Forecasting time series using a methodology based on autoregressive integrated moving average and genetic programming

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Forecasting time series using a methodology based on autoregressive

integrated moving average and genetic programming

Yi-Shian Lee

*

_{, Lee-Ing Tong}

Department of Industrial Engineering and Management, National Chiao Tung University, 1001, Ta-Hsuch Rd., Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history: Received 15 March 2010

Received in revised form 26 June 2010 Accepted 14 July 2010

Available online 17 July 2010

Keywords: ARIMA Hybrid model Genetic programming Forecasting

Artiﬁcial neural network

a b s t r a c t

The autoregressive integrated moving average (ARIMA), which is a conventional statistical method, is employed in many fields to construct models for forecasting time series. Although ARIMA can be adopted to obtain a highly accurate linear forecasting model, it cannot accurately forecast nonlinear time series. Artificial neural network (ANN) can be utilized to construct more accurate forecasting model than ARIMA for nonlinear time series, but explaining the meaning of the hidden layers of ANN is difficult and, more-over, it does not yield a mathematical equation. This study proposes a hybrid forecasting model for non-linear time series by combining ARIMA with genetic programming (GP) to improve upon both the ANN and the ARIMA forecasting models. Finally, some real data sets are adopted to demonstrate the effective-ness of the proposed forecasting model.

1. Introduction

Many approaches for forecasting time series have been devel-oped. Of conventional statistical methods, the autoregressive integrated moving average (ARIMA) is extensively utilized in constructing a forecasting model. For instance, Kumar and Jain

[1] employed ARIMA to develop a model for forecasting trafﬁc-noise time series. Ediger and Akar[2]applied ARIMA model to fore-cast demand for fuel in Turkey. However, ARIMA cannot be utilized to produce an accurate model for forecasting nonlinear time series. In recent years, the artiﬁcial neural network (ANN) and the support vector machines (SVM) have been successfully utilized to develop a nonlinear model for forecasting time series [3–9]. These approaches usually yield better results than the ARIMA model in nonlinear time series. Zhang et al. [10] reviewed forecasting models using ANN for time series.

Since determining whether a linear or nonlinear model should be ﬁtted to a real-world data set is difﬁcult, several investigations have developed some hybrid forecasting models that combine dif-ferent methods to reduce the forecast error. Zhang[11]developed a hybrid forecasting model that combines ARIMA with ANN to forecast the Canadian lynx time series more accurately than either of the models used separately. Pai and Lin[12]employed a hybrid ARIMA and SVM to construct a model for forecasting stock price. Chen and Wang [13] presented a hybrid seasonal time series

ARIMA (SARIMA) and SVM to forecast the production values of the machinery industry in Taiwan. Like Zhang[11], Aladag et al.

[14]developed a hybrid model that combined ARIMA and Elman’s recurrent neural networks (ERNN) to forecast Canadian lynx time series.

The above hybrid models[11–14]can be employed to combine the linear and nonlinear forecasting system with high overall fore-casting accuracy. The hybrid models can be expressed as follows:

yt¼ Ltþ Nt; ð1Þ

where ytrepresents the original positive time series at time t; Lt represents the linear component, and Ntis the nonlinear component of the model, respectively. The residuals can be obtained using the ARIMA model:

rt¼ yt ^Lt; ð2Þ

where rtis estimated using such nonlinear methods as ANN, SVM, or ERNN. ^Ltis the forecasted value of Ltand is estimated using the ARIMA model. Accordingly, the residual can be rewritten as follows: rt¼ f ðrt1;rt2; . . . ;rtnÞ þ

e

t; ð3Þ

where f(rt1, rt2, . . ., rtn) represents the nonlinear function that is constructed using ANN, SVM, or ERNN and

e

tis the random error term. The hybrid model for forecasting time series is:

^

yt¼ ^Ltþ ^Nt: ð4Þ

Although these hybrid models exhibited favorable overall forecasting performance, the hidden layers in ANN are difﬁcult to

*Corresponding author. Tel.: +886 3 5712121x57356; fax: +886 3 5722392. E-mail addresses:bill.net.tw@yahoo.com.tw(Y.-S. Lee),litong@cc.nctu.edu.tw

(L.-I. Tong).

Contents lists available atScienceDirect

Knowledge-Based Systems

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explain and the relationship between the input variables and out-put variable(s) in ANN or SVM cannot be expressed by a mathe-matical equation. Furthermore, the ANN model needs large data sets to train a robust network model[15]. Accordingly, this study proposes a novel hybrid model for forecasting time series that combines the ARIMA model with genetic programming (GP). The proposed hybrid model takes the advantages of the ARIMA and GP models in linear or nonlinear modeling and f(rt1, rt2, . . ., rtn) in Eq. (3) can be obtained using GP. Furthermore, unlike ANN, which requires for large data sets to train an appropriate network model, GP can perform well even with small data sets[15]. Thus, the proposed hybrid model can easily be constructed in practice for either large or small data sets. This study is organized as fol-lows. Section2describes the procedure of combining the ARIMA and GP model to construct the proposed hybrid model. Section3

employs some real-world data sets to demonstrate the effective-ness of the proposed method and the proposed method is also compared with other time series forecasting models. Section 4

draws conclusions.

2. The model development

Box and Jenkins presented the ARIMA model in 1970[16]. The method has been widely used in financial, economic and social sci-entific fields[17]. In the ARIMA(p, d, q) model, p is the order of auto-regression, d is the order of differencing, and q is the order of the moving average process[16]. Generally speaking, the ARIMA model can be represented as a linear combination of the past observations and past errors as follows:

ð1 /1B /2B 2 /pB p Þð1 BÞdyt ¼ d þ ð1 h1B h2B2 hqBqÞet; t ¼ 2; 3; . . . ; ð5Þ

where ytis the actual value, B is the backward shift operator, d is the constant item,

e

tis the random error at time t, /pand hqare the coefﬁcients of the model and can be estimated utilizing the least square method. Furthermore, the model has following setups: mod-el identiﬁcation, parameter estimation, and modmod-eling diagnosis. The appropriate ARIMA(p, d, q) model is obtained by applying the Akaike Information Criterion (AIC) rule[18,19]. Although the ARIMA model can have high forecasting performance in large or linear data set, it cannot obtain a robust forecasting ability in small or nonlin-ear data set. Hence, some improving ARIMA models have been pro-posed to solve the nonlinear or small data[11–14].

Recently, some nonlinear methods such as ANN, SVM, and ERNN are usually utilized to ﬁt nonlinear time series. Both theoret-ical and empirtheoret-ical analyses have shown that forecasting by a hy-brid ARIMA forecasting model that combines two forecasting methods is more accurate than forecasting using just a single fore-casting method[11–14]. However, a hybrid forecasting model that is constructed by combining two forecasting methods cannot typ-ically be expressed by a mathematical forecasting equation and needs large data sets to construct the appropriate model. To solve this problem, GP is utilized to ﬁt a nonlinear forecasting time series model.

Koza[20]developed GP as a new algorithm for computer pro-grams that exploits the concept of evolution to solve model struc-ture identiﬁcation problems and perform symbolic regression[21]. The basic concepts of GP are similar those of genetic algorithms (GAs), and include mutation, crossover and reproduction[22]. Un-like GAs, GP uses the generic parse-tree representation to replace the logic number of the genetic state (0 and 1). Hence, GP has be-come more popular than conventional linear forecasting methods because it can be employed to search complex nonlinear spaces. Notably, GP is also widely utilized in practical applications such as in a real-time prediction of coastal algal blooms[23], the

con-struction of credit scoring models[15,24], emulating the rainfall-runoff process[25], and forecasting electric power demand[22].

Functions or statements in GP have operators ({+, , , , log, and exp}), a trigonometric function ({sin, cos, and tan}), and condi-tional statements (if, then). Hence, a GP parse tree (Fig. 1) can be applied to a simple example: cos[9x] tan[5y]. Furthermore, GP system can yield an effective function for predicting the value of the dependent variable. When selecting input variables, GP auto-matically ﬁnds the variables that contribute most to the model

[23] and then constructs an equation [22,23,25]. Moreover, GP does not have any restriction on the data size as compared to that of the ANN[15,24].

This study proposes a novel hybrid forecasting model, which combines ARIMA to model the linear component (Lt) of a time ser-ies and the GP to model the nonlinear component (Nt), to improve the accuracy of ARIMA forecasting. Since utilizing only linear mod-els or nonlinear modmod-els to forecast time series data may not obtain satisfactory results. To improve the forecasting accuracy, a hybrid forecasting system that possesses both linear and nonlinear mod-eling abilities can be utilized. Moreover, utilizing GP to model the nonlinear component of time series can obtain a mathematical equation than ANN and SVM model no matter data sets are large or small. In practice, the forecasting values utilizing GP can be veriﬁed through the mathematical equation. For ANN and SVM models, although the application of these models is easy, the relation be-tween the input and output variables are difﬁcult to explain and cannot verify the forecasting value through the mathematical equation. Therefore, the proposed hybrid approach is as follows:

Step 1. The ARIMA model is utilized to model the linear compo-nent of time series. That is, ^Ltis obtained by using the ARIMA model.

Step 2. From Step 1, the residuals from the ARIMA model can be obtained. The residuals are modeled by the GP model in Eq.(3). That is, ^Ntis the forecast value of Eq.(3)by using GP.

Step 3. Using Eq.(4), forecasts of the hybrid model are obtained by adding the forecasted values of linear and nonlinear compo-nents, yield in Step 1 and Step 2, respectively.

3. Empirical results 3.1. Data sets

In this study, to demonstrate the effectiveness of the proposed hybrid forecasting model, three data sets are utilized in this study to examine the performance of the proposed hybrid model. More-over, two literature hybrid models, developed by combining ARI-MA and ANN models[11]; and by ARIMA and SVM models[12], are utilized as benchmark models. Through compared with other hybrid ARIMA models, it will be clear to see the forecasting accu-rately among different hybrid ARIMA models. The ﬁrst data, the Canadian lynx data, are adopted as an example. The data are the

-cos 9 tan x * * 5 y

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annual number of trapped lynxes in the Mackenzie River district of Northern Canada from 1821 to 1934. The data set has been ana-lyzed in some of the literature on hybrid time series forecasting models[11,14]. The data that are plotted inFig. 2reveal a period-icity of around ten years. In this data, Zhang[11]and Aladag et al.

[14]adopted ARIMA and SARIMA to estimate the linear component (Lt), respectively. Hence, to compare different hybrid ARIMA mod-els, this study adopts ARIMA model to estimate the linear compo-nent of this time series. The Canadian lynx data are grouped into a training set (100 data observations) and a testing set (last 14 observations).

The second data, the energy consumption data of China are uti-lized from 1957 to 2007, giving a total of 51 observations (see

Fig. 3). The study of energy issue is important for policy makers and related organizations [26,27]. The energy consumption data is regarded as nonlinear and is utilized to demonstrate the effec-tiveness of nonlinear models. In this data analysis, China’s energy consumption data are grouped into a training set (43 data points) and a testing set (last 6 data points). Finally, the US quarterly GDP financial data are utilized from 1947 to 2003, giving 228 data points in the time series (seeFig. 4). In econometric field, the finan-cial data is often used to forecast future trend through time series models. The numbers of training set and testing set about this data are first 210 data points and last 14 data points, respectively. Hence, the three data sets are utilized to evaluate effectiveness of the proposed hybrid forecasting model.

3.2. Results

In this study, all ARIMA modeling is obtained using the SPSS sta-tistical package. The results reveal that the best ARIMA model in-volved the application of the AIC [18,19]. Besides these hybrid ARIMA models (i.e., ARIMA–ANN model, ARIMA–SVM model, and proposed hybrid model), the ANN model and GP model are also added to forecast the three data sets. Furthermore, in compare with these forecasting models, this study uses three evaluation indices. The ﬁrst index is root mean square error (RMSE), which compares forecast value with real value. Notably, RMSE is deﬁned as:

RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN t¼1 ðft otÞ2 N v u u t ; ð6Þ

where ftis the forecast value for the tth year, otis the real value for the tth year, and N is the number of observations. The second index is mean absolute percentage error (MAPE). This index measures the accuracy of time series data ﬁtted using a statistical method. Nota-bly, MAPE is deﬁned as:

MAPE ¼1 N XN t¼1 ft ot ot 100%: ð7Þ

Similar with the MAPE, the third index is mean absolute error (MAE). This index is deﬁned as:

0 1000 2000 3000 4000 5000 6000 7000 8000 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109

Fig. 2. Canadian lynx data (1821–1934).

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MAE ¼X N

t¼1 jft otj

N : ð8Þ

Notably, the literature about how to determine the residual lagged variables (i.e., rt1, rt2, . . ., rtn) are not consistent. For example, Zhang[11]adopted the given network structure or trial and error to determine an appropriate setting in his analyzed cases; Aladag et al.[14]also adopted trial and error to determine the appropriate residual network model, and Chen and Wang

[13]adopted their given residual lagged variables to estimate the residual value. In this study, trial and error is utilized to determine the appropriate forecasting residual model.

In the Canadian lynx data, following other time series studies

[11,28], the logarithms (base 10) of the data are utilized to reduce

the degree of asymmetry very greatly (a not unusual result with biological observations) in the original data. In order to stabilize the variance and render it stationary, the ﬁrst-order differencing is utilized. The derived appropriate model, ARIMA(2, 1, 1), satisﬁes statistical assumptions, according to Box–Pierce and White Tests. In the ANN model, the settings of nodes of input layer, hidden layer, and output layer are 7 5 1 following Zhang[11]. As for the GP model, the input, output variables are (yt1, yt2, . . ., yt7) and yt, respectively. Like the ANN modeling, the network structure of nonlinear component of ARIMA–ANN model is 7 5 1. Simi-larly with GP, the input variables of nonlinear components of pro-posed model (ARIMA–GP) are (rt1, rt2, . . ., rt7). Finally, the ARIMA–SVM model must consider the nonlinear component parameter settings (i.e., kernel function type, C,

r

2_{, and}

_e

_{). Chen} and Wang[13]pointed that no standard procedure exists to deter-mine C,

r

2_{, and}

_e

_{parameters. Some studies} _[12,13,29] _utilized Gaussian kernel function type can yield better prediction perfor-mances. In this lynx data, the parameters settings are given as (Gaussian kernel function, C = 10,

r

2_{= 0.413, and}

_e

_{= 0.5) can} ob-tain an appropriate forecasting model. In the energy consumption data, the appropriate linear ARIMA model is found to be ARI-MA(0, 1, 0). This means that this time series will be stationary sit-uation utilizing ﬁrst differencing. A neural network of 2 1 1 is

Fig. 4. US quarterly GDP data series (1947/Q1–2003/Q4).

Table 1

The appropriate parameter settings of GP.

Parameter Value

Population size 100

Maximum number of generation 1000

Crossover rate 0.9

Mutation rate 0.01

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utilized to model the data set. As for the GP model, the input, out-put variables are (yt1, yt2) and yt, respectively. Similarly with the settings of nonlinear component of Canadian lynx data, the network structure of nonlinear component of ARIMA–ANN model is 2 1 1; the input variables of nonlinear components of

ARIMA–GP are (rt1, rt2). Finally, the ARIMA–SVM model, the parameters settings which are given as (Gaussian kernel function, C = 3,

r

2_{= 0.4, and}

_e

_{= 0.3) can obtain an appropriate forecasting} model. In the US quarterly GDP data, the appropriate linear ARIMA model is found to be ARIMA(2, 2, 1). A neural network of 4 3 1

Fig. 6. Actual and ﬁtted values for energy consumption of China.

Fig. 7. Actual and ﬁtted values for US quarterly GDP time series.

Table 2

Canadian lynx data forecasting results. Time Actual

value

ARIMA(2, 1, 1) GP ANN ARIMA–ANN ARIMA–SVM Proposed

Model value Errora

1921 229 278.48 21.61 231.61 1.14 565.50 146.94 319.36 39.46 945.24 312.77 250.52 9.4 1922 399 633.56 58.79 332.75 16.6 843.47 111.40 747.54 87.35 607.51 52.26 410 2.76 1923 1132 785.92 30.57 907.13 19.86 1330.91 17.57 801.64 29.18 1286.00 13.60 1089.24 3.78 1924 2432 2233.54 8.16 2136.79 12.14 2047.87 15.79 2475.45 1.79 2534.00 4.19 2315.67 4.78 1925 3574 3008.92 15.81 3135.55 12.27 3308.55 7.43 3127.43 12.49 3486.70 2.44 3528.14 1.28 1926 2935 2936.89 0.06 2680.17 8.68 3721.21 26.79 3058.85 4.22 2837.00 3.34 2814.59 4.1 1927 1537 1686.05 9.7 1300.68 15.38 1863.11 21.22 1798.99 17.05 2076.50 35.10 1520.67 1.06 1928 529 793.88 50.07 600.76 13.57 646.28 22.17 878.93 66.15 693.00 31.00 480.74 9.12 1929 485 289.56 40.3 586.96 21.02 206.88 57.34 369.56 23.80 755.12 55.69 518.75 6.96 1930 662 564.75 14.69 644.94 2.58 448.20 32.30 613.98 7.25 740.42 11.85 640.25 3.29 1931 1000 929.26 7.07 932.31 6.77 791.46 20.85 1017.07 1.71 1230.12 23.01 980.78 1.92 1932 1590 1313.26 17.41 1277.87 19.63 1485.18 6.59 1398.70 12.03 1475.23 7.22 1480.65 6.88 1933 2657 1851.84 30.3 2580.56 2.88 2241.52 15.64 1969.42 25.88 2153.39 18.95 2578.54 2.95 1934 3396 2698.72 20.53 3157.96 7.01 3207.75 5.54 2814.40 17.13 3104.98 8.57 3205.89 5.6 RMSE 367.62 213.05 347.66 328.42 317.28 80.79 MAPE (%) 23.22 11.39 36.25 24.67 41.43 4.56 MAE 282.29 171.69 304.86 259.72 254.11 62.51 a ER ¼j^ytytj yt 100%.

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is utilized to model the data set. As for the GP model, the input, output variables are (yt1, yt2, yt3, yt4) and yt, respectively. The network structure of nonlinear component of ARIMA–ANN model is 4 3 1; the input variables of nonlinear components of ARI-MA–GP are (rt1, rt2, rt3, rt4). Finally, the ARIMA–SVM model, the parameters settings which are given as (Gaussian kernel func-tion, C = 11,

r

2_{= 0.25, and}

_e

_{= 0.16) can obtain an appropriate} fore-casting model.

To reduce the forecast error of nonlinear component of ARIMA– GP model, the objective function of GP can be expressed as: Minimize : X n t¼1 ^rt rt ð Þ j j; ð9Þ

where rtrepresents the actual residual value, and the ^rtrepresents the forecasted value of rt. In the operation of GP, the operators {+, , , , log(base = e), sin, cos, and exp} are adopted. The parameters of the GP model in the three residual data sets from ARIMA model are determined from the lagged values of rt1to rtn, and are used to predict rt.Table 1presents the appropriate parameter settings for GP utilized to estimate these residual data sets. Following Huang et al.[24], the appropriate setting values of GP are obtained.

These above all time series models are utilized to compare the forecasting accuracy in training set of each data set.Figs. 5–7plot these forecasting trends among three data sets.Tables 2–4present the forecasts and errors with all forecasting models in these data sets. From the MAPE index, the proposed model has lower values than other time series models in Canadian lynx data, China’s

energy data, and US quarterly GDP ﬁnancial data, respectively (i.e., 4.56%, 0.45%, and 0.12%). These results reveal that the pro-posed hybrid model has lower model errors than other models. Furthermore, the hybrid models have better forecasting accuracy than utilizing only ARIMA model. This proves that combining dif-ferent nonlinear method can improve the forecasting performance of utilizing only linear time series model. Compared with other developed hybrid ARIMA models, the proposed improved ARIMA model has highly forecasting accuracy. Moreover, the proposed hy-brid forecasting model can display a mathematical form in given period t. For example, the forecasting value of Canadian lynx data in 1921 year can be display as follows:

^

y1921¼ 0:0012 þ 1:388y1920 0:733y1919 0:998e1920 þ ðr1917 r1914Þ r1916

r1914þ cosðcosðr1917ÞÞ þ cosðr1914Þ

; ð10Þ

where the Eq.(10)is composed of linear component and nonlinear component. The linear component (i.e., ^Lt¼1921) is obtained by 0.0012 + 1.388y1920 0.733y1919 0.998

e

1920; the nonlinear com-ponent (i.e., ^Nt¼1921) is obtained byr1914þcosðcosðrðr1917r19141917ÞrÞÞþcosðr1916 1914Þ.

Similarly, the forecasting value of China’s energy data in 2002 year can be expressed as follows:

^

y2002¼ 2839:8333 þ y2001þ exp½expðcosðr2000ÞÞþexpðsinðr2000ÞÞ: ð11Þ

Finally, the forecasting value of US quarterly GDP data in 2000 Q3 can be expressed as follows:

Table 3

China’s energy consumption data forecasting results (unit: MT tons of SCE).

Year Actual value ARIMA(0, 1, 0) GP ANN ARIMA–ANN ARIMA–SVM Proposed

Model value Errora

Model value Errora 2002 151,797 146038.83 3.79 148076.89 2.45 138193.19 8.96 149206.29 1.71 149761.76 1.34 151678.56 0.08 2003 174,990 148878.67 14.92 160626.89 8.21 152792.78 12.68 155822.05 10.95 154800.65 11.54 174830.86 0.09 2004 203,227 151718.50 25.35 198414.89 2.37 193067.89 5.00 185641.91 8.65 174834.81 13.97 203880.14 0.32 2005 224682 154558.33 31.21 231695.89 3.12 234853.03 4.53 232850.88 3.64 219285.91 2.40 224228.47 0.20 2006 264,270 157398.17 40.44 246368.89 6.77 254192.53 3.81 256609.32 2.90 258743.02 2.09 260415.17 1.46 2007 265,583 160238.00 39.67 304089.89 14.50 285628.84 7.55 270267.10 1.76 278311.98 4.79 267100.83 0.57 RMSE 71652.59 18689.21 15208.71 11766.58 15489.57 1724.14 MAPE (%) 25.89 6.23 7.09 4.94 6.02 0.45 MAE 60953.08 14386.2 14375.75 9976.58 12378.14 1126.15 a _{ER ¼}j^ytytj yt 100%. Table 4

US quarterly GDP (unit: in billion of current dollars).

Time Actual value

ARIMA(2, 2, 1) GP ANN ARIMA–ANN ARIMA–SVM Proposed

Model value Errora

Model value Errora 2000/Q3 9862.1 9962.64 1.02 9979.22 1.19 9805.16 0.58 9987.22 1.27 9984.17 1.24 9868.8 0.07 2000/Q4 9953.6 10102.51 1.5 9971.51 0.18 9918.03 0.36 10043.89 0.91 10055.72 1.03 9947.58 0.06 2001/Q1 10024.8 10238.03 2.13 10057.92 0.33 10007.47 0.17 10105.40 0.80 10117.94 0.93 10036.95 0.12 2001/Q2 10088.2 10373.38 2.83 10092.52 0.04 10086.57 0.02 10149.14 0.60 10168.54 0.80 10097.26 0.09 2001/Q3 10096.2 10508.8 4.09 10163.29 0.66 10136.96 0.40 10216.59 1.19 10223.36 1.26 10074.7 0.21 2001/Q4 10193.9 10644.69 4.42 10145.36 0.48 10183.05 0.11 10221.95 0.28 10230.68 0.36 10197.95 0.04 2002/Q1 10329.3 10781.13 4.37 10251.52 0.75 10228.91 0.97 10296.64 0.32 10289.84 0.38 10330.35 0.01 2002/Q2 10428.3 10918.19 4.7 10409.2 0.18 10282.23 1.40 10427.07 0.01 10412.94 0.15 10460.23 0.31 2002/Q3 10,542 11055.86 4.87 10535.26 0.06 10340.89 1.91 10552.40 0.10 10540.08 0.02 10555.29 0.13 2002/Q4 10623.7 11194.16 5.37 10653.76 0.28 10420.64 1.91 10670.14 0.44 10662.82 0.37 10611.8 0.11 2003/Q1 10735.8 11333.09 5.56 10719.6 0.15 10496.41 2.23 10772.97 0.35 10774.78 0.36 10758.69 0.21 2003/Q2 10846.7 11472.66 5.77 10835.74 0.1 10566.12 2.59 10891.70 0.41 10905.02 0.54 10872.51 0.24 2003/Q3 11,107 11612.85 4.55 10945.86 1.45 10635.57 4.24 11016.47 0.82 11045.27 0.56 11106.37 0.01 2003/Q4 11,262 11753.68 4.37 11258.37 0.03 10724.49 4.77 11210.20 0.46 11238.10 0.21 11250.15 0.11 RMSE 448.19 63.09 234.05 69.14 71.02 15.72 MAPE (%) 3.97 0.42 1.55 0.57 0.59 0.12 MAE 418.43 43.84 167.33 58.62 60.03 12.77 a ER ¼j^ytytj yt 100%.

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^

y2000;q3¼ 0:6320 0:0246y2000;q2þ 0:1175y2000;q1 0:7450e2000;q2 0:1578e2000;q1þ ðr2000;q1 r2000;q2Þ: ð12Þ

Through the algorithm of ARIMA–GP model, the forecasting values from the ARIMA–GP model can be obtained inTables 2–4.

Hence, from the above analysis, it is obviously that the proposed hybrid forecasting model has more accuracy in forecasting the Canadian lynx data, China’s energy consumption data, and US quarterly GDP data than the other methods.

4. Conclusions

The traditional statistical forecasting methods can effectively model linear time series, but to accurately forecast nonlinear time series is difﬁcult. Recently, ANN and SVM time series models have been developed to enhance the forecasting accuracy. However, when dealing with the real-world problems, it is not easy to judge whether linear or nonlinear structure is appropriate. In this case, the hybrid methodology can be a valid way to enhance the fore-casting performance. This study is motivated by evidence that dif-ferent forecasting models can complement each other in modeling data sets, and proposed a novel hybrid methodology which com-bines the ARIMA and GP models. From the empirical results, the proposed hybrid methodology is more outperform than other fore-casting models. In future studies, the proposed method will be ap-plied to forecast more different time series to demonstrate the universality of the proposed hybrid forecasting model.

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