1、1The Application of Multiplicative HoltWinters Model in Prediction of Railway Passenger FlowaAbstract. The railway passenger volume has an obvious seasonal behavior， and has volatility and instability. The features of it brings a lot of difficulties to forecast the volume in short-term. To investiga
2、te the trends and seasonal variations of railway passenger flow， we discuss multiplicative Holt-Winters model as a method. This paper states the basic theory and algorithm of the model， and provides the experimental results by using the data of China railway passenger volume in 2006 to 2010. The res
3、ult shows the feasibility and effectiveness of the proposed approaches. Key words： Holt-Winter Model， Railway， Passenger Volume， Prediction. 1. Introduction Prediction of railway passenger volume is an important foundation for the work of the railway transport organization sector， the accuracy of it
4、s results has a direct impact on the decisions and arrangements of the relevant sector. Railway 2passenger volume influence by time， and has certain regularity and periodicity. In order to predict the railway passenger volume accurately， we must consider the impact of various factors in the process
5、of establishing appropriate model. At present， widely used methods of establishing prediction model include time series method， grey system method， neural network method， and so on. Huijing Wang applied gray forecast theory to develop passenger traffic volume forecast program， and made experiment to
6、 prove the availability of gray forecast theory.1 Dabin Zhang and Hou Zhu presented a gray forecast model based on genetic algorithm to modify factors， and made study on the efficiency of the model.2 These methods can get accurate results while the time series is smooth and steady or has a notable c
7、hange in trend， and often be used in long-term predictions. Therefore， these methods are used to long-term forecast in railway passenger volume. In recent years， railways passenger flow increase a lot， and influenced by the seasonal factors greatly. The railway passenger flow has a obvious volatilit
8、y and instability， short-term forecast is needed to the railway transport organization sector. So， we discussed a short-term forecast method with high accuracy to deal with this situation. Common 3seasonal short-term prediction methods are Holt-Winter model and seasonal ARIMA model. For seasonal ARI
9、MA model is quit complex and the accuracy of it is not as high as Holt-Winter model 3， we applied Holt-Winters model to forecast railway passenger flow， and hope to get better prediction result. 2. Holt-Winters Model Holt-Winters model is a relatively common form of time series model； it is Holt lin
10、ear model with a extended cycle item. This model is used to solve the exponential smoothing for data with a tread and seasonal behavior. 4 The main idea is study on linear tread， stochastic volatility and seasonal variation respectively， and combines the result with exponential smoothing method. By
11、using this model can deal with the data with both trends and seasonal variations， and can filter effects of stochastic volatility properly. 5 2.1 Equations 2.2 Forecast Equation The forecast equation of Holt-Winters multiplicative model is ： （4） Where is the number of time intervals from current tim
12、e to prediction time； is the predicted value of moment 2.3 Getting Initial Values 4We need at least one complete seasonal cycle to initialize the values of level（ ） ， trend（ ） and seasonal（ ）. In fact， if we want get a better result， two complete seasonal cycles is needed. After getting the initial
13、values， we use the formula 4 above to make forecasts. The equations gives the arithmetic mode of initial value of level， trend and seasonal. is the average of the observational data in one cycle； is the initial value of increment， is the average of the value gets from the second cycle data minus the
14、 first cycle corresponding data； is the seasonal change of the first cycle. 2.4 Determination of Optimal Smoothing Coefficient 2.5 Algorithm of Multiplicative Holt-Winters model . Calculate the value of level trend and seasonal in accordance with the initial equation formula 1， 2， and 3.According to
15、 the observed value of the second cycle（ ） ， we can calculate the appropriate value of ， and . Compute the appropriate value of level， trend and seasonal until no more observed value is available. Record the value of level trend and seasonal of the last cycle for the subsequent processing. 5Accordin
16、g to the values we recorded in step 5， calculate the predicted value in accordance with the Forecast Equation （formula 4） 3. Algorithm Verification In order to validate the prediction results of Multiplicative Holt-Winters model， we use national railway passenger volume in 2006-2010 as observation d
17、ata. The data comes from National Bureau of Statistics of China. Table 1 shows the factual data of railway passenger volume Fig.1 Sequence diagram of National railway passenger volume in 2006-2010 By using matlab， we plotted the time sequence diagram of national railway passenger volume in 2006-2010
18、. Figure 1 is the Sequence diagram of the data from table 1， we can learn that railway passenger volume is rising in the overall trend， and has instability and volatility obviously. At the same time， the seasonal factors have a great influence on the railway passenger flow. Multiplicative Holt-Winte
19、rs model is quite suitable for this kind of data. in interval 0.01， 0.99， and compute the predicted value and error rate of railways passenger flow in 2009-2010. we calculated the quadratic sum of all combinations error rate respectively to specify the optimal smoothing coefficient. 6By the compute
20、of Matlab， we determined the optimal smoothing coefficient as and ， the error rate between the predicted value and actual value is about 2.5% mostly. We got the initial values and the optimal smoothing coefficient， we calculate the value of level tread and seasonal of 2010， to forecast the railways
21、passenger volume in 2011. 4. Result Analysis After the compute of Matlab program， we got the prediction result of the multiplicative Holt-Winters model we built above. The exact value of prediction is showed in table 2， and the actual value and error are given in the table， too In the same inputted
22、data， the prediction result of the seasonal ARIMA model is showed in table 2，and the actual value and error are given in the table， too The actual value and the predicted values of the two models we discussed above are all showed in figure 2. The full line gives the actual value of railway passenger
23、 volume in 2011； the dotted line gives the predicted value of multiplicative Holt-Winters model； the dot dash line gives the predicted value of seasonal ARIMA model The prediction errors of the two models we discussed above are all showed in figure 3. The full line gives the prediction 7error of mul
24、tiplicative Holt-Winters model； the dotted line gives the prediction error of seasonal ARIMA model From figure 3 we learn that the predictions error rate of multiplicative Holt-Winters model and seasonal ARIMA model are all about 5%， within the acceptable range of values. With multiplicative Holt-Wi
25、nters forecast model， maximum of prediction error is 8.5%， minimum of prediction error is 0.01%； with seasonal ARIMA forecast model， maximum of prediction error is 12.5%， minimum of prediction error is 0.08%. In conclusion， multiplicative Holt-Winters model has a more accurate predicted result than
26、seasonal ARIMA model； it is more suitable for short-term prediction of railway passenger volume. 5. Conclusion This paper discussed the application of multiplicative Holt-Winters model in the prediction of railway passenger volume， and given the experiment result by using the data of national railwa
27、y passenger volume in 2006-2010. The test result indicates that multiplicative Holt-Winters model can fit the tread and seasonal behavior of time series， the error rate between the forecasted and actual values is acceptable. So， the method we discussed above id suitable to the prediction of railway
28、passenger volume in short-term， and can provide 8support for the decisions of the railway transport organization sector. Still， the railway passenger volume is affected by many other factors， the treatment of outliers and reduction of predictions error are still need further studies References 1Huij
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30、me”. Statistics and Decision， vol.24， pp.24-26， 2009 3 Li_Zhang， Shifeng_Yan. “Comparison of Holt-Winters and Arima Methods for Forecasting Charge of China Airline Passengers”. Journal of Shanghai University of Engineering Science， vol.9， pp. 280-283， 2006. 4Yonghong_Du， Jian_Wang. “Seasonal Time Se
31、ries Theory and Application”. Tianjin：Nankai UP， 2008. 5Mingrong_Tong， Hengxin_Xue， Lin_Lin. “Study on the Forecast of Railway Freight Traffic Volume Based on Holt-Winter model”. Railway Transport and Economy， vol.29， no.1， pp.79-81， 2007. 96Yuan_Ding， Bo_Yu. “The Application of Holt-Winters Model of railway freight volume forecasting”. Railway Freight Transport， vol.12， pp.19-21， 2010
