PSO-based High Order Time Invariant Fuzzy Time Series Method_stock Exchange Rate

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(2009), Egrioglu et al. (2009a,b), Yolcu et al. (2013) and Aladag

Economic Modelling xxx (2014) xxxxxx

ECMODE-03235; No of Pages 7

Contents lists available at ScienceDirect

M

e lsSong and Chissom (1993b). There have been a lot of studies abouttime invariant fuzzy time series in the literature. But there have beena limited number of studies about time variant fuzzy time series.When fuzzy time series methods are examined, it can be said thatthey consist of three stages: fuzzication, determining fuzzy relationand defuzzication. Fuzzy time series methods are based on differentforecasting models. The forecasting models can be rst order or highorder. When the rst order models are used, it is assumed that fuzzytime series is caused by one order lagged fuzzy time series. Similarly,

methods which are used to determine fuzzy relations didn't take intoconsideration 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 membershipvalues of fuzzy sets.

In this study, a novel fuzzy time series method is proposed. The pro-posedmethod uses the fuzzy c-meanmethod in fuzzication stage, andthe particle swarm optimization method in the determining fuzzy rela-tion stage. The proposed method is based on the high order fuzzy timewhen nth order fuzzy time series forecastintime series are caused by 1,2,,n order lagged

In the literature, manymethods are used fotions. These methods are using fuzzy logic grocial neural networks, fuzzy relation matricfuzzy set operations, particle swarm opalgorithms. Chen (1996, 2002), Lee et al. (2

http://dx.doi.org/10.1016/j.econmod.2014.02.0170264-9993/ 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Egrioglu, E., PSO-bModel. (2014), http://dx.doi.org/10.1016/j.ehissom (1993a) denedriant and time invariant.ethod was proposed in

(2010), Egrioglu et al. (2009a,b, 2010), Aladag et al. (2009), Chen(2013), Qiu et al. (2013), and Jilani and Burney (2008) studies arebased on the high order fuzzy time series forecasting model. Sometwo different fuzzy time series types: time vaThe rst time invariant fuzzy time series mmethods are probabilistic methods, andtions. Moreover, probabilistic methodfuzziness. However, some real life timeof this fact, various fuzzy time series merature. Fuzzy time series methods donormality and linearity.

Fuzzy time series methods were r(1993a). First denitions and methodsand some fuzzy set operations. Songeed some strict assump-take into considerationontain fuzziness. Becausewere proposed in the lit-ed any assumptions like

ed in Song and Chissomased on fuzzy set theory

Chissom (1993b, 1994) used a fuzzy relation matrix obtained fromsome fuzzy set operations. Egrioglu (2012) used a fuzzy relation matrixobtained from a genetic algorithm and Aladag et al. (2012) used a fuzzyrelationmatrix obtained fromparticle swarmoptimization. Aladag et al.(2012) and Egrioglu (2012)methods are based on rst order fuzzy timeseries forecastingmodels. The high order models are needed to forecastmany 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.from probabilistic statistical methods. Classicthey nal time series analysis (2013) used some type of articial neural networks. Song and

Fuzzy time series methods have different approaches to uncertaintyPSO-based high order time invariant fuzzyexchange data

Erol EgriogluOndokuz Mays University, Department of Statistics, Turkey

a b s t r a c ta r t i c l e i n f o

Article history:Accepted 18 February 2014Available online xxxx

Keywords:Fuzzy time seriesParticle swarm optimizationFuzzy c-meansForecastingDene fuzzy relation

Fuzzy time series methods aon fuzzy set theory. In the earecent years, articial intelligthis paper, a novel fuzzy timorder fuzzy time series forection of fuzzy relations is perfposed method is compared tthat the proposed method h

1. Introduction

Economic

j ourna l homepage: www.g model is used, fuzzyfuzzy time series.r determining fuzzy rela-up relation tables, arti-es obtained from sometimization and genetic007, 2008), Duru et al.

ased high order time invarianconmod.2014.02.017me series method: Application to stock

ffective techniques to forecast time series. Fuzzy time series methods are basedears, classical fuzzy set operationswere used in the fuzzy time seriesmethods. Ine techniques have been used in different stages of fuzzy time series methods. Inries method which is based on particle swarm optimization is proposed. A highng model is used in the proposed method. In the proposed method, determina-ed by estimating the optimal fuzzy relation matrix. The performance of the pro-me methods in the literature by using three real world time series. It is shownetter performance than other methods in the literature.

2014 Elsevier B.V. All rights reserved.

(2010), Lee et al. (2013), Uslu et al. (2013), Bulut (2014) and Chenand Chen (2014) used fuzzy logic group relation tables. Aladag et al.

odelling

ev ie r .com/ locate /ecmodseries forecasting model. The proposed method is an improved versionof the Aladag et al. (2012) method. Aladag et al. (2012) was based onthe rst order fuzzy time series forecasting model as distinct from theproposed method. Particle swarm optimization is summarized in thesecond section of this paper. In the third section, the particulars of theproposedmethodare given. The application results are given in the fourthsection. The results are discussed in the last section of the paper.

t fuzzy time series method: Application to stock exchange data, Econ.

2. Particle swarm optimization

Particle swarm optimization, which is an articial intelligence tech-nique, was rstly proposed by Kenedy and Eberhart (1995). There havebeen 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 coefcients in their algorithm. Analgorithmwhich uses time varying inertiaweight and a time varying ac-celeration coefcient is given below. We called this algorithmmodiedparticle swarm optimization. This algorithm was rstly used in Aladaget al. (2012).

Algorithm 1. The modied particle swarm optimization

Step 1. Positions of each kth (k = 1,2, , pn) particle's positions arerandomly determined and kept in a Xk given as follows:

Xk xk ;1; xk ;2;; xk ;dn o

; k 1;2;; pn 1

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 po-sitions in a particle, respectively.

Step 2. Velocities are randomly determined and stored in a vector Vkgiven below.

n o

Step 4. Let c1 and c2 represent cognitive and social coefcients, respec-tively, and w is the inertia parameter. Let (c1i, c1f), (c2i, c2f), and(w1,w2) be the intervals which include possible values for c1, c2andw, respectively. In each iteration, these parameters are cal-culated by using the formulas given in Eqs. (5), (6) and (7).

c1 c1 fc1i t

maxt c1i 5

c2 c2 fc2i maxtt

maxt c2i 6

w w2w1 maxttmaxt

w1 7

where maxt and t represent the maximum iteration numberand the current iteration number, respectively.

Step 5. Values of velocities and positions are updated by using the for-mulas given in Eqs. (8) and (9), respectively.

vt1i; j w vti; j c1 rand1 pi; jxti; j

c2 rand2 pg; jxti; j h i

8

2 E. Egrioglu / Economic Modelling xxx (2014) xxxxxxVk vk;1; vk;2;; vk;d ; k 1;2;;pn: 2

Step 3. According to the evaluation function, Pbest and Gbest particlesgiven in Eqs. (1) and (2), respectively, are determined.

Pbestk pk;1;pk;2;;pk;d

; k 1;2;;pn 3

Gbest pg;1;pg;2;;pg;d

4

where Pbestk is a vector stores the positions corresponding tothe kth particle's best individual performance, and Gbest repre-sents the best particle, which has the best evaluation functionvalue found so far.Fig. 1. Flow chart of the

Please cite this article as: Egrioglu, E., PSO-based high order time invarianModel. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017xt1i; j xti; j vt1i; j 9

where rand1 and rand2 are generated random values from theinterval [0,1].

Step 6. Steps 3 to 5 are repeated until a predetermined maximum iter-ation number (maxt) is reached.

3. The proposed method

There have been a lot of studies about fuzzy time series methods inthe literature. The most important differences in fuzzy time seriesmethods from classical methods aremembership values and the advan-tages of membership values. Although the defuzzication process isperformed in the fuzzy time series methods, obtaining fuzzy forecastsproposed method.


is still a good advantage because ofmembership values. In the literature,some studies didn't take into consideration these membership values inthe determination of fuzzy relation stage. Aladag et al. (2012) proposeda fuzzy time series method which is based on particle swarm optimiza-tion. The Aladag et al. (2012) method used the rst order fuzzy time se-ries forecasting method. Better quality forecasts can be obtained fromhigh order models instead of rst order models. The high order fuzzytime series forecasting model is dened as below.

Fig. 2. Positions of

3E. Egrioglu / Economic Modelling xxx (2014) xxxxxxDenition. Let F(t) be a time invariant fuzzy time series. If F(t) is causedby F(t 1), F(t 2),, and F(t n) then this fuzzy logical relationshipis represented by

F tn ;; F t2 ; F t1 F t 10

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 in-

tersection operations. After R fuzzy relation matrix is obtained, fuzzyforecasts can be calculated by using Eq. (11).

F t F tn F t2 F t1 R 11

where is maxmin composition. R matrix was obtained by usingmaxmin compositions and union operations in Song and Chissom(1993b). These operations were very complex and time consuming inSong and Chissom (1993b).

Model (10) is used in the proposed novel fuzzy time series forecast-ing method. The novel method is an improved version to high ordermodels of Aladag et al. (2012). The proposed method is using thefuzzy c-mean method that was proposed in Bezdek (1981) in thefuzzication stage, and the particle swarm optimization method in thedetermining fuzzy relation stage. Some advantages of the proposedmethod are listed below:

Because of using fuzzy c-means in fuzzication stage, there is no needfor subjective decisions like determining interval length.

The proposed method takes into consideration membership values. Because R relation matrix is obtained from particle swarm optimiza-tion, there is no necessity for complex and time consuming matrixoperations.

Because the proposed method is based on the high order fuzzy timeseries forecasting model, the better quality forecasts can be obtainfrom the proposed method for real life time series.Fig. 3. The sequence chart of IMKB data.

Please cite this article as: Egrioglu, E., PSO-based high order time invarianModel. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017The proposedmethod is given in Algorithm 2 and a ow chart of theproposed method is given in Fig. 1.

Algorithm 2.

Step 1. The parameters of the proposed method are determined. Theseparameters are:

pn: Particle number of swarm[c1i, c1f]: Cognitive coefcient interval[c2i, c2f]: Social coefcient intervalmaxt:Maximum iteration numberfsn: Number of fuzzy setntest: Observation number of testn: Model order.The root ofmean square error (RMSE) is used as a tness function in theproposed method. RMSE is calculated according to Eq. (12).

RMSE 1n

Xnt1 yty^t

2r

12

where yt, y^t, and n represent crisp time series, defuzzied forecasts, andthe number of forecasts, respectively.Step 2. The fuzzy c-meanmethod is applied to the training data of time

series. The cluster centers of fsn fuzzy sets Lr (r= 1,2,,fsn) andmembership values of training data observations are obtainedby the fuzzy c-mean method. The fuzzy sets are redesigned ac-cording to the ascending ordered centers. The membershipvalues of test data observations are obtained from cluster cen-ters which were determined for training data by fuzzy c-mean. Fuzzy c-mean method is iteratively applied according tothe Bezdek (1981) procedure. First, the initial cluster centersare simulated by the interval on which time series is dened.The memberships are calculated according to Eq. (14).Eqs. (13) and (14) are consecutively used.

vi

Xnj1

uij x j

Xnj1

uij

13

uij 1

Xfsnk1

d xj; vi

d xj; vk

0@

1A

2= 1 14

where is fuzziness indices and d(.) is Euclidean distance, x1, x2,, xn are observations of training data and uij is membershipvalue of xj to ith fuzzy set. At the end of the FCM application pro-cesses, cluster centers vi(i= 1, 2,, c) and membership valuesof training data observations to all fuzzy sets (uij, i= 1, 2,, c;j= 1, 2,, n) are obtained. The cluster centers are sorted into

one particle.an ascending order and the membership values are arrangedby the sort of orders.

Step 3. Generate a random initial positions and velocities.In the proposed method, positions are generated by uniformdistribution with (0,1) parameters. Velocities are generatedby uniform distribution with (1,1). There are pn particles


Table 1Forecasting results for IMKB data set.

Date Test set Song and Chissom(1993b)

Chen (1996) Huarng(2001)a

23.12.2008 26,294 26,410 26,400 26,20024.12.2008 26,055 26,410 26,400 26,20025.12.2008 26,059 26,410 26,400 26,20026.12.2008 26,499 26,410 26,400 26,20029.12.2008 26,424 26,410 26,400 26,20030.12.2008 26,411 26,410 26,400 26,20031.12.2008 26,864 26,410 26,400 26,200

RMSE 261.01 259.76 310.47MAPE 0.75% 0.75% 0.96%

4 E. Egrioglu / Economic Modelling xxx (2014) xxxxxxand velocities in the swarm. One particle has d positions. Inthe proposed method, positions of a particle are elements ofR fuzzy relation matrix. R fuzzy relation matrix has fsn col-umns and fsn rows and d= fsn fsn. Each fuzzy relation ma-trix (Ri, i = 1, 2, , pn) is obtained from each particle.

Step 4. Fitness (RMSE) values of the particles are calculated. In theproposed method, Steps 4.1 and 4.4 are applied to calculatethe RMSE value for each particle.

Step 4.1. Ri fuzzy relation matrix is constituted from particle posi-tions. The ith particle is shown in Fig. 2.Then R matrix is designed from ith particle as below:

Ri xi;1 xi;2 xi;fsn

xi;fsn1 xi;fsn2 xi;2fsn

xi; fsn1 fsn1 xi; fsn1 fsn2 xi;fsnfsn

2664

3775:

Step 4.2. Fuzzy forecasts for training data are calculated byusing 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

R 1 0:5 0:50:1 0 10:1 0 1

24

35:

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

F t F t2 F t1 R F t2 F t1 min 0:7;0:5 ; min 0:3;0:5 ; min 0;0 0:5 0:3 0

1 0:5 0:52 3

MAE 197.14 198.57 254

a Distribution based method.b Average based method.F^ t 0:5;0:3;0 0:1 0 10:1 0 1

4 5

6000

6500

7000

7500

Fig. 4. The sequence chart of TAIFEX data.

Please cite this article as: Egrioglu, E., PSO-based high order time invarianModel. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017forecasts and RMSE value for test data are calculated byusing Roptimal and applying Steps 4.2 and 4.4. max min 0:5;1 ; min 0:3;0:1 ; min 0;0:1 max min 0:5;0:5 ; min 0:3;0 ; min 0;0 max min 0:5;1 ; min 0:3;0:1 ; min 0;0:1

max 0:5;0:1;0 max 0:5;0;0 max 0:5;0:1;0 F^ t 0:5 0:5 0:5 :

Step 4.3. Defuzzied forecasts are obtained. The ordered clustercenters of fuzzy sets and membership values of fuzzyforecasts are used for the defuzzication stage.

If themembership values of the fuzzy forecast have only onemax-imum, then take the center value of this set as the defuzzied fore-casted value.

If membership values of fuzzy forecast have two ormore consecu-tive maximums, then select the arithmetic mean of the centers ofthe corresponding clusters as the defuzzied forecasted value.

Otherwise, standardize the fuzzy output and use the center of thefuzzy 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 coefcient c1, social coefcient c2, and the in-

ertia 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 byusing 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

Huarng(2001)b

Huarng and Yu(2006)

Cheng et al.(2008)

Yolcu et al.(2013)

Proposedmethod

26,100 26,091 26,390 26,274 26,34226,367 26,091 26,390 26,273 26,34226,100 26,091 26,390 26,339 26,34226,100 26,091 26,390 26,337 26,34226,500 26,608 26,390 26,565 26,34226,500 26,608 26,390 26,429 26,34226,500 26,091 26,390 26,460 26,639251.24 354.72 258.87 219.27 189.600.80% 0.98% 0.76% 0.67% 0.62%210.71 261.85 200 177.57 164.424. The application

In the literature, there are many studies about stock exchange fore-casting.Wei (2013) and Cheng et al. (2013) proposed newhybrid ANFIS(adaptive network fuzzy inference system) methods to forecast TAIEXdata. Cheng and Wei (2014) proposed a hybrid method to forecastTAIEX. In this study, the proposed method's performance is comparedwith some methods by using three different sets of the stock indextime series. The application results are given in the subsections.

4.1. IMKB application

The rst time series is the data of Index 100 for the stocks and bondsexchangemarket of Istanbul (IMKB). Observations of IMKB are obtaineddaily between 03/October/2008 and 31/December/2008. A sequencechart of IMKB is given in Fig. 3. The time series has 59 observations.


Table 2Forecasting 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 6709.75 6621.43 6917.40 6850.00 6745.45 6750 6778 682611.09.1998 6726.50 6677.48 6852.23 6850.00 6757.89 6750 6778 6741.14.09.1998 6774.55 6709.63 6805.71 6850.00 6731.76 6850 6778 674115.09.1998 6762.00 6732.02 6762.37 6850.00 6722.54 6850 6778 674116.09.1998 6952.75 6753.38 6793.06 6850.00 6753.72 6850 6778 696317.09.1998 6906.00 6756.02 6784.40 6850.00 6761.54 6850 6856 696318.09.1998 6842.00 6804.26 6970.74 6850.00 6857.27 6850 6925 689419.09.1998 7039.00 6842.04 6977.22 6850.00 6898.97 6850 6856 689421.09.1998 6861.00 6839.01 6874.46 6850.00 6853.07 6950 6856 689422.09.1998 6926.00 6897.33 7126.05 6850.00 6951.95 6850 6856 689423.09.1998 6852.00 6896.83 6862.49 6850.00 6896.84 6850 6856 689424.09.1998 6890.00 6919.27 6944.36 6850.00 6919.94 6850 6856 689425.09.1998 6871.00 6903.36 683,188 6850.00 6884.99 6850 6856 689428.09.1998 6840.00 6895.95 6843.24 6850.0029.09.1998 6806.00 6879.31 6858.45 6850.00

s acc

5E. Egrioglu / Economic Modelling xxx (2014) xxxxxxThe rst 52 and the last 7 observations are used as the training and thetest sets, respectively.

In Yolcu et al. (2013), IMKB data set was forecasted by Song andChissom (1993b), Chen (1996), and Huarng (2001) distribution and av-erage based methods, and Huarng and Yu (2006), and Cheng et al.(2008) methods. The forecasts and RMSE, mean absolute percentageerror (MAPE) and mean absolute error (MAE) values of these methodsare given in Table 1. MAPE and MAE values are calculated by using Eqs.(15)(16).

MAPE 1n

Xnt1

yty^tyt

15

MAE 1n

Xnt1 yty^tj j: 16

The best forecasts are obtained from thesemethods in the following

30.09.1998 6787.00 6878.34 6825.64 6850.00RMSE 93.5 102.96 83.58MAPE 1.09% 1.14% 0.96%MAE 74.62 78.08 65.62

if Table 2 is examined, it is clear that the proposed method outperforms the other methodsituations: 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 neuronsis 5. In the Huarng (2001) distribution based method, length of intervalis 800; in average basedmethod, length of interval is 200.Moreover, the

5100

5600

6100

6600

7100

7600

TAIEX

Fig. 5. The sequence graph of TAIEX Data.

Please cite this article as: Egrioglu, E., PSO-based high order time invarianModel. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017best result obtained from the proposedmethod is given in Table 1. Ifvefuzzy sets and secondordermodel are used in the proposedmethod, thebest forecast result can be obtained from IMKB data set. In this situation,it obtained the optimal R matrix given below.

R

0;8278 0;7949 0;8168 0;5580 0;00130;2988 0;8823 0;2765 0;3085 0;00000;4846 0;7052 0;8618 0;0877 0;85600;0000 0;4821 0;5991 0;6784 0;50650;5778 0:2249 0;4929 0;4238 0;7642

266664

377775

If Table 1 is examined, it is clear that the proposed method is betterthan the others according to RMSE and MAPE criteria.

4.2. Taiwan future exchange application

Secondly, the proposed method is applied to Taiwan future ex-change (TAIFEX) data whose observations are between 03.08.1998

6894.10 6850 6856 68946866.17 6850 6856 68266865.06 6750 6778 692680.02 72.55 74.94 66.080.87% 0.82% 0.75% 0.73%60.19 56.37 52.05 49.78

ording to RMSE, MAPE and MAE criteria.and 30.09.1998. The time series has 47 observations. The rst 31 andthe last 16 observations are used as the training and the test sets, re-spectively. The graph of TAIFEX is given in Fig. 4.

TAIFEX data is forecasted by the proposed method. TAIFEX data isalso 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 inAladag et al. (2009), Hsu et al. (2010), Aladag (2013) and Aladag et al.(2012) were taken from corresponding papers. When the proposed

Table 3The results obtained from all methods.

Method RMSE

Song and Chissom (1993b) 77.86Chen (1996) 77.18Chen (2002) 71.98Huarng and Yu (2006) 63.57Huarng et al. (2007) 72.35Yu and Huarng (2008) 67.00Aladag et al. (2009) 69.80Chen and Chen (2011) 57.30Proposed method 51.14


method is applied to TAIFEX data, the best forecasts are obtained fromsecond order model and ve fuzzy sets. All forecasted results are givenin Table 2.

4.3. Taiwan Stock Exchange CapitalizationWeighted Stock Index Application

Finally, the proposed method is applied to Taiwan Stock ExchangeCapitalization Weighted Stock Index (TAIEX) data between 01.01.2004and 31.12.2004. The sequence chart of the time series is shown inFig. 5. The rst 205 observations are used as training set and the last45 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 andHuarng (2008), Aladag et al. (2009) and Chen and Chen (2011)methods were taken from corresponding papers. When the proposedmethod is applied to TAIEX data, the best forecasts are obtained fromthe second ordermodel andwhen seven fuzzy sets are used. All forecast

Lee, L.W., Wang, L.H., Chen, S.M., 2007. Temperature prediction and TAIFEX forecastingbased on fuzzy logical relationships and genetic algorithms. Expert Syst. Appl. 33,

6 E. Egrioglu / Economic Modelling xxx (2014) xxxxxxresults are given in Table 3.Moreover, theMAPE value of the proposedmethod for TAIEX data is

0.0069. It can be concluded that the proposed method outperforms theother method for TAIEX data according to RMSE criterion. Also, theMAPE value of the proposed method is very small. The sequence chartof forecasts and test data is given in Fig. 6.

5. Conclusion and discussions

Determination of the fuzzy relation stage in the fuzzy time seriesmethods is very important for forecast performance. Aladag et al.(2012) proposed a rst order fuzzy time series method. In thispaper, this method is successfully improved for a high order fuzzytime series forecasting model. According to the application results,the proposed method has better forecasting performance thanmany other methods in the literature. Because the proposed methodis based on the high order fuzzy time series forecasting model, reallife time series can bewell forecasted. Moreover, the proposedmeth-od takes into consideration all membership vales. It should not beforgotten that the performance of the proposed method can changefor different data sets. It is not easy to say it will outperform othermethods for every data set. As a result of implementation, it can beseen that the proposed method can produce good forecasts for thethree stock exchange data sets. Although the proposedmethod is im-proved to a high order form, the order selection is an importantproblem for it. In future studies, order selection for the proposedmethod can be achieved by using optimization techniques. If somenew techniques applied in the fuzzication and defuzzicationstages, a better forecasting performance could be obtained from theproposed method. In the future, proposed method can be easilymodied for better forecasting performance and multivariate fuzzy

5600

5700

5800

5900

6000

6100

6200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Proposed Method

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

Please cite this article as: Egrioglu, E., PSO-based high order time invarianModel. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017539550.Lee, L.W., Wang, L.H., Chen, S.M., 2008. Temperature prediction and TAIFEX forecasting

based on high-order fuzzy logical relationships and genetic simulated annealing tech-niques. Expert Syst. Appl. 34, 328336.

Lee, M.H., Sadaei, H.J., Suhartono, 2013. Introducing polynomial fuzzy time series. J. Intell.Fuzzy Syst. 25 (1), 117128.

Ma, Y., Jiang, C., Hou, Z., Wang, C., 2006. The formulation of the optimal strategies for theelectricity producers based on the particle swarm optimization algorithm. IEEE Trans.time series models. Although fuzzy time series methods can producegood forecasts, the condence intervals for forecasts cannot be ob-tained. It can be said that this is a very big challenge for non-probabilistic forecasting methods. Obtaining condence intervals offorecasts for the proposed method will be considered in futurestudies.

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7E. Egrioglu / Economic Modelling xxx (2014) xxxxxxPlease cite this article as: Egrioglu, E., PSO-based high order time invarianModel. (2014), http://dx.doi.org/10.1016/j.econmod.2014.02.017t fuzzy time series method: Application to stock exchange data, Econ.

PSO-based high order time invariant fuzzy time series method: Application to stock exchange data1. Introduction2. Particle swarm optimization3. The proposed method4. The application4.1. IMKB application4.2. Taiwan future exchange application4.3. Taiwan Stock Exchange Capitalization Weighted Stock Index Application

5. Conclusion and discussionsReferences

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PSO-based High Order Time Invariant Fuzzy Time Series Method_stock Exchange Rate