Finding an optimal interval length in high order fuzzy time series Erol Egrioglu a , Cagdas Hakan Aladag b, * , Ufuk Yolcu a , Vedide R. Uslu a , Murat A. Basaran c a Department of Statistics, Ondokuz Mayis University, Samsun 55139, Turkey b Department of Statistics, Hacettepe University, Ankara 06532, Turkey c Department of Mathematics, Nigde University, Nigde 51000, Turkey article info Keywords: Forecasting Fuzzy sets High order fuzzy time series forecasting model Length of interval Optimization abstract Univariate fuzzy time series approaches which have been widely used in recent years can be divided into two classes, which are called first order and high order models. In the literature, it has been shown that high order fuzzy time series approaches improve the forecasting accuracy. One of the important parts of obtaining high accuracy forecasts in fuzzy time series is that the length of interval is very vital. As men- tioned in the first-order models by Egrioglu, Aladag, Basaran, Uslu, and Yolcu (2009), the length of inter- val also plays very important role in high order models too. In this study, a new approach which uses an optimization technique with a single-variable constraint is proposed to determine an optimal interval length in high order fuzzy time series models. An optimization procedure is used in order to determine optimum length of interval for the best forecasting accuracy, we used optimization procedure. In the optimization process, we used a MATLAB function employing an algorithm based on golden section search and parabolic interpolation. The proposed method was employed to forecast the enrollments of the University of Alabama to show the considerable outperforming results. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Fuzzy time series approaches have been successfully applied to the data such as stock exchange, temperature and enrollment which include uncertainty. Fuzzy time series approaches have found many diversified application areas since it differs from con- ventional approaches in many respects. The most important is that it does not require the check of theoretical assumptions. Fuzzy time series approach was firstly proposed by Song and Chissom (1993a, 1993b, 1994). Sullivan and Woodall (1994) proposed a method based on Markov model. Chen (1996) also proposed a new method which is simple since it does not require matrix operations. Huarng (2001) pointed out that the interval length influences the forecasting performance and proposed two methods which are based on the average and the distribution, for defining the length. Egrioglu, Aladag, Basaran, Uslu, and Yolcu (2009) introduced a new approach based on the optimization of the interval length. The studies mentioned above can be catego- rized under the name of first-order fuzzy time series model. Since first-order fuzzy time series models have a simple struc- ture, they can generally be insufficient to explain more complex relationships. For this reason, Chen (2002) proposed a new method which analyzes a high order fuzzy time series forecasting model. Aladag, Basaran, Egrioglu, Yolcu, and Uslu (2009) introduced a new approach, which uses a feed-forward neural network for defining fuzzy relations and is based on a high order fuzzy time series forecasting model. In this study, a new approach is proposed so that it analyzes a high order fuzzy time series forecasting model by optimizing the interval length which plays an important role in partitioning the universe of discourse of the time series. The optimization of the interval length is executed by a MATLAB function called ‘‘fminbnd” which uses the polynomial interpolation together with golden section search. The proposed method is applied to the data of enrollments of Alabama University. The obtained results have been compared to the results from the first and high order ap- proaches available in the literature. In Section 2, the fundamental definitions about fuzzy time ser- ies are presented. In Section 3, the method of Chen (2002) is given. In subsequent section, the proposed method and its application re- sults are presented. The final section provides a brief conclusion. 2. Fuzzy time series The definition of fuzzy time series was firstly introduced by Song and Chissom (1993a, 1993b). In fuzzy time series approaches, the validation of theoretical assumptions does not needs to be checked just as in conventional time series procedures. The most important advantage of fuzzy time series approaches is to be able 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.12.006 * Corresponding author. Tel.: +90 3122977900. E-mail address: chaladag@gmail.com (C.H. Aladag). Expert Systems with Applications 37 (2010) 5052–5055 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa