A Study on Fuzzy Cognitive Map Optimization Using Metaheuristics
Fuzzy Cognitive Maps (FCMs) are a framework based on weighted directed graphs which can be used for system modeling. The relationships between the concepts are stored in graph edges and they are expressed as real numbers from the \([-1,1]\) interval (call
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Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland {a.cislak,a.jastrzebska}@mini.pw.edu.pl Faculty of Economics and Informatics in Vilnius, University of Bialystok, Kalvariju g. 135, LT-08221 Vilnius, Lithuania [email protected] Abstract. Fuzzy Cognitive Maps (FCMs) are a framework based on weighted directed graphs which can be used for system modeling. The relationships between the concepts are stored in graph edges and they are expressed as real numbers from the [−1, 1] interval (called weights). Our goal was to evaluate the effectiveness of non-deterministic optimization algorithms which can calculate weight matrices (i.e. collections of all weights) of FCMs for synthetic and real-world time series data sets. The best results were reported for Differential Evolution (DE) with recombination based on 3 random individuals, as well as Particle Swarm Optimization (PSO) where each particle is guided by its neighbors and the best particle. The choice of the algorithm was not crucial for maps of size roughly up to 10 nodes, however, the difference in performance was substantial (in the orders of magnitude) for bigger matrices.
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Introduction
Real-world phenomena modeling requires a framework that would not be hindered a by variety and diversity of relevant information. Standard methods, for instance for time series modeling, are predominantly numerical and are not well-fitted to process data in a form different than a sequence of numbers. An impressive range of fuzzy and granular models has emerged as a remedy for such issues. Fuzzy Cognitive Maps (FCMs) have been proposed by Kosko [11] in 1986 as an alternative framework for phenomena modeling. FCMs represent knowledge in the form of a directed graph. Phenomena are stored in vertices, while edges represent their relationships. These relationships are expressed as real numbers from the [−1, 1] interval. Weight matrix (or connection matrix) is a formal representation of each FCM as it gathers all weights in the map. In our research we focus on the application of FCMs to time series modeling, a domain relatively new as it has emerged in the 2000s [20]. In this paper, we present a study on a very important aspect of modeling with FCMs, namely on weight matrix learning procedures. In general, the core of each FCM, the weight matrix, can be constructed in three ways: (a) manually, c IFIP International Federation for Information Processing 2016 Published by Springer International Publishing Switzerland 2016. All Rights Reserved K. Saeed and W. Homenda (Eds.): CISIM 2016, LNCS 9842, pp. 577–588, 2016. DOI: 10.1007/978-3-319-45378-1 51
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by human experts; (b) automatically, using optimization algorithms; (c) with the combination of the two aforementioned options. The first and the last option often turn out to be inapplicable, as they require human expert knowledge. A convenient alternative is offered by automated approaches that are able to determine the shape of the weight matr
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