Adaptive hierarchical cluster analysis by Self-Organizing Box Maps

The present paper aims at an extension of {\sc Kohonen's} Self-Organizing Map (SOM) algorithm to be called Self-Organizing Box Map (SOBM) algorithm; it generates box codebooks in lieu of point codebooks. Box codebooks just like point codebooks indirectly define a Voronoi tessellation of the input space, so that each codebook vector represents a unique set of points. Each box codebook vector comprises a multi-dimensional interval that approximates the related partition of the Voronoi tessellation. Upon using the automated cluster identification method that has recently been developed by the authors, the codebook vectors can be grouped in such a way that each group represents a point cluster in the input space. Since the clustering usually depends on the size of the SOM, one cannot be sure, whether the clustering comes out to be optimal. Refinement of part of the identified clusters would often improve the results. This paper presents the concept of an adaptive multilevel cluster algorithm that performs such refinements automatically. Moreover the paper introduces a concept of essential dimensions and suggests a method for their identification based on our herein suggested box codebooks. Applications of the algorithm to molecular dynamics will be described in a forthcoming paper.