Multilevel fuzzy relational systems: structure and identification

 Existing fuzzy relational equations (FRE) typically possess an evident single-level structure, where no consequence part of the rule being modeled, is used as a fact to another rule. Corresponding to multistage fuzzy reasoning, a natural extension of traditional fuzzy relational systems (FRS) is to introduce some intermediate levels of processing governed by enhanced FRE's so that the structure resulted becomes multilevel or multistage. Three basic multilevel FRS structures, namely, incremental, aggregated, and cascaded, are considered in this paper and they correspond to different reasoning mechanisms being frequently used by human beings in daily life. While the research works on multilevel FRS are sparse and our ability to solve a system of multilevel FRE's in a purely analytical manner is very limited, we address the identification problem from an optimization approach and introduce three fuzzy neural models. The proposed models consist of single-level FRS modules that are arranged in different hierarchical manners. Each module can be realized by Lin and Lee's fuzzy neural model for implementing the Mamdani fuzzy inference. We have particularly addressed the problem of how to distribute the input variables to different (levels of) relational modules for the incremental and aggregated models. In addition, the new models can learn a complete multistage fuzzy rule set from stipulated data pairs using structural and parameter learning. The effectiveness of the multilevel models has been demonstrated through various benchmarking problems. It can be generally concluded that the new models are distinctive in learning, generalization, and robustness.