Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 935815, 12 pages http://dx.doi.org/10.1155/2013/935815 Research Article An ARMA Type Fuzzy Time Series Forecasting Method Based on Particle Swarm Optimization Erol Egrioglu, 1 Ufuk Yolcu, 2 Cagdas Hakan Aladag, 3 and Cem Kocak 4 1 Department of Statistics, Ondokuz Mayıs University, 55139 Samsun, Turkey 2 Department of Statistics, Ankara University, 06100 Ankara, Turkey 3 Department of Statistics, Hacettepe University, 06100 Ankara, Turkey 4 Medical High School, Hitit University, 19000 C ¸ orum, Turkey Correspondence should be addressed to Erol Egrioglu; erole@omu.edu.tr Received 18 April 2013; Revised 21 June 2013; Accepted 22 June 2013 Academic Editor: Ming Li Copyright © 2013 Erol Egrioglu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the literature, fuzzy time series forecasting models generally include fuzzy lagged variables. Tus, these fuzzy time series models have only autoregressive structure. Using such fuzzy time series models can cause modeling error and bad forecasting performance like in conventional time series analysis. To overcome these problems, a new frst-order fuzzy time series which forecasting approach including both autoregressive and moving average structures is proposed in this study. Also, the proposed model is a time invariant model and based on particle swarm optimization heuristic. To show the applicability of the proposed approach, some methods were applied to fve time series which were also forecasted using the proposed method. Ten, the obtained results were compared to those obtained from other methods available in the literature. It was observed that the most accurate forecast was obtained when the proposed approach was employed. 1. Introduction Fuzzy time series frstly proposed by Song and Chissom [1] can be divided into two subclasses time variant and time invariant. Fuzzy time series method generally embodies three stages such as fuzzifcation, determination of fuzzy relations, and defuzzifcation stages. In the fuzzifcation stage, obser- vations of time series are fuzzifed. Fuzzy relations between the observations are defned in the stage of determination of fuzzy relations. Finally, the calculated fuzzy forecasts are defuzzifed in the defuzzifcation phase. If one or more of these stages can be improved, the performance of the method will increase. Terefore, new fuzzy time series approaches have been proposed by making contributions to these stages in the literature. In the literature, various methods have been proposed to fuzzify observations. While fxed interval lengths are used in Song and Chissom [1–3], Chen [4], Huarng [5], Chen [6], Tsaur et al. [7], Singh [8], and Egrioglu et al. [9, 10], dynamic length of interval lengths is employed in Huarng and Yu [11], Davari et al. [12], Yolcu et al. [13], Kuo et al. [14, 15], Park et al. [16], Hsu et al. [17], and Huang et al. [18] in order to partition the universe of discourse. Also, Cheng et al. [19], Li et al. [20], Egrioglu et al. [21], Chen and Tanuwijaya [22], and Bang and Lee [23] used some methods based on clustering algorithms. Song and Chissom [3] exploited feed forward neural networks to defuzzify fuzzy forecasts. Chen [4], Huarng [5], and Huarng and Yu [11] utilized centroid method in the defuzzifcation stage. Besides, Cheng et al. [24] and Aladag et al. [25] used a diferent technique based on adaptive expectation and centroid methods for fuzzifcation. Establishing fuzzy relations plays important role in the forecasting performance of the method. In this phase, Song and Chissom [1] utilized fuzzy relationship matrix, and Sullivan and Woodall [26] used transition matrices based on Markov chain instead of fuzzy relationship matrix. Chen [4] suggested an approach in which fuzzy logic group relation- ship tables are employed to defne fuzzy relations, and Cheng et al. [24], Huarng [5], Huarng and Yu [11], Yu [27], and Egrioglu et al. [21] also used fuzzy logic group relationship tables. Huarng and Yu [28] and Aladag et al. [29] preferred to utilize feed forward artifcial neural networks in this stage.