A VERHULST MODEL ON TIME SERIES ERROR CORRECTED FOR PORT THROUGHPUT FORECASTING

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1 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 A VERHUS MODE ON IME SERIES ERROR CORRECED FOR POR HROUGHPU FORECASING Zijia GUO Associate Professor School of Civil ad Hydraulic Egieerig Dalia Uiversity of echology iggog Road, Gajigzi District, Dalia 604, iaoig, P. R. Chia Fax: zjguo@dlut.edu.c Xiagu SONG Associate Professor School of Civil ad Hydraulic Egieerig Dalia Uiversity of echology iggog Road, Gajigzi District, Dalia 604, iaoig, P. R. Chia Fax: sogx_0@sia.com Jia YE Graduate School of Civil ad Hydraulic Egieerig Dalia Uiversity of echology iggog Road, Gajigzi District, Dalia 604, iaoig, P. R. Chia Fax: yjdlut@6.com Abstract: he grey theory maily works o systems aalysis with poor, icomplete or ucertai messages. he popular grey model, GM(, is efficiet for log-term port throughput forecastig. However, it is imperfect whe the throughput icreases i the curve with S type or the icremet of throughput is i the saturatio stage. I this case, the throughput forecastig error of grey system model will become larger ad the result is uaccepted i the real world. o solve this problem, we propose the grey Verhulst model o time series error corrected for the port throughput forecastig. By applyig this Verhulst model to the port throughput forecastig, it shows that the grey Verhulst model o time series error corrected is applicable, especially, whe the throughput icreases accordig to the curve with S type, ot oly higher forecastig accuracy ca be obtaied, but also the superiority ad the features of grey system model ca be reserved. Key Words: throughput forecastig, Verhulst model, time series error. INRODUCION hroughput forecastig is the foudatio of the research i the port developmet tactic ad it s importat for port plaig ad buildig. Basig o the forecastig result of port throughput, we ca decide the directio of port developmet, the amout of port ivestmet, the selectio of berths locatio ad the maagemet of port operatio etc. here are also may methods for forecastig port throughput, ad the grey system model is oe of them. Because the grey system model eeds little origi data, has simple calculate process ad higher forecastig accuracy, it has bee widely used i the predictio of a lot of research fields. I the predictio of port throughput, we usually use the grey GM(, model. However, it is imperfect whe the throughput icreases i the curve with S type, or the icremet of throughput is i the saturatio stage. I this case, the throughput forecastig error of grey system model will become larger ad the result is uaccepted i the real world. o solve this problem, we eed a ew grey model for the port throughput forecastig. 88

2 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 I this paper, we itroduce a grey Verhulst model o time series error corrected for the port throughput forecastig. By applyig this Verhulst model to the port throughput forecastig, it shows that the grey Verhulst model o time series error corrected is applicable, especially, whe the throughput icreases accordig to the curve with S type, ot oly higher forecastig accuracy ca be obtaied, but also the superiority ad the features of grey system model ca be reserved.. IERAURE REVIEW Grey theory, origially developed by Deg (98, focuses o model ucertaity ad iformatio isufficiecy i aalyzig ad uderstadig systems via research o coditioal aalysis, predictio ad decisio-makig. I the field of iformatio research, deep or light colors represet iformatio that is clear or ambiguous, respectively. Meawhile, black idicates that the researchers have absolutely o kowledge of system structure, parameters ad characteristics, while white represets that the iformatio is completely clear. Colors betwee black ad white idicate systems that are ot clear, such as social, ecoomic or weather systems. he fields covered by grey theory iclude systems aalysis, data processig, modelig, predictio, decisio makig ad cotrol. he grey theory maily works o systems aalysis with poor, icomplete or ucertai messages. he Grey method has umerous applicatios, as ay issue of the Joural of Grey System will testify. Extesive research has bee doe to attempt to explai the pheomeo of geography, geology, agriculture ad earthuakes (ee 986; Sog 99. Meawhile, other researches have studied social pheomeo icludig fiacial operatig performace, stock markets, supply ad demad for electroic power (Morita et al. 996, the market for air travel (Hsu et al. 998 ad maagemet decisios (Mo et al Numerous works have examied scietific techologies such as military weapos (Wu 994, the textile idustry (u et al. 995 ad medicies (Chew 995 ad have applied the Grey forecastig model, GM (, or GM (,N to these areas. he GM (,N model is suitable for applicatio to systems, aalysis, data processig, modelig, predictio, decisio-makig ad cotrol. he GM (, model uses the most up-to-date data to predict future values, ad poor forecastig may result whe the data are radom with cetral symmetry. Verhulst model was first proposed by Germay biologist Verhulst to describe some icreasig process like S curve which has saturatio. It has bee extesively used i umerous applicatios to explai the pheomeo of populatio icreasig, livig creature breedig ad its idividual growth. he grey Verhulst model is a special kid of model withi the grey system. Researchers have examied scietific techologies such as disease icidece forecast, time predictio of ladslide, load forecast, predictio of groud displacemet ad deformatio, predictio of buildig s subsidece ad have applied the Verhulst model to these areas (i 004; Zhag et al. 003; uo 000; Guo

3 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , MEHODOOGY 3. raditioal Predictio Method he fuctio of liear time series is xˆ t = a + bt, a = ( x b t /, b = ( tx t x /( t ( t, where t deotes time, represets umber of data, x is the sample value ad predictio value of period t. xˆ t deotes the he logistic fuctio is x ˆ t = + bt ae, where is upper boud of x, a > 0, b > 0, which t ca be estimated by the followig euatio: where D = ad m = / 3 b = (l D m m m i= xi i= m+ xi D D m l D, = m /, a = i xi D D = D m 3m, D = x, the results oly take iteger umber. D ( e c b b, c = b i= m+ i i= m+ xi e e, / 3 m =, 3. GM (, for ime Series Forecastig he GM(, is oe of the most freuetly used grey forecastig model. his model is a time series forecastig model, ecompassig a group of differetial euatios adapted for parameter variace, rather tha a first order differetial euatio. Its differece euatios have structures that vary with time rather tha beig geeral differece euatios. Although it is ot ecessary to employ all the data from the origial series to costruct the GM(,, the potecy of the series must be more tha four. I additio, the data must be take at eual itervals ad i cosecutive order without bypassig ay data (Deg 986. he GM(, model costructig process is described below: Deote the origial data seuece by x = ( x, x (, x (3,..., x (, where is the umber of years observed. he AGO formatio of x is defied as: x = ( x, x (, x (3,..., x (, ( where 883

4 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 k x = x, ad x ( k = x ( m, k =,3,..., (3 m= he GM(, model ca be costructed by establishig a first order differetial euatio for x ( k as: dx ( k / dk + ax ( k = b. (4 herefore, the solutio of E. (4 ca be obtaied by usig the least suare method. hat is, where xˆ ( k = ( x bˆ e ( k bˆ +, (5 ad [, bˆ] = ( B B B X (6 0.5( x + x ( 0.5( x ( + x (3 B =, (7 M M 0.5( x ( - + x ( X ] = [ x (, x (3, x (4,..., x ( (8 We obtaied ˆx from E. (5. et ˆx be the fitted ad predicted series, x ˆ = ( xˆ, xˆ (, xˆ (3,..., xˆ (,..., (9 ˆ (0 where x = x. Applyig the iverse AGO, we the have ˆ ( 0 b ˆ ( k x ˆ ( k = ( x ( e a e, k =,3,..., (0 where x ˆ, xˆ (,..., xˆ ( are called the GM(, fitted seuece, while ˆ ˆ x ( 0 ( +, x ( 0 ( +,..., are called the GM(, forecast values. he GM(, model is relatively applicable to describe the mootoous variety process. However, it is imperfect whe the throughput icreases i the curve with S type or the icremet of throughput is i the saturatio stage. I this case, the throughput forecastig 884

5 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 error of grey system model will become larger ad the result is uaccepted i the real world. o solve this problem, we itroduce the grey Verhulst model for the port throughput forecastig. 3.3 Grey Verhulst Model Grey Verhulst model is also a time series forecastig model, ad we ca costruct the Grey Verhulst model just as the above GM(, by establishig a first order differetial euatio for x ( k as: dx ( k / dk + ax ( k = b( x ( k. ( herefore, the solutio of E. ( ca be obtaied by usig the least suare method. hat is, where ax ˆ x ˆ ( k = ( ( ˆ ˆ ( k bx + a bx e ad [, bˆ] = ( B B B X (3 0.5( x + x ( 0.5( x ( + x (3 B = M 0.5( x ( - + x ( 0.5( x + x ( 0.5( x ( + x (3, (4 M 0.5( x ( + x ( X ] = [ x (, x (3, x (4,..., x ( (5 We obtaied ˆx from E. (7. et ˆx be the fitted ad predicted series, x ˆ = ( xˆ, xˆ (, xˆ (3,..., xˆ (,..., (6 where x ˆ = x. Applyig the iverse AGO, we the have ˆ ( k ax ˆ (ˆ a bx ( e e x ˆ ( k =, k =,3,... (7 ( (ˆ ˆ ( k ( (ˆ ˆ ( k bx + a bx e bx + a bx e where x ˆ, xˆ (,..., xˆ ( are called the GM(, fitted seuece, while ˆ ˆ x ( 0 ( +, x ( 0 ( +,..., are called the GM(, forecast values. 885

6 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 Euatio ( ad (7 are costructed by the origial data seuece, we call this model as x -Verhulst model. I actual applicatio, if the origial data seuece icreases i the curve with S type, we ca also use it as the x to costruct Verhulst euatio directly. 3.4 Residual Modificatio of Grey Verhulst Model o ime Series Error Corrected It s ievitable that there exits throughput forecastig error whe we use the historical port throughput data seuece to establish grey Verhulst model (GVM. o improve forecastig precisio, there are two kid methods we ca take. he usual adopted method is to modify the x residual series as most of issue of joural of grey system. O the other had, we ca also correct the time series o the hypothesis that the port throughput series, x, is true ad oly the time series exits error. I this case, we establish a improved Verhulst model o time series error corrected. We deote the residual time series as : = ( (, (3, (4,... ( (8 where ad kˆ 0 ˆ ( k = k k ( k, k =,3,..., (9 0 ˆ x ( bx ( k ( k = + l, k =,3,..., (0 ˆ ( ( ˆ ˆ a x k a bx Accordig to, we use GM(, to modify the residual time series. E.( deotes the residual time series GM(,. he value of a or u is estimated usig OS. uˆ ( 0 ( k ˆ ( k = ( ( e e, k =,3,... ( ˆ (0 where =. Combiig Euatio (9 ad (, we ca get the corrected time series of grey Verhulst model by GM(, ˆ uˆ ( k k r ( k = k ( ( e e, =,3,... k ( Combiig Euatio (7 ad ( yields residual modificatio of grey Verhulst model o time series error corrected. 886

7 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , Fourier residual modificatio model We use grey Verhulst model to forecast primary series treds, ad modify the time series error by the Fourier series. he key purpose of Fourier series is to ehace the forecastig predictio. Whe deotig the residual time series, the differece betwee the real time k ad the ˆ 0 k model-fitted ( 0 ( k is obtaied from E. (0 as ( k, ˆ ( k = k k0 ( k, k =,3,..., E. (3 ca be expressed i the Fourier series as E. (4. z πi πi ( k a0 + ai cos( k bi si( k, k =,3,, + i = (3 (4, z = (( / where = E. (4 ca be rewritte usig E.(5: ( k PC, the result oly take iteger umber. (5 = P cos( cos(3 cos( π π π si( si(3 si( π π π cos( cos(3 cos( π π π si( si(3 si( π π π cos( cos(3 cos( πz πz πz si( si(3 si( πz πz πz, C = [ a0, a, b, a, b,, a, b ]. he solutio of coefficiet of matrix C is calculated usig the OS, which yields the followig euatio: ˆ C = ( P P P (6 Substitutig for ai or bi from E. (4 ito E. (6 yields the value ˆ ( 0 ( k. herefore, the Fourier series ca be deoted by the followig fuctio. ˆ (0 k F ( k = k ˆ ( k, k =,3, (7 Combiig Euatio (7 ad (7 yields Fourier residual modificatio of grey Verhulst model o time series error corrected. 3.6 Model Accuracy Examiatio o examie the accuracy of the model performace, we employ two evaluatio stadards. 887

8 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 First, relative percetage error (RPE compares the real ad forecast values to evaluate the results. RPE is defied as x ( k xˆ ( k RPE = 00 %, (8 x ( k where RPE is the absolute value of error rate, x ( k is the actual value ad x ( 0 ( k is the predicted value (E.(8. Secod, the mea absolute percetage error (MAPE ad the root mea suare error (RMSE are used, which ca be calculated usig the followig fuctios (E. (9 ad E. (30: x ( k xˆ ( k MAPE =, k = x ( k (9 RMSE i= = ( x ( k xˆ ( k ˆ. (30 RMSE weighted suare every error values, as it ca icrease the precisio of comparisos amog models. 4. EMPIRICA ANAYSIS o demostrate the effectiveess of the improved Verhulst model o time series error corrected, we use the cotaier throughput forecastig of a chia port as a illustratig example. I this study, we use the historical aual cotaier throughput from 989 to 004 as our research data. here are 6 observatios, where are used for model fittig ad are reserved for ex post testig. For the purposes of compariso, we also use the same umber of observatios to formulate the traditioal time series model ad the origial GM(, model. he predicted results obtaied by the liear time series model, the logistic model, the origial GM(, model, the residual grey Verhulst model (GVM ad the residual Fourier model are show i able ad Fig.. he predicted results for out-of-sample forecasts of the value of the RMSE are illustrated i able. From able we ca see, the mea absolute percetage error (MAPE of the liear time series model, the logistic model, the origial GM(, model, the residual GVM model ad the residual Fourier model from 00 to 004 are 6.%, 0.5%,.79%, 3.38% ad.56%, respectively. From able, we kow that the root mea suare error (RMSE of the liear time series model, the logistic model, the origial GM(, model, the residual GVM model ad the residual Fourier model from 00 to 004 are.5, 5.03, 47.63, 7.5 ad 5.35, respectively. Accordig to the results show above, the improved Verhulst model o time series error 888

9 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 corrected by Fourier series obtais the lowest post forecastig errors amog these models. It is idicated that the modificatio of our improved Verhulst model is applicable, especially, whe the throughput icreases accordig to the curve with S type, ot oly higher forecastig accuracy ca be obtaied, but also the superiority ad the features of grey system model ca be reserved. able Model Values ad Forecast Errors (uit: 0 5 EU Year iear ogistic GM(, GVM residual Fourier residual Real RPE RPE RPE RPE RPE value Forecast Forecast Forecast Forecast Forecast (% (% (% (% (% Mape( Mape( EU ogistic iear GVM Residual Fourier Residual Real Values GM(, Model Fittig 0 00 Posterior Forecastig year Fig. Real Values ad Model Values for Port Cotaier hroughput from 989 to 004 able Out-of-sample Forecastig Value of he RMSE ( Model iear ogistic GM(, GVM residual Fourier residual RMSE CONCUSION 889

10 Joural of the Easter Asia Society for rasportatio Studies, Vol. 6, pp , 005 he GM(, model is relatively applicable to describe the mootoous variety process. However, it is imperfect whe the throughput icreases i the curve with S type or the icremet of throughput is i the saturatio stage. I this paper, we itroduce the grey Verhulst model for the port throughput forecastig. By modifyig the residual time series, we proposed the improved Verhulst model o time series error corrected. We have applied this improved grey Verhulst model to the port throughput forecastig. Our study results show that the modified grey Verhulst model o time series error corrected ca yield more accurate results tha the traditioal model ad the GM(, model i the predictio about port throughput i the saturatio stage. hrough this study, the grey Verhulst model o time series error corrected is applicable, especially, whe the throughput icreases accordig to the curve with S type, ot oly higher forecastig accuracy ca be obtaied, but also the superiority ad the features of grey system model ca be reserved. *his work was supported by the Natioal Natural Sciece Foudatio of Chia (No REFERENCES Chew, J.M., i, Y.H., ad Che, J.Y. (995 he Grey predictio cotrol i iverted pedulum system, J. Chia Ist. echol. Commer, Vol., 7 6. Deg J.. (98 Grey System Fudametal Method. Press of Huazhog Uiversity of Sciece ad echology, Wuha. Deg, J.. (986 Grey Predictio Ad Decisio, Press of Huazhog Uiversity of Sciece ad echology, Wuha. Guo, M. (000 Usig GM(, ad Verhulst model to predict buildig s subsidece, Geotechical Egieerig World, Vol. 3, No. 0, Hsu, C.I., ad We, Y.U. (998 Improved Grey predictio models for tras-pacific air passeger market, rasp. Pla. echol, Vol., ee, C. (986 Grey system theory wi applicatio o earthuake forecastig, J. Seismol, Vol. 4, No., 7 3. i, D.H. (004 Verhulst model to predicate groud displacemet ad deformatio. Coal Sciece ad echology, Vol. 3, No. 3, i,.b., ad Che, M.D. (996 ime predictio o ladslide usig Verhulst iverse-fuctio model, Joural of Geological Hazards ad Eviromet Preservatio, Vol. 7, No. 3, 3-7. u, Y.Q., Wu, K.Z., Wag,.J., Zhao, C.G. ad Ya, G.S. (995 Grey predictig the demad of techicias i textile idustry, J. Grey Syst. heory, Vol. 7, No.,

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