Designing interpretable Hierarchical Fuzzy Systems

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Abstract
Complexity is a typical criteria to analyze interpretability of fuzzy systems: number of variables, terms, rules, etc. Hierarchical fuzzy systems have shown a great potential to reduce fuzzy systems complexity. On the other hand, the counterpart to complexity reduction is the presence of synthetic variables generated at intermediate levels of the hierarchy. Those synthetic variables are generally affected by an absolute absence of semantics, minimizing their interpretability. As a consequence, when analyzing interpretability in hierarchical fuzzy systems, complexity should only be a part of the equation, since semantics should also be considered. And particularly, semantics of the intermediate variables added to reduce complexity. The use of hierarchical fuzzy systems will only produce an effective interpretability improvement when the design of the hierarchical structure was driven by the semantics of the intermediate variables. In other words, intermediate variables should be interpretable in terms of the problem under analysis. This consideration, made in the framework of hierarchical fuzzy systems, could be extended to any kind of hierarchical system defined under the umbrella of explainable artificial intelligence. The present paper will consider different aspects of interpretability concerning hierarchical fuzzy systems, and will explore the role of intermediate variables in accordance with the previous comments. The appropriate selection of those intermediate variables will drive to a process where subsystems will be decoupled for design and interpretation. Under this assumption we will finally consider the measurement of interpretability in hierarchical fuzzy systems as a process where the interpretability of each component (fuzzy system) of the hierarchy is first evaluated and then aggregated to achieve an overall interpretability evaluation.

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