New aspects of the elastic net algorithm for cluster analysis.

The elastic net algorithm formulated by Durbin-Willshaw as a heuristic method and initially applied to solve the traveling salesman problem can be used as a tool for data clustering in n-dimensional space. With the help of statistical mechanics, it is formulated as a deterministic annealing method, where a chain with a fixed number of nodes interacts at different temperatures with the data cloud. From a given temperature on the nodes are found to be the optimal centroids of fuzzy clusters, if the number of nodes is much smaller than the number of data points. We show in this contribution that for this temperature, the centroids of hard clusters, defined by the nearest neighbor clusters of every node, are in the same position as the optimal centroids of the fuzzy clusters. The same is true for the standard deviations. This result can be used as a stopping criterion for the annealing process. The stopping temperature and the number and sizes of the hard clusters depend on the number of nodes in the chain. Test was made with homogeneous and nonhomogeneous artificial clusters in two dimensions. A medical application is given to localize tumors and their size in images of a combined measurement of X-ray computed tomography and positron emission tomography.