A Two-Phase Development of Fuzzy Rule-Based Model and Their Analysis

Fuzzy rule-based models form a commonly encountered category of fuzzy models. As such they have enjoyed a great deal of conceptual and algorithmic developments followed by numerous case studies. This paper contributes to this area by bringing forward a two-phase design of fuzzy rules completed on the basis of experimental data. This design directly reflects upon the nature of the rules vis-à-vis the data used in their construction. First, information granules (fuzzy sets) standing in condition and conclusion parts of the individual rules are formed following a commonly used clustering technique of Fuzzy C-Means (FCM). The results of fuzzy clustering are directly used to build a collection of fuzzy sets of conditions and conclusions forming the individual rules. Some optimization aspects are raised in this context by expressing the performance of the condition and conclusion fuzzy sets in terms of the reconstruction abilities of the data captured by the rules. Second, fuzzy sets present in the rules (which are typically described by membership functions having infinite support) are transformed into interval-valued information granules of finite support that capture the essential (core) relationships between the regions in the input and output spaces strongly supported by the experimental data. In this way, the proposed rule-based model exhibits a two-tier architecture built in two successive phases. Subsequently, the proposed architecture invokes two fundamentally different modes of reasoning: 1) a recall mode in case when a new datum is positioned within the interval-valued information granules and 2) approximation mode where we invoke an aggregation of the individual rules given their activation levels in case a new datum does not belong to the core structure of the rules. These two modes produce granular results (represented as intervals). A way of assessing the quality of the obtained results is provided. Along with these two modes, we offer a characterization of the quality of results as well as the quality of the rules (expressed in terms of coverage, specificity of condition space and specificity of conclusion space). Experimental results are reported to illustrate the design process and the performance of the constructed model.