the application of multiplicative holt—winters model in prediction of railway passenger flowa

1、<p>　　The Application of Multiplicative Holt—Winters Model in Prediction of Railway Passenger Flowa</p><p>　　Abstract. The railway passenger volume has an obvious seasonal behavior， and has volatility a

2、nd instability. The features of it brings a lot of difficulties to forecast the volume in short-term. To investigate the trends and seasonal variations of railway passenger flow， we discuss multiplicative Holt-Winters mo

3、del 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. </p><p>　　Key words：

4、 Holt-Winter Model， Railway， Passenger Volume， Prediction. </p><p>　　1. Introduction </p><p>　　Prediction of railway passenger volume is an important foundation for the work of the railway trans

5、port organization sector， the accuracy of its results has a direct impact on the decisions and arrangements of the relevant sector. Railway passenger volume influence by time， and has certain regularity and periodicity.

6、In order to predict the railway passenger volume accurately， we must consider the impact of various factors in the process of establishing appropriate model. </p><p>　　At present， widely used methods of esta

7、blishing 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 experim

8、ent to 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 meth

9、ods can get accurate </p><p>　　In recent years， railways passenger flow increase a lot， and influenced by the seasonal factors greatly. The railway passenger flow has a obvious volatility and instability， sh

10、ort-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 seasonal short-term prediction methods are Ho

11、lt-Winter model and seasonal ARIMA model. For seasonal ARIMA model is quit complex and the accuracy of i</p><p>　　Holt-Winters model is a relatively common form of time series model； it is Holt linear model

12、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 resp

13、ectively， and combines the result with exponential smoothing method. By using this model can deal with the data with both trends and seasonal variations， and can filter effects of stochas</p><p>　　2.1 Equati

14、ons </p><p>　　2.2 Forecast Equation </p><p>　　The forecast equation of Holt-Winters multiplicative model is ： （4） </p><p>　　Where is the number of time intervals from current time t

15、o prediction time； is the predicted value of moment </p><p>　　2.3 Getting Initial Values </p><p>　　We need at least one complete seasonal cycle to initialize the values of level（ ）， trend（ ） and

16、 seasonal（ ）. In fact， if we want get a better result， two complete seasonal cycles is needed. After getting the initial values， we use the formula 4 above to make forecasts. </p><p>　　The equations gives th

17、e 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

18、 the first cycle corresponding data； is the seasonal change of the first cycle. </p><p>　　2.4 Determination of Optimal Smoothing Coefficient </p><p>　　2.5 Algorithm of Multiplicative Holt-Winter

19、s model </p><p><b>　　. </b></p><p>　　③Calculate the value of level trend and seasonal in accordance with the initial equation formula 1， 2， and 3.According to the observed value of t

20、he second cycle（ ）， we can calculate the appropriate value of ， and . </p><p>　?、蹸ompute the appropriate value of level， trend and seasonal until no more observed value is available. </p><p>　　⑤

21、Record the value of level trend and seasonal of the last cycle for the subsequent processing. </p><p>　　⑥According to the values we recorded in step 5， calculate the predicted value in accordance with the Fo

22、recast Equation （formula 4） </p><p>　　3. Algorithm Verification </p><p>　　In order to validate the prediction results of Multiplicative Holt-Winters model， we use national railway passenger volu

23、me in 2006-2010 as observation data. The data comes from National Bureau of Statistics of China. Table 1 shows the factual data of railway passenger volume </p><p>　　Fig.1 Sequence diagram of National railwa

24、y passenger volume in 2006-2010 　　By using matlab， we plotted the time sequence diagram of national railway passenger volume in 2006-2010. Figure 1 is the Sequence diagram of the data from table 1， we can learn that rai

25、lway 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-Winters

26、 model is quite sui</p><p>　　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

27、respectively to specify the optimal smoothing coefficient. </p><p>　　By the compute of Matlab， we determined the optimal smoothing coefficient as and ， the error rate between the predicted value and actual v

28、alue 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 passenger volume in 2011. </p><p

29、>　　4. Result Analysis </p><p>　　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

30、2， and the actual value and error are given in the table， too </p><p>　　In the same inputted data， the prediction result of the seasonal ARIMA model is showed in table 2，and the actual value and error are gi

31、ven in the table， too </p><p>　　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 volume in

32、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 </p><p>　　The prediction errors of the two models we

33、discussed above are all showed in figure 3. The full line gives the prediction error of multiplicative Holt-Winters model； the dotted line gives the prediction error of seasonal ARIMA model </p><p>　　From fi

34、gure 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-Winters forecast model， max

35、imum 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-W

36、inters model has a more accurate predicted result tha</p><p>　　This paper discussed the application of multiplicative Holt-Winters model in the prediction of railway passenger volume， and given the experimen

37、t result by using the data of national railway 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 betwee

38、n the forecasted and actual values is acceptable. So， the method we discussed above id suitable to the prediction of railway passenger v</p><p>　　References </p><p>　　[1]Huijing_Wang. “Research

39、on Railway Traffic Volume Forecast Based on Grey Forecast Model”. Railway Transport and Economy， vol. 28， no. 6， pp.79-81， 2006 </p><p>　　[2]Dabin_Zhang， Hou_Zhu， Wei_Li， Jingguang_Zhang. “A New Grey Model B

40、ased on Genetic Algorithm and Its Application in Prediction of Railway Passenger Volume”. Statistics and Decision， vol.24， pp.24-26， 2009 </p><p>　　[3] Li_Zhang， Shifeng_Yan. “Comparison of Holt-Winters and

41、Arima Methods for Forecasting Charge of China Airline Passengers”. Journal of Shanghai University of Engineering Science， vol.9， pp. 280-283， 2006. </p><p>　　[4]Yonghong_Du， Jian_Wang. “Seasonal Time Series

42、Theory and Application”. Tianjin：Nankai UP， 2008. </p><p>　　[5]Mingrong_Tong， Hengxin_Xue， Lin_Lin. “Study on the Forecast of Railway Freight Traffic Volume Based on Holt-Winter model”. Railway Transport and

43、 Economy， vol.29， no.1， pp.79-81， 2007. </p><p>　　[6]Yuan_Ding， Bo_Yu. “The Application of Holt-Winters Model of railway freight volume forecasting”. Railway Freight Transport， vol.12， pp.19-21， 2010</p&g