Multidimensional descriptors

In the multidimensional descriptor space, each compound can be described by a point, with the coordinates along the descriptor axes equal to the measured values of the corresponding property descriptor, see Fig. 15.2. 

We have seen that additional atom properties allow a discrimination of molecules beyond the three-dimensional structure. However, we will find cases where the information content of enhanced RDF descriptors is still not sufficient for a certain application. In particular, if the problem to be solved depends on more than a few parameters, it may be necessary to divide information that is summarized in the onedimensional RDF descriptors. Though the RDF descriptors introduced previously are generated in one dimension, it is generally possible to calculate multidimensional descriptors. In this case, we can extend the function into a new property dimension by simply introdncing the property into the exponential term... 

RDF descriptors may be used in any combination to fit the required task. For instance, it is possible to calculate a multidimensional descriptor based on bond-path distances and restricted to nonhydrogen atoms in the shape of a frequency pattern. Consequently, more than 1,400 different descriptors are available. A final summary of RDF descriptor types, their properties, and applications is given in Table 5.1. This section summarizes typical applications, some of which are described in detail in the next chapter. 

Fig. 3 Projection of the multidimensional descriptor space of a set of compounds usually reveals distinct clustering which reflects chemical similarity. Here, each compound is depicted as circle. According views can be obtained either from principal component analysis or cluster analysis. The centroids of each cluster (filled circles) are representative samples and can be used as test set. The remaining compounds form the training set, because they cover the available descriptor space adequately.

Geometric Methods Convex Hull AD. AD is defined as a convex hull of points in the multidimensional descriptor space (Fechner et al. 2008). 

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