1063-6706 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2014.2300134, IEEE Transactions on Fuzzy Systems TFS-2013-0379.R1 1 Abstract— In this study, we propose a cluster–oriented development of fuzzy models. An overall design process is focused on an efficient usage of fuzzy clustering, Fuzzy C–Means (FCM), in particular, to form information granules–clusters used in the construction of the fuzzy model. Fuzzy models are regarded as mappings from information granules expressed in the input and output spaces. This position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules. The study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages. The augmented FCM method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and in this way the performance of the resulting fuzzy model is optimized. It is shown that the cluster–oriented modeling gives rise to the Mamdani–like fuzzy rules and a zero–order Takagi–Sugeno model (under a certain decoding scheme). We identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding–decoding (or granulation–degranulation) encountered in processing fuzzy sets and Granular Computing, in general. Further refinements of zero–order fuzzy models are investigated leading to first–order fuzzy models with linear functions standing in the conclusions of the rules. A series of experiments is reported where we used synthetic and real–world data in which an issue of generalization capabilities is elaborated in detail. Index Terms— fuzzy modeling, information granules, Fuzzy C–Means, augmented clustering, rule–based model I. INTRODUCTION UZZY models and fuzzy modeling dwell upon a concept of information granules – fuzzy sets. Fuzzy sets form a blueprint of any fuzzy model and contribute to the key features of fuzzy models such as their interpretability. Fuzzy clustering as an essential mechanism of building information granules plays a pivotal role in fuzzy modeling. Fuzzy C– Means (FCM) [4] is well documented as a design vehicle of fuzzy sets. In fuzzy modeling, especially in rule–based models, fuzzy sets appear always in the condition and Manuscript received June 10, 2013. This work was supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Research Chair Program, the Alberta Innovates – Technology Futures and Alberta Advanced Education & Technology. W. Pedrycz is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2V4, with the Department of Electrical and Computer Engineering Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia, and with the System Research Institute, Polish Academy of Sciences, Warsaw 00-716, Poland (e-mail: wpedrycz@ualberta.ca). H. Izakian is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2V4 (e-mail: izakian@ualberta.ca). sometimes in the conclusion parts of the rules. When being considered directly in the setting of fuzzy models, fuzzy clustering leads to the formation of fuzzy sets in the input and output space. A construction of these fuzzy sets has to be completed having in mind their role being played in fuzzy models. In other words, one has to be aware that there is a mapping from the input to the output space and fuzzy clusters are involved in the realization of this granular mapping. The input and output space (and fuzzy sets therein) are to be considered together with some provisions to incorporate some flexibility to clustering being cognizant of the directionality of the mapping under construction. There are several essential contributions of this study and those along with well-motivating factors all together bring forward a certain facet of originality: • We clearly identify a role of information granules–fuzzy sets in the construction of fuzzy models. In this context, it becomes imperative to associate a way in which information granules are constructed with an integral role they play in the formation of the model. This linkage is revealed and directly exploited in the re-formulation of the objective function used in clustering. The modified objective function brings an important feature of directionality to the clustering process. Clustering, in its virtue, is inherently direction-free, viz. no distinction is being made among input and output variables when forming clusters. By introducing a carefully structured distance function, we articulate relationships between information granules formed in the spaces of input and output variables. • We recall a fundamental concept of information granulation-degranulation [5][6][7] and demonstrate that this idea is exploited in Mamdani-like fuzzy models. In essence, fuzzy models can be sought as constructs that are formed directly on a basis of linkages between information granules. • Further refinements of fuzzy models are realized through a Taylor-like expansion where conclusions of the rules are completed around linearization points being the prototypes of the clusters. This makes the resulting refined constructs easily interpretable. Proceeding with more details, we propose a cluster–centric development methodology of fuzzy models. We develop a detailed design process involving an augmented FCM algorithm endowed with an ability to construct interrelated clusters that effectively realize granular mapping. From a functional point of view, the cluster–centric models realize a Mamdani–like fuzzy models or are equivalent to the zero– order Takagi–Sugeno (T–S) fuzzy models (assuming a certain format of the decoding – defuzzification scheme). From the design perspective, the process revolves around Cluster–Centric Fuzzy Modeling Witold Pedrycz, Fellow, IEEE, and Hesam Izakian, Student Member, IEEE F