PSO-based high order time invariant fuzzy time series ...

ECMODE-03235; No of Pages 7 Economic Modelling xxx (2014) xxx–xxx

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

PSO-based high order time invariant fuzzy time series method: Application to stock exchange data Erol Egrioglu Ondokuz Mayıs University, Department of Statistics, Turkey

a r t i c l e

i n f o

Article history: Accepted 18 February 2014 Available online xxxx Keywords: Fuzzy time series Particle swarm optimization Fuzzy c-means Forecasting Define fuzzy relation

a b s t r a c t Fuzzy time series methods are effective techniques to forecast time series. Fuzzy time series methods are based on fuzzy set theory. In the early years, classical fuzzy set operations were used in the fuzzy time series methods. In recent years, artificial intelligence techniques have been used in different stages of fuzzy time series methods. In this paper, a novel fuzzy time series method which is based on particle swarm optimization is proposed. A high order fuzzy time series forecasting model is used in the proposed method. In the proposed method, determination of fuzzy relations is performed by estimating the optimal fuzzy relation matrix. The performance of the proposed method is compared to some methods in the literature by using three real world time series. It is shown that the proposed method has better performance than other methods in the literature. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Fuzzy time series methods have different approaches to uncertainty from probabilistic statistical methods. Classical time series analysis methods are probabilistic methods, and they need some strict assumptions. Moreover, probabilistic methods don't take into consideration fuzziness. However, some real life time series contain fuzziness. Because of this fact, various fuzzy time series methods were proposed in the literature. Fuzzy time series methods do not need any assumptions like normality and linearity. Fuzzy time series methods were first defined in Song and Chissom (1993a). First definitions and methods were based on fuzzy set theory and some fuzzy set operations. Song and Chissom (1993a) defined two different fuzzy time series types: time variant and time invariant. The first time invariant fuzzy time series method was proposed in Song and Chissom (1993b). There have been a lot of studies about time invariant fuzzy time series in the literature. But there have been a limited number of studies about time variant fuzzy time series. When fuzzy time series methods are examined, it can be said that they consist of three stages: fuzzification, determining fuzzy relation and defuzzification. Fuzzy time series methods are based on different forecasting models. The forecasting models can be first order or high order. When the first order models are used, it is assumed that fuzzy time series is caused by one order lagged fuzzy time series. Similarly, when nth order fuzzy time series forecasting model is used, fuzzy time series are caused by 1,2,…,n order lagged fuzzy time series. In the literature, many methods are used for determining fuzzy relations. These methods are using fuzzy logic group relation tables, artificial neural networks, fuzzy relation matrices obtained from some fuzzy set operations, particle swarm optimization and genetic algorithms. Chen (1996, 2002), Lee et al. (2007, 2008), Duru et al.

(2010), Lee et al. (2013), Uslu et al. (2013), Bulut (2014) and Chen and Chen (2014) used fuzzy logic group relation tables. Aladag et al. (2009), Egrioglu et al. (2009a,b), Yolcu et al. (2013) and Aladag (2013) used some type of artificial neural networks. Song and Chissom (1993b, 1994) used a fuzzy relation matrix obtained from some fuzzy set operations. Egrioglu (2012) used a fuzzy relation matrix obtained from a genetic algorithm and Aladag et al. (2012) used a fuzzy relation matrix obtained from particle swarm optimization. Aladag et al. (2012) and Egrioglu (2012) methods are based on first order fuzzy time series forecasting models. The high order models are needed to forecast many real life time series. Chen (2002), Lee et al. (2007, 2008), Kuo et al. (2009, 2010), Park et al. (2010), Chen and Chung (2006), Hsu et al. (2010), Egrioglu et al. (2009a,b, 2010), Aladag et al. (2009), Chen (2013), Qiu et al. (2013), and Jilani and Burney (2008) studies are based on the high order fuzzy time series forecasting model. Some methods which are used to determine fuzzy relations didn't take into consideration membership values of fuzzy sets. Song and Chissom (1993b), Yolcu et al. (2013), Yu and Huarng (2010), Egrioglu (2012) and Aladag et al. (2012) papers took into consideration membership values of fuzzy sets. In this study, a novel fuzzy time series method is proposed. The proposed method uses the fuzzy c-mean method in fuzzification stage, and the particle swarm optimization method in the determining fuzzy relation stage. The proposed method is based on the high order fuzzy time series forecasting model. The proposed method is an improved version of the Aladag et al. (2012) method. Aladag et al. (2012) was based on the first order fuzzy time series forecasting model as distinct from the proposed method. Particle swarm optimization is summarized in the second section of this paper. In the third section, the particulars of the proposed method are given. The application results are given in the fourth section. The results are discussed in the last section of the paper.

http://dx.doi.org/10.1016/j.econmod.2014.02.017 0264-9993/© 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Egrioglu, E., PSO-based high order time invariant fuzzy time series method: Application to stock exchange data, Econ. Model. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017

2

E. Egrioglu / Economic Modelling xxx (2014) xxx–xxx

2. Particle swarm optimization Particle swarm optimization, which is an artificial intelligence technique, was firstly proposed by Kenedy and Eberhart (1995). There have been different versions of particle swarm optimization in the literature. Shi and Eberhart (1999) used time varying inertia weight and Ma et al. (2006) used time varying acceleration coefficients in their algorithm. An algorithm which uses time varying inertia weight and a time varying acceleration coefficient is given below. We called this algorithm modified particle swarm optimization. This algorithm was firstly used in Aladag et al. (2012).

Step 4. Let c1 and c2 represent cognitive and social coefficients, respectively, and w is the inertia parameter. Let (c1i, c1f), (c2i, c2f), and (w1, w2) be the intervals which include possible values for c1, c2 and w, respectively. In each iteration, these parameters are calculated by using the formulas given in Eqs. (5), (6) and (7). c1 ¼ c1 f −c1i

t þ c1i maxt

ð5Þ

maxt−t þ c2i c2 ¼ c2 f −c2i maxt

ð6Þ

maxt−t þ w1 maxt

ð7Þ

Algorithm 1. The modified particle swarm optimization Step 1. Positions of each kth (k = 1,2, …, pn) particle's positions are randomly determined and kept in a Xk given as follows: n o X k ¼ xk;1 ; xk;2 ; …; xk;d ; k ¼ 1; 2; …; pn

ð1Þ

where xk,i (i = 1,2,…,d) represents ith position of kth particle. pn and d represent the number of particles in a swarm and positions in a particle, respectively. Step 2. Velocities are randomly determined and stored in a vector Vk given below. n o V k ¼ vk;1 ; vk;2 ; …; vk;d ; k ¼ 1; 2; …; pn:

ð2Þ

Step 3. According to the evaluation function, Pbest and Gbest particles given in Eqs. (1) and (2), respectively, are determined. Pbest k ¼ pk;1 ; pk;2 ; …; pk;d ; k ¼ 1; 2; …; pn

ð3Þ

Gbest ¼ pg;1 ; pg;2 ; …; pg;d

ð4Þ

where Pbestk is a vector stores the positions corresponding to the kth particle's best individual performance, and Gbest represents the best particle, which has the best evaluation function value found so far.

w ¼ ðw2 −w1 Þ

where maxt and t represent the maximum iteration number and the current iteration number, respectively. Step 5. Values of velocities and positions are updated by using the formulas given in Eqs. (8) and (9), respectively. h i tþ1 t t t vi; j ¼ w vi; j þ c1 rand1 pi; j −xi; j þ c2 rand2 pg; j −xi; j

ð8Þ

tþ1

t

tþ1

xi; j ¼ xi; j þ vi; j

ð9Þ

where rand1 and rand2 are generated random values from the interval [0,1]. Step 6. Steps 3 to 5 are repeated until a predetermined maximum iteration number (maxt) is reached. 3. The proposed method There have been a lot of studies about fuzzy time series methods in the literature. The most important differences in fuzzy time series methods from classical methods are membership values and the advantages of membership values. Although the defuzzification process is performed in the fuzzy time series methods, obtaining fuzzy forecasts

Fig. 1. Flow chart of the proposed method.



3

… Fig. 2. Positions of one particle.

is still a good advantage because of membership values. In the literature, some studies didn't take into consideration these membership values in the determination of fuzzy relation stage. Aladag et al. (2012) proposed a fuzzy time series method which is based on particle swarm optimization. The Aladag et al. (2012) method used the first order fuzzy time series forecasting method. Better quality forecasts can be obtained from high order models instead of first order models. The high order fuzzy time series forecasting model is defined as below. Definition. Let F(t) be a time invariant fuzzy time series. If F(t) is caused by F(t − 1), F(t − 2), …, and F(t − n) then this fuzzy logical relationship is represented by F ðt−nÞ; …; F ðt−2Þ; F ðt−1Þ→ F ðt Þ

ð10Þ

and it is called the nth order fuzzy time series forecasting model. To obtain forecasts from a high order model (10) can be used in intersection operations. After R fuzzy relation matrix is obtained, fuzzy forecasts can be calculated by using Eq. (11).

F ðt Þ ¼ ð F ðt−nÞ∩…∩ F ðt−2Þ∩ F ðt−1Þ∘RÞ

ð11Þ

where “°” is max–min composition. R matrix was obtained by using max–min compositions and union operations in Song and Chissom (1993b). These operations were very complex and time consuming in Song and Chissom (1993b). Model (10) is used in the proposed novel fuzzy time series forecasting method. The novel method is an improved version to high order models of Aladag et al. (2012). The proposed method is using the fuzzy c-mean method that was proposed in Bezdek (1981) in the fuzzification stage, and the particle swarm optimization method in the determining fuzzy relation stage. Some advantages of the proposed method are listed below: • Because of using fuzzy c-means in fuzzification stage, there is no need for subjective decisions like determining interval length. • The proposed method takes into consideration membership values. • Because R relation matrix is obtained from particle swarm optimization, there is no necessity for complex and time consuming matrix operations. • Because the proposed method is based on the high order fuzzy time series forecasting model, the better quality forecasts can be obtain from the proposed method for real life time series.

Fig. 3. The sequence chart of IMKB data.

The proposed method is given in Algorithm 2 and a flow chart of the proposed method is given in Fig. 1. Algorithm 2. Step 1. The parameters of the proposed method are determined. These parameters are: pn: Particle number of swarm [c1i, c1f]: Cognitive coefficient interval [c2i, c2f]: Social coefficient interval maxt: Maximum iteration number fsn: Number of fuzzy set ntest: Observation number of test n: Model order. The root of mean square error (RMSE) is used as a fitness function in the proposed method. RMSE is calculated according to Eq. (12). RMSE ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Xn ^t Þ2 ðy −y t¼1 t n

ð12Þ

^t , and n represent crisp time series, defuzzified forecasts, and where yt, y the number of forecasts, respectively. Step 2. The fuzzy c-mean method is applied to the training data of time series. The cluster centers of fsn fuzzy sets Lr (r = 1,2,…,fsn) and membership values of training data observations are obtained by the fuzzy c-mean method. The fuzzy sets are redesigned according to the ascending ordered centers. The membership values of test data observations are obtained from cluster centers which were determined for training data by fuzzy cmean. Fuzzy c-mean method is iteratively applied according to the Bezdek (1981) procedure. First, the initial cluster centers are simulated by the interval on which time series is defined. The memberships are calculated according to Eq. (14). Eqs. (13) and (14) are consecutively used. n X

vi ¼

β

uij x j

j¼1 n X

β

ð13Þ

uij

j¼1

uij ¼

1 12 = 0 ðβ−1Þ fsn d x j ; vi X @ A d x j ; vk k¼1

ð14Þ

where β is fuzziness indices and d(.) is Euclidean distance, x1, x2, …, xn are observations of training data and uij is membership value of xj to ith fuzzy set. At the end of the FCM application processes, cluster centers vi(i = 1, 2, …, c) and membership values of training data observations to all fuzzy sets (uij, i = 1, 2, …, c; j = 1, 2, …, n) are obtained. The cluster centers are sorted into an ascending order and the membership values are arranged by the sort of orders. Step 3. Generate a random initial positions and velocities. In the proposed method, positions are generated by uniform distribution with (0,1) parameters. Velocities are generated by uniform distribution with (− 1,1). There are pn particles


4


Table 1 Forecasting results for IMKB data set. Date

Test set

Song and Chissom (1993b)

Chen (1996)

Huarng (2001)a

Huarng (2001)b

Huarng and Yu (2006)

Cheng et al. (2008)

Yolcu et al. (2013)

Proposed method

23.12.2008 24.12.2008 25.12.2008 26.12.2008 29.12.2008 30.12.2008 31.12.2008

26,294 26,055 26,059 26,499 26,424 26,411 26,864 RMSE MAPE MAE

26,410 26,410 26,410 26,410 26,410 26,410 26,410 261.01 0.75% 197.14

26,400 26,400 26,400 26,400 26,400 26,400 26,400 259.76 0.75% 198.57

26,200 26,200 26,200 26,200 26,200 26,200 26,200 310.47 0.96% 254

26,100 26,367 26,100 26,100 26,500 26,500 26,500 251.24 0.80% 210.71

26,091 26,091 26,091 26,091 26,608 26,608 26,091 354.72 0.98% 261.85

26,390 26,390 26,390 26,390 26,390 26,390 26,390 258.87 0.76% 200

26,274 26,273 26,339 26,337 26,565 26,429 26,460 219.27 0.67% 177.57

26,342 26,342 26,342 26,342 26,342 26,342 26,639 189.60 0.62% 164.42

a b

Distribution based method. Average based method.

and velocities in the swarm. One particle has d positions. In the proposed method, positions of a particle are elements of R fuzzy relation matrix. R fuzzy relation matrix has fsn columns and fsn rows and d = fsn × fsn. Each fuzzy relation matrix (Ri, i = 1, 2, …, pn) is obtained from each particle. Step 4. Fitness (RMSE) values of the particles are calculated. In the proposed method, Steps 4.1 and 4.4 are applied to calculate the RMSE value for each particle. Step 4.1. Ri fuzzy relation matrix is constituted from particle positions. The ith particle is shown in Fig. 2. Then R matrix is designed from ith particle as below: 2 6 Ri ¼ 6 4

xi;1

xi;2

xi;fsnþ1 ⋮

xi;fsnþ2 ⋮

xi;ðfsn−1Þfsnþ1

xi;ðfsn−1Þfsnþ2

3 … xi;fsn … xi;2fsn 7 7: 5 ⋮ ⋮ … xi;fsnfsn

Step 4.2. Fuzzy forecasts for training data are calculated by using Eq. (11). For example, let model order be 2, fsn = 3, F(t − 1) = [0.7 0.3 0], F(t − 2) = [0.5 0.5 0] and 2

1 0:5 R ¼ 4 0:1 0 0:1 0

3 0:5 1 5: 1

Then, fuzzy forecast for t time is calculated as below:

F ðt Þ ¼ F ðt−2Þ∩ F ðt−1Þ R

F ðt−2Þ∩F ðt−1Þ ¼ ½ minð0:7; 0:5Þ; minð0:3; 0:5Þ; minð0; 0Þ ¼ ½0:5 0:3 0

2 ^F ðt Þ ¼ ½0:5; 0:3; 0

4

1 0:5 0:1 0 0:1 0

3 0:5 1 5 1

¼ maxð minð0:5; 1Þ; minð0:3; 0:1Þ; minð0; 0:1ÞÞ maxð minð0:5; 0:5Þ; minð0:3; 0Þ; minð0; 0ÞÞ maxð minð0:5; 1Þ; minð0:3; 0:1Þ; minð0; 0:1ÞÞ ¼ ½ maxð0:5; 0:1; 0Þ maxð0:5; 0; 0Þ maxð0:5; 0:1; 0Þ ^F ðt Þ ¼ ½0:5 0:5 0:5:

Step 4.3. Defuzzified forecasts are obtained. The ordered cluster centers of fuzzy sets and membership values of fuzzy forecasts are used for the defuzzification stage. • If the membership values of the fuzzy forecast have only one maximum, then take the center value of this set as the defuzzified forecasted value. • If membership values of fuzzy forecast have two or more consecutive maximums, then select the arithmetic mean of the centers of the corresponding clusters as the defuzzified forecasted value. • Otherwise, standardize the fuzzy output and use the center of the fuzzy sets as the forecasted value. Step 4.4. RMSE value is calculated according to Eq. (12). Step 5. According to RMSE, the Pbest and Gbest particles which are given in Eqs. (3) and (4), respectively, are determined. Step 6. Update cognitive coefficient c1, social coefficient c2, and the inertia parameter w at each iteration by using the formulas (5), (6) and (7), respectively. Step 7. New velocities and positions of the particles are calculated by using the formulas given in Eqs. (8) and (9). Step 8. Repeat Step 4 to Step 8 until maximum iteration bound (maxt) is reached. Step 9. Gbest gives optimal fuzzy relation matrix (Roptimal). The forecasts and RMSE value for test data are calculated by using Roptimal and applying Steps 4.2 and 4.4. 4. The application In the literature, there are many studies about stock exchange forecasting. Wei (2013) and Cheng et al. (2013) proposed new hybrid ANFIS (adaptive network fuzzy inference system) methods to forecast TAIEX data. Cheng and Wei (2014) proposed a hybrid method to forecast TAIEX. In this study, the proposed method's performance is compared with some methods by using three different sets of the stock index time series. The application results are given in the subsections.

7500 7000 6500

4.1. IMKB application

6000

Fig. 4. The sequence chart of TAIFEX data.

The first time series is the data of Index 100 for the stocks and bonds exchange market of Istanbul (IMKB). Observations of IMKB are obtained daily between 03/October/2008 and 31/December/2008. A sequence chart of IMKB is given in Fig. 3. The time series has 59 observations.



5

Table 2 Forecasting results for TAIFEX data set. Date

Test set

Lee et al. (2007)

Lee et al. (2008)

Aladag et al. (2009)

Hsu et al. (2010)

Aladag (2013)

Aladag et al. (2012)

Proposed method

10.09.1998 11.09.1998 .14.09.1998 15.09.1998 16.09.1998 17.09.1998 18.09.1998 19.09.1998 21.09.1998 22.09.1998 23.09.1998 24.09.1998 25.09.1998 28.09.1998 29.09.1998 30.09.1998

6709.75 6726.50 6774.55 6762.00 6952.75 6906.00 6842.00 7039.00 6861.00 6926.00 6852.00 6890.00 6871.00 6840.00 6806.00 6787.00 RMSE MAPE MAE

6621.43 6677.48 6709.63 6732.02 6753.38 6756.02 6804.26 6842.04 6839.01 6897.33 6896.83 6919.27 6903.36 6895.95 6879.31 6878.34 93.5 1.09% 74.62

6917.40 6852.23 6805.71 6762.37 6793.06 6784.40 6970.74 6977.22 6874.46 7126.05 6862.49 6944.36 683,188 6843.24 6858.45 6825.64 102.96 1.14% 78.08

6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 6850.00 83.58 0.96% 65.62

6745.45 6757.89 6731.76 6722.54 6753.72 6761.54 6857.27 6898.97 6853.07 6951.95 6896.84 6919.94 6884.99 6894.10 6866.17 6865.06 80.02 0.87% 60.19

6750 6750 6850 6850 6850 6850 6850 6850 6950 6850 6850 6850 6850 6850 6850 6750 72.55 0.82% 56.37

6778 6778 6778 6778 6778 6856 6925 6856 6856 6856 6856 6856 6856 6856 6856 6778 74.94 0.75% 52.05

6826 6741 6741 6741 6963 6963 6894 6894 6894 6894 6894 6894 6894 6894 6826 6926 66.08 0.73% 49.78

if Table 2 is examined, it is clear that the proposed method outperforms the other methods according to RMSE, MAPE and MAE criteria.

The first 52 and the last 7 observations are used as the training and the test sets, respectively. In Yolcu et al. (2013), IMKB data set was forecasted by Song and Chissom (1993b), Chen (1996), and Huarng (2001) distribution and average based methods, and Huarng and Yu (2006), and Cheng et al. (2008) methods. The forecasts and RMSE, mean absolute percentage error (MAPE) and mean absolute error (MAE) values of these methods are given in Table 1. MAPE and MAE values are calculated by using Eqs. (15)–(16). MAPE ¼

MAE ¼

^t 1 Xn yt −y t¼1 n yt

ð15Þ

1 Xn ^t j: jy −y t¼1 t n

ð16Þ

The best forecasts are obtained from these methods in the following situations: In Song and Chissom (1993b), the number of fuzzy sets is 12; in Chen (1996), length of interval is 1200; in Huarng and Yu (2006) ratio based method, ratio sample percentile is 0.5; in Cheng et al. (2008), the number of fuzzy sets is 5; in Yolcu et al. (2013) method, the number of fuzzy sets is 11 and the number of hidden layer neurons is 5. In the Huarng (2001) distribution based method, length of interval is 800; in average based method, length of interval is 200. Moreover, the

best result obtained from the proposed method is given in Table 1. If five fuzzy sets and second order model are used in the proposed method, the best forecast result can be obtained from IMKB data set. In this situation, it obtained the optimal R matrix given below. 2

0; 8278 6 0; 2988 6 R¼6 6 0; 4846 4 0; 0000 0; 5778

0; 7949 0; 8823 0; 7052 0; 4821 0:2249

0; 8168 0; 2765 0; 8618 0; 5991 0; 4929

0; 5580 0; 3085 0; 0877 0; 6784 0; 4238

3 0; 0013 0; 0000 7 7 0; 8560 7 7 0; 5065 5 0; 7642

If Table 1 is examined, it is clear that the proposed method is better than the others according to RMSE and MAPE criteria. 4.2. Taiwan future exchange application Secondly, the proposed method is applied to Taiwan future exchange (TAIFEX) data whose observations are between 03.08.1998 and 30.09.1998. The time series has 47 observations. The first 31 and the last 16 observations are used as the training and the test sets, respectively. The graph of TAIFEX is given in Fig. 4. TAIFEX data is forecasted by the proposed method. TAIFEX data is also forecasted by using methods proposed by Lee et al. (2007, 2008), Aladag et al. (2009), Hsu et al. (2010), Aladag (2013) and Aladag et al. (2012). The forecast results produced by the methods proposed in Aladag et al. (2009), Hsu et al. (2010), Aladag (2013) and Aladag et al. (2012) were taken from corresponding papers. When the proposed

7600 7100 TAIEX

6600 6100 5600 5100

Fig. 5. The sequence graph of TAIEX Data.

Table 3 The results obtained from all methods. Method

RMSE

Song and Chissom (1993b) Chen (1996) Chen (2002) Huarng and Yu (2006) Huarng et al. (2007) Yu and Huarng (2008) Aladag et al. (2009) Chen and Chen (2011) Proposed method

77.86 77.18 71.98 63.57 72.35 67.00 69.80 57.30 51.14


6


method is applied to TAIFEX data, the best forecasts are obtained from second order model and five fuzzy sets. All forecasted results are given in Table 2. 4.3. Taiwan Stock Exchange Capitalization Weighted Stock Index Application Finally, the proposed method is applied to Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) data between 01.01.2004 and 31.12.2004. The sequence chart of the time series is shown in Fig. 5. The first 205 observations are used as training set and the last 45 observations are used as a test set. The forecast results produced by Song and Chissom (1993b), Chen (1996, 2002), Huarng and Yu (2006), Huarng et al. (2007), Yu and Huarng (2008), Aladag et al. (2009) and Chen and Chen (2011) methods were taken from corresponding papers. When the proposed method is applied to TAIEX data, the best forecasts are obtained from the second order model and when seven fuzzy sets are used. All forecast results are given in Table 3. Moreover, the MAPE value of the proposed method for TAIEX data is 0.0069. It can be concluded that the proposed method outperforms the other method for TAIEX data according to RMSE criterion. Also, the MAPE value of the proposed method is very small. The sequence chart of forecasts and test data is given in Fig. 6. 5. Conclusion and discussions Determination of the fuzzy relation stage in the fuzzy time series methods is very important for forecast performance. Aladag et al. (2012) proposed a first order fuzzy time series method. In this paper, this method is successfully improved for a high order fuzzy time series forecasting model. According to the application results, the proposed method has better forecasting performance than many other methods in the literature. Because the proposed method is based on the high order fuzzy time series forecasting model, real life time series can be well forecasted. Moreover, the proposed method takes into consideration all membership vales. It should not be forgotten that the performance of the proposed method can change for different data sets. It is not easy to say it will outperform other methods for every data set. As a result of implementation, it can be seen that the proposed method can produce good forecasts for the three stock exchange data sets. Although the proposed method is improved to a high order form, the order selection is an important problem for it. In future studies, order selection for the proposed method can be achieved by using optimization techniques. If some new techniques applied in the fuzzification and defuzzification stages, a better forecasting performance could be obtained from the proposed method. In the future, proposed method can be easily modified for better forecasting performance and multivariate fuzzy

6200

Proposed Method TAIEX Test Set

6100 6000 5900 5800 5700 5600 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Fig. 6. The sequence chart of TAIEX data and forecasts of proposed method.

time series models. Although fuzzy time series methods can produce good forecasts, the confidence intervals for forecasts cannot be obtained. It can be said that this is a very big challenge for nonprobabilistic forecasting methods. Obtaining confidence intervals of forecasts for the proposed method will be considered in future studies.

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