Regression-Type Models for Kohonen's Self-Organizing Networks

In: R.Decker, W. Gaul (eds): Classification and Data Analysis. Proc. 23th Annual Conference of the Gesellschaft für Klassifikation, Univ. Bielefeld, 10-12 March 1999. Springer-Verlag, Heidelberg, 2000 (submitted).

Kohonen 'self-organizing' maps (SOMs) visualize high-dimensional data x₁, x₂, ... Î R^p essentially by combining two steps: (1) clustering the data points into classes, and (2) displaying 'neighbouring' classes by 'neighbouring' vertices in a given low-dimensional lattice (map). This paper generalizes the classical approach where each class is represented by a typical point in R^p (class center) to the case where (a) the data u₁ = (x₁, y₁), (x₂, y₂), ... Î R^p+q are partitioned into an explanatory part x_k Î R^pand a response part y_k Î R^q and (b) each class is represented by a regression hyperplane. The class-specific linear regression functions can be combined to a global one, and classes with similar response functions are visualized as neighbours in the map. In analogy to classical cluster analysis, we define an optimum regression-type SOM by a (discrete or continuous) generalized clustering criterion (K-criterion) and propose SOM algorithms derived from k-means, MacQueen's sequential approach, or from stochastic approximation. The resulting SOMs can be applied, e.g., in segmented credit scoring or for response surface approximation.

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