Stochastic Manipulations and Graphical Representations of the Similarity Matrix

STOCHASTIC MANIPULATIONS AND GRAPHICAL REPRESENTATIONS OF THE SIMILARITY MATRIX 

The similarity matrix Z obtained here, can naturally be further manipulated in a number of ways. One option that has been explored in some detail is the stochastic transformation, which implies that all elements in a certain column of the matrix are divided by the sum of all elements in this column. Denoting the sum of the elements of a column I as Zi , the stochastic matrix is given by 

It is immediately clear that the resulting matrix S is no longer a symmetrical matrix. On the other hand, the sum of all elements in a column of S belong to a unit shell vector semispace (see below), meaning that Vf (s ) 1. It is clear that the columns of the stochastic matrix become a new descriptor set for the 

We have already used similarity matrices to cluster molecules. As such, they provide the necessary data to investigate the construction of a molecular set taxonomy. The most common techniques to do so include the molecular point clouds previously described. There, the columns of the molecular quantum similarity matrix yielded coordinates of the molecules in the N-dimensional space. Often, the N dimensionality cannot yet be used for a graphical representation. However, several techniques exist to reduce the dimensionality of the data, which allow it to be represented graphically on common devices like a computer screen or a plotter. In addition to these plots, several instances have involved Kruskal trees and other algorithms. ° 

Such dendrograms can reveal interesting additional information about the similarity not only between two molecules in pair-wise comparisons, but also between clusters of molecules. 

---