Seasonal Fuzzy Time Series, Fuzzy C-means, Artificial Neural Network, Membership Degree, Air Pollution American Journal of Intelligent Sy stems 2013, 3(1): 13-19 DOI: 10.5923/j.ajis.20130301.02 A Novel Seasonal Fuzzy Time Series Method to the Forecasting of Air Pollution Data in Ankara Ozge Cagcag1, Ufuk Yolcu2, Erol Egrioglu1,*, Cagdas Hakan Aladag3 1Dep artment of Statistics, University of OndokuzM ay is, Samsun, 55139, Turkey 2Dep artment of Statistics, Giresun University, Giresun, 28000, Turkey 3Dep artment of Statistics, Hacettep eUniversity, Ankara, 06800, Turkey Abstract Fu zzy time series forecasting methods have been widely studied in recent years. This is because fuzzy time series forecasting methods are co mpatib le with fle xib le ca lculat ion techniques and they do not require constraints that exist in conventional time series approaches. Most of the real life time series e xh ibit periodical changes arising fro m seasonality. These variations are ca lled seasonal changes. Although, conventional time series approaches for the analysis of time series which have seasonal effect are abundant in literature, the number of fuzzy t ime series approaches is limited. In a lmost all of these studies, me mbership values are ignored in the analysis process. This affects forecasting performance of the approach negatively due to the loss of information as well as posing a situation that is incompatible with the basic features of fuzzy set theory. In this study, for the first time in literature, a new seasonal fuzzy time series approach which considers me mbe rship values in both identificat ion of fuzzy relat ions and defuzzification steps was proposed. In the proposed method, we used fuzzy C-means clustering method in fuzzification step and artificia l neural networks (ANN) in identification of fu zzy relation and defuzzification steps which consider me mbe rship values. The proposed method was applied to various seasonal fuzzy time series and obtained results were compared with some conventional and fuzzy time series approaches. In consequence of this evaluation, it was determined that forecasting performance of the proposed method is satisfactory. Keywords Seasonal Fuzzy T ime Se ries, Fuzzy C-means, Artific ia l Neura l Network, Me mbe rship Degree, Air Po llution 1. Introduction Nowadays, it is of vital impo rtance to make predict ions about the future in terms of planning and strategy formulat ion. This can be realized by accurate and realistic analysis of in formation and data that have e merged fro m past to present. This analysis can also be termed as time series analysis. Many different approaches have been proposed in literature for the analysis of time series. Each of these approaches has pros and cons. This leads to emergence of alternative methods which may enhance forecasting performance of the method namely fuzzy t ime series approaches which have a superior forecasting performance and which do not require hypothesis that are found in conventional approaches. On the other hand, due to the uncertainty that they contain, most of the time series encountered in real life should be considered as fuzzy time series. Fuzzy set theory proposed by Zadeh provides a basis for many studies as well as the fuzzy t ime series approaches [1]. The concept of fu zzy t ime series was first introduced by * Corresponding author: erole1977@y ahoo.com(Erol Egrioglu) Published online at http://journal.sapub.org/ajis Copyright Β© 2013 Scientific & Academic Publishing. All Rights Reserved Song and Chissom[2]. Fro m that day to this, fuzzy time series have been studied intensively and applied to many fie ld such as informat ion technologies, economy, finance, environment and hydrology. Since fuzzy time series approaches do not require constraints such as model assumption, the number of observations and normal distribution which e xist in conventional approaches and they are accordant with the use of fle xib le calcu lation methods, these approaches are becoming increasingly popular. Fu zzy time series approaches consist of three ma in steps as fuzzification, identification of fu zzy relat ions and defuzzificat ion. In literature, various approaches have been proposed for the improve ment of these steps. While some of these studies involve first order fuzzy time series forecasting models (such as[2-6]), others involve high order fuzzy time series forecasting models ([7-11]). A lthough, there are numerous approaches in literature fo r the analysis of fu zzy time series involving first and high degree fu zzy relat ions, few approaches have been proposed for the analysis of fuzzy time series involv ing seasonal fu zzy relat ions. However, most of the time series encountered in real life involve seasonal components. The use of seasonal fuzzy time series in the analysis of this type of fuzzy t ime series would be more rea listic and would provide superior forecas ting performance. In literature, the first approach for the analysis of seasonal 14 Ozge Cagcag et al. : A Novel Seasonal Fuzzy Time Series M ethod to the Forecasting of Air Pollution Data in Ankar a fuzzy t ime series was introduced by Song but this was not applied to any data[12]. In orde r to analy ze seasonal fuzzy time series, Egrioglu et al. proposed a hybrid fuzzy time series approach based on SARIMA and artificia l neural networks[13]. A lthough, the method proposed by Egrioglu et al. has some advantages, it uses universal set fragmentation in fuzzification step. These subjective judgments have negative impact on forecas ting performance of the method. In order to eliminate this proble m, Uslu et al. proposed an approach which does not require universal set frag mentation and which uses Fuzzy C-Means (FCM) in fuzzification step[14]. In all of these studies, only the fuzzy set having the highest me mbership value was considered in the analysis process and other fuzzy sets having lower me mbership values were ignored. This affects forecasting performance of the approach negatively due to the loss of informat ion as we ll as posing a situation that is incompatib le with the basic features of fuzzy set theory. In this study, we a imed to overcome the above mentioned factors which affect the forecasting performance of the method negatively in the analysis of a seasonal fu zzy time series. For this purpose, we proposed a new seasonal fuzzy time series fo recasting model wh ich considers me mbership values of each observations belonging to all fu zzy sets in both identification of fu zzy relat ion and defuzzificat ion steps. In the proposed model, we used FCM in fu zzification step and avoided subjective judgments and determined me mbe rship values with a systematic approach. We ut ilized ANN which considers all me mbe rship values in identification of fuzzy relat ion and prevented loss of informat ion and made use of fle xib le ca lculation ability of ANN. In defu zzificat ion step, artific ial neura l network which uses all me mbe rship values as input and real (crisp) values of time series as target was used for the first time in literature. The proposed method was applied to the amount of sulfur dio xide in Ankara and was co mpared with so me conventional time series approaches as well as fuzzy time series forecasting methods. In the second chapter, SARIMA models which were used in determining the model order, FCM wh ich was used in fuzzification step and ANN wh ich was used in determination of fu zzy relat ion and defuzzfication step will be introduced. Third chapter will deal with basic fu zzy t ime series concept and definitions. In the fourth chapter, proposed method and its algorith m will be g iven. In the fifth chapter, the proposed method will be applied to a rea l seasonal time series and obtained results will be presented with the other results obtained from other methods. In the last chapter, obtained results will be evaluated and discussed. 2. Review 2.1. SARIMA When a time series with πœ‡πœ‡ mean, than the model is e xpressed in equation (1) t s t Dsds aBBZBBBB )()()()1()1)(()( Θ=βˆ’βˆ’βˆ’Ξ¦ θ¡φ (1) Model parameters can be given as follows; )1()( 1 p p BBB φφφ βˆ’βˆ’βˆ’=  (2) )1()( 1 q q BBB ΞΈΞΈΞΈ +++=  (3) )1()( 1 sP P s BBB Ξ¦βˆ’βˆ’Ξ¦βˆ’=Ξ¦  (4) )1()( 1 sQ Q s BBB Θ++Θ+=Θ  (5) Detailed information on the model which is called seasonal autoregressive integrated moving average (SARIMA) and which is e xpressed as sQ)D,q)(P,d,SARIMA(p, can be obtained fro m Bo x and Jenkins[15]. 2.2. The Fuzz y C-Means (FCM) Clustering Technic al FCM clustering technical method is first introduced by Be zdek[16]. This is a most wide ly used fuzzy c lustering algorith m. FCM partit ions sets of n observation and each fuzzy c luster has a set center jv )1( ncc << . The me mbe rships of the observations are described by a fuzzy matrix Β΅ with n rows and c columns in which n is the number of data objects and c is the number of clusters. ijΒ΅ , the ele ment in the 𝑖𝑖ith row and jth colu mn in Β΅ , indicates the degree of association or me mbership function value of the ith object with the jth cluster. The characters of Β΅ are as follows: [ ]0,1 1, 2, , ; 1, 2, ,ij i n j cΒ΅ ∈ βˆ€ = βˆ€ =  (6) 1 1 1, 2, , c ij j i nΒ΅ = = βˆ€ =βˆ‘  (7) 1 0 1, 2, , c ij j n j cΒ΅ = < < βˆ€ =βˆ‘  (8) The objective function of FCM algorith m is to minimize the equation (9) 1 1 ( , , ) c n ij ij j i J X V dΞ²Β΅ Β΅ = = = βˆ‘βˆ‘ (9) where, ij i jd x v= βˆ’ (10) in wh ich, )1( >Ξ²Ξ² is a scalar termed the weighting e xponent and controls the fuzziness of the resulting clusters and ijd is the Euclid ian distance from object ix to the cluster center jv . In this method, minimizing is done by an iterative a lgorith m. In each repetition the values of ijΒ΅ and jv are updated by the formulas given in equation (11) and equation (12). American Journal of Intelligent Sy stems 2013, 3(1): 13-19 15 1 1 n ij j j n ij i i x v Ξ² Ξ² Β΅ Β΅ = = = βˆ‘ βˆ‘ (11) ( ) 2 1 1 1 ij c ij ikk d d Ξ² Β΅ βˆ’ = =  ο£Ά  ο£· ο£­ ο£Έ βˆ‘ (12) 2.3. Fee d For war d Ne ural Network Artificia l neural networks (ANN) can be defined as the mathe matica l a lgorith m that is inspired by the biological neural networks[17]. Art ific ial neural networks are much more d ifferent than biological ones in terms of their structure and ability[18]. Artific ia l neural networks compose of a mathe matica l mode l[19]. The learning capability of an artific ial neuron is achieved by ad justing the weights in accordance to the chosen learning algorith m. The basic architecture consists of three types of neuron layers: input, hidden, and output layers. In feed-forward networks, the signal flow is fro m input to output units, strictly in a feed-forward direct ion. Artific ial neura l network architectures are characterized by the follo wing attributes: Number of Layers: The a rtific ial neurons are arranged in an input layer, one or mo re h idden layers, and an output layer. Number of Neurons: The art ificia l neural network has to learn the features of the series for the analysis and forecasting of a fu zzy t ime series. As the number of neurons in the input and output layers are determined by the tra ining patterns, the number of neurons in the hidden layers can then be chosen arbitrarily (see Fig. 1). More a rtific ial neurons implies more we ighting matrices. Thus, fro m c lassical fie lds of application of artificia l neural networks (e.g., pattern recognition), the well-known proble m of over fitting must be considered. Fi gure 1. Archit ect ure of mult ilayer feed forward neural net work Activation Function: The proper selection of activation function that enables curvilinear matching between input and output units, significantly affect the performance of the network. Method of Training: The learning situations in neural networks may be classified into three d istinct sorts. These are supervised learning, unsupervised learning, and reinforce ment learn ing. In supervised learning, an input vector is presented at the inputs together with a set of desired responses, one for each node, at the output layer. The most wide ly used one is Back Propagation algorith m which updates weights based on the difference between ava ilab le data and the output of the network. Lea rning para meter which is used in back propagation algorithm and which can be taken fixedly or updated in the algorith m dyna mica lly, plays an important role in reaching optimal results. 3. Fuzzy Time Series The definition of fu zzy time series was firstly introduced by Song and Chissom[2]. Basic definitions of fuzzy time series not including constraints such as linear model and observation number can be given as follo ws; Definiti on 1 Fuzzy t ime series. Let ),2,1,0,)(( =ttY , a subset of real numbers, be the universe of discourse by which fuzzy sets )(tf j are defined. If )(tF is a collection of ),...(),( 21 tftf then )(tF is called a fuzzy t ime series defined on )(tY . Definiti on 2 First order seasonal fuzzy time series forecasting model. Let )(tF be a fu zzy t ime series. Assume there e xists seasonality in { })(tF , first order seasonal fuzzy t ime series forecasting model: )()( tFmtF β†’βˆ’ (13) where m denotes the period. Definiti on 3 High order fu zzy t ime series forecasting model. Let )(tF be a fu zzy t ime series. If )(tF is caused by )2(),1( βˆ’βˆ’ tFtF ,…, and )( ntF βˆ’ , then this fuzzy logical relationship is represented by )()1(),2(),...,( tFtFtFntF β†’βˆ’βˆ’βˆ’ (14) and it is called the nth order fuzzy t ime series forecasting model. Definiti on 4 First order b ivariate fu zzy t ime series forecasting model. Let F and G be two fu zzy time series. Suppose that ( 1) iF t Aβˆ’ = , kBtG =βˆ’ )1( and jAtF =)( . A bivariate fuzzy logica l re lationship is defined as iA , jk AB β†’ , where ki BA , are re ferred to as the left hand side and jA as the right hand side o f the b ivariate fu zzy logical re lationship. Therefore, first order bivariate fu zzy time series forecasting model is as follo ws: )()1(),1( tFtGtF β†’βˆ’βˆ’ (15) 16 Ozge Cagcag et al. : A Novel Seasonal Fuzzy Time Series M ethod to the Forecasting of Air Pollution Data in Ankar a Definiti on 5 High order partia l bivariate fu zzy time series forecasting model. Let F and G be two fu zzy time series. If )(tF is caused by 1 1( ),.., ( ), ( ),k kF t m F t m F t mβˆ’βˆ’ βˆ’ βˆ’ 1 1( ),.., ( ),lG t n G t n βˆ’βˆ’ βˆ’ )( lntG βˆ’ , where im ),..,2,1( ki = and jn ),..,2,1( lj = are integers kmm <<≀ ...1 1 , lnn <<≀ ...1 1 then this FLR is represented by; )( )(),(),...,( ),(),(),...,( 11 11 tF ntGntGntG mtFmtFmtF ll kk β†’ ο£Ύ ο£½ ο£Ό βˆ’βˆ’βˆ’ βˆ’βˆ’βˆ’ βˆ’ βˆ’ (16) 4. Proposed Method Although, there are nume rous fuzzy t ime series approaches in literature, the number of approaches aiming at analyzing seasonal fuzzy time series which are frequently encountered in real life and wh ich inc lude seasonal components are limited. The first model proposed by Song involves one variable which belongs to only one period[12]. The approach proposed by Egrioglu et al. determines me mbe rship values of each observation belonging to fuzzy sets objectively[13]. A lthough, Uslu et a l. proposed a subjective judgment-free approach, she used set number representing the fuzzy set having the highest me mbership value of observations in identification of fu zzy re lations and defuzzificat ion[14]. Th is poses a situation which is incompatib le with fu zzy set theory as well as a ffecting the forecasting performance of the method negatively due to the loss of information. In this study, we proposed a new seasonal fuzzy time series forecasting model which does not require subjective judgments in all analysis processes and which uses SARIMA in determination of the model, FCM in fu zzification and ANN in defu zzificat ion steps. The advantages of the proposed model are as follows; β€’ The proble m of determining the mode l o rder was eliminated by using SARIMA and delayed variab les in the model was determined systematically. β€’ Subjective judgments were avoided by using FCM in fuzzification step and me mbership values which are compatible with the model we re determined by a systematic infrastructure. β€’ Again, in the determination of fuzzy relat ions, the problem re lated to the number of input of ANN was eliminated by co-clustering of delayed variables data set and was limited by the set number. β€’ Input and target values of ANN which a re used in the determination of fuzzy relat ion are not the set number but the me mbe rship values obtained from FCM. Thus, the approach becomes more realistic in e xposing fuzzy re lations in fuzzy time series. β€’ In order to prevent loss of information, for the first time in literature me mbership values were used in fuzzification step. The algorithm of the proposed method in this study is given below; Algorithm Step 1 The model order is defined by SARIMA The time series concerned is analyzed by Bo x-Jen kins method after the model order is defined. Then we obtain residuals series )( ia . As an illustration let us suppose we have defined the model as SARIMA (1,1,0)(0,1,1)12 via Bo x-Jenkins method. This implies that tX will be a linear combination of the corresponding lagged variables. That is, ),,,,,( 1214131221 βˆ’βˆ’βˆ’βˆ’βˆ’βˆ’= ttttttt aXXXXXfX (17) Therefore, ),( lk representing the order of the model and the parameters kmm ,,1  and 1, , ln n are determined based on the inputs of the SARIMA model. Accordingly k and l are defined as 5 and 1 respectively. Then the model will be thlk ),( -order partia l b ivariate fu zzy time series forecasting model and the fuzzy relat ionship can be given as follow; )( )12(),14(),...,13( ),12(),2(),...,1( tF tGtFtF tFmtFtF β†’ ο£Ύ ο£½ ο£Ό βˆ’βˆ’βˆ’ βˆ’βˆ’βˆ’ (18) This imp lies ,14,13,12,2,1 54321 ===== mmmmm ,121 =n )(tF denotes the fuzzified time series tX and )(tG denotes the fuzzified residual series ta . Step 2 Data set of lagged variab les is created. Depending on the model order defined in previous step, for each time series wh ich should be included in the model tX , and residual series ta for each lagged variab les are lagged less than order of lagged variables and data set is created. In other words, when a model given in equation (18) is considered, lagged variables data set will include 111312111 ,,,,, βˆ’βˆ’βˆ’βˆ’βˆ’ tttttt aXXXXX . Step 3 Data set of lagged variab les is clustered via FCM. The number of fuzzy set is determined with c where nc ≀≀2 and n is the nu mber of observation. Data set which covers the delays in times series is clustered via FCM clustering method. Thus, fuzzy set centers for each lagged variables constituting data set and me mbership values showing order of observations belonging to fuzzy sets for each observation are obtained. In this step, fuzzy sets are sorted according to set centers represented with , 1, 2, ,rv r c=  and , 1, 2, ,rL r c=  fu zzy sets are obtained. Step 4 Fu zzy re lations are determined via Feed Forward Artificia l Neura l Net works (ANN). The number of neurons in input and output layer of feed forwa rd artific ia l neural network used in determining fu zzy relations equals to number of fu zzy set )(c . The number of neurons in hidden layer is determined by trial and error. He re, American Journal of Intelligent Sy stems 2013, 3(1): 13-19 17 the point to take into consideration is that hidden layer unit number should be selected in a way that not losing generalization ab ility of feed forward a rtific ial neural network. The a rchitecture of feed forwa rd artificia l neural network having two hidden layers for a model including seven sets is presented in Figure 2. In Figure 2, ))(( tDS rL Β΅ representsthe me mbership value of lagged data set belonging to thi fuzzy set at t time. Moreover, while me mbership value of observation of lagged data set belonging to c number fu zzy set at t time constitutes the inputs of ANN; me mbe rship value of observation of lagged data set belonging to c number fu zzy set at t time constitutes the outputs of ANN. In all layers of feed forwa rd artific ia l neural networks which is used in determining fu zzy re lation and whose architectural structure is e xe mp lified above, logistic activation function given in (19) equation is used. 1( ) (1 exp( ))f x x βˆ’= + βˆ’ (19) Feed forwa rd artific ial neura l networks are trained according to Levenberg-Marquardt learning a lgorith m and optima l weights are obtained. Trained artificia l neural network lea rned the relation between consecutive time series observations and me mbership values of sets. Fi gure 2. Archit ect ure of feed forward art ificial neural net work for three set s Step 5 De fuzzification of forecasts. In order to obtain real forecasts of fuzzy time series at t time, me mbership values of observations belonging to fuzzy sets at t time depending on , 1, 2, ,rv r c=  fuzzy set center which was obtained from FCM method were determined and then these me mbership values were entered to feed fo rwa rd art ific ial neural networks as inputs and thus outputs of feed forward art ificia l neura l networks are created. These outputs represent forecast of observation at t time . A architecture of feed forward a rtific ial neural network for three sets is given Figure 3. Fi gure 3. A archit ect ure of feed forward art ificial neural net work in defuzzificat ion 5. Application The proposed method was applied to time series of β€œthe amount of sulfur dio xide in Ankara p rovince between March 1994 and April 2006 (ANSO)”. The graph of ANSO time series is presented in Figure 4. In order to evaluate the performance of the proposed method, the last 10 observations were taken as test set and obtained results were co mpared with some conventional and alternative time series methods. In the applicat ion, in order to determine the order of fu zzy time series forecasting model, crisp time series is analyzed using Box-Jenkins method and optima l SA RIMA model is determined and residual time series ( )ta as well as tX time series are obtained. In this step, optimal model fo r ANSO t ime series was SARIMA (1,1,0)(0,1,1)12. As a linear function of tX , this mode l can be e xpressed as; 1 2 12 13 14 12( , , , , , )t t t t t ttX f X X X X X aβˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’= (20) Thus, the model will be (5,1)th order partia l high order fuzzy time series forecasting model where 5=k and 1=l . This model can be e xpressed as; )( )12(),14(),...,13( ),12(),2(),...,1( tF tGtFtF tFmtFtF β†’ ο£Ύ ο£½ ο£Ό βˆ’βˆ’βˆ’ βˆ’βˆ’βˆ’ (21) Fi gure 4. The t ime series dat a of t he amount of SO2 in Ankara After determining the model order o f partia l high order model, lagged variables data set for each lagged variable that should be included in the model is created. Lagged variables data set for (5,1)th order partia l model is created using 1 11 12 13 11, , , , ,t t t t t tX X X X X aβˆ’ βˆ’ βˆ’ βˆ’ βˆ’ lagged variables. Here, it must be noted that lagged variables data set consists of one step leaded variable in partia l high order fuzzy time series forecasting model given in (20). Created data set is clustered via FCM. Clustering is applied to all lagged variable data sets together. In this step, data set is clustered by shifting the number of sets 5 to 15. Me mbership values of observations belonging to each fuzzy set are also determined via FCM method. The re lationship between these me mbership values, in other words, the nu mber of neurons in hidden layer of feed forwa rd artific ia l neuron network wh ich is used in determining fu zzy relat ion were shifted between 1 and 15. In the light of this informat ion, 1651511 =Γ— different analyses were done and Root Mean Squared Error (RMSE) 0 50 100 150 18 Ozge Cagcag et al. : A Novel Seasonal Fuzzy Time Series M ethod to the Forecasting of Air Pollution Data in Ankar a and Mean Absolute Percentage Error (MAPE) were used as performance evaluation criteria . ( ) 1 1 Λ† T t t t RMSE x x T = = βˆ’βˆ‘ (22) 1 Λ†1 T t t t t x x MAPE T x= βˆ’ = βˆ‘ (23) Where Λ†tx , and Λ†tx , T represent crisp time series, defuzzified forecasts, and the number of forecasts, respectively. The algorith m of the proposed method is coded in Matlab version 7.9. In consequence of all analyses, the best forecasting performance was obtained in the case in which the nu mber of set is 14, the hidden layer unit number is 6 in the determination of fuzzy relat ion stage and the hidden layer unit number is 2 in the defu zzification stage. Results obtained fro m the proposed method and results of some other methods are summarized in Table 1. Table 1 c learly shows the superior performance of the proposed method in co mparison with conventional time series approaches as well as seasonal fuzzy t ime series approaches with respect to three criteria. Additionally, graph of the forecasts obtained fro m the proposed model with real values are given in Figure 5. When Table 1 and Figure 5 a re analyzed together, a ll the advantages as well as the superior forecasting performance of the proposed method can be seen easily. Table 1. Result of methods Data Set S ARI W ME [12] [13] [14] Th e 21 22.93 15.40 41.6 20.0 22.7 24.1713 27 22.35 16.11 27.5 30.0 22.7 24.1326 25 23.61 17.77 41.6 20.0 22.7 24.0786 28 28.81 25.12 41.6 30.0 22.7 24.1375 38 46.97 41.11 41.6 30.0 42.0 38.6718 45 54.62 46.12 46.7 50.0 42.0 39.4255 38 58.13 49.80 45.0 40.0 42.0 39.4255 36 46.99 44.24 46.7 30.0 42.0 39.4255 24 37.85 31.96 46.7 30.0 22.7 20,7401 22 24.76 18.39 27.5 20.0 22.7 26,5369 RMSE 9.62 7.11 12.7 4.56 3.66 3.32 MAP E 0.23 0.22 0.40 0.13 0.11 0.10 Fi gure 5. The graph of the result s obt ained from the proposed method and real t ime series 6. Discussion and Conclusions Diffe rent approaches have been proposed for the forecasting problemswh ich constitute an important ro le in future planning and strategy formulat ion. Fuzzy t ime series forecasting methods have attracted much attention in recent years. Although, numerous first and high order fuzzy time series forecasting models have been proposed in literature, these models are insuffic ient in the analysis of seasonal time series which are frequently encountered in real life. The approaches proposed in literature have specific outstanding features as well as some insufficiencies. The most significant disadvantage of these models is that they require subjective judgments and ignore me mbership values representing the degree of observations belonging to fuzzy sets in analysis process. This affects forecasting performance of these approaches negatively as well as posing a situation that is incompatib le with the basic features of fuzzy set theory. Seasonal fuzzy t ime series forecasting method proposed in this study eliminates this problem by considering the me mbe rship value in the determination of fuzzy relat ion and defuzzificat ion stages and presents fuzzy re lations mo re realistically. It is evident that partial high order seasonal fuzzy time series forecasting method which is proposed in this study and in which model order was determined via SARIMA and ANN was used in determining fu zzy re lations and defuzzification stages has some advantages and exhib its superior forecasting performance.It should be noted that these results are obtained for the para meter sets given above and ANSO t ime series e xa mined in the study. For instance, if the length of test set is shifted, the results can change or similarly if these parameter sets are used for other time series, the obtained results can change. Therefore, the obtained results are valid for only these parameter sets and this time series. In orde r to reach general results, a co mprehensive simu lation study has to be made. However, it is very hard to perform such a simulation study since there are many types of time series and many para meter co mb inations. REFERENCES [1] Zadeh, L. A., 1965, Fuzzy Sets, Inform and Control, 8, 338-353. [2] Son g Q., and Ch issom, B. S., 1993, Fuzzy time series and its models, Fuzzy Sets and Sy stems, 54, 269-277. [3] Son g Q., and Chissom, B. S., 1993, Forecasting enrollments with fuzzy time series- Part I, Fuzzy Sets and Sy stems, 54, 1-10. [4] Son g Q., and Chissom, B. S., 1994, Forecasting enrollments with fuzzy time series Part II, Fuzzy Sets and Sy stems, 62, 1-8. [5] Chen, S. M ., 1996, Forecasting enro llments based on fuzzy time-series, Fuzzy Sets and Sy stems, 81, 311-319. 20 25 30 35 40 45 1 2 3 4 5 6 7 8 9 10 data set the proposed method American Journal of Intelligent Sy stems 2013, 3(1): 13-19 19 [6] Yolcu, U., Egr io glu, E., Uslu, V. R., Basar an, M . A., and Aladag C. H., 2009, A New App roach for Determinin g the Len gth of Intervals for Fuzzy Time Ser ies, App lied Soft Comp uting, 9, 647-651. [7] Chen, S. M ., 2002, Forecasting enrollments based on high order fuzzy time series, Cy bernetics and Sy stems, 33, 1-16. [8] Aladag, C. H., Basaran, M . A., Egrio glu, E., Yolcu, U., and Uslu, V.R., 2009, Forecasting in high ord er fuzzy time series by using neural networks to define fuzzy relations, Exp ert Sy stems with App lications, 36, 4228-4231. [9] Egr io glu, E., Aladag, C. H., Yolcu, U., Uslu, V. R., and Basaran, M .A., 2009, A new app roach based on artificial neural networks for high order multivariate fuzzy time series, Exp ert Sy stems with App lications, 36, 10589-10594. [10] E. Egr io glu, V. R. Uslu, U. Yolcu, M . A. Basaran, and C. H. Aladag, A new app roach based on artificial neural networks for high order biv ariate fuzzy time series, J.M ehnen et al. (Eds.): App lications of Soft Comp uting, AISC 58, Sp ringer- Ver lag Ber lin Heid elber g, 265-273, 2009. [11] Egr io glu, E., Aladag, C. H., Yolcu, U., Uslu, V. R., and Basaran, M .A., 2010, Finding an op timal interval length in high ord er fuzzy time series, Exp ert Sy stems with App lications, 37, 5052-5055. [12] Son g, Q., 1999, Seasonal forecasting in fuzzy time series, Fuzzy Sets and Sy stems, 107(2), 235. [13] Egr io glu, E., Alad ag, C. H., Yolcu, U., Basaran, M .A., and Uslu, V. R., 2009, A new hy brid app roach based on SARIMA and p artial high order bivariate fuzzy time series forecasting model, Exp ert Sy stems with App lications, 36, 7424-7434. [14] V. R. Uslu, C. H. Alad ag, U. Yolcu, and E. Egr io glu, A n ew hy brid app roach for forecasting a season al fuzzy time series, 1st International Sy mp osium on Comp uting In Science & Engineer in g, IzmΔ±r –Turkey , 2010. [15] G. E. P. Box and G. M. Jenkins, Time series analy sis: Forecasting and control, San Francisco: CA: Holdan-Day , 1976. [16] J. C. Bezdek, Pattern recogn ition with fuzzy objective function algorithms, NY: Plenu m Press, 1981. [17] S. Gunay , E. Egrio glu E, and C. H. Alad ag, Introduction to single variable time series analy sis, Ankara, Turkey : Hacettep e University Press., 2007. [18] J. M . Zurada, Introduction of artificial neur al sy stems, St. Paul: West Publishin g, 1992. [19] Zhang, G., Patuwo, B. E., Hu, Y. M ., 1998, Forecasting with artificial neural n etworks: the state of the art, International Journal of Forecasting, 14, 35-62. 1. Introduction 2. Review 3. Fuzzy Time Series 4. Proposed Method 5. Application 6. Discussion and Conclusions