IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 1, FEBRUARY 2008 61 Numerical and Linguistic Prediction of Time Series With the Use of Fuzzy Cognitive Maps Wojciech Stach, Member, IEEE, Lukasz A. Kurgan, Member, IEEE, and Witold Pedrycz, Fellow, IEEE Abstract—In this paper, we introduce a novel approach to time- series prediction realized both at the linguistic and numerical level. It exploits fuzzy cognitive maps (FCMs) along with a recently pro- posed learning method that takes advantage of real-coded genetic algorithms. FCMs are used for modeling and qualitative analysis of dynamic systems. Within the framework of FCMs, the systems are described by means of concepts and their mutual relationships. The proposed prediction method combines FCMs with granular, fuzzy-set-based model of inputs. One of their main advantages is an ability to carry out modeling and prediction at both numerical and linguistic levels. A comprehensive set of experiments has been carried out with two major goals in mind. One is to assess quality of the proposed architecture, the other to examine the influence of its parameters of the prediction technique on the quality of predic- tion. The obtained results, which are compared with other predic- tion techniques using fuzzy sets, demonstrate that the proposed ar- chitecture offers substantial accuracy expressed at both linguistic and numerical levels. Index Terms—Fuzzy cognitive maps (FCMs), fuzzy systems, lin- guistic prediction, prediction methods, time series. I. INTRODUCTORY COMMENTS AND MOTIVATION T HIS paper proposes a novel application of fuzzy cognitive maps (FCMs) to time-series analysis. Although applica- tions of FCMs include a wide range of research and industrial areas, with specific examples including diagnosis of language impairment (SLI) [5], analysis of electrical circuits [28] and failure modes effects [18], fault management in distributed network environment [14], modeling and analysis of business performance indicators [10], supervisory control systems [29], software development projects [22], [26], virtual worlds [4], plant control [7], representation of political affairs [12], and ge- ographic information systems [19], their use in analysis of time series has not been considered so far. In the proposed approach, FCMs along with their recently introduced genetic algorithm- based learning mechanism are aimed to provide the following: 1) a description of a given time series at a certain abstraction level and 2) numerical and linguistic predictions (explained in Section III). This paper introduces a complete, highly modular Manuscript received November 18, 2005; revised April 30, 2006 and January 10, 2007. This work was supported in part by the Alberta Ingenuity, the Alberta Informatics Circle of Research Excellence (iCORE), the Natural Sciences & Engineering Research Council of Canada (NSERC), and the Canada Research Chair (CRC) program. The authors are with the Electrical and Computer Engineering Department, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: wstach@ece. ualberta.ca; lkurgan@ece.ualberta.ca; pedrycz@ece.ualberta.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2007.902020 architecture of this prediction system, as well as provides results of carefully performed experiments that validate the usefulness of the design. This includes experiments performed with diverse configurations of the system, tests on various data sets, and analysis of impact of the statistical characteristics of data sets on the prediction accuracy. Our results are compared with other state-of-the-art prediction methods, which are based on fuzzy sets. The main objectives and contribution of this paper, including the motivation for choosing FCMs, are the following. 1) Application of FCMs to time-series prediction. The moti- vation behind using this particular technique comes from its simple and comprehensive structure. It consists of con- cepts connected by mutual relationships and is adaptable to a given domain. FCMs are capable of capturing behavior of a given dynamic system. Recently introduced learning al- gorithm based on genetic optimization (genetic algorithm) allows for automated development of the FCM from his- torical data. This learning approach is flexible with respect to the input data, i.e., each two observations at the succes- sive time points and can be used to learn the map. For instance, if some observations in the historical data are missing, all the remaining pairs of points can be still suc- cessfully used for learning. 2) Design and development of the highly modular prediction system based on FCMs that is able to perform prediction at two levels, i.e., numerical and linguistic. Fig. 1 highlights the key design phases of the proposed system. The proposed architecture falls within the realm of fuzzy modeling and consists of three well-delineated and func- tionally distinct modules. They are as follows: 1) input in- terface, 2) processing core formed by an FCM, and 3) the output interface. The modeling and prediction activities supported by the FCM are realized at the linguistic level as opposed to the numerical one at which the experimental data become available. Therefore, in contrast to classical time-series prediction systems that predict only numerical values, the proposed system can also carry out prediction at the linguistic level. Let us briefly elaborate on the role of each of these mod- ules. The dynamics of a given numerical time series is cap- tured through its amplitude and change of the amplitude, say ( ). These values are transformed through a collection of predefined linguistic descriptors (Fig. 1, step 1), and become available in the form of their activation levels. Here, the encoding (fuzzification) process entails the determination of the membership values of the respec- tive fuzzy sets. The computations here are straightforward 1063-6706/$25.00 © 2007 IEEE