# Visualization

## Topology based visualization

As in practical data mining problems high-dimensional data has to be analyzed, it is very informative to map and visualize the hidden structure of the complex data set in a low-dimensional space. However, the data set to be analyzed often includes lower-manifolds that are nonlinearly embedded in a higher-dimensional vector space. In this project we suggested a new graph based visualization methods to unfold the hidden structure of data. The proposed method combines the main benefits of the topology representing networks and the multidimensional scaling.

A. Vathy-Fogarassy, J. Abonyi, Local and globalmappings of topology representing networks, Information Sciences, Elsevier (21) pp. 3791-3803. 2009.

A. Vathy-Fogarassy, A. Kiss, J. Abonyi, Topology representing network map– A new tool for visualization of high-dimensional data, LNCS Transactions on Computational Science, pp. 61-84, 2008.

A. Vathy-Foggarasy, J. Abonyi, Topology representingnetworks based visualizations of manifolds, Journal of mathematical modelling and algorithms, pp. 351-370. 2008.

## Visualization of fuzzy clustering results

Since in practical data mining problems high-dimensional data are clustered, the resulting clusters are high-dimensional geometrical objects, which are difﬁcult to analyze and interpret. Cluster validity measures try to solve this problem by providing a single numerical value. As a low dimensional graphical representation of the clusters could be much more informative than such a single value, this paper proposes a new tool for the visualization of fuzzy clustering results. By using the basic properties of fuzzy clustering algorithms, this new tool maps the cluster centers and the data such that the distances between the clusters and the data-points are preserved. During the iterative mapping process, the algorithm uses the membership values of the data and minimizes an objective function similar to the original clustering algorithm. Comparing to the original Sammon mapping not only reliable cluster shapes are obtained but the numerical complexity of the algorithm is also drastically reduced. The developed tool has been applied for visualization of reconstructed phase space trajectories of chaotic systems. The case study demonstrates that proposed FUZZSAMM algorithm is a useful tool in user-guided clustering.

B. Feil, B. Balasko, J. Abonyi, Visualization of fuzzy clusters by fuzzy sammon mapping projection – Application to the analysis of phase space trajectories, special issue of "Soft Computing" on "Soft Computing for Information Mining", 11, (5), 479-488,. 2006 (IF: 0.33)

J. Abonyi, R. Babuska, FUZZSAM - Visualization of fuzzy clustering results by modified Sammon mapping, 13th International Conference on Fuzzy Systems, Budapest, Hungary, 2004.07.25-2004.07.29. New York: IEEE Press, 2004. pp. 365-370.

## Visualization and Complexity Reduction of Neural Networks

The identification of the proper structure of nonlinear neural networks (NNs) is a difficult problem, since these black-box models are not interpretable. The aim of the paper is to propose a new approach that can be used for the analysis and the reduction of these models. It is shown that NNs with sigmoid transfer function can be transformed into fuzzy systems. Hence, with the use of this transformation NNs can be analyzed by human experts based on the extracted linguistic rules. Moreover, based on the similarity of the resulted membership functions the hidden neurons of the NNs can be mapped into a two dimensional space. The resulted map provides an easily interpretable figure about the redundancy of the neurons. Furthermore, the contribution of these neurons can be measured by orthogonal least squares technique that can be used for the ordering of the extracted fuzzy rules based on their importance. A practical example related to the dynamic modeling of a chemical process system is used to prove that synergistic combination of model transformation, visualization and reduction of NNs is an effective technique, that can be used for the structural and parametrical analysis of NNs.

Kenesei, Tamas, Balazs Feil, Janos Abonyi, Visualization and Complexity Reduction of Neural Networks, Applications of Soft Computing. Springer Berlin Heidelberg, 2009. 43-52.

## Node Similarity Based Graph Clustering and Visualization

The basis of the presented methods for the visualization and clustering of graphs is a novel similarity and distance metric, and the matrix describing the similarity of the nodes in the graph. This matrix represents the type of connections between the nodes in the graph in a compact form, thus it provides a very good starting point for both the clustering and visualization algorithms. Hence visualization is done with the MDS (Multidimensional Scaling) dimensionality reduction technique obtaining the spectral decomposition of this matrix, while the partitioning is based on the results of this step generating a hierarchical representation. A detailed example is shown to justify the capability of the described algorithms for clustering and visualization of the link structure of Web sites.

Erdelyi M, Abonyi J, Node Similarity-based Graph Clustering and Visualization, 7th International Symposium of Hungarian Researchers Computational Intelligence, Budapest, Hungary, 2006.11.24-2006.11.25. Budapest: Budapesti Műszaki Főiskola; Magyar Fuzzy Társaság, 2006. pp. 1-12.