Self-organizing maps

After that, all model vectors or a subset of them that belong to nodes centered around node c = c x) are updated as 

As the data items are mapped to those units on the map that have the closest reference vectors, nearby units will have similar data items mapped on them. This property requires that the map be regarded as an ordered display, which facilitates the understanding of the structures in the data set. This map display may also be used as an ordered groundwork, on which the original data variables, as well as other information related to the data cases, may be displayed. 

An interesting feature of the SOM is its ability to deal with missing data. Some of the components of the data vectors may not be available for all data items, or may not be applicable or defined. The simplest solution when dealing with such incomplete components would be to discard the incomplete variables or incomplete data items completely, but in this way we will lose useful information. In the case of the SOM, the problem of missing data may be treated as follows When choosing the winning unit, the input vector can be compared with the reference vectors nti using only those components that are 

The outliers are data items situated very far from the main body of data. This may be due to measurement errors, typographical errors, or may simply be correct data that is strikingly different from the rest. In the case of the maps generated by the SOM, each outlier affects only one map unit and its neighborhood, while the rest of the map is available for the rest of the analysis. Furthermore, the outliers can be detected in the map due to the much lower density of the space near them. 

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An observation of the results of cross-validation revealed that all but one of the compounds in the dataset had been modeled pretty well. The last (31st) compound behaved weirdly. When we looked at its chemical structure, we saw that it was the only compound in the dataset which contained a fluorine atom. What would happen if we removed the compound from the dataset The quahty ofleaming became essentially improved. It is sufficient to say that the cross-vahdation coefficient in-CTeased from 0.82 to 0.92, while the error decreased from 0.65 to 0.44. Another learning method, the Kohonen s Self-Organizing Map, also failed to classify this 31st compound correctly. Hence, we had to conclude that the compound containing a fluorine atom was an obvious outlier of the dataset. 

This format was developed in our group and is used fruitfully in SONNIA, software for producing Kohonen Self Organizing Maps (KSOM) and Coimter-Propaga-tion (CPG) neural networks for chemical application [6]. This file format is ASCII-based, contains the entire information about patterns and usually comes with the extension "dat . 

Now, one may ask, what if we are going to use Feed-Forward Neural Networks with the Back-Propagation learning rule Then, obviously, SVD can be used as a data transformation technique. PCA and SVD are often used as synonyms. Below we shall use PCA in the classical context and SVD in the case when it is applied to the data matrix before training any neural network, i.e., Kohonen s Self-Organizing Maps, or Counter-Propagation Neural Networks. 

Initially the dataset contained 818 compounds, among which 31 were active (high TA, low USE), 157 inactive (low TA, high USE), and the rest intermediate. When the complete dataset was employed, none of the active compounds and 47 of the inactives were correctly classified by using Kohonen self-organizing maps (KSOM). 

The Kohonen Self-Organizing Maps can be used in a. similar manner. Suppose Xj., k = 1,. Nis the set of input (characteristic) vectors, Wy, 1 = 1,. l,j = 1,. J is that of the trained network, for each (i,j) cell of the map N is the number of objects in the training set, and 1 and j are the dimensionalities of the map. Now, we can compare each with the Wy of the particular cell to which the object was allocated. This procedure will enable us to detect the maximal (e max) minimal ( min) errors of fitting. Hence, if the error calculated in the way just mentioned above is beyond the range between e and the object probably does not belong to the training population. 

Kohonen networks, also known as self-organizing maps (SOMs), belong to the large group of methods called artificial neural networks. Artificial neural networks (ANNs) are techniques which process information in a way that is motivated by the functionality of biological nervous systems. For a more detailed description see Section 9.5. 

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

Figure 15.4 Self-organizing maps showing the distribution of selectivity values [Ki (Ai)/kii (A2a)] of the initial 153-member library 1, and position of the most selective compound from the secondary combinatorial library [50].

Kohonen T. Self-organizing maps, 3rd edition. New York Springer Verlag, 2000. 

Schneider G, Nettekoven M. Ligand-based combinatorial design of selective purinergic receptor (A2A) antagonists using self-organizing maps. J Comb Chem 2003 5 233-7. 

The relationships between the molecular structure of environmental pollutants, such as polychlorinated biphenyls (PCBs), and their rate of biodegradation are still not well understood, though some empirical relationships have been established. Self-organizing maps (SOMs) have been used to rationalize the resistance of PCBs to biodegradation and to predict the susceptibility to degradation of those compounds for which experimental data are lacking.3 The same technique has been used to analyze the behavior of lipid bilayers, following a... 

Murtola, T., et al., Conformational analysis of lipid molecules by self-organizing maps, /. Chem. Phys., 125, 054707, 2007. 

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