High-dimensional linear maps

Also nonlinear methods can be applied to represent the high-dimensional variable space in a smaller dimensional space (eventually in a two-dimensional plane) in general such data transformation is called a mapping. Widely used in chemometrics are Kohonen maps (Section 3.8.3) as well as latent variables based on artificial neural networks (Section 4.8.3.4). These methods may be necessary if linear methods fail, however, are more delicate to use properly and are less strictly defined than linear methods. 

Foct, F 2 > AE, and Log P) were used in the cluster analysis. The results are shown in Fig. 3. This is a non-linear map of the descriptors in their four-dimensional space projected into two-dimensions. Nine different clusters can be identified. However, one cluster plus one member of a nearby cluster (the enclosed area within the map) contains all active compounds except for the 3-hydroxy nitroso-piperidine. That is, highly carcinogenic compounds cluster together while the "non-carcinogenic" cyclic nitrosamines are scattered about in the four-dimensional descriptor space. 

SupportVector Machines (a) linearclassifier, (b) nonlinear classifier, (c) nonlinear classifier mapped to a high-dimensional space, where the classification becomes linear. Crosses and circles denote preclassified points on which the support... 

SVR) maximizes the prediction accuracy of the classifier (regression) model while simultaneously escaping from data overfitting. In SVM, the inputs are first nonlin-early mapped into a high-dimensional feature space (O) wherein they are classified using a linear hyperplane (Fig. 3.4). 

In practice, a non-linear model is often required for adequate data fitting. In the same manner as the non-linear support vector classification approach, a non-linear mapping can be used to map the data into a high dimensional feature space where linear regression can be used (see Fig. 2.10). As noted in the previous subsection, the complete SVM can be described in terms of dot products between the data. The nonlinear SVR solution, using an f-insensitive loss function (2.54) is given by solving the problem ... 

SVM can also be used to separate classes that cannot be separated with a linear classifier (Figure 2, left). In such cases, the coordinates of the objects are mapped into a feature space using nonlinear functions called feature functions ( ). The feature space is a high-dimensional space in which the two classes can be separated with a linear classifier (Figure 2, right). 

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