1、Keywords:series is strong; however, if the seasonality is weak, different network structures may bemore suitable.C211 2008 Elsevier Inc. All rights reserved.neuralpapers,examineing conclusions. While some researchers claim that ANN is unsuccessful in finding out seasonal effect in the data structure
2、,others state that they reach better results with ANN. The literature given below shows this contradiction. Kolarik andRudorfer 21, Nelson et al. 25, Hill et al. 17, Nelson et al. 26 and Zhang and Qi 38 state that ANN is not appropriatefor modeling the seasonal effect directly; hence to obtain a bet
3、ter ANN forecasting, seasonal effect should be removed fromthe raw data. On the other hand, Tang et al. 34, Sharda and Patil 30, Tang and Fishwick 33, Nam and Schaefer 24,0020-0255/$ - see front matter C211 2008 Elsevier Inc. All rights reserved.* Tel.: +90 372 2574010; fax: +90 372 2574057.E-mail a
4、ddress: coskunhkaraelmas.edu.trInformation Sciences 178 (2008) 45504559Contents lists available at ScienceDirectInformation Sciencesdoi:10.1016/j.ins.2008.07.024banking crises; Wang and Chien 37 presented a forecasting model that predicts innovation performance using technicalinformational resources
5、 and clear innovation objectives by back-propagation neural network; Singh and Deo 31 and Sahooand Ray 29 used ANN for flow prediction; Pai and Hong 27 and Palmer et al. 28 used ANN for prediction of tourism timeseries; Barbounis and Teocharis 3 and Cadenas and Rivera 7 predicted wind speed via ANN;
6、 Catalao et al. 8 predictedshort-term electricity prices; Szen et al. 32 and Hamzaebi 15 used ANN to forecast sectoral energy consumption. Thisvast body of literature is still growing.Most time series consist of trends and seasonal effects. Seasonality is especially observable in the economic and bu
7、sinesstime series. In the literature, trend and seasonality have been modeled using different approaches. One of the best-knownapproaches is BoxJenkins Model. ANN is also used to study seasonal time series, although many of the studies reach differ-Artificial neural networksSeasonal time seriesSeaso
8、nal BoxJenkins modelHoltWintersForecasting1. IntroductionSince the late of 1980s, artificialliterature of ANN time series forecasting.in the literature. Since these two reviewFor example, elik and Karatepe 10networks (ANNs) have been used for time series forecasting. There is a vastZhang et al. 40 a
9、nd Adya and Collopy 1 reviewed the related works existingnumerous other ANN applications from different areas have been published.d the performance of neural networks use in evaluating and forecasting ofImproving artificial neural networks performance in seasonal timeseries forecastingCosC223kun Ham
10、zaebi*Department of Business Administration, Zonguldak Karaelmas University, Incivez, 67100 Zonguldak, Turkeyarticle infoArticle history:Received 11 September 2007Received in revised form 2 June 2008Accepted 31 July 2008abstractIn this study, an artificial neural network (ANN) structure is proposed
11、for seasonal timeseries forecasting. The proposed structure considers the seasonal period in time series inorder to determine the number of input and output neurons. The model was tested for fourreal-world time series. The results found by the proposed ANN were compared with theresults of traditiona
12、l statistical models and other ANN architectures. This comparisonshows that the proposed model comes with lower prediction error than other methods.It is shown that the proposed model is especially convenient when the seasonality in timejournal homepage: Hamzaebi/Information Sciences 178 (2008) 455
13、04559 4551and output neuron numbers are vital for increasing the performance of ANN forecasting.Cybenko 9, Hornik et al. 18 and Funahashi 14 have shown that ANN is a universal function approximator. In a sea-sonal time series, the recurrence of some recognizable patterns after some regular interval
14、as seen in Fig. 1 reminds that aseasonal time series forecasting problem can be thought of as a function approximation problem. So, it could be said thatANN can learn seasonality in the data structure without removing the seasonal effect from the series and with a properANN structure it can make a s
15、uccessful forecasting. Using the s” parameter for determining the input and output neuronsnumber may help to make better predictions. The s” parameter presents the series structure, such as; monthly, quarterly,etc. For monthly time series s = 12 and for quarterly time series s = 4. Representing the
16、number of input and output neuronswith parameter s” can increase prediction performance of ANN in seasonal time series forecasting. In this kind of networkFranses and Draisma 13 and Alon et al. 2 report that ANN is capable of modeling the seasonal and trend effects in datastructure without even remo
17、ving seasonal effects.Improving accuracy in time series forecasting is an important yet often difficult task. Combining multiple models or usingensemble methods can be an effective way to improve forecasting performance 42. In this paper, a new ANN structure issuggested to improve ANN forecasting pe
18、rformance without any preprocessing of raw data. The organization of the remain-der of the paper is as follows: Section 2 presents BoxJenkins models. Important parameters of the ANN models are pre-sented in Section 3. Section 4 explains the proposed model. Section 5 discusses a sample application, a
19、nd the last sectionconcludes the paper.2. BoxJenkins modelsThe most common and best-known statistical technique that is used to forecast time series is BoxJenkins Models. Thistechnique is based on discrete linear stochastic processes. BoxJenkins models are also known as ARIMA(p,d,q) models.Sometimes
20、 there can be seasonal or cyclic components in a time series. Series recorded on a monthly or quarterly basis oftendisplay annual seasonal patterns. BoxJenkins models used for seasonal time series are SARIMA(p,d,q) C2 (P,D,Q)s. In a sea-sonal time series, there are two types of variations: The first
21、 type is between consecutive observations, while the second typeis between pairs of corresponding observations belonging to consecutive seasons. For seasonal time series, ARIMA(p,d,q)models can be constructed to depict the relationship between consecutive observation values, whereas ARIMA(P,D,Q)smod
22、-els can be formed to show the relationship between corresponding observation values of consecutive seasons. The param-eters P, D, and Q in the seasonal ARIMA(P,D,Q)smodel, like in ARIMA(p,d,q), stand for autoregressive, degree of differencingand moving average processes, respectively and s” is the
23、number of periods between two corresponding observations ofsuccessive seasons. The s” parameter means that the time series at time t, Yt, depends on YtC0s,YtC02s,YtC03s,., and otherrelated lags values.3. Artificial neural networksMost widely used ANNs in forecasting problems are multilayer perceptro
24、ns (MLPs). In an MLP built for forecast purposes,a sigmoid or a hyperbolic tangent function is used as the hidden layer activation function; while in the output layer comeswith a linear activation function is generally preferred. In an MLP designed for time series forecasting, determining the vari-a
25、bles, such as number of input, hidden and output neurons, is highly important. However, these parameters are subject tochange with respect to the problem under consideration. Since the number of input neurons helps reveal the relation be-tween observations, it is quite effective on the performance o
26、f the network. Other important parameters are the numberof hidden layers and the number of hidden neurons (HN). Hidden neurons are the ones that bring out the defining propertieswithin data and that help establish the nonlinear relationship between input and output. There can be more than one hidden
27、layer in an ANN; however, only one hidden layer is generally preferred 19,40. Another factor that may affect performance ofMLPs constructed for time series forecasting is number of output neurons. If the forecasting horizon involves one period,then the number of output neurons is 1. In multi-periodi
28、c forecasts, the number of output neurons varies according tothe approach being used. If the direct forecast method is adopted, the number of output neurons can take the same valueas the forecast horizon. On the other hand, if the iterative forecast method is adopted, the number of output neurons is
29、 equalto 1. The iterative method can only be used in making single-period forecasts. The predicted value is used as an input for thesuccessive period prediction. This process is repeated until the end of the forecast horizon. This way of forecasting is thesame approach that is used in BoxJenkins mod
30、els. In multi-period time series forecasting via ANN, Zhang 41 stated thatthe direct method gives better results than iterative method. This conclusion is also supported by Zhang et al. 40, Kline 20,and Hamzaebi et al. 16.4. ANN for the seasonal time seriesAlthough the SARIMA model has been successf
31、ul in many forecasting applications, it suffers from limitation because of itslinear form. Due to its linearity, SARIMA model cannot capture any nonlinear. Linear models are not always suitable for com-plex real-world problems 39. However, ANNs satisfactory prediction is dependent on proper network
32、structure beingformed. On the other hand, it is difficult to determine the appropriate net structure. As mentioned before, input, hiddenFig. 1. The recurrence of some recognizable patterns after some regular interval.4552 C. Hamzaebi/Information Sciences 178 (2008) 45504559Here Yt+l(l =1,2,.,s), rep
33、resents the predictions for the future s periods; YtC0i(i =0,1,2,.,s C0 1) are the observations of theprevious s periods, vij(i =0,1,2,.,s C0 1; j =1,2,.,m) are weights of connections from input layer neurons to hidden layerneurons, wjl(j =1,2,.,m; l =1,2,.,s) are weights of connections from hidden
34、layer neurons to output layer neurons, al(l =1,2,.,s) and hj(j =1,2,.,m) are weights of bias connections and f is the activation function.According to the proposed SANN, the number of input and output neurons should be 12 for monthly time series and 4 forquarterlyYtl alXmj1wjlfXsC01i0vijYtC0i hj; l
35、1; 2; .; s: 1structure ith seasonal period observations are values of input neurons and (i + 1)th seasonal period observations output neu-rons values. Each seasonal period is composed of a number of observations. In a way such that the ith period is made up of anumber of data values as shown in Fig.
36、 1. The suggested model is called seasonal artificial neural network (SANN). SANNstructure is shown in Fig. 2. Eq. (1) gives the mathematical expression of output of the SANN:!time series for better forecasting. The number of hidden neurons can be determined by experimental design. TheFig. 2. Propos
37、ed ANN structure for seasonal time series.modelsstudyto selectseriesC. Hamzaebi/Information Sciences 178 (2008) 45504559 4553Box et al. 4 state that the best model that fits the airline data is SARIMA(0,1,1) C2 (0,1,1) after making logarithmictransformation. Faraway and Chatfield 12 tried to obtain
38、the best ANN estimator by experimenting on 13 different ANNstructures in their studies. The authors compare their results with the SARIMA model results. They used the data for19491959 for training and the data for 1960 for testing. When forecasting with SANN, the data for 19491958 was usedfor traini
39、ng, the data for 1959 for validation while the data for 1960 was used for testing. The test set is the same as thatused by Faraway and Chatfield 12. The activation function is a sigmoid function for the hidden layer and a linear functionfor the output layer.Table 1 shows the best result of ANNs stud
40、ied by Faraway and Chatfield 12 and the results of SARIMA(0,1,1) C2 (0,1,1)12model. Table 2 shows the results of SANN and Fig. 4 shows SANN outputs and the real values for the test set. The hiddenneuron number is denoted as HN and the parameter number as Par in Table 1 and in the latter tables.5.2.
41、The Taiwan machinery industry time series (TMITS)This series studied by Tseng et al. 35 consists of data obtained for the period between January 1991 and December 1996.As shown in Fig. 5, the series is affected by trend and seasonal effects. The authors used the data for January 1991December1995 for
42、 training and the data for 1996 for testing. The authors compared their SARIMABP models results with two differentis affected by increasing trend and seasonal changes.12t1MSE 1nXnt1YtC0 Ft2; 6where Ytis the actual and Ftis the forecasted value of period t, and n is the number of total observation.5.
43、1. The airline passenger data set (APTS)The airline passenger data set was first used by Brown 6 and then by Box and Jenkins 5. The airline passenger data setshows the airline passenger numbers between January 1949 and December 1960 on a monthly basis. As shown in Fig. 3, theor traditional methods.
44、Although SSE and MSE are similar for judging performance, both of them were used in thisbecause of the use of their use in studies employed for comparison. Also, MAPE, MAE, and MSE were used in this studythe best network structure as previously mentioned. These criteria are given below:MAPE 1nXnt1Yt
45、C0 FtYtC12C12C12C12C12C12C12C12C3100; 3MAE 1nXnt1jYtC0 Ftj; 4SSE XnYtC0 Ft2; 5number of network parameters is equal to the number of connections between the neurons and the bias terms. The numberof network parameters should be adjusted according to the training set size to avoid memorization instead
46、 of learning.5. An applicationIn order to examine whether SANN gives better results or not, the data set of airline passengers, the data set of the totalproduction value of Taiwan machinery industry, the data set of the sales of soft drinks and the data set of the quarterly salesfrom the literature
47、were used. When experimenting with time series, dividing the series into three parts (trainingvalida-tiontest) may help the researchers. The validation set helps select the best network structure from several ANN models thatwere constructed. The ANN model that produces the lowest value of performanc
48、e criterion on the validation set was used asa prediction model.One important point to consider when constructing the ANN is overfitting. Overfitting may occur when the number ofparameters in the network is not much smaller then the total number of data in the training set. The number of parametersi
49、n the network is equal to total number of connections between the neurons and bias terms. In such situations to avoid over-fitting, the training algorithm performance criterion may be chosen as msereg” and is shown asmsereg lC3mse1C0lC3msw; 2where msereg denotes the regularized performance function; mse is the value of squared errors for the network output;msw denotes the value of squared network weigh
