1063-6706 (c) 2017 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.2018.2793904, IEEE Transactions on Fuzzy Systems IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 1, NO. 1, AUGUST 9999 1 Two-Stage Learning based Fuzzy Cognitive Maps Reduction Approach Mikl´ os F. Hatw´ agner, Engin Yesil, Senior Member, IEEE, M. Furkan Dodurka, Member, IEEE, Elpiniki Papageorgiou Member, IEEE, Leon Urbas Member, IEEE, and L´ aszl´ o T. K´ oczy Senior Member, IEEE Abstract—In this study, a new two-stage learning based reduc- tion approach for Fuzzy Cognitive Maps (FCM) is introduced in order to reduce the number of concepts. FCM is a graphical modeling technique that follows a reasoning approach similar to the human reasoning and decision-making process. The FCM model incorporates the available knowledge and expertise in the form of concepts and in the direction and strength of the interactions among concepts. One of the modeling problems of FCMs is that over-sized FCM models suffer from interpretability problems. An over-sized FCM may contain concepts that are semantically similar and affects the other concepts in a similar way. This new study introduces a two-stage model reduction approach, and both static and dynamic analysis are considered without losing essential information. In the first stage the number of concepts is reduced by merging similar concepts into clusters, while in the second stage the transformation function parameters of concepts are optimized. In order to show the benefit of using the proposed reduction approach, two sets of studies are conducted. First, a huge set of synthetic FCMs are generated, and the results of these statistical analyses are presented via various tables and figures. Subsequently, suggestions to the decision makers are given. Second, experimental studies are also presented to show the decision parameters and procedure for the proposed approach. The results show that after using the concept reduction approach presented in this study, the interpretability of FCM increases with an acceptable amount of information loss in the output concepts. Index Terms—Fuzzy Cognitive Maps, Concept Reduction, Unsupervised and supervised reduction, Clustering, Big Bang- Big Crunch Optimization. I. I NTRODUCTION F UZZY Cognitive Map (FCM) is a graphical modeling technique that mimics a reasoning approach similar to human reasoning and human decision-making [1], [2]. FCM is a signed (c.f. bipolar [3]) fuzzy graph that consists of con- cepts (nodes) and relationships (weighted arcs). Each concept reflects a key factor, variable, or state of the modeled system. M. F. Hatw´ agner and L. T. K´ oczy are with the Department of Infor- mation Technology, Sz´ echenyi Istv´ an University, Gy˝ or, Hungary, e-mail: miklos.hatwagner@sze.hu, koczy@sze.hu; L. T. K´ oczy is also affiliated to the Department of Telecommunication and Media Informatics, Bu- dapest University of Technology and Economics, Budapest, Hungary, e-mail: koczy@tmit.bme.hu E. Yesil and M. F. Dodurka are with Chair for Process Control Sys- tems Engineering, Technische Universit¨ at Dresden, Dresden, Germany, and Getron Corp., 221 River Street, Hoboken, New Jersey 07030, USA, e-mail: engin.yesil@getron.com, furkan.dodurka@getron.com E. Papageorgiou is with Department of Computer Engineering, Technolog- ical Education Institute (TEI) of Central Greece, 3rd Km Old National Road Lamia-Athens, Lamia 35100, Greece, e-mail: epapageorgiou@teilam.gr L. Urbas is with Chair for Process Control Systems Engineering, Technische Universit¨ at Dresden, Dresden, Germany, e-mail: leon.urbas@tu-dresden.de Manuscript received April 19, 9999; revised August 26, 9999. The relationships between concepts indicate cause and effect relationships and express their degrees as well [4]. FCM models are used for both static and dynamic analysis. The first FCM paper by Kosko [1] proposed static analysis based on fuzzy causality. Because of the cyclic and non- feedforward structure, a concept may affect other concepts indirectly. In [5], Kosko suggested to use FCMs also for dynamic analysis, where each concept in the FCM has a value. Using the initial values of the concepts, a “what-if” analysis is performed using an inference formula. Such a dynamic analysis is crucial for decision makers since it helps them to see the future of decision concepts under certain conditions. The methodology for developing manual FCMs is described analytically in [6], [7]. As this construction of FCMs is mainly based on experts’ knowledge, there is a drawback as experts may be uncertain about the exact importances of the factors in the model. Often they include less important concepts and so the complexity of the model further increases (the number of connections is quadratic in terms of the number of concepts). The reduction of over-complicated FCMs is unambiguously an important research problem. A short literature review of the subject is provided in the next section. In this paper a novel FCM reduction approach will be proposed. Two goals are defined and thus a two-stage learning methodology is presented. The first goal (an extension of [8]) is a new ap- proach that copes with model reduction of FCMs for decision- making under uncertainty. Our reduction approach decreases the original model gradually in an automated way, until it reaches the required accuracy. The second objective is to keep the behavior of the original FCM intact, i.e. the output concepts of the reduced model should have similar values at the end of the simulation than the original model for the same initial concept values. (Output concepts are affected by other concepts but are not affecting any other concept.) To succeed with the first goal, an enhanced reduction ap- proach based on merging similar concepts into fuzzy clusters by using fuzzy tolerance relations with new calculations of the weights to be assigned to the edges among the new (merged) concepts is proposed. The approach uses a design parameter (ǫ) during merger. The reduced models are constructed with four different operators and compared with the original one by simulations. The idea of combining concepts into clusters in FCM is in itself not entirely new. Dickerson and Kosko proposed nested FCMs in [9]. However, the proposed new technique is entirely different. In contrary to nested FCMs, this technique does not use disjoint crisp clusters, thus the same original concept