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Forecasting nonlinear time series with a hybrid methodology

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Applied Mathematics Letters 22 (2009) 1467–1470
Contents lists available at ScienceDirect
Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
Forecasting nonlinear time series with a hybrid methodology
Cagdas Hakan Aladag
a,
∗
, Erol Egrioglu
b
, Cem Kadilar
a
a
Department of Statistics, Hacettepe University, Ankara, Turkey
b
Department of Statistics, Ondokuz Mayis University, Samsun, Turkey
a r t i c l e i n f o
Article history:
Received 4 February 2009Accepted 4 February 2009
Keywords:
ARIMACanadian lynx dataHybrid methodRecurrent neural networksTime series forecasting
a b s t r a c t
In recent years, artificial neural networks (ANNs) have been used for forecasting in timeseriesintheliterature.Althoughitispossibletomodelbothlinearandnonlinearstructuresin time series by using ANNs, they are not able to handle both structures equally well.Therefore, the hybrid methodology combining ARIMA and ANN models have been usedintheliterature.Inthisstudy,anewhybridapproachcombiningElman’sRecurrentNeuralNetworks(ERNN)andARIMAmodelsisproposed.Theproposedhybridapproachisappliedto Canadian Lynx data and it is found that the proposed approach has the best forecastingaccuracy.
©
2009 Elsevier Ltd. All rights reserved.
1. Introduction
In recent years, the artificial neural networks (ANN) have been applied to many areas of statistics. One of these areas istime series forecasting [1]. Since ANN can model both nonlinear and linear structures of time series, using neural networksin forecasting can give better results than the other methods. Zhang et al. [2] review the literature of forecasting time series
using ANN.Both theoretical and empirical findings in the literature show that combining different methods can be an affective andefficientwaytoimproveforecasts.Therefore,hybridARIMAandANNsmethodshavebeenusedformodelingbothlinearandnonlinear patterns equally well. Pai and Lin [3] proposed hybrid ARIMA and support vector machines model. Tseng et al. [4]
combined seasonal time series ARIMA model and feedforward neural network (FNN). Zhang [5] proposed a hybrid ARIMA
and FNN model, composed of linear and nonlinear components as follows:
y
t
=
L
t
+
N
t
,
(1)where
y
t
denotes srcinal time series,
L
t
denotes the linear component and
N
t
denotes the nonlinear component. Linearcomponent is estimated by ARIMA model and residuals obtained from the ARIMA model
e
t
=
y
t
−ˆ
L
t
,
(2)are estimated by FNN. Here
ˆ
L
t
is the forecasting value for time
t
of the time series
y
t
by ARIMA. Zhang [5] claims that any
ARIMA model can be selected for the data as this does not affect the final forecast accuracy.With
n
input nodes, the ANN model for the residuals can be written as
e
t
=
f
(
e
t
−
1
,
e
t
−
2
, . . . ,
e
t
−
n
)
+
ε
t
,
(3)
∗
Corresponding author. Tel.: +90 312 2992016.
E-mail addresses:
aladag@hacettepe.edu.tr (C.H. Aladag), erole@omu.edu.tr (E. Egrioglu), kadilar@hacettepe.edu.tr (C. Kadilar).
0893-9659/$ – see front matter
©
2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.aml.2009.02.006
1468
C.H. Aladag et al. / Applied Mathematics Letters 22 (2009) 1467–1470
context units11x[k]y[k]output neuronshidden neuronsexternal input neuronsu[k]
Fig. 1.
Structure of an ERNN model [9].
where
f
is a nonlinear function determined by the FNN and
ε
t
is the random error. The estimation of
e
t
by (3) will yieldthe forecasting of nonlinear component of time series,
N
t
. By this way, forecasting values of the time series are obtained asfollows:
ˆ
y
t
= ˆ
L
t
+ ˆ
N
t
.
(4)In the next section, we modify Zhang’s hybrid approach mentioned above. To obtain
ˆ
N
t
, we propose to use ERNN instead of FNN. In Section 3, the proposed hybrid method is applied to Canadian lynx data which is also used in Zhang [5] and Kajitani
et al. [1]. By this way, we can compare the forecasting accuracy of the proposed method with the alternative methods. In
the last section, we discuss the results of the application.
2. The proposed hybrid method
ARIMA and seasonal ARIMA (SARIMA) models were introduced by Box and Jenkins [6] and these models have recently
beenusedsuccessfullyinforecastinglineartimeseries.However,itiswellknownthattheapproximationofARIMAmodelstocomplexnonlinearproblemsisnotadequate[5].Therefore,nonlineartimeserieshavebeenforecastedbyusingnonlinear
methods like ANNs. Although FNN has been used in many applications of ANNs, it is also possible to use recurrent neuralnetworks. One type of recurrent neural networks is ERNN which was introduced by Elman [7]. According to the generalprinciple of the recurrent networks, there is a feedback from the outputs of some neurons in the hidden layer to neuronsin the context layer which seems to be an additional input layer. In the case of comparison with other type of multilayerednetwork,themostimportantadvantageofERNNisarobustfeatureextractionability,whichprovidesfeedbackconnectionsfrom the hidden layer to a context layer [8]. The structure of an ERNN is illustrated in Fig. 1.
Zhang[5]’shybridapproachusesFNNtoestimate
N
t
in(1).SinceERNNcontainsthecontextlayer,itiscertainthatusing
ERNN, instead of FNN, can improve forecasting accuracy. Therefore, we propose a new hybrid approach as follows:
Step
1. Box–Jenkins models are used to analyze the linear part of the problem. That is,
ˆ
L
t
is obtained by using Box–Jenkinsmethod.
Step
2. ERNN model is developed to fit the residuals from the Box–Jenkins models. That is,
ˆ
N
t
is obtained by using ERNN.
Step
3. Using (4), forecasts of the hybrid method are obtained by adding the estimates of linear and nonlinear components
of the time series, found in Step 1 and Step 2, respectively.
3. Application
The proposed hybrid method is applied to Canadian lynx data consisting of the set of annual numbers of lynx trappingsin the Mackenzie River District of North–West Canada for the period from 1821 to 1934. Canada lynx data, which is plottedinFig.2,wasalsoexaminedbyZhang[5]andKajitanietal.[1],beyondtheothervariousstudiesinthetimeseriesliterature.
We would like to note that we use the logarithms (to the base 10) of the data in the analysis.The proposed hybrid method is applied to the data as follows:Firstly,Box–Jenkinsmethodisusedforestimatinglinearpartoftheproblem.TheCanadianlynxdatashowsaperiodicityof approximately 10 years. Because of this, the data is fitted by
SARIMA
(
2
,
0
,
0
)
×
(
0
,
1
,
1
)
10
model. We check that thismodel satisfies all statistical assumptions such as no autocorrelation, homoskedasticity, etc. using Box–Pierce and WhiteTests. Secondly, residuals obtained from
SARIMA
(
2
,
0
,
0
)
×
(
0
,
1
,
1
)
10
model are estimated by the ERNN model. Note thattheresidualsaredividedintotrainingset(100datapoints)andtestset(last14datapoints).Numberofinputnodesisvariedfrom1to12,numberofhiddenlayernodesisalsovariedfrom1to12andbythisway114architecturesareexaminedtotally.We find that the most appropriate ERNN architecture is 4
×
4
×
1. Thirdly, forecasts of last 14 years were obtained using
C.H. Aladag et al. / Applied Mathematics Letters 22 (2009) 1467–1470
1469
1 11 21 31 41 51 61 71 81 91 101 111010002000300040005000600070008000
Fig. 2.
Canadian lynx data series (1821–1934).
Fig. 3.
Hybrid prediction of Canadian lynx data.
Table 1
Canadian lynx data forecasting results.Method MSEFNN 0.020Zhang [5] Hybrid 0.017Kajitani [1] SETAR 0.014
Proposed Hybrid 0.009
the proposed hybrid method. Finally, these forecasting values for last 14 years are shown in Fig. 3. Solid line represents the
srcinal time series data and dot line represents the forecasts.Themeansquareerror(MSE)valuesforthelast14observationsoftheproposedapproach,Zhang[5]andKajitanietal.[1]
are summarized in Table 1.It is observed from Table 1 that the MSE of the proposed method is the smallest. Thus, it is concluded that the proposedapproach has the best forecasting values for this widely used data.
4. Conclusions
Since artificial neural networks (ANN) can model both nonlinear and linear structures of time series, using ANN cangive better results than other methods in forecasting. Therefore, in the literature, there have been many studies in whichtime series are solved by using ANN in recent years [10,2,11]. One type of ANN is recurrent neural network and one of the
recurrent nets is ERNN.Statisticians have studied to obtain better forecasts for long years and by these studies hybrid methods have beenimproved in the literature. In this paper, we consider that using ERNN instead of FNN in Zhang’s hybrid method shouldimprovetheforecastingaccuracy.Therefore,weproposeahybridARIMAandrecurrentneuralnetworkmodel.Itisobservedthat the proposed method yields better result than other methods for Canadian lynx data. It is well known that forecastingaccuracy of ERNN is better than FNN, because of containing a context layer. Since ERNN is used in the proposed hybridapproach,asexpectedthisapproachisfoundbetterthanZhang[5]’shybridapproach.Inthefutureworkwehopetoincrease
theforecastingaccuracybychangingthetypeofANNusedinhybridmethodssuchasJordanrecurrentneuralnetworks[12].
References
[1] Y. Katijani, W.K. Hipel, A.I. Mcleod, Forecasting nonlinear time series with feedforward neural networks: A case study of Canadian lynx data, Journalof Forecasting 24 (2005) 105–117.[2] G. Zhang, B.E. Patuwo, Y.M. Hu, Forecasting with artificial neural networks: The state of the art, International Journal of Forecasting 14 (1998) 35–62.
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[3] P.F. Pai, C.S. Lin, A hybrid ARIMA and support vector machines model in stock price forecasting, The International journal of Management Science 33(2005) 497–505.[4] F.M. Tseng, H.C. Yu, G.H. Tzeng, Combining neural network model with seasonal time series ARIMA model, Technological Forecasting & Social Change69 (2002) 71–87.[5] G. Zhang, Time series forecasting using a hybrid ARIMA and neural network model, Neurocomputing 50 (2003) 159–175.[6] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Holdan-Day, San Francisco, CA, 1976.[7] J.L. Elman, Finding structure in time, Cognitive Science 14 (1990) 179–211.[8] S. Seker, E. Ayaz, E. Turkcan, Elman’s recurrent neural network applications to condition monitoring in nuclear power plant and rotating machinery,Engineering Applications of Artificial Intelligence 16 (2003) 647–656.[9] D.T. Pham, D. Karaboga, Training Elman and Jordan networks for system identification using genetic algorithms, Artificial Intelligence in Engineering13 (1999) 107–117.[10] T.Y. Kim, K.J. Oh, C. Kim, J.D. Do, Artificial neural networks for non-stationary time series, Neurocomputing 61 (2004) 439–447.[11] G.Zhang,B.E.Patuwo,Y.M.Hu,Asimulationstudyofartificialneuralnetworkfornonlineartimeseriesforecasting,Computers&OperationsResearch28 (2001) 381–396.[12] M.I. Jordan, Attractor dynamics and parallelism in a connectionist sequential machine, in: Conference of the Cognitive Science Society (1986).

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