Augmentation of rule-based models with a granular quantification of results

This study is concerned with the augmentation of rule-based models or fuzzy rule-based models by associating the numeric results produced by them with granular characterization in the form of prediction intervals.The role of prediction intervals encountered in regression analysis is well emphasized in the literature, especially for "monolithic" models such as linear regression or neural networks. However, there have not been comprehensive and algorithmically complete studies devoted to the conceptualization and determination of prediction intervals for rule-based models, Boolean (two-valued) and fuzzy ones. While the results generated by rule-based models are formed by aggregating partial outcomes resulting from the individual rules, the prediction intervals adhere to the same way of aggregation. In this sense, one can regard the rule-based model augmented by the associated prediction intervals as a granular rule-based model. The construction of rule-based models is, in a nutshell, the optimization process being commonly guided by the sum of squared errors (say, RMSE or alike), and the quality of the granular counterpart of the rule-based models is assessed by looking at the quality of prediction intervals and introducing the two pertinent quality measures focused on the granular nature of the obtained results, namely coverage and specificity along with their combined index. In this study, we analyze an impact of the rules and the parameters of fuzzy clustering on the quality of the numeric and granular performance of the models. A series of experimental results is presented to help quantify the performance of granular outputs (prediction intervals) constructed for rule-based models.