-based method is that one model is built for each class separately, that is, disjoint class modeling is performed. The algorithm starts by determining the optimal number of PCs for each individual model with CV. The resulting PCs are then used to define a hypervolume for each class. The boundary around one group of objects is then the confidence limit for the residuals of all objects determined by a statistical T-test (30, 31). The direction of the PCs and the limits established for these PCs define the model of a class (Fig. 13.13). 

Disjoint principal components modelling [266] and SIMCA (soft independent modelling of class analogy) [261,262,267] are examples of PCR wherein principal components models are developed for individual groups of responses within a data set. For these methods, classification is based on quality of fit of an unknown response pattern to the model developed for a given analyte [268-270]. This approach differs from standard PCR, where principal components are derived from the data matrix as a whole. 

This approach was originally developed by Wold (1976) under the name disjoint principal components models, later termed simple modelling of class analogy (SIMCA) (see also Wold and Sjostrom, 1977 Wold et al., 1983). While biological applications of SIMCA have been limited (e.g. Wold, 1976 Dahl et al., 1984), the technique exhibits some of the attributes of much more advauced neural-net architectures (see following discussion). Moreover, because of its basis in... 

Similarly to classes, a property can be equivalent to or disjoint with other properties. Object properties can have a set of property characteristics that add additional meaning to the ontological model. Some of the property characteristics used in this book are as follows ... 

Consistency checking detects any logical contradictions within an ontological model. For example, if two classes A and B are declared disjoint, and then an instance is added that is of both types A and B, this leads to an inconsistent ontology. This could be the case for classes mo Pump and mo Engine in our example domain, which can be declared disjoint. If an instance is asserted as an instance of both these classes, then the ontology model becomes inconsistent. 

Both spline kernels have a remarkable flexibility in modeling difficult data. This characteristic is not always useful, especially when the classes can be separated with simple nonlinear functions. The SVM models from Figure 38 (a, spline b, B spline, degree 1) show that the B spline kernel overfits the data and generates a border hyperplane that has three disjoint regions. 

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