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Ahighorderfuzzytimeseriesforecastingmodelbasedonadaptiveexpectationandartificialneuralnetworks.

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Mathematics and Computers in Simulation 81 (2010) 875–882

Original articles

A high order fuzzy time series forecasting model based on adaptiveexpectation and artificial neural networks

Cagdas Hakan Aladag a,∗, Ufuk Yolcu b, Erol Egrioglu b

a Department of Statistics, Hacettepe University, Ankara 06532, Turkeyb Department of Statistics, Ondokuz Mayis University, Samsun 55139, Turkey

Received 27 February 2009; received in revised form 10 September 2010; accepted 18 September 2010Available online 29 September 2010

Abstract

Many fuzzy time series approaches have been proposed in recent years. These methods include three main phases such asfuzzification, defining fuzzy relationships and, defuzzification. Aladag et al. [2] improved the forecasting accuracy by utilizing feedforward neural networks to determine fuzzy relationships in high order fuzzy time series. Another study for increasing forecastingaccuracy was made by Cheng et al. [6]. In their study, they employ adaptive expectation model to adopt forecasts obtained fromfirst order fuzzy time series forecasting model. In this study, we propose a novel high order fuzzy time series method in orderto obtain more accurate forecasts. In the proposed method, fuzzy relationships are defined by feed forward neural networks andadaptive expectation model is used for adjusting forecasted values. Unlike the papers of Cheng et al. [6] and Liu et al. [14], forecastadjusting is done by using constraint optimization for weighted parameter. The proposed method is applied to the enrollments ofthe University of Alabama and the obtained forecasting results compared to those obtained from other approaches are available inthe literature. As a result of comparison, it is clearly seen that the proposed method significantly increases the forecasting accuracy.© 2010 IMACS. Published by Elsevier B.V. All rights reserved.

Key words: Adaptive expectation model; Feed forward neural networks; Forecasting; Fuzzy relations; Fuzzy time series

1. Introduction

In recent years, fuzzy time series approach introduced by Song and Chissom [16,17] has been used widely. In theliterature, many studies have been made to improve forecasting accuracy in fuzzy time series model. Chen [4] proposeda method which is simpler than the method proposed by Song and Chissom [16,17] in forecasting fuzzy time series.The method proposed by Chen [4] does not include complex matrix operations in defining fuzzy relation. Huarng [11]pointed out that the interval length influences the forecasting performance and proposed two methods, which are basedon the average and the distribution, for defining the length of interval. Egrioglu et al. [8] suggested a new approachwhich is based on the optimization of the interval length. Cheng et al. [6] introduced a method based on adaptiveexpectation model. In the method proposed by Cheng et al. [6], the forecasts obtained from the first order fuzzy timeseries model are adjusted by employing adaptive expectation model. Cheng et al. [6] and Liu et al. [14] introduceda method based on adaptive expectation model. In the methods proposed by Cheng et al. [6] and Liu et al. [14], the

∗ Corresponding author. Tel.: +90 312 2977900; fax: +90 312 2977913.E-mail addresses: aladag@hacettepe.edu.tr, chaladag@gmail.com (C.H. Aladag).

0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2010.09.011

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forecasts obtained from the first order fuzzy time series model are adjusted by employing adaptive expectation model.The studies mentioned above use first order fuzzy time series forecasting model.

Since first order fuzzy time series models have got a simple structure, they can be often insufficient to explain morecomplex relationships. For this reason Chen [5] proposed a new method which analyze a high order fuzzy time seriesforecasting model. The implementation of Chen’s approach becomes more difficult when the order of fuzzy time seriesincreases. However, neural networks can be used easily for high order fuzzy time series. Aladag et al. [2] introduced anew approach, which uses feed forward neural networks for defining fuzzy relationships and is based on a high orderfuzzy time series forecasting model.

In this study, a new high order fuzzy time series forecasting model based on feed forward neural networks andadaptive expectation model is proposed to gain better forecasting accuracy. The forecasts are produced by employing ahigh order fuzzy time series model in which fuzzy relationships are defined by utilizing feed forward neural networks.Then, these obtained forecasts are adapted by using adaptive expectation model in order to reach more accuracy. Theproposed method is employed to forecast the enrollments of the University of Alabama to show the considerableoutperforming results. This well known time series is also analyzed by using the methods proposed by Chen [4],Huarng [11], Chen [5], Huarng and Yu [12] and Aladag et al. [2]. In addition, the enrollment data is analyzed by usingthe conventional methods which are exponential smoothing, ARIMA and time series regression methods. The resultsobtained from the proposed method are compared with the results from the methods mentioned above.

Section 2 includes the definitions of first and high order time series forecasting models. Section 3 gives the briefinformation related to neural networks. The new proposed method is introduced and the implementation results of theenrollment data are given in Sections 4 and 5, respectively. Final section concludes the paper.

2. Fuzzy time series

The definition of fuzzy time series was firstly introduced by Song and Chissom [16,17]. In fuzzy time series approach,there is no need for various theoretical assumptions just as in conventional time series procedures. The most importantadvantage of fuzzy time series approach is its ability to work with a very small set of data and no requirement forlinearity assumption. Some general definitions of fuzzy time series are given as follows:

Let U be the universe of discourse, where U ={u1, u2, . . ., ub}. A fuzzy set Ai of U is defined as Ai = fAi (u1)/u1 +fAi (u2)/u2 + · · · + fAi (ub)/ub, where fAi is the membership function of the fuzzy set Ai; fAi : U → [0, 1]. ua is ageneric element of fuzzy set Ai; fAi (ua) is the degree of belongingness of ua to Ai; fAi (ua) ∈ [0, 1] and 1 ≤ a ≤ b.

Definition 1. Fuzzy time series Let Y(t) (t = . . . , 0, 1, 2, . . . ) a subset of real numbers, be the universe of discourse bywhich fuzzy sets fj(t) are defined. If F(t) is a collection of f1(t), f2(t), . . . then F(t) is called a fuzzy time series definedon Y(t).

Definition 2. Let F(t) be a fuzzy time series. If F(t) is a caused by F(t − 1), then this fuzzy logical relationship isrepresented by

F (t − 1) → F (t) (1)

and it is called first order fuzzy time series forecasting model.

Definition 3. Let F(t) be a fuzzy time series. If F(t) is caused by F(t − 1), F(t − 2), . . ., F(t − m), then this fuzzylogical relationship is represented by

F (t − m), . . . , F (t − 2), F (t − 1) → F (t) (2)

and it is called the mth order fuzzy time series forecasting model.

3. Artificial neural networks

Artificial neural networks were originally motivated by the biological structures in the brains of humans and animals,which are extremely powerful for tasks such as information processing, learning and adaptation. The most important

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Fig. 1. A broad feed forward neural network architecture.

characteristics of neural networks are large number of simple units, highly parallel units, strongly connected units,robustness against the failure of single units, and learning from data [15].

Some main elements compose artificial neural networks. Determining the elements of the artificial neural networkshas an important effect on performance of artificial neural networks. Elements of the artificial neural networks aregenerally given as network architecture, learning algorithm and activation function [10].

There are various types of artificial neural networks. One of them is called as feed forward neural networks. Thefeed forward neural networks have been used successfully in many studies [10]. In the feed forward neural networks,there are no feedback connections. The broad feed forward neural network architecture that has single hidden layerand single output is given as an illustration [1] (Fig. 1).

One critical decision is to determine the appropriate architecture, that is, the number of layers, number of nodesin each layers and the number of arcs which interconnect with the nodes [18]. Aladag et al. [2] employ feed forwardneural network for defining fuzzy relationships. Therefore, our focus is on the feed forward networks. Determiningarchitecture depends on the basic problem. Since, in the literature, there are no general rules for determining the bestarchitecture, many architecture should be examined for the correct results [7].

Learning of artificial neural networks for a specific task is equivalent to finding all the values of the weights such thatthe desired output is generated to the corresponding input. Various training algorithms have been used for determiningthe optimal weight values [7].

The activation function is another element that affects the performance of artificial neural networks. It shows therelationship between inputs and outputs of a node and a network. In general, the activation function introduces a degreeof the nonlinearity that is valuable for the most artificial neural networks applications [1].

4. The proposed method

In the literature, it is observed that the high order time series models can produce better forecasts than those obtainedfrom the first order fuzzy time series models. In implementations, however, it is hard to use the high ordered modelssince they require usage of fuzzy relationship tables and complex matrix operations. When the artificial neural networksare used for determining fuzzy relations in the high order models, there is no need to employ these tables and operations.Therefore, utilizing the artificial neural networks make the process easier. Besides, better results can be obtained.

In stock markets, investors usually make their investment decision according to recent stock information such aslate market news, stock technical indicators, or price fluctuations. Reasonable investors will modify their forecasts withrecent forecasting errors [6]. Cheng et al. [6] and Liu et al. [14] showed that the forecasting accuracy can be increasedby applying adaptive expectation model to defuzzyfied forecasts. At the same time, their studies are based on the firstorder fuzzy time series forecasting models.

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Table 1Notations for second order fuzzy time series.

Observation no. Ft−2 Ft−1 Ft Input-1 Input-2 Target

1 – – A6 – – –2 – A6 A2 – – –3 A6 A2 A3 6 2 34 A2 A3 A7 2 3 75 A3 A7 A4 3 7 46 A7 A4 A2 7 4 2

A novel high ordered fuzzy time series forecasting approach in which fuzzy relationships are defined by theartificial neural networks and forecasts are adjusted by employing adaptive expectation model is proposed in this study.The proposed method’s advantages and properties different from other approaches available in the literature can besummarized as follows:

• The proposed method is more useful since instead of utilizing fuzzy relationships group tables and complex matrixoperations for defining fuzzy relationships, artificial neural networks are used.

• The artificial neural networks have nonlinear structure. Therefore, the fuzzy relationships can be defined moreaccurately so more accurate forecasts can be obtained.

• This is the first time, adaptive expectation model is applied to high ordered fuzzy time series model.• Unlike the studies of Cheng et al. [6] and Liu et al. [14], forecast adjusting is done by using constraint optimization

for weighted parameter at the last step of the proposed approach. Therefore, the forecasting accuracy is increased.

In order to construct high order fuzzy time series model, various feed forward neural network architectures are employedto define fuzzy relationships in the implementation. The feed forward neural networks architecture, which includesone hidden layer and one output, is used to define fuzzy relationships. Back propagation learning algorithm is usedto train neural network models and logistic activation function is employed in all neurons. The stages of the proposedmethod based on feed forward neural networks and adaptive expectation model are given below.

Stage 1. Define and partition the universe of discourse.The universe of discourse for observations, U = [starting, ending], is defined. After the length of intervals, l, is

determined, the U can be partitioned into equal-length intervals u1, u2, . . ., ub, b = 1, . . . and their correspondingmidpoints m1, m2, . . ., mb, respectively.

ub = [starting + (b − 1) × l, starting + b × l], mb = [starting + (b − 1) × l, starting + b × l]

2

Stage 2. Define fuzzy sets.Each linguistic observation, Ai, can be defined by the intervals u1, u2, . . ., ub.

Ai = fAi (u1)

u1+ fAi (u2)

u2+ · · · + fAi (ub)

ub

(3)

Stage 3. Fuzzify the observations.For example, a datum is fuzzified to Ai, if the maximal degree of membership of that datum is in Ai.Stage 4. Establish the fuzzy relationship with feed forward neural network.An example will be given to explain stage 4 more clearly for the second order fuzzy time series. Because of dealing

with second order fuzzy time series, two inputs are employed in neural network model, so that lagged variables Ft−2and Ft−1 are obtained from fuzzy time series Ft. These series are given in Table 1. The index numbers (i) of Ai ofFt−2 and Ft−1 series are taken as input values whose titles are input-1 and input-2 in Table 1 for the neural networkmodel. Also, the index numbers of Ai of Ft series are taken as target values whose title is target in Table 1 for the neuralnetwork model. When the third observation is taken as an example, inputs values for the learning sample [A6, A2] are6 and 2. Then, target value for this learning sample is 3.

Stage 5. Defuzzify results.

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The defuzzyfied forecasts are middle points of intervals which correspond to fuzzy forecasts obtained by neuralnetworks in the previous stage.

Stage 6. Adopt the defuzzyfied forecasts.The forecasts obtained in the previous stage are adapted by utilizing adaptive forecasting equation given in (4).

Adaptive Forecast( t

α

)= P(t − 1) + α(F (t) − P(t − 1)) (4)

where P(t − 1) represents the observed value at time t − 1 and F(t) is the defuzzyfied forecast calculated in Stage 5for time t. α value (α ∈ (0,1)) is determined by constrained single variable optimization. The root mean square error(fRMSE(α) or RMSE) over the adaptive forecasts and the observed values is calculated as follows:

fRMSE(α) =(∑T

t (P(t) − Adaptive Forecast(t/α))2

T

)1/2

where T is the number of the test data. Thus, an α value makes fRMSE(α) minimum which is chosen by solving theoptimization problem defined below.

minα

fRMSE(t) (5)

subject to 0 < α < 1In the optimization process, we used a MATLAB function called “fminbnd” to minimize RMSE value. The function

“fminbnd” is used to find minimum of single-variable function on fixed interval. It finds a minimum for a problemspecified by

minx

f (x)

subject to x1 < x < x2x, x1, and x2 are scalars and f(x) is a function that returns a scalar. In MATLAB, x̂ = fminbnd(f (x), x1, x2) returns

a value x̂ that is a local minimum of the scalar valued function f(x) in the interval x1 < x < x2. In other words, to find theminimum of the function f(x) on the interval (x1, x2),

a = fminbnd(f (x), x1, x2)

can be used in MATLAB. f(a) gives the local minimum value in the interval (x1, x2). The algorithm used by fminbndis based on golden section search introduced by Kiefer [13] and parabolic interpolation. Unless the left endpoint x1 isvery close to the right endpoint x2, fminbnd never evaluates f(x) at the endpoints, so f(x) need only be defined for x inthe interval x1 < x < x2. If the minimum actually occurs at x1 or x2, fminbnd returns an interior point at a distance ofno more than 2 × TolX from x1 or x2, where TolX is the termination tolerance. See Brent [3] or Forsythe et al. [9] fordetails about the algorithm.

To solve the optimization problem given in (5), we use the function “fminbnd” as follows:

α∗ = fminbnd(fRMSE(α), 0, 1)

This function used in MATLAB finds an α* that makes fRMSE(α) minimum in the interval 0 < α < 1.

5. Application to the enrollment data

The proposed method is applied to the enrollment data of University of Alabama which is shown in Table 2. Theenrollment observations from 1971 to 1988 were used for the estimation, while the observations from 1989 to 1992were used as a test data. Second, third, and fourth order fuzzy time series model are used in the implementation. Inthe first stage of the proposed method, as Huarng [11] did, the lengths of intervals are chosen as 200, 300, 400, 500,600, 700, 800, 900 and 1000. After following stages 2 and 3 in the method given in Section 4, the number of neuronsof hidden layer is altered 1 through 4 not to lose generalization ability of neural network model. Table 3 gives resultsof the proposed method for second through fourth high order fuzzy time series and rounded RMSE values for the testdata. For each RMSE value given in Table 3, a corresponding optimal α value exists. According to Table 3, by using

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Table 2The enrollment data.

Years Actual Years Actual

1971 13,055 1982 15,4331972 13,563 1983 15,4971973 13,867 1984 15,1451974 14,696 1985 15,1631975 15,460 1986 15,9841976 15,311 1987 16,8591977 15,603 1988 18,1501978 15,861 1989 18,9701979 16,807 1990 19,3281980 16,919 1991 19,3371981 16,388 1992 18,876

Table 3RMSE values obtained from the proposed method.

Order Hidden layer number Length of interval

200 300 400 500 600 700 800 900 1000

2 1 499.2 455.7 472.8 383.8 500.4 246.0 239.7 503.1 466.42 503.1 269.2 503.2 444.5 225.0 503.2 239.7 492.8 467.43 225.0 269.2 202.6 444.5 225.0 503.2 459.3 503.3 283.04 503.3 503.2 503.3 503.2 501.5 502.8 503.2 503.2 496.4

3 1 332.2 503.3 503.2 80.3* 499.0 317.9 503.2 420.1 503.32 225.0 269.2 202.6 212.0 225.0 317.9 239.7 336.2 283.03 503.4 503.3 503.3 503.3 503.3 503.3 503.3 503.3 503.24 502.0 503.2 104.5 212.0 462.7 482.1 434.2 503.3 488.3

4 1 503.3 503.3 502.2 503.3 503.2 317.9 503.2 503.3 283.02 503.2 503.3 202.6 503.3 502.0 317.9 413.7 503.3 503.33 494.7 501.9 502.2 212.0 503.3 492.0 503.3 336.2 503.34 503.2 494.0 202.6 212.0 503.3 494.8 239.7 492.8 494.4

Table 4The comparison of the results.

Method Order RMSE

Chen [4] 1 575.0Huarng [11]a 1 529.4Huarng [11]b 1 500.0Chen [5] 2 738.4Huarng and Yu [12] 1 396.1Aladag et al. [2] 3 153.1The proposed method 3 80.3

a Average based method.b Distribution based method.

the proposed model, the best forecasts are obtained when 3rd order model, 500 length of interval and, 1 unit in hiddenlayer are employed. For this case, obtained RMSE and optimal α value are 80.3 and 0.7761, respectively.

For the aim of a comparative study, methods that Chen [4], Huarng [11], Chen [5], Huarng and Yu [12] and, Aladaget al. [2] are also applied to the data of the enrollments of the University of Alabama and the obtained results aresummarized in Table 4. Except from the method proposed by Huarng and Yu [12], the length of interval is changedbetween 200 and 1000 for the rest of all the methods. In addition, when the methods proposed by Chen [5] and Aladaget al. [2] are employed, the order of models is taken as 2, 3 and 4.

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Table 5The results obtained from conventional time series forecasting methods.

Method RMSE

Brown exponential smoothing 2,469,727Logarithmic regression 2,332,806Inverse regression 3,001,306Quadratic regression 2,865,191Cubic regression 2,412,646Compound regression 1,605,017Power regression 3,009,670S-curve regression 1,605,010Growth regression 1,605,017Exponential regression 1,605,017ARIMA(0,1,0) method 477,468

Fig. 2. The forecasts obtained from the proposed method and the enrollment data.

The enrollment data also forecasted by employing conventional time series methods such as exponential smoothing,ARIMA and various time series regression methods. In calculations, SPSS 15.0 is used. RMSE values calculated overthe test set are presented in Table 5.

According to Tables 4 and 5, the result of our proposed method has the smallest RMSE value when comparedwith the other methods. In other words, the most accurate forecasts are obtained when the proposed fuzzy time seriesapproach is used. Thus, it can be said that the new proposed method produces better forecasts. In addition, to show theforecasting results visually, the forecasts obtained from the proposed method and original data are drawn in Fig. 2.

6. Conclusions

An important issue in fuzzy time series approaches is to determine fuzzy relationships. Defining fuzzy relationshipsin high order fuzzy time series approach is more complicated and computationally more expensive than that in firstorder fuzzy time series approach when fuzzy logical relationship tables are utilized. In order to avoid this complexity,artificial neural networks can be used to define fuzzy relationships in high order fuzzy time series. Aladag et al. [2]proposed a high order fuzzy time series approach in which fuzzy relationships are determined by feed forward neuralnetworks. They also show that forecasting accuracy is increased when the feed forward neural networks are employed.

Cheng et al. [6] improved the forecasting accuracy by using the adaptive expectation model to adopt forecastsobtained from the first order fuzzy time series approach. In this study, we propose a new high order fuzzy time seriesforecasting method in which fuzzy relationships are defined by feed forward neural networks and forecasted valuesare adopted by the adaptive expectation model. The proposed method is employed to forecast the well known dataof the enrollments of the University of Alabama to show the considerable outperforming results. Also, the methodsproposed by Chen [4], Huarng [11], Chen [5], Huarng and Yu [12], Aladag et al. [2] and conventional time seriesmethods are applied to the data for comparison. In the end of the comparison, it is obviously seen that the proposedmethod produces better forecasts than those of other methods. These results indicate that utilizing feed forward neural

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networks for defining fuzzy relations and the adaptive expectation model for adjusting forecasts significantly improvethe forecasting accuracy in high order fuzzy time series.

Acknowledgment

We would like to thank the referee for the valuable comments, which provided insights that helped improve thepaper.

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