Multi-Dimensional Scaling applied to Hierarchical Rule Systems

This paper presents an approach for visualizing highdimensional fuzzy rules arranged in a hierarchy together with the training patterns they cover. A standard multi-dimensional scaling method is used to map the rule centers of the top hierarchy level to one coherent picture. Rules of the underlying levels are projected relatively to their parent level(s). In addition to the rules, all patterns are mapped onto the two-dimensional projection in relation to the positions of the corresponding rule centers. Visualization is further extended by showing hierarchical relationships between overlapping rules of different levels, which are generated by a hierarchical rule learner. This delivers interesting insights into the rule hierarchy and offers better explorative properties. Additionally, rules can be highlighted interactively emphasizing the subsequent rules at all underlying levels together with the patterns they cover. We demonstrate that this technique allows investigation of interesting rules at different levels of granularity, which makes this approach applicable even for a large number of rules. The proposed technique is illustrated and discussed based on a number of hierarchical rule model visualizations generated from well-known benchmark data sets