Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 935815, 12 pages

http://dx.doi.org/10.1155/2013/935815

Research Article

An ARMA Type Fuzzy Time Series Forecasting Method Based on Particle Swarm Optimization

Erol Egrioglu,¹ Ufuk Yolcu,² Cagdas Hakan Aladag,³ and Cem Kocak⁴

¹Department of Statistics, Ondokuz Mayıs University, 55139 Samsun, Turkey

²Department of Statistics, Ankara University, 06100 Ankara, Turkey

³Department of Statistics, Hacettepe University, 06100 Ankara, Turkey

⁴Medical High School, Hitit University, 19000 Çorum, Turkey

Received 18 April 2013; Revised 21 June 2013; Accepted 22 June 2013

Academic Editor: Ming Li

Copyright © 2013 Erol Egrioglu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In the literature, fuzzy time series forecasting models generally include fuzzy lagged variables. Thus, these fuzzy time series models have only autoregressive structure. Using such fuzzy time series models can cause modeling error and bad forecasting performance like in conventional time series analysis. To overcome these problems, a new first-order fuzzy time series which forecasting approach including both autoregressive and moving average structures is proposed in this study. Also, the proposed model is a time invariant model and based on particle swarm optimization heuristic. To show the applicability of the proposed approach, some methods were applied to five time series which were also forecasted using the proposed method. Then, the obtained results were compared to those obtained from other methods available in the literature. It was observed that the most accurate forecast was obtained when the proposed approach was employed.

1. Introduction

Fuzzy time series firstly proposed by Song and Chissom [1] can be divided into two subclasses time variant and time invariant. Fuzzy time series method generally embodies three stages such as fuzzification, determination of fuzzy relations, and defuzzification stages. In the fuzzification stage, observations of time series are fuzzified. Fuzzy relations between the observations are defined in the stage of determination of fuzzy relations. Finally, the calculated fuzzy forecasts are defuzzified in the defuzzification phase. If one or more of these stages can be improved, the performance of the method will increase. Therefore, new fuzzy time series approaches have been proposed by making contributions to these stages in the literature.

In the literature, various methods have been proposed to fuzzify observations. While fixed interval lengths are used in Song and Chissom [1–3], Chen [4], Huarng [5], Chen [6], Tsaur et al. [7], Singh [8], and Egrioglu et al. [9, 10], dynamic length of interval lengths is employed in Huarng and Yu [11], Davari et al. [12], Yolcu et al. [13], Kuo et al. [14, 15], Park et al. [16], Hsu et al. [17], and Huang et al. [18] in order to partition the universe of discourse. Also, Cheng et al. [19], Li et al. [20], Egrioglu et al. [21], Chen and Tanuwijaya [22], and Bang and Lee [23] used some methods based on clustering algorithms.

Song and Chissom [3] exploited feed forward neural networks to defuzzify fuzzy forecasts. Chen [4], Huarng [5], and Huarng and Yu [11] utilized centroid method in the defuzzification stage. Besides, Cheng et al. [24] and Aladag et al. [25] used a different technique based on adaptive expectation and centroid methods for fuzzification.

Establishing fuzzy relations plays important role in the forecasting performance of the method. In this phase, Song and Chissom [1] utilized fuzzy relationship matrix, and Sullivan and Woodall [26] used transition matrices based on Markov chain instead of fuzzy relationship matrix. Chen [4] suggested an approach in which fuzzy logic group relationship tables are employed to define fuzzy relations, and Cheng et al. [24], Huarng [5], Huarng and Yu [11], Yu [27], and Egrioglu et al. [21] also used fuzzy logic group relationship tables. Huarng and Yu [28] and Aladag et al. [29] preferred to utilize feed forward artificial neural networks in this stage. Aladag et al. [30] used a different type of artificial neural networks which is Elman recurrent neural networks. In the determination of fuzzy relations, Yu and Huarng [31] and Yolcu et al. [32] proposed different approaches in which feed forward artificial neural networks using membership values are used. Eğrioğlu [33] utilized genetic algorithms to define fuzzy relations. Moreover, many soft computing techniques have been used for forecasting in the literature. Yang et al. [34], and Yang et al. [35], and Yang et al. [36] are some of them.

In all of the studies mentioned previously, fuzzy time series forecasting models have autoregressive (AR) structure. However, many real life fuzzy time series can contain both autoregressive and moving average (MA) structures. Using forecasting models that have only AR structure for these time series can produce insufficient results. To analyze such time series, it is needed to use an ARMA type fuzzy time series forecasting model that includes both AR and MA structures. In this study, a novel fuzzy time series forecasting approach based on ARMA type fuzzy time series forecasting model is proposed to increase forecasting accuracy. Some fuzzy time series approaches in which fuzzy logic group relationship tables are employed disregard the fuzzy set theory since fuzzy sets’ elements whose membership value is 1 are only taken into account in the fuzzification phase. Therefore, information loss occurs, and it is expected to decrease the explanation power of the model. In order to overcome these problems, the particle swarm optimization method is employed in the proposed approach to establish fuzzy relations. In addition, the computational cost of the proposed approach is very reasonable since it does not need to perform complex matrix operations when the particle swarm optimization method is used. Another advantage of the proposed method is that it uses the fuzzy -means clustering method in the fuzzification phase. In the fuzzification phase of the fuzzy time series approach, there are some common problems such as the decision of the number of intervals, arbitrary determination of interval length, and arbitrary choice of degrees of membership. Since the fuzzy -means clustering technique is employed in the proposed approach, it works in a systematic way.

In the real life, some time series have long-range dependence. Long-range-dependent time series is forecasted by using special linear models. ARFIMA models, which are used to forecast long-range-dependent time series, were firstly introduced by Granger and Joyeux [37]. General properties of ARFIMA models were given by Hosking [38, 39] and Beran [40]. First studies are concerned with estimation of fractional differencing parameter in fractional white noise processes. statistic was proposed in Hurst [41]. Egrioglu and Gunay [42], and Li [43], Li and Zhao [44] are important papers about fractional processes. In this study, time series are examined by using hypothesis test to determine long-range dependence in the application. The performance of proposed method is examined according to long-range dependence.

The next section presents basic definitions of fuzzy time series and fuzzy- means clustering method. In Section 3, the modified particle swarm optimization algorithm is given. Section 4 introduces the proposed approach based on ARMA type fuzzy time series forecasting model and particle swarm optimization. The implementation is given in Section 5. Finally, the last section concludes the paper.

2. Fuzzy Time Series Basic Definitions and Fuzzy -Means Clustering

The fuzzy time series was firstly introduced by Song and Chissom [1]. The fundamental definitions of fuzzy time series, time variant, and time invariant fuzzy time series definitions are presented in what follows.

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

Definition 2. Suppose that is caused by only; that is, . Then, this relation can be expressed as , where is the fuzzy relationship between and and is called the first-order model of . “” represents max-min composition of fuzzy sets.

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

Song and Chissom [2] firstly introduced an algorithm based on the first-order model for forecasting time invariant . In Song and Chissom [2], the fuzzy relationship matrix is obtained by many matrix operations. The fuzzy forecasts are obtained based on max-min composition as in what follows:

The dimension of matrix depends on the number of fuzzy sets. The number of fuzzy sets equals the number of intervals that is composed of universe of discourse. The more fuzzy sets are used, the more matrix operations are needed to obtain matrix. When the number of fuzzy set is high, using the method proposed by Song and Chissom [2] considerably increases the computational cost.

For fuzzification, partition of universe of discourse method is used in the method proposed by Song and Chissom [1]. However, there are several problems related to the decomposition of universe of discourse. These problems are the determination of the number of intervals, arbitrarily choice of interval length, and membership degrees. In order to deal with these problems, Cheng et al. [19] and Li et al. [20] used fuzzy -means clustering method for fuzzification. The fuzzy -means clustering method was firstly introduced by Bezdek [45]. This clustering method is the most widely used one. In this method, fuzzy clustering is conducted by minimizing the least squared errors within groups. Let be the membership values, the center of cluster, the number of variables, and the number of clusters. Then, the objective function, which is minimized in fuzzy clustering, is

where is a constant and called the fuzzy index. is a similarity measure between an observation and the center of corresponding fuzzy cluster. The objective function is minimized subject to constraints given in what follows:

In fuzzy -means clustering method, to solve the minimization problem given previously, an iterative algorithm is used. In each iteration, the values of and are updated by using the formulas given in (4) and (5), respectively:

3. The Particle Swarm Optimization

Particle swarm optimization, which is a population-based heuristic algorithm, was firstly proposed by Kennedy and Eberhart [46]. Distinguishing feature of this heuristic algorithm is that it simultaneously examines different points in different regions of the solution space to find the global optimum solution. Local optimum traps can be avoided because of this feature.

In a few fuzzy time series studies, particle swarm optimization method has been exploited in fuzzification phase. While the particle swarm optimization method was employed by Davari et al. [12] for fuzzification in the first-order fuzzy time series forecasting model, Kuo et al. [14] utilized the method in high-order models. In the fuzzification phase, Kuo et al. [15] utilized the particle swarm optimization method in both the first- and the high-order models. Park et al. [16] used the same method for fuzzification in a two-factor high-order model. However, the particle swarm optimization method has never been used to establish fuzzy logic relations in the literature. In time variant fuzzy time series forecasting method proposed in this study, the modified particle swarm optimization whose algorithm is given later is exploited. The modified particle swarm optimization algorithm has time varying inertia weight like in Shi and Eberhart [47]. In a similar way, this algorithm also has time varying acceleration coefficient like in Ma et al. [48].

Algorithm 4. The algorithm of the modified particle swarm optimization.

Step 1. Positions of each th particles’ positions are randomly determined and kept in a vector given as follows:

where represents th position of th particle. and represent the number of particles in a swarm and positions, respectively.

Step 2. Velocities are randomly determined and stored in a vector given in what follows:

Step 3. According to the evaluation function, and particles given in (8) and (9), respectively, are determined:

where is a vector that stores the positions corresponding to the th particle’s best individual performance and represents the best particle, which has the best evaluation function value, found so far.

Step 4. Let and represent cognitive and social coefficients, respectively, and is the inertia parameter. Let , and be the intervals which include possible values for , , and , respectively. At each iteration, these parameters are calculated by using the formulas given in (10):

where and represent maximum iteration number and current iteration number, respectively.

Step 5. Values of velocities and positions are updated by using the formulas given in (11) and (12), respectively.

where and are random values from the interval [].

Step 6. Steps 3 to 5 are repeated until a predetermined maximum iteration number () is reached.

We would like to note that the aim of this study is not to propose a new particle swarm optimization algorithm. In the literature, it was shown that using some time varying parameters can increase the convergence speed of the algorithm (Shi and Eberhart [47]; Ma et al. [48]). Therefore, we just used both the time varying acceleration coefficient (Ma et al. [48]) and the time varying inertia weight (Shi and Eberhart [47]) in the standard particle swarm optimization as mentioned previously. Then, this modified particle swarm optimization method was utilized to calculate membership values in the fuzzy relationship matrix as addressed in the next section. It is not claimed in the paper that a better particle swarm optimization algorithm was improved.

4. The Proposed Approach

The fuzzy time series forecasting methods in the literature have only AR structure that includes lagged fuzzy variables. On the other hand, it will be wiser to use a model which includes both AR and MA structures in order to forecast real life time series that contain both structures. In the literature, we firstly define a new model which includes both AR and MA terms in Definition 5.

Definition 5. Let fuzzy time series be caused by lagged fuzzy time series and lagged absolute fuzzy error series ; then the relationship can be expressed as

This model given in (13) is called an ARMA(1,1) fuzzy time series forecasting model. Thus, definition of fuzzy relations for this model is presented in Definition 6.

Definition 6. Suppose that is caused by only and ; that is, , . Then, this relation can be expressed as

where is the fuzzy relationship matrix between and and is the fuzzy relationship matrix between and .

In this study, a new fuzzy time series forecasting approach is suggested to forecast ARMA(1,1) fuzzy time series forecasting model in (13). The proposed approach provides some advantages which can be given as follows.

(i)Since the model given in (13) is employed in the proposed approach, it is more proper to use this method for forecasting real life time series that contain both AR and MA structures.(ii)Since fuzzy -means clustering method is utilized for fuzzification in the proposed approach, the number of intervals, the interval length, and the degrees of memberships is not arbitrarily determined in the fuzzification phase, so the proposed approach is a systematic method.(iii)The methods in the literature that exploit fuzzy logic group relationship tables disregard the fuzzy set theory since only some elements of fuzzy sets are taken into account in the fuzzification phase. Thus, information loss occurs and it is expected to decrease the explanation power of the model. However, in the proposed method, particle swarm optimization method is used to calculate membership values in the fuzzy relationship matrix, so information loss and decrease in explanatory power of the model are prevented.(iv)In the proposed method, instead of using centroid method, a method which considers all membership degrees is employed. Therefore, information loss that occurs in the defuzzification stage is avoided.(v)Since the proposed approach provides the advantages given previously, it is expected that the suggested method has high forecasting accuracy.

The algorithm of the proposed method is given later step by step.

Algorithm 7. The proposed method’s algorithm.

Step 1. Fuzzify time series by using the fuzzy -means clustering, and define fuzzy sets for absolute values of errors.

Let be the number of fuzzy set, such that . The fuzzy -means clustering algorithm in which the number of fuzzy sets is is applied to the time series which consists of crisp values. After this application, the center of each fuzzy set is determined. Then, the degrees for each observation, which denote a degree of belonging to a fuzzy set for that observation, are calculated with respect to the obtained values of center of fuzzy sets. Finally, ordered fuzzy sets, , are obtained according to the ascending ordered centers, which are denoted by .

Fuzzy sets for absolute values of errors are defined in accordance with predetermined length of interval. Subintervals that contain absolute values of errors are generated. The number of clusters is taken as which equals the number of cluster of time series. Thus, the number of subintervals is . The upper bound of the last subinterval is always open. While the first subinterval has the minimum absolute error, the subsequent subintervals have comparatively greater absolute errors. Generally, subintervals can be defined by

where represents the length of interval. In accordance with these subintervals, fuzzy sets for absolute errors are defined as follows:

Step 2. Determine the parameters of particle swarm optimization.

Some parameters of particle swarm optimization are possible intervals for social coefficient , cognitive coefficient , and inertia weight . The other parameters are swarm size () and maximum iteration number (). Evaluation function is RMSE criterion which is computing as in what follows:

where is crisp time series, is defuzzified forecasts, and is the number of forecasts.

Step 3. Generate a random initial population.

Elements of each particle are elements of and fuzzy relation matrices given in Definition 6. Since both and matrices consist of elements , the number of elements in a particle equals to . For the number of fuzzy sets , representation of a particle is illustrated in Figure 1.

In this step, values of all particles are randomly generated from the interval [] to generate a random initial population.

Figure 1: The structure of a particle.

Step 4. Calculate the evaluation function values of all particles in the current swarm.

The method for calculating RMSE for any particle is given in Steps  4.1,  4.2, and 4.3.

Step  4.1. and fuzzy relation matrices are designed from positions of particles. Each position sets are the rows of and matrices.

Step  4.2. Fuzzy forecasts are calculated by using (14). After and relation matrices are estimated, fuzzy forecasts can be obtained by using fuzzy errors and fuzzy observations. When the first forecast is calculated, it is assumed that the error equals to zero since this assumption is also valid in conventional time series analysis.

For instance, some observations of a time series , corresponding membership values of these observations, and the center values for each fuzzy set are given in Table 1.

Table 1: An example time series .

Date

Membership values

1971
13055
0.7
0.2
0.1
0
0

1972
13563
0.2
0.7
0.1
0
0

1973
13867
0.1
0.4
0.5
0
0

Let the relations matrices be as follows:

Thus, fuzzy forecast for 1972 can be calculated as follows:

After this procedure, fuzzy forecasts are defuzzified by using the rules given in what follows.

(i)If the memberships of fuzzy forecast have only one maximum, then select the center of this cluster as the defuzzified forecasted value. (ii)If memberships of fuzzy forecast have two or more consecutive maximums, then select arithmetic mean of centers of corresponding clusters as the defuzzified forecasted value. (iii)Otherwise, standardize the fuzzy output, and use the center of the fuzzy sets as the forecasted value.

For example, defuzzified forecast for 1972 can be calculated as follows. Since the memberships of fuzzy forecast have only one maximum, defuzzified forecast value is taken as the center value of the first fuzzy set which has the maximum membership value. This can be expressed by

Then, absolute error value for 1972 is calculated given as follows:

Absolute error value is fuzzified by using corresponding fuzzy set which has the maximum membership value for the interval which includes this absolute error value. For instance, for the time series given in Table 1, when the interval length of absolute error is taken as 50, 5 subintervals are as follows:

Since the absolute error value 55 is in the second interval , defuzzified absolute error is . Thus, for 1972, the absolute error is obtained given as follows:

Then, the forecast for 1973 can be calculated as follows:

In the same way, other forecasts for other years can be obtained.

Step  4.3. RMSE is computed using the formula given in (17).

Step 5. Update cognitive coefficient , social coefficient , and inertia parameter at each iteration by using formulas in (10).

Step 6. New positions of the particles are calculated by using the formulas given in (11) and (12).

Step 7. Repeat Steps 4 to 7 until maximum iteration bound () is reached.

5. Application of the Proposed Method

Firstly, the proposed method was applied to Alabama University enrollment data (1971–1992) that is well-known data in the literature. The enrollment data is presented in Table 2.

Table 2: The enrollment data.

Years
Actual
Years
Actual

1971
13055
1982
15433

1972
13563
1983
15497

1973
13867
1984
15145

1974
14696
1985
15163

1975
15460
1986
15984

1976
15311
1987
16859

1977
15603
1988
18150

1978
15861
1989
18970

1979
16807
1990
19328

1980
16919
1991
19337

1981
16388
1992
18876

The algorithm of the proposed method was coded in MATLAB 7.9 version. In the application of the proposed method, we used seven fuzzy sets () in Step 1 since the same number has been employed in other studies available in the literature (Cheng et al., 2008). The centers of clusters and the membership values which are obtained from FCM algorithm in Step 1 are given in Tables 3 and 4, respectively. In the fuzzification process of the absolute error, if the interval length is too small, aggregation will arise in the last fuzzy set. If it is too big, aggregation will occur in the first fuzzy set. Therefore, the length of interval for absolute error is picked as 300. Then, 7 fuzzy sets for absolute errors can be defined as follows:

Table 3: The centers of clusters obtained from FCM algorithm in Step 1.

13458.81
14700.56
15379.31
15904.31
16412.94
16889.11
19075.31

Table 4: The memberships of the observations obtained from FCM algorithm in Step 1.

Years
Cluster 1
Cluster 2
Cluster 3
Cluster 4
Cluster 5
Cluster 6
Cluster 7

1971
0.8768
0.0528
0.0265
0.0176
0.0127
0.0097
0.0039

1972
0.9839
0.0083
0.0032
0.0019
0.0013
0.0010
0.0004

1973
0.7128
0.1709
0.0519
0.0286
0.0183
0.0130
0.0044

1974
0.0000
0.9999
0.0000
0.0000
0.0000
0.0000
0.0000

1975
0.0015
0.0107
0.9463
0.0312
0.0068
0.0030
0.0005

1976
0.0013
0.0121
0.9679
0.0128
0.0037
0.0018
0.0003

1977
0.0063
0.0354
0.5767
0.3178
0.0440
0.0174
0.0024

1978
0.0003
0.0014
0.0079
0.9824
0.0060
0.0017
0.0002

1979
0.0006
0.0014
0.0031
0.0078
0.0410
0.9448
0.0012

1980
0.0001
0.0002
0.0004
0.0009
0.0035
0.9948
0.0002

1981
0.0001
0.0002
0.0006
0.0026
0.9939
0.0025
0.0001

1982
0.0007
0.0052
0.9769
0.0127
0.0029
0.0013
0.0002

1983
0.0029
0.0193
0.8823
0.0737
0.0146
0.0063
0.0010

1984
0.0133
0.1919
0.6905
0.0658
0.0236
0.0125
0.0025

1985
0.0118
0.1599
0.7306
0.0622
0.0219
0.0115
0.0022

1986
0.0009
0.0036
0.0163
0.9388
0.0324
0.0073
0.0006

1987
0.0001
0.0002
0.0004
0.0010
0.0045
0.9936
0.0002

1988
0.0176
0.0325
0.0504
0.0767
0.1281
0.2432
0.4516

1989
0.0004
0.0006
0.0009
0.0012
0.0017
0.0025
0.9928

1990
0.0018
0.0029
0.0040
0.0053
0.0073
0.0104
0.9684

1991
0.0019
0.0031
0.0042
0.0056
0.0077
0.0110
0.9664

1992
0.0013
0.0022
0.0032
0.0044
0.0064
0.0098
0.9728

In the literature, there are no general rules to determine parameter values of particle swarm optimization. Parameter values for this method have been generally specified intuitively due to the data in most of the applications. Therefore, the parameters of particle swarm optimization used in this study were intuitively determined like in other studies available in the literature (Ma et al., 2006). The parameters of the modified particle swarm optimization are determined according to trial and error method. The different values of the parameters were employed. The best parameters of the modified particle swarm optimization are determined as follows: , and . The optimal and matrices obtained from our methods are given later. The fuzzy and defuzzified forecasts of the proposed method are given in Table 5:

Table 5: The forecasts produced by the proposed methods.

Years
Fuzzy forecasts
Defuzzified forecasts

1972
0.8768
0.5000
0.1475
0.0458
0.5824
0.0000
0.4540
13458.81

1973
0.9839
0.5000
0.1475
0.0458
0.5824
0.0000
0.4540
13458.81

1974
0.5000
0.7128
0.1709
0.1709
0.5000
0.5000
0.5000
14700.56

1975
0.9999
0.2359
0.9999
0.0458
0.4169
0.0000
0.4540
15395.91

1976
0.7084
0.3526
0.9463
0.0458
0.0312
0.0000
0.4540
15379.31

1977
0.7084
0.3526
0.9679
0.0458
0.0128
0.0000
0.4540
15379.31

1978
0.5767
0.3526
0.5767
0.0458
0.3178
0.0000
0.4540
15673.54

1979
0.5122
0.5000
0.7212
0.0079
0.9824
0.0000
0.4540
16412.94

1980
0.0078
0.0410
0.2108
0.0031
0.5000
0.0078
0.5000
17217.39

1981
0.0009
0.0035
0.2108
0.0004
0.9019
0.0000
0.4540
16412.94

1982
0.0026
0.5000
0.9939
0.0006
0.1144
0.0000
0.0026
15379.31

1983
0.7084
0.3526
0.9769
0.0458
0.0127
0.0000
0.4540
15379.31

1984
0.7084
0.3526
0.8823
0.0458
0.0737
0.0000
0.4540
15379.31

1985
0.6905
0.3526
0.6905
0.0458
0.1919
0.0000
0.4540
15517.4

1986
0.5000
0.3526
0.5000
0.5000
0.1599
0.5000
0.5000
15987.15

1987
0.5122
0.5000
0.7212
0.0163
0.9388
0.0000
0.4540
16412.94

1988
0.0010
0.0045
0.2108
0.0004
0.5000
0.0010
0.5000
17318.67

1989
0.3198
0.1281
0.2108
0.0504
0.2432
0.0767
0.4516
19075.31

1990
0.3198
0.0017
0.1446
0.0009
0.0025
0.0000
0.4540
19075.31

1991
0.3198
0.0073
0.1446
0.0040
0.0104
0.0000
0.4540
19075.31

1992
0.3198
0.0077
0.1446
0.0042
0.0110
0.0000
0.4540
19075.31

The mean square error (MSE) value of the proposed method is given in Table 6. For comparison, the forecasting results of some other well-known fuzzy time series forecasting approaches are also presented in Table 6. When Table 6 was examined, it is seen that the proposed method gives the most accurate forecasts in terms of MSE criterion.

Table 6: The obtained results for the enrollment data.

Method
MSE

[2]
412499

[26]
386055

[4]
407507

[49] ()
333274

[49] (1998) ()
299634

[49] (1998) ()
321418

[7]
134923

[8]
133700

[19]
228918

[24]
192086

The proposed method
85089

Secondly, to examine the forecasting performance of the proposed approach on test set, it was also applied to four different data sets. The first data set is index 100 in stocks and bonds exchange market of İstanbul (ISBEMI) whose daily observations are between May 20, 2008 and September 29, 2008. This data is called ISBEMI Set 1. The second data is index 100 in stocks and bonds exchange market of İstanbul (ISBEMI) whose daily observations are between October 3, 2008 and December 31, 2008. The second data is called ISBEMI Set 2. The third one is ISBEMI Set 3, and its daily observations are from October 01, 2009 to December 31 2009. ISBEMI Set 1, ISBEMI Set 2, and ISBEMI Set 3 have 95, 59, and 63 observations, respectively. For comparison, these time series were also forecasted using other well-known fuzzy time series method proposed by Song and Chissom [2], Chen [4], Chen [6], Huarng [5], Huarng and Yu [11], and Aladag et al. [29]. In the implementation, the last 15 of ESBEMI Set 1, 7 observations of ISBEMI Set 2, and 7 observations and 15 observations of ISBEMI Set 3 were used for test sets.

It is a well-known fact that there are no general rules to determine the parameters such as the length of interval or the number of fuzzy sets for some methods used in the implementation. Thus, related parameters were determined using trial and error method like in other studies available in the literature. To find the best length of interval for the methods introduced in [4], Chen [6], and Aladag et al. [29], the values between 200 and 2500 were examined with increment 100. The interval lengths in the methods given in Huarng [5] and Huarng and Yu [11] were determined when the methods were progressing because of the nature of these methods. The number of fuzzy sets was experienced between 5 and 25 for the proposed approach and Song and Chissom [2]. When the methods proposed by Chen [5] and Aladag et al. [29] were employed, models order 2, 3, 4, and 5 were examined since time series have daily observations. Finally, the number of neurons in hidden layer was changed from 1 to 10 when the approach proposed by Aladag et al. [29] was applied. After practicing, the forecasts obtained from the case where we have got the best result for the test data and the error criteria related to those forecasts are presented in Tables 7 and 8 for ISBEMI Sets 1 and 2, respectively.

Table 7: The obtained results for ISBEMI Set 1.

Date
The test data
[2]
[4]
[5] distribution-based method
[5] average-based method
[11]
[32]
The proposed method

23.12.2008
26294
26410
26400
26200
26100
26091
26274
26 343

24.12.2008
26055
26410
26400
26200
26367
26091
26273
26 292

25.12.2008
26059
26410
26400
26200
26100
26091
26339
25 983

26.12.2008
26499
26410
26400
26200
26100
26091
26337
26 255

29.12.2008
26424
26410
26400
26200
26500
26608
26565
26 513

30.12.2008
26411
26410
26400
26200
26500
26608
26429
26 192

31.12.2008
26864
26410
26400
26200
26500
26091
26460
26 513

RMSE
261.0109
259.7625
310.4688
251.2390
354.7194
219.2734
207.91

MAPE
0.0075
0.0075
0.0096
0,0080
0.0098
0.0067
0.0067

DA
0.6667
0.6667
0.8333
0.5000
0.5000
0.6667
0.6667

Table 8: The obtained results for ISBEMI Set 2.

Date
The test data
[2]
[4]
[6]
[5] distribution-based method
[5] average-based method
[11]
[29]
The proposed method

09.09.2008
40124.6
39958.4
40550
39700
40000
39967
40547
40500
40457.3

10.09.2008
39295.0
40294.2
40550
37500
40000
39800
40126
39500
40457.3

11.09.2008
37388.1
39431.8
38650
37500
40000
39100
39603
39500
39335.3

12.09.2008
37033.9
37656.1
36750
37500
37200
37500
37602
38500
39335.3

15.09.2008
35081.4
37272.4
36750
37500
37200
37100
36955
37500
36339.1

16.09.2008
33736.4
35024.0
34850
35300
34533
34700
34881
35500
33234.8

17.09.2008
32727.6
33954.4
33900
33100
34400
33700
34037
33500
33234.8

18.09.2008
32216.4
33638.4
33900
33100
32400
32700
32266
32500
33234.8

19.09.2008
36370.2
37746.4
33900
35300
32400
32300
32221
35500
36339.1

22.09.2008
36183.6
36450.8
36750
35300
36800
37700
36711
35500
36339.1

23.09.2008
35454.2
36166.4
36750
35300
36800
36100
35225
35500
36339.1

24.09.2008
35177.1
35072.6
34850
35300
35600
35100
35634
35500
33234.8

25.09.2008
36361.8
35024.0
34850
35300
34533
34700
34881
35500
36339.1

26.09.2008
36556.6
36450.8
36750
35300
36800
37700
36711
37500
36339.1

29.09.2008
36051.3
36450.8
36750
37500
36800
35900
36711
35500
36339.1

RMSE
1155.3
1229.7
1143.6
1579.1
1483.0
1484.9
1145.9
1107.2

MAPE
0.0268
0.0299
0.0261
0.0327
0.0308
0.0298
0.0255
0.0233

DA
0.4286
0.5000
0.5714
0.4286
0.7143
0.4286
0.6429
0.5714

For Table 7, the cases in which the superlative results were obtained are

(i)the number of the fuzzy sets was 12 for the Song-Chissom method [2];(ii)the interval length was 1200 for the Chen method [4];(iii)the interval length was 800 for the Huarng distribution-based method [5];(iv)the interval length was 200 for the Huarng average-based method [5];(v)the ratio sample percentile was 0.5 for the Huarng and Yu ratio-based method [11];(vi)the number of the fuzzy sets was five for the Cheng et al. method [19];(vii)the number of the fuzzy sets was 11 and the number of the neurons in the hidden layer was five for Yolcu et al. [32];(viii)the number of the fuzzy sets was 5 in the proposed method.

For Table 8, the cases in which the superlative results were obtained are

(i)when the number of fuzzy sets is 12, for the method of Song and Chissom [2];(ii)when the length of interval is 1900 for the method of [4];(iii)when the length of interval is 2200 and the model order is 2 for the method of Chen [6];(iv)when the length of interval is 800 for the method of Huarng distribution based [5];(v)when the length of interval is 200 for the method of Huarng average based [5];(vi)when the ratio sample percentile is 0.5 for the method of Huarng and Yu ratio based [11];(vii)when the length of interval is 1000, the model order is 2, and number of neurons in hidden layer is 7 for the method proposed by Aladag et al. [29];(viii)when the number of fuzzy sets is 5 in the proposed method.

The best cases for the results given in Table 9 resulted when

(i)Song-Chissom method [2] was applied with nine fuzzy sets;(ii)interval length was 1300 for the Chen method [4];(iii)interval length was 800 for the Huarng distribution-based method [5];(iv)interval length was 200 for the Huarng average-based method [5];(v)ratio sample percentile was 0.50 for the Huarng and Yu ratio-based method [11];(vi)the number of the fuzzy sets was 13 and the number of the neurons in the hidden layer was seven for Yolcu et al. [32];(vii)when the number of fuzzy sets is 5 in the proposed method.

Table 9: The obtained results (last 7 observations are test set) for ISBEMI Set 3.

Date
The test data
[2]
[4]
[5] distribution-based method
[5] average-based method
[11]
[32]
The proposed method

23.12.2009
51162
51137
52150
51900
51573
51033
51317
51684

24.12.2009
51461
51137
50850
50700
50373
51033
51317
51109

25.12.2009
51661
51137
50850
50700
51240
51033
51317
51684

28.12.2009
51619
51137
52150
51900
51573
51033
51317
51684

29.12.2009
51786
51137
52150
51900
51573
51033
51317
51684

30.12.2009
51668
51137
52150
51900
51773
52004
51317
51684

31.12.2009
52825
51137
52150
51900
51573
51033
51317
51684

RMSE
771.0155
666.5039
659.9450
671.4803
830.4429
640.4131
494.5180

MAPE
0.0116
0.0123
0.0111
0.0097
0.0128
0.0090
0.0061

DA
0.3333
0.3333
0.3333
0.3333
0.1667
0.5000
0.6667

The best cases for the results given in Table 10 resulted when

(i)Song-Chissom [2] method was applied with nine fuzzy sets;(ii)interval length was 1500 for the Chen method [4];(iii)interval length was 800 for the Huarng distribution-based method [5];(iv)interval length was 200 for the Huarng average-based method [5];(v)ratio sample percentile was 0.50 for the Huarng and Yu ratio-based method [11];(vi)the number of the fuzzy sets was seven and the number of the neurons in the hidden layer was three for Yolcu et al. [32];(vii)the number of the fuzzy sets was five in the proposed method.

Table 10: The obtained results (last 15 observations are test set) for ISBEMI Set 3.

Date
The test data
[2]
[4]
[5] distribution-based method
[5] average-based method
[11]
[32]
The proposed method

11.12.2009
49386
49872
50250
49500
49100
49748
49516
49982.39

14.12.2009
50198
48606
48750
49500
49300
49316
50064
49982.39

15.12.2009
50450
49872
50250
49900
50500
50405
50942
50833.82

16.12.2009
50817
50294
50250
49900
48900
48886
50217
50833.82

17.12.2009
49963
50294
50250
50300
50900
48886
49641
50833.82

18.12.2009
50138
49872
50250
49900
49900
49748
49619
49613.83

21.12.2009
51281
49872
50250
49900
50500
50405
50932
51390.68

22.12.2009
51533
51137
51000
50300
50967
50625
50817
51390.68

23.12.2009
51162
51137
51000
51900
51500
51065
51100
51390.68

24.12.2009
51461
51137
51000
50300
50550
50625
50646
51390.68

25.12.2009
51661
51137
51000
50300
51500
51065
51073
51390.68

28.12.2009
51619
51137
51000
51900
51700
51065
51117
51390.68

29.12.2009
51786
51137
51000
51900
51700
51065
51114
51390.68

30.12.2009
51668
51137
51000
51900
51700
51963
51119
51390.68

31.12.2009
52825
51137
51000
51900
51700
51065
51117
51390.68

RMSE
810.8493
820.5748
815.9879
760.7710
917.1070
662.3497
522.0852

MAPE
0.0128
0.0133
0.0134
0.0110
0.0147
0.0106
0.0075

DA
0.2857
0.5000
0.2857
0.5000
0.3571
0.5000
0.6429

When Tables 6–10 are examined, it is obvious that the most accurate forecasts are obtained for all data sets when the proposed forecasting approach is used.

6. Conclusion and Discussion

In this study, an ARMA type fuzzy time series forecasting model is firstly introduced in the literature. In addition, to analyze this model, a novel approach based on fuzzy -means and particle swarm optimization methods is suggested in this study. To show the forecasting performance of the proposed forecasting approach, the enrollments of Alabama University are applied to a well-known data and obtained forecasting results are compared to those produced by other fuzzy time series forecasting methods available in the literature. In addition, to evaluate the forecasting performance of the proposed forecasting approach on test set, two different time series are forecasted by utilizing the proposed method and other approaches available in the literature. As a result of the comparison, it is observed that the proposed approach gives the most accurate forecasts.

In the literature, the proposed method is the first method based on a fuzzy time series forecasting model which contains both AR and MA terms. In the linear stochastic ARMA models, the AR(∞) model is equal to MA(1) model. But this result is only valid for linear ARMA models. There is no such kind of result for fuzzy time series model, artificial neural network model, and nonlinear stochastic models. Because of this, adding MA terms to fuzzy time series forecasting model is really important. Also, in the proposed method, particle swarm optimization method is used to calculate membership values in the fuzzy relationship matrix, so information loss and decrease in explanatory power of the model are prevented. Besides, since fuzzy -means clustering method is employed in the fuzzification process, the number of intervals, the interval length, and the degrees of memberships are not arbitrarily determined, so the proposed approach works in a systematic way. In addition, information loss which occurs in the defuzzification stage is avoided by using a method which takes all membership degrees into account instead of using centroid method. Therefore, the proposed method has high forecasting accuracy.

Moreover, long-range dependence was determined for all time series by using tests, and results are given in the appendix. All of the time series are short-range dependent. In the future studies, the performance of the proposed method will be able to research for long-range time series. But in the proposed fuzzy time series method, any linear model was not used. Fuzzy time series methods can be used to forecast nonlinear time series. In the fuzzy time series methods, the relations are employed instead of functions.

The proposed forecasting approach is suggested to forecast a first-order model including both AR and MA terms. In future studies, high-order models can be defined. Then, it is possible to extend the proposed approach to forecast high order models.

Appendix

test was applied to five time series to determine long-range dependence. The test results are given in Table 11. As a result of hypotheses tests, all data sets are short-range dependent. The tests were applied in S-PLUS package program.

Table 11: Results of test.

Data
R/S statistics

ISBEMI Set 1
1.2966
No

ISBEMI Set 2
1.5937
No

ISBEMI Set 3
1.6523
No

Enrollment data
1.0969
No

Acknowledgments

The authors would like to thank the reviewers for their helpful comments and opinions. This work was supported by “The Scientific and Technological Research Council of Turkey (TUBITAK),” Turkey, under Project no. 210T150.

References

Q. Song and B. S. Chissom, “Fuzzy time series and its models,” Fuzzy Sets and Systems, vol. 54, no. 3, pp. 269–277, 1993.

Q. Song and B. S. Chissom, “Forecasting enrollments with fuzzy time series—part I,” Fuzzy Sets and Systems, vol. 54, pp. 1–10, 1993.

Q. Song and B. S. Chissom, “Forecasting enrollments with fuzzy time series—part II,” Fuzzy Sets and Systems, vol. 62, no. 1, pp. 1–8, 1994.

S. M. Chen, “Forecasting enrollments based on fuzzy time series,” Fuzzy Sets and Systems, vol. 81, no. 3, pp. 311–319, 1996.

K. Huarng, “Effective lengths of intervals to improve forecasting in fuzzy time series,” Fuzzy Sets and Systems, vol. 123, no. 3, pp. 387–394, 2001.

S. Chen, “Forecasting enrollments based on high-order fuzzy time series,” Cybernetics and Systems, vol. 33, no. 1, pp. 1–16, 2002.

R.-C. Tsaur, J. C. O. Yang, and H.-F. Wang, “Fuzzy relation analysis in fuzzy time series model,” Computers & Mathematics with Applications, vol. 49, no. 4, pp. 539–548, 2005.

S. R. Singh, “A simple method of forecasting based on fuzzy time series,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 330–339, 2007.

E. Egrioglu, C. H. Aladag, U. Yolcu, V. R. Uslu, and M. A. Basaran, “Finding an optimal interval length in high order fuzzy time series,” Expert Systems with Applications, vol. 37, no. 7, pp. 5052–5055, 2010.

E. Egrioglu, C. H. Aladag, M. A. Basaran, U. Yolcu, and V. R. Uslu, “A new approach based on the optimization of the length of intervals in fuzzy time series,” Journal of Intelligent and Fuzzy Systems, vol. 22, no. 1, pp. 15–19, 2011.

K. Huarng and T. H. Yu, “Ratio-based lengths of intervals to improve fuzzy time series forecasting,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 36, no. 2, pp. 328–340, 2006.

S. Davari, M. H. F. Zarandi, and I. B. Turksen, “An improved fuzzy time series forecasting model based on particle swarm intervalization,” in Proceedings o the Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS '09), Cincinnati, Ohio, USA, June 2009.

U. Yolcu, E. Egrioglu, V. R. Uslu, M. A. Basaran, and C. H. Aladag, “A new approach for determining the length of intervals for fuzzy time series,” Applied Soft Computing Journal, vol. 9, no. 2, pp. 647–651, 2009.

I. H. Kuo, S. J. Horng, T. W. Kao, T. L. Lin, C. L. Lee, and Y. Pan, “An improved method for forecasting enrollments based on fuzzy time series and particle swarm optimization,” Expert Systems with Applications, vol. 36, no. 3, pp. 6108–6117, 2009.

I. H. Kuo, S. J. Horng, Y. H. Chen et al., “Forecasting TAIFEX based on fuzzy time series and particle swarm optimization,” Expert Systems with Applications, vol. 37, no. 2, pp. 1494–1502, 2010.

J. Park, D. Lee, C. Song, and M. Chun, “TAIFEX and KOSPI 200 forecasting based on two-factors high-order fuzzy time series and particle swarm optimization,” Expert Systems with Applications, vol. 37, no. 2, pp. 959–967, 2010.

L. Hsu, S. Horng, T. Kao et al., “Temperature prediction and TAIFEX forecasting based on fuzzy relationships and MTPSO techniques,” Expert Systems with Applications, vol. 37, no. 4, pp. 2756–2770, 2010.

Y. Huang, S. Horng, T. Kao et al., “An improved forecasting model based on the weighted fuzzy relationship matrix combined with a PSO adaptation for enrollments,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 7A, pp. 4027–4046, 2011.

C. Cheng, G. Cheng, and J. Wang, “Multi-attribute fuzzy time series method based on fuzzy clustering,” Expert Systems with Applications, vol. 34, no. 2, pp. 1235–1242, 2008.

S.-T. Li, Y.-C. Cheng, and S.-Y. Lin, “A FCM-based deterministic forecasting model for fuzzy time series,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3052–3063, 2008.

E. Egrioglu, C. H. Aladag, U. Yolcu, V. R. Uslu, and N. A. Erilli, “Fuzzy time series forecasting method based on Gustafson-Kessel fuzzy clustering,” Expert Systems with Applications, vol. 38, no. 8, pp. 10355–10357, 2011.

S. Chen and K. Tanuwijaya, “Fuzzy forecasting based on high-order fuzzy logical relationships and automatic clustering techniques,” Expert Systems with Applications, vol. 38, no. 12, pp. 15425–15437, 2011.

Y. Bang and C. Lee, “Fuzzy time series prediction using hierarchical clustering algorithms,” Expert Systems with Applications, vol. 38, no. 4, pp. 4312–4325, 2011.

C. Cheng, T. Chen, H. J. Teoh, and C. Chiang, “Fuzzy time-series based on adaptive expectation model for TAIEX forecasting,” Expert Systems with Applications, vol. 34, no. 2, pp. 1126–1132, 2008.

C. H. Aladag, U. Yolcu, and E. Egrioglu, “A high order fuzzy time series forecasting model based on adaptive expectation and artificial neural networks,” Mathematics and Computers in Simulation, vol. 81, no. 4, pp. 875–882, 2010.

J. Sullivan and W. H. Woodall, “A comparison of fuzzy forecasting and Markov modeling,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 279–293, 1994.

H. Yu, “Weighted fuzzy time series models for TAIEX forecasting,” Physica A, vol. 349, no. 3-4, pp. 609–624, 2005.

K. Huarng and T. H. Yu, “The application of neural networks to forecast fuzzy time series,” Physica A, vol. 363, no. 2, pp. 481–491, 2006.

C. H. Aladag, M. A. Basaran, E. Egrioglu, U. Yolcu, and V. R. Uslu, “Forecasting in high order fuzzy times series by using neural networks to define fuzzy relations,” Expert Systems with Applications, vol. 36, no. 3, pp. 4228–4231, 2009.

C. H. Aladag, E. Egrioğlu, S. Gunay, and U. Yolcu, “High order fuzzy time series model and stock exchange application,” Anadolu Üni,” Anadolu Üniversitesi Bilim ve Teknoloji Dergisi, vol. 11, no. 2, pp. 95–101, 2010.

T. H. Yu and K. Huarng, “A neural network-based fuzzy time series model to improve forecasting,” Expert Systems with Applications, vol. 37, no. 4, pp. 3366–3372, 2010.

U. Yolcu, C. H. Aladag, E. Egrioglu, and V. R. Uslu, “Time series forecasting with a novel fuzzy time series approach: an example for Istanbul stock market,” Journal of Statistical Computation and Simulation, vol. 83, no. 4, 2013.

E. Eğrioğlu, “A new time-invariant fuzzy time series forecasting method based on genetic algorithm,” Advances in Fuzzy Systems, vol. 2012, Article ID 785709, 6 pages, 2012.

X. H. Yang, Z. F. Yang, and Z. Y. Shen, “GHHAGA for environmental systems optimization,” Journal of Environmental Informatics, vol. 5, no. 1, pp. 36–41, 2005.

X. Yang, Z. Yang, X. Yin, and J. Li, “Chaos gray-coded genetic algorithm and its application for pollution source identifications in convection-diffusion equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1676–1688, 2008.

X. H. Yang, D. X. She, Z. F. Yang, Q. H. Tang, and J. Q. Li, “Chaotic bayesian method based on multiple criteria decision making (MCDM) for forecasting nonlinear hydrological time series,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 11-12, pp. 1595–1610, 2009.

C. W. J. Granger and R. Joyeux, “An introduction to long-memory time series models and fractional differencing,” Journal of Time Series Analysis, vol. 1, no. 1, pp. 15–29, 1980.

J. R. M. Hosking, “Fractional differencing,” Biometrika, vol. 68, no. 1, pp. 165–176, 1981.

J. R. M. Hosking, “Modeling persistence in hydrological time series using fractional differencing,” Water Resources Research, vol. 20, no. 12, pp. 1898–1908, 1984.

J. Beran, Statistics for Long-Memory Processes, vol. 61, Chapman and Hall, New York, NY, USA, 1994.

H. E. Hurst, “Long-term storage capacity of reservoirs,” Transactions of the American Society of Civil Engineers, vol. 116, pp. 770–779, 1951.

E. Egrioglu and S. Gunay, “Two new Mmethods for bayesian model selection in ARFIMA models,” Expert Systems with Applications, vol. 37, pp. 8359–8364, 2010.

M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010.

M. Li and W. Zhao, “On 1/f noise,” Mathematical Problems in Engineering, vol. 2012, Article ID 673648, 23 pages, 2012.

J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, NY, USA, 1981.

J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, pp. 1942–1948, IEEE Press, Piscataway, NJ, USA, December 1995.

Y. Shi and R. C. Eberhart, “Empirical study of particle swarm optimization,” in Proceedings of the Congress on Evolutionary Computation (CEC '99), vol. 3, pp. 1950–1945, 1999.

Y. Ma, C. Jiang, Z. Hou, and C. Wang, “The formulation of the optimal strategies for the electricity producers based on the particle swarm optimization algorithm,” IEEE Transactions on Power Systems, vol. 21, no. 4, pp. 1663–1671, 2006.

J. Hwang, S. Chen, and C. Lee, “Handling forecasting problems using fuzzy time series,” Fuzzy Sets and Systems, vol. 100, no. 1–3, pp. 217–228, 1998.