Multi-dimensional scaling applied to hierarchical rule systems

This paper presents an approach for visualizing high- dimensional 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 rel- atively 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 high- lighted interactively emphasizing the subsequent rules at all under- lying levels together with the patterns they cover. We demonstrate that this technique allows investigation of interesting rules at differ- ent 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 visual- izations generated from well-known benchmark data sets.