1. Introduction

The methods of fuzzy time series are based on the theory of fuzzy set of Zadeh [28]. In the literature, Song and Chissom [21] defined several forecasting methods of fuzzy time series for the first time and classified the fuzzy time series into two types as time variant and time invariant. Also, Song and Chissom [22] proposed an analysis method based on resultant actions for the time invariant fuzzy time series. In Song and Chissom’s study [23], a similar analysis method was proposed for the time variant fuzzy time series. Although time invariant fuzzy time series were proposed for the analysis in most of the studies in the literature, there are also methods offered for the analysis of variant fuzzy time series. These studies were followed by the methods based on the tables of fuzzy group relation developed by Chen [2] which depends on Markov model presented by Sullivan and Woodall [24]. Furthermore, Hwang et al. [15], Huarng and Yu [12], Yu [26] and Yu [27] and Chen and Hwang [4] proposed first order forecasting model of fuzzy time series. In the literature, high order forecasting model of fuzzy time series have also been proposed. The first high order forecasting model of fuzzy time series was introduced by Chen [3]. Some other high order forecasting models of fuzzy time series in the literature can be listed as Aladag et al. [1] and Egrioglu et al. [8].

Huarng and Yu [13] introduced the first order forecasting models of fuzzy time series in the step of determination of fuzzy relation, as well as fuzzy time series of artificial neural networks used for the first time in the step of defuzzification in literature by Song and Chissom [23]. Moreover, Aladag et al. [1] and Egrioglu et al. [7] utilized artificial neural networks in the analysis of high order and multivariate models respectively in the step of determination of fuzzy relation. As the usage of tables of fuzzy groups relation in high order and multivariate models require quite intensive and complex actions, the usage of artificial neural networks in the step of the determination of fuzzy relation removes the complexity especially in the high order and multivariate models.

The most of the studies in the literature are those in which interval length is determined intuitively, diverse interval lengths are tried and then the best forecasts are achieved. Huarng [11] showed in his study that the selection of the interval length used in the division of the universal set into equal intervals is a critical decision and directly affects the forecasting performance, and determined the optimal interval length by recommending two separate approaches based on average and distribution for the selection of the interval length. In this way, the analysis was performed in a single application without trying diverse interval lengths. Huarng [11]’s approaches base on average and rate brought systematicity to the selection of the interval length, but could not guarantee the achievement of the best forecasting performance. For this reason, there are many studies [5, 6, 8, 13, 18, 25] for the determination of the optimal interval length in a single application by using statistical theory.

All of the methods of fuzzy time series developed in the literature have used forecasting models depend on AR (Autoregressive) variables except for only two studies [9,16] until year 2013. These primary studies [9, 16] are first order fuzzy ARMA type models that one of them [9] is based on particle swarm optimization and the other [16] is based on fuzzy logic relation tables. Proposed method in this study also is first-order ARMA type model which is based on artificial neural network. The main difference of the proposed method is using artificial neural network for determining fuzzy relations. The proposed method is the first method employee artificial neural network for a fuzzy ARMA type models.

In the modeling of most of the real life time series, MA (Moving Avarage) variables are also required. In this regard, using only AR variables where MA variables are also required for the analysis of fuzzy time series, could remain insufficient for both in the determination of fuzzy relations and forecasting performance. Moreover, the usage of MA variables with AR variables in the analysis of fuzzy time series can increase forecasting performance because of the fact that it could give opportunity to make application with much more knowledge.

In this study, for forecasting time invariant fuzzy time series, a new forecasting method of fuzzy time series based on a model including MA variables is proposed. The advantages of proposed algorithm are stated as;

• The analysis depend on the forecasting model of fuzzy time series is more real method for real life time series, as it also include MA variables,

• The usage of artificial neural networks in the determination of fuzzy relation remove the complexity of fuzzy logic group relation tables,

• In comparison with the other methods developed in the literature, it has a higher forecasting performance.

• The proposed method has better forecasting performance than other Fuzzy ARIMA methods [9, 16].

• Because of using artificial neural network, unknown patterns can be easily forecasted.

• It is difficult that the methods based on fuzzy relation tables are designed for high order models but the proposed method can be easily extended for high order models like Aladag et al. [1].

In section 2, the main definitions related with fuzzy time series and in Section 3, main information about artificial neural networks are presented. In section 4, the forecasting model of fuzzy time series proposed is defined and the analysis of algorithm is stated. In section 5, the results of the application of proposed method with this study and other methods to Istanbul Stock Exchange (IMKB) national 100 index time series and the prices of gold (Turkey daily gold prices series can be obtained from the website of Central Bank of The Turkish Republic) are presented. In section 6, the results are discussed.

2. Definition of Fuzzy Time Series

The definition of fuzzy time series was firstly introduced by Song and Chissom [21, 22]. General definitions of fuzzy time series are given as follows:

Definition 1. Let , a subset of real numbers, be the universe of discourse on which fuzzy sets are defined. If F(t) is a collection of then F(t) is called a fuzzy time series defined on

Definition 2. Fuzzy time series relationships assume that F(t) is caused only by F(t –1), the model can be expressed as:

(1)

This is called as a first-order fuzzy time series forecasting model. Then this relation can be expressed as where is the fuzzy relationship between and

Definition 3. Suppose is a first-order model of If for any t, is independent of t, then is called a time-invariant fuzzy time series otherwise it is called a time-variant fuzzy time series.

Definition 4. Fuzzy time series relationships assume that F(t) is caused only by F(t –1), then the model can be expressed as:

(2)

This is called as a m^th-order fuzzy time series forecasting model [3].

3. Feed Forward Artificial Neural Networks

The technology of artificial neural networks is an information processing mechanism which emerges in the simulation of human neuron and nervous system at the computer environment. The most important feature of the artificial neural networks is its ability of learning from the samples. Although artificial neural networks have simpler structure than human nervous system, it is successful in solving lots of problem such as forecasting, pattern recognition and classification [10, 20, 29].

Although there are several types of artificial neural networks in the literature, only feed forward artificial neural network is used for the problem of forecasting fuzzy time series. Feed forward artificial neural network is formed of input layer, hidden layer or layers and output layer. The sample of architecture of feed forward artificial neural network is shown in Figure 1. Each layer is formed of units named as neuron and there is no relationship between the neurons belonging to the same layer. The neurons of different layers are connected to each other by their weights. In Figure 1, each weight is shown by directional arrows. In the feed forward artificial neural network, the connections shown by directional arrows are forward and one-way. In both the hidden layer and output layer of the feed forward artificial neural network, an activation function is used for each neuron. When the outputs of the neurons of the previous layer are multiplied with the related weights and these multiplication results are added to each other, the inputs that come to the neurons of hidden and output layers are formed. The information come to these neurons are passed through an activation function and the neuron output is formed. Activation function provides non-linear match. For this reason non-linear activation functions are used for the units of the hidden layer [10].

Figure 1. Multilayer feed forward artificial neural network with one output neuron

Learning in the feed forward artificial neural network is the determination of weights which produce the nearest outputs towards the target values corresponding to the outputs of artificial neural networks. Learning is provided by the optimization of the total error according to weights. There are many training algorithm proposed in the literature for the learning of feed forward artificial neural network. One of these widespread training algorithms that are employed in also this study is Levenberg-Marquardt (LM) algorithm. The detailed information about artificial neural network can be attained from [10, 17].

4. The Proposed Method

The forecasting models of fuzzy time series proposed are the fuzzy AR (Autoregressive) type models except for only two studies [9, 16] in the literature. However, It is necessary that ARMA type models are used to many fuzzy time series in the real life. The definition of first-order ARMA (autoregressive moving average) type fuzzy time series model made by Egrioglu et al. [9] and Kocak [16] is defined below.

Definition 5. Fuzzy time series relationships assume that F(t) is caused by F(t –1) and ε(t –1), then the relationship can be expressed as:

(3)

This is called as an first-order fuzzy time ARMA (1, 1) fuzzy time series forecasting model.

In this study, an algorithm is proposed to analyze the ARMA (1, 1) model of fuzzy time series as defined in Eq. (3). In the proposed algorithm, firstly a fuzzy AR(1) defined in Eq. (1) is obtained. Then, the errors are calculated by subtracting the observed values and deffuzzified forecasts with using the results obtained from this fuzzy AR(1) model. The ARMA (1, 1) model given in Eq. (3) is forecasted by using time series and errors. The algorithm of the proposed approach is given below.

The algorithm of the proposed method

Step 1. The universe of discourse and the partitions of U are defined.

The beginning and the ending points of universe of discourse for time series are determined. These points are selected in the way that will include the possible values of time series. Then, universe of discourse is divided into sub-intervals (partitions) according to proper interval length. In this method, the determination of the interval length depends on the researcher. It should not be forgotten that the interval length determined is effective on the number of fuzzy sets.

Step 2. The fuzzy sets are defined in accordance with universe of discourse (U) and the sub-intervals

(4)

Where,

(5)

Step 3. The observations are made fuzzy.

Sub-interval is determined for each observation. The fuzzy set that has the highest membership value of the determined sub-interval is determined. The fuzzy value of the observation is this determined fuzzy set.

Step 4. Fuzzy relations are determined by the feed forward artificial neural network (FANN).

In the determination of relation by the feed forward artificial neural network, one lagged fuzzy variable is the input of artificial neural network, is target value and is the output of artificial neural network. The learning samples of artificial neural network consist of the sequence number of fuzzy sets. For instance, the observations of fuzzy time series are respectively stated as A5, A1, A4, A3, A2. In Table 1, the input and the target values of artificial neural network for this example is shown. In the determination of fuzzy relation, the architectural structure of artificial neural network is given in Figure 2.

Table 1. An example for using of FANN for determination of fuzzy relationship

Figure 2. Architecture of ANN for determination of fuzzy relationship

Step 5. Fuzzy forecasts are obtained.

The output of the feed forward artificial neural network that has the architectural structure given in Figure 2 is the sequence numbers of fuzzy sets and the fuzzy sets that have these sequence numbers form the fuzzy forecasts. For instance, when the output of artificial neural network is rounded up as 5, the fuzzy forecast is

Step 6. Deffuzzified forecasts are obtained for the time series.

In this step centralization method is utilized. When the fuzzy forecast is deffuzzified forecast is the middle point of interval which has the highest membership value in fuzzy set.

Step 7. Errors are calculated.

When the deffuzzified forecast values obtained in the Step 6 are subtracted from the values observed in the time series the errors are obtained. The error series is calculate as,

(6)

Step 8. The universe of discourse and its sub-intervals are defined for the errors.

The universe of discourse for the errors is defined and the universe of discourse is divided in accordance with a determined interval length.

Step 9. For errors, the fuzzy sets are defined based on universe of discourse (V) and sub-intervals

(7)

Where

(8)

Step 10. The error series is made fuzzy.

Sub-interval is determined for each observation. The fuzzy set that has the highest membership value of the determined sub-interval is determined. The fuzzy value of the observation is this determined fuzzy set.

Step 11. The fuzzy relations are determined by the feed forward artificial neural network (FANN).

In the determination of relation by the feed forward artificial neural network, one lagged fuzzy variable and one lagged fuzzy error series are the inputs of artificial neural network, is target value and is the output of artificial neural network. The learning samples of artificial neural network consist of the sequence number of fuzzy sets. For instance, the observations of fuzzy time series are respectively stated as A5, A1, A4, A3, A2 and the observations of fuzzy error series are respectively stated as B1, B3, B7, B1, B4. In the table below, the input and the target values of artificial neural network for this example is shown.

Table 2. An example for using of FANN for determination of fuzzy relationship

In the determination of fuzzy relation the architectural structure of artificial neural network is given in Figure 3.

Figure 3. Architecture of FANN for determination of fuzzy relationship

Step 12. Fuzzy forecasts are obtained.

The output of the feed forward artificial neural network that has the architectural structure given in Figure 3 is the sequence number of fuzzy sets and the fuzzy sets that have these sequence numbers form the forecasts. For instance, when the output of artificial neural network is rounded up as 5, the fuzzy forecast is

Step 13. Defuzzified forecasts are obtained.

In this step centralization method is used. When the fuzzy forecast is defuzzified forecast is the middle point of interval which has the highest membership value in fuzzy set.

5. Application

In the application, two different real world data sets and two simulated chaotic data sets were used. The real world time series are data of Istanbul Stock Exchange and the prices of Gold. In the application of the proposed method and the methods in the literature, the fuzzy set number of the time series was increased by 1 unit between 5 and 35 and 31 different fuzzy set numbers were tried. If the method requires calculating according to interval length, the interval lengths are determined by using the formula given in the way that the fuzzy set numbers are between 5 and 35.

(9)

At step 8 of the algorithm, 31 different fuzzy set numbers were tried also by increasing the fuzzy set number of the error series by 1 unit between 5 and 35. The hidden layer unit number in the artificial neural network used at step 4 and step 11 of the algorithm of the proposed method was modified between 1 and 10. In training the artificial neural network, the Levenberg-Marquardt method was used. In the hidden layer units and output layer, logistic activation function given in Eq. (10) was used.

(10)

The forecasts that minimize the Root of Mean Square Error (RMSE) given in the Eq. (11) were considered the best result of the method.

(11)

For the best results of all methods as per RMSE, MAPE (mean absolute percentage error) and DA (direction accuracy) values are calculated as in Eqs. (12) and (13) as well.

(12)

(13)

5.1. Analysis of IMKB Data in the Period of 01.10.2010 - 23.12.2010

The first data used for comparing the future forecasting performances of the methods is IMKB close prices time series with 53 observations between the dates of 01.10.2010 - 23.12.2010 as seen in Table 3 and Figure 4.

Table 3. IMKB time series in the period of 01.10.2010 - 29.12.2010

Figure 4. IMKB time series graph in the period of 01.10.2010 - 23.12.2010

Two different applications were performed by taking the last 7 observations as the test set in the first analysis of IMKB data and the last 15 observations as the test set in the second analysis as seen in Table 3. For both applications performed,

• In Song and Chissom [21] and Egrioglu et al. [9]’s method, the test set forecast having the minimum RMSE value from 31 different cases obtained by increasing the fuzzy set number by 1 unit between 5 and 35 was determined as the best result of the method.

• For each of Chen [2] ,Chen [3], Aladag et al. [1] and Kocak [16] methods, RMSE values of the different cases, which were obtained by increasing the interval length by 100 units between 300 and 2300 (in the way that the fuzzy set number is between 5 and 35 according to Eq. (9)), was found. From these forecasts, the test set forecasts having the minimum RMSE value were determined as the best results of the methods.

• By using the optimal interval lengths calculated by the distribution-based approach and average-based approach of Huarng [11], the analysis was performed with the first order fuzzy time series method of Chen [2]. In this way, the best results of the distribution-based approach and average-based approach for the test set were obtained in a single application.

• In the application of rate-based approach of Huarng and Yu [14], the best result of the method for the test set was obtained in a single application by taking the alpha parameter as 0.50.

For two different applications performed by taking the test set number as 7 and 15, in the analysis of IMKB data in the period of 01.10.2010 - 23.12.2010 by using the fuzzy ARMA(1,1) first order forecasting method which proposed;

• In the fuzzification step, 21 different numbers were tried by increasing the interval length by 100 units between 300 and 2300 (so that the fuzzy set number is between 5 and 35 according to Eq. (9)).

• Different forecasts were obtained by increasing the hidden layer unit number in the artificial neural network by 1 unit between 1 and 10 (Step 4) for each of 21 different interval lengths used.

• The error value of the first observation of data has been assumed as 0 and the error values of other observations have been calculated with the Eq. (6) for 21 different forecasts obtained.

• In the fuzzification step of the errors, 31 different numbers were tried by increasing the fuzzy set number of e(t) error series by 1 unit between 5 and 35.

• When determining the fuzzy relations in step 11 for the second time, feed-forward artificial neural network with 2 inputs and 1 output as seen in Figure 3 was used.

• When determining the fuzzy relations for the second time, forecasts having different performances were obtained by increasing the interval length of the time series by 100 between 300 and 2300 and the hidden layer unit number in the artificial neural network by 1 between 1 and 10 for each of the fuzzy set numbers (between 5 and 35) of the error series.

• From these forecasts, the forecast of the test set having the minimum RMSE value was determined as the best result of the proposed method.

In the applications of the methods by taking the last 7 observations of IMKB time series as the test set, the fuzzy set numbers that give the best results were determined as 9 in Song and Chissom method [21], 37 (the interval length is 300) in Chen 's method [2], 11 (the interval length is 1000) in the distribution-based approach and 55 (the interval length is 200) in the average-based approach of Huarng [11], which are the first order methods. In the analyses performed by using the high order models, the fuzzy set numbers that give the best forecasts were found as 5 (the interval length is 2200) in the 3rd order model of Chen [3] 18 (the interval length is 600) in the 2nd order model of Aladag et al. [1], 7 fuzzy sets were used in Egrioglu et al. [9] and 2200 length of interval was use in Kocak [16]. In the application of the proposed method. The case in which the best results were obtained, when the interval length of time series is 900, the fuzzy set number of errors is 33 and the hidden layer unit number of the artificial neural network is 10. The architecture achieved for the value 33 of the fuzzy set number of error series that gives the best result of the proposed method is submitted in Table 4. As a result of the analysis of IMKB data in the period of 01.10.2010 - 23.12.2010, the best forecasts and forecasting performances of all methods for the test set with 7 observations are summarized in Table 5.

When Table 5 is examined, it is seen that the proposed method has the best forecasting performance with minimum RMSE value of 706.84, minimum MAPE value of 0.769% and maximum direction accuracy value of 66.67% as compared with the other methods. That even RMSE value of 928.70, which was calculated by Chen method [3] having the best performance following the proposed method, is a value that is considerably higher than RMSE value of 706.84 of the proposed method indicates that the proposed method has a significantly higher performance than the other methods in the literature except Kocak [16]. The graphs of the test set forecasts obtained by using the last 7 observations of IMKB data in the period of 01.10.2010-23.12.2010 and proposed method are given together in Figure 5.

Table 5. The forecasts and performances of 7-observation test set for IMKB data

Figure 5. The graph of 7-observation test set and the forecasts obtained by using proposed method for IMKB data

In the applications of methods by taking the last 15 observations of IMKB time series as the test set, the fuzzy set numbers that give the best results were determined as 20 by using Song and Chissom [21] method, 12 (the interval length is 900) by using Chen [2] method, 11 (the interval length is 1000) with the distribution-based approach and 55 (the interval length is 200) with the average-based approach of Huarng [11], which are the first order methods. In the analysis performed by using the high order models, the fuzzy set numbers that give the best forecasts were found as 8 (the interval length is 1400) in the 2nd order model of Chen [3], 7 (the interval length is 1500) in the 2nd order model of Aladag et al. [1], 7 fuzzy sets were used in Egrioglu et al. [9] and 1400 length of interval was used in Kocak [16]. In the application of the proposed method, The case in which the best results were obtained, When the interval length of time series is 2100, the fuzzy set number of errors is 28 and the hidden layer unit number of the artificial neural network is 8. As a result of the analysis of IMKB data in the period of 01.10.2010 - 23.12.2010, the best forecasts and forecasting performances of all methods for the test set with 15 observations are summarized in Table 6.

When Table 6 is examined, it is seen that the proposed method has the best forecasting performance with minimum RMSE value of 761.91, minimum MAPE value of 0.894% and maximum direction accuracy value of 85.714% as compared with the other methods. That even RMSE value of 896.96, which was calculated by the rate-based approach of Huarng and Yu [14] method having the best performance following the proposed method, is a value that is considerably higher than RMSE value of 761.91 of the proposed method indicates that the proposed method has a significantly higher performance than the other methods in the literature. The graphs of the test set forecasts obtained by using the last 15 observations of IMKB data in the period of 01.10.2010-23.12.2010 and the first order time series model of the proposed method are given together in Fig 6.

Table 6. The forecasts and performances of 15-observation test set for IMKB data

Figure 6. The graph of 15-observation test set and the forecasts obtained by using proposed method for IMKB data

5.2. Analysis of Gold Prices

The second data used for comparing the future forecasting performances of the methods is the time series of the gold prices with 248 observations between the dates of 02.01.2009 - 31.12.2009, which were taken from the website of the Central Bank of the Republic of Turkey as seen in Table 7 and Figure 7.

Table 7. The gold prices of 2009

Figure 7. The gold prices graph of 2009

Two different applications were performed by taking the last 15 observations as the test set in the first analysis and the last 30 observations as the test set in the second analysis of the gold prices data of 2009 as seen in Table 8. For both applications performed, 31 different situations were tried by increasing the fuzzy set number of the time series and the fuzzy set number of the error series by 1 unit between 5 and 35. If the method is analyzed into the interval lengths, the interval lengths were determined in a way to being in the range of 5-35 according to Eq. (9).

In the analysis of the methods by taking the last 15 observations of the gold prices of 2009 as the test set, the fuzzy set numbers that give the best results were determined as 33 by using Song and Chissom [21] method, 29 (the interval length is 600) by using Chen [2] method, 43 (the interval length is 400) with the distribution-based approach and 86 (the interval length is 200) with the average-based approach of Huarng [11] which are the first order methods. In the analyses performed by using the high order models, the fuzzy set numbers that give the best forecasts were found as 22 (the interval length is 800) in the 5th order model of Chen [3] and 19 (the interval length is 900) in the 2nd order model of Aladag [1], 7 fuzzy sets were used in Egrioglu et al [9] and 300 length of interval was used in Kocak [16]. In the application of the proposed method, The case in which the best result were obtained when the interval length of time series is 1000, the fuzzy set number of errors is 28 and the hidden layer unit number of the artificial neural network is 6. As a result of the analysis of the time series of the gold prices of 2009, the best forecasts and forecasting performances of all methods for the test set with 15 observations are summarized in Table 8.

When Table 8 is examined, it is seen that the proposed method has the best forecasting performance with minimum rmse value of 404.24, minimum mape value of 0.563% and maximum direction accuracy (da) value of 71.43% as compared with the other methods. The graphs of the test set and forecasts the proposed method are given together in Figure 8.

Table 8. The forecasts and performances of 15-observation test set for Gold Prices

Figure 8. The graph of 15-observation test set and the forecasts obtained by using proposed method for gold prices

In the analysis of the methods by taking the last 30 observations of the gold prices of 2009 as the test set, the fuzzy set numbers that give the best results were determined as 10 by using Song and Chissom [21] method, 29 (the interval length is 600) by using Chen [2] method, 43 (the interval length is 400) with the distribution-based approach and 86 (the interval length is 200) with the average-based approach of Huarng [11], which are the first order methods. In the analyses performed by using the high order models, the fuzzy set numbers that give the best forecasts were found as 22 (the gap length is 800) in the 3td order model of Chen [3], 29 (the interval length is 600) in the 5th order model of Aladag [1], 7 fuzzy sets were used in Egrioglu et al. [9] and 800 length of interval was used in Kocak [16]. In the application of the proposed method, The case in which the best result were obtained when the interval length of time series is 1700, the fuzzy set number of errors is 16 and the hidden layer unit number of the artificial neural network is 9. As a result of the analysis of the time series for the gold prices of 2009, the best forecasts and forecasting performances of all methods for the test set with 30 observations are summarized in Table 9.

When Table 9 is examined, it is seen that the proposed method has the best forecasting performance with minimum RMSE value of 714.61, minimum MAPE value of 1.132% and maximum direction accuracy value of 51.72% as compared with the other methods. That even RMSE value of 857.34, which was calculated by using Chen (2002)’ method having the best performance following the proposed method, is a value that is considerably higher than RMSE value of 761.91 of the proposed method indicates that the proposed method has a significantly higher performance than the other methods in the literature. The graphs of the test set and forecasts of proposed method are given together in Figure 9.

Table 9. The forecasts and performances of 30-observation test set for gold prices

Figure 9. The graph of 30-observation test set and the forecasts obtained by using proposed method for gold prices

Moreover, two simulated time series were used to compare proposed methods with other alternatives. Simulated time series are chaotic series and they generated by following formula [19]:

(14)

Number of Observations of these series are 250. The last 45 data was taken as test set, other observations were used as training data. Similar applications were made for both of simulated series and the obtained results of applications were given Table 10 and 11. When Table 10 and 11 are examined, it is clearly seen that the proposed method produces the best result for both simulated series.

Table 10. Application results for first simulated time series

Table 11. Application results for second simulated time series

6. Discussion and Conclusions

The models used in the classical time series methods include not only Autoregressive (AR) variables, but also Moving Average (MA) variables. However, the forecasting models of the fuzzy time series proposed in the literature do not included MA variables except for two studies. In this study, an analysis algorithm based on the artificial neural networks was proposed by identifying a new fuzzy time series forecasting model of ARMA(1,1) type that includes the fuzzy MA variables along with the fuzzy AR variables. In the proposed method, finding MA variable reveals a more realistic approach since the majority of the real life time series should be analyzed by using the models including MA variables.

Furthermore, it was seen as a result of the application that the inclusion of MA variables into the model increases the forecasting performance. Moreover, it is remarkable that the proposed method has considerably high direction accuracy (DA) as well as it provides lower RMSE and MAPE values than the other methods in the literature. In the proposed method, the use of feed forward artificial neural networks at the step of determining the fuzzy relation make easer the process of determining the fuzzy relation. In the future studies, it can be ensured that the proposed method has more systematical and higher forecasting performance through the improvements to be made at various steps.