On Hybrid Granular Min-Max Fuzzy-Neuro Relational Learners: Conception and Validation

This paper comprises two parts, the first deals with the conception of a class of Hybrid Granular Min-Max Fuzzy-Neuro Relational Learners, for which a learning scheme was devised that uses an exhaustive search over the fuzzy partitions of involved variables, automatic fuzzy hypotheses generation, formulation and testing, and successive approximation procedure of Min-Max relational equations. The main idea is to start learning from coarse fuzzy partitions of the involved input variables and proceed progressively toward fine-grained partitions until finding the appropriate partitions that fit the data. According to the complexity of the problem at hand, it learns the whole structure of the fuzzy system, i.e. conjointly appropriate fuzzy partitions, appropriate fuzzy production rules, their number and their associated membership functions. The fuzzy relational calculus in the context of approximation of fuzzy relations equations, constitutes a good candidate tool in machine learning, and is especially useful for dealing with inverse problems. The second part deals with verification and validation issues of such learners, validation brings us to a systematic study of value approximation performed during the inference (recall) phase. We provide a rigorous formal mathematical proof that Min-Max rule preserves the property of approximation when it is applied to entities characterized by approximately equal fuzzy values. Hence, using standard Min-Max is a suitable choice in building Hybrid Granular Fuzzy-Neuro or Neuro-Fuzzy Relational Learners, as it is accepted that generalization capability is proportional to value approximation. Keywords— Hybrid Granular Min-Max Fuzzy-Neuro Relational Learners, Importance, Approximation of Min-Max relational equations, Verification and Validation, Value approximation,