The publicatio appeared i Szoste R.: Modificatio of Holt s model exemplified by the trasport of goods by ilad waterways trasport, Publishig House of Rzeszow Uiversity of Techology No. 85, Maagemet ad Maretig z. 9(4), Rzeszow, pp. 6-68 MODIFICATION OF HOLT S MODEL EXEMPLIFIED BY THE TRANSPORT OF GOODS BY INLAND WATERWAYS TRANSPORT Roma Szoste, Ph.D., Eg. Departmet of Quatitative Methods Rzeszow Uiversity of Techology ul. Wicetego Pola, 35-959 Rzeszów rszoste@prz.rzeszow.pl Abstract: I the paper there were preseted the modificatios of Holt s method. Firstly, it was assumed that the values of the parameters appearig i Holt s model do ot eed to be limited, as is commoly assumed, to the rage of [, ]. Secodly, it was proposed the more precise way of forectig of rage series for more distat time. The aim of the paper is to propose some modificatios to Holt s model which will allow for better forecastig. The paper presets a idea ad the effects of the proposed modificatios exemplified by the data o the trasport of goods by ilad waterways trasport i Polad. Keywords: Holt s method, optimizatio, forecastig.. INTRODUCTION Holt's method is oe of expoetial smoothig methods. It is based o the smoothig of the aalyzed time series by meas of a movig average ad it is used to smooth the time series i which there is the tedecy of radom fluctuatios. Thas to the series smoothig it is obtaied a piece of iformatio about its properties used to determie the forecast. Expoetial smoothig may be based o differet models, respectively matched to the course of the series. I additio to Holt's model it is used a simple model of expoetial smoothig as well as Witer s model.. HOLT S MODEL Holt's model allows for smoothig time series, i which there is the tred ad radom fluctuatios. The values of the forecasted series are idicated by symbols x, x,..., x -. This model has two parameters α ad β ad the followig form: F S S F = x, () x x =, () = t xt t + ( α)( F + t S ) α, (3) t = ( Ft Ft ) + ( β ) St β, (4) where: t =,,, F t the smoothed value of time series S t the smoothed value of the tred icremet o the momet t, α, β model parameters.
The values of F t ad S t are calculated i a recursive maer. Forecast of future values of a series are calculated as follows: x, (5) * + = F + S where: =,, 3, Holt s model parameters α ad β are chose as to miimize the errors of the expired forecast. For this purpose the values of ay specific parameter are assumed ad determied i accordace with (5) for = of the expired forecast x (6) * = + t Ft St for the momet of time t (t =, 3,, ) o the basis of the value of the time series from the earlier period (x, x,..., x t ). This forecast ca be compared with the actual values of a x t series. The resultig differeces are the errors of the expired forecast which gives the model for the adopted parameters α ad β. As a measure of the quality of the method it should be assumed the average from the errors of the expired forecas. It ca be a average mea or qudratic mea 6 t= 6 J = Ft + St xt (7) = ( Ft + St xt ) 6 t= 6 J. (8) Fially, amog all the possible values of the parameters α ad β there should be foud those that give the smallest error value J or J. I this way, there are determied the optimum values of model parameters, i.e. its optimizatio is carried out. The value of J * is a measure of forecast error determied by the model. It is commoly assumed that α [, ] ad β [, ]. It seems, however, that this limitatio is uecessary, therefore i the carried out calculatios it was omitted. I this way it was assumed that the best model parameters are such for which the model determies the expired forecast i the best way irrespectively whether these parameters are higher or lower tha the uity. The evaluatio of the optimum values of parameters α ad β is coducted by meas of umarical methods. 3. DATA ANALYSIS The aalysed time series was preseted i table. It cotais the iformatio o the data o the trasport of goods by ilad waterways trasport i Polad i aual periods. The values of the optimum parameters of Holt s model are differet depedig o the applied quality measure. The parameters grid was checed with the accuracy of.. The forecasts were determied for the ext future periods. Fo the liear meaure (7) the optimum are parameters α =.48 ad β = -.336. (9) For the squared measure (8) the optimum are parameters α =.3884 ad β = -.45. () Table. Passegers air trasport i Polad Year Trasport Trasport Year (thousad toes) (thousad toes) 433 6 97 55 7 979 779 8 89 3 7968 9 5655 4 8747 54 5 967 593 Source: Cetral Statistical Office (www.stat.gov.pl) [][]. Time series together with the forecast for the measure (7) was preseted i fig..
Fig.. Forecast, whe i model α =.48 ad β = -.336. 8-3 4 5 6 7 8 9 3 4 X(t) F(t) S(t) X*[t] Time series together with the forecast for the measure (8) was preseted i fig.. Fig.. Forecast, whe i model α =.3884 ad β = -.45. 8-3 4 5 6 7 8 9 3 4 X(t) F(t) S(t) X*[t] For compariso, whe the parameters value of the model α ad β are limited to the rage [, ], the the optimum values of these parameters for the liear measure (7) are whereas for the squared measure (8) they are α=. ad β=., () α=. ad β=.37. () Time series together with the forecast for the case whe α =. ad β =. was show i fig. 3. Fig.3. Forcast, whe i model α =. ad β =.. 8-3 4 5 6 7 8 9 3 4 X(t) F(t) S(t) X*[t] 3
Table. Calculatio results. Compariso of three solutios t X(t) α=.48 β=-.336 α=.3884 β=-.45 α=, β=, F(t) S(t) X*[t] F(t) S(t) X*[t] F(t) S(t) X*[t] 433 - - - - - - - - - 55 55-78 - 55-78 - 55-78 - 779 758.9-8. 77 687. -3.7 77 779-78 77 3 7968 883.9-7. 779. 8465.8-56.5 6686.3 7968-78 755 4 8747 89. -45.5 866.8 897. -65.4 839.3 8747-78 779 5 967 98. -8.6 8766.7 9939. -8.6 875.7 967-78 8569 6 97 983.7-65.6 963.5 98.4-7.8 9756.7 97-78 949 7 979 9979.9-97.9 98. 34.7-9.6 899.6 979-78 993 8 89 77.8-8. 978. 7396. -53.6 9944. 89-78 964 9 5655 588.9-47.9 7574.8 538.4 -.7 74.6 5655-78 793 54 54. -47.9 54. 58. -6. 496.7 54-78 5477 593 593. -47.9 593. 589.5-6. 5. 593-78 4963 6 545. 4963.5 495. 7 forecast: 4997.3 forecast: 4837.6 forecast: 4737. 8 4949.5 47.5 4559. li. error 787.7 837.4 85.7 squ. error 96.4 76.5 6. I table there were show the results of calculatios for the above three cases. The optimum values of quality measures were show i bold i the table. Limitatio of the values of the parameters α ad β to the rage [, ] maes that to the large family of a time series it is assiged the same Holt s model with parameter α =. (or β =.). That is so for the cosidered rage. I this cotext, a wide variety of series is described by the same model. At the same time the parameters of the Holt s model ca be selected i order to determie the expired forecast i a more accurate way. These parameters should be used for forecastig. 4. FORECAST FOR FURTHER INSTANTS OF TIME I the traditioal approach preseted i the form of equatios ()-(5) the optimum parameters of the model are based o the expired forecast calculated for oe step forewards. If, however, by usig the model the forecast for steps forwards eeds to be determied, it should be performed the optimizatio model taig ito accout the expired forecast for steps forwards. Thus, it was suggested that for every istat of time which the forecast is to be determied, it should be determied aother model, each time o the basis of differet quality measure. If the predictio is to be performed o the a -th step forwards, the the model parameters should be determied by miimizig a measure of the quality of or where: =,, 3, J ( ) = Ft + St xt (3) 5 t= 5+ ( ) = ( Ft + St xt ) 5 t= 5+ J, (4) The calculatios were made with a applicatio of C++ of the followig code: it mai(it argc, char *argv[]) {double X[]={433, 55, 779, 7968, 8747, 967, 97, 979, 89, 5655, 54, 593; double F[], S[], F_opt, S_opt; // the table eeds to be of the size double J, J, a_opt, b_opt, J_mi=999999999, J_mi=999999999; double a, b, ro=.; it t, =, =; // meas o how may steps forwards is the forecast, =,, 3,... F[]=X[]; S[]=X[]-X[]; for (a=.; a<=.; a=a+ro) { for (b=.; b<=.; b=b+ro) { J=; J=; for (t=; t<; t++) 4
{ F[t]=a*X[t]+(-a)*(F[t-]+S[t-]); S[t]=b*(F[t]-F[t-])+(-b)*S[t-]; if (t>=(5+)) { J=J+pow((F[t-]+*S[t-]-X[t]),); //squared error J=J+fabs(F[t-]+*S[t-]-X[t]); //liear error if (J < J_mi) //miimizatio of liear error { J_mi=J; J_mi=J; a_opt=a; b_opt=b; F_opt=F[-]; S_opt=S[-]; J_mi=J_mi/(-5-); //liear error J_mi=sqrt(J_mi/(-5-)); //squared error cout << "optimum a = " << a_opt << edl; cout << "optimum b = " << b_opt << edl; cout << "forecast for the momet " << +- << " = " << F_opt+*S_opt << edl; cout << "liear error J = " << J_mi << edl; cout << "squared error J = " << J_mi << edl; system("pause"); Forecast value for steps forwards is calculated accordig to equatio (5). The results of calculatios for the quality measure (3) are preseted i table. 3 ad i figure 4. I the case of the modified method the forecasts do ot have to be a straight lie, as it is always i case of the covetioal method. Fig. 4. Forecast for the modified quality measure J () 8 3 4 5 6 7 8 9 3 X(t) X*[t] Thas to the preseted modificatio there were obtaied lower average errors of expired forecast. They are summarized i table. 4 (for liear quality measure). These errors for the traditioal method were determied for a model i which α ad β have the values (9), but accordig to the equatio (3). Yet, for the modified method the errors of expired forecast come from table 3. Table 3. Calculatio results for the modified quality measure J () = = =3 t X(t) α=.48 β=-.336 α=.749 β=-.68 α=.6 β=47.43 F(t) S(t) X*[t] F(t) S(t) X*[t] F(t) S(t) X*[t] 433 - - - - - - - - - 55 55. -78. - 55. -78-55. -78. - 779 758.9-8. 77. 66.9 -. - 73. -355. - 3 7968 883.9-7. 779. 9454.8-69.4 9899. 975.3-487. - 4 8747 89. -45.5 866.8 8356.7-53.8 587. 97.5-53.4 97. 5 967 98. -8.6 8766.7 64.8-94.5 96. 875.5-455.3 97.9 5
6 97 983.7-65.7 963.5 843.6-6.9 849. 85.8-378.3 854. 7 979 9979.9-97.9 98. 895.5-5. 35.9 7876.6-33.6 7657. 8 89 77.8-8. 978. 637.7-3. 88.8 7643.7-98.5 7339.6 9 5655 588.9-47.9 7574.8 537. -7. 485.5 744.4-333.5 76.9 54 54. -47.9 54. 59. -5. 5977.3 75.8-48.9 775.8 593 593. -47.8 593. 577.8-8.5 59.8 66.4-597.4 748.4 6 progoza: 545. 7 progoza: 494.9 8 progoza: 489. Forecast error. J () = 787.7 J () =. J (3) = 555.4 Table 4. Compariso of average errors of expired forecast for liear quality measuremet Forecast for steps forwards = = =3 Traditioal method 787.7 464. 665.8 Modified method 787.7. 555.4 Based o the criterio which is the average error of expired forecast, oe ca coclude that the modified method allows for the determiatio of more reliable forecasts. 5. CONCLUSIONS I the classic applicatios of Holt s model the values of its parameter are restricted to the rage [, ]. This approach has bee used for example i a Statistica pacage. I the paper it was proposed ot to restrict the parameters of the model to the rage [, ]. Such models ca better determie the expired forecast. So they are a better way to determie future forecasts. The limitatio of the parameters values α ad β to the rage [, ] probably resulted from the idea that the values of series F t ad S t are i a certai percetage of the previous values of these series, ad the rest of the values of aother factor (i accordace with (3) ad (4 )). Resigatio of reducig the value of these parameters is a atural geeralizatio of the method. I the cases where i the model the parameter α> was obtaied, the series F t was ot always the smoothig of x t series. For some of the aalyzed examples the values of F t series oscillated at aroud the value of x t series. I that case F t series did ot smooth but o the cotrary sharpeed the fluctuatio of the origial series. However, always modified models determied the expired forecast i a better way, so they are a better tool to determie future forecasts. I the paper it was also proposed a modified way of Holt s model optimizatio. It cosists i idepedet optimizatio of some models, oe for each forecast for the -th period forwards. I this way there are determied so may Holt s models o how may steps forwards the forecast is made. Such a modified model determies the expired forecast with less average error. It is, therefore, a better way to determie future forecast. Thus obtaied forecast doe ot eed to be a straight lie (Fig. 4), as is always i the classic approach. 6. REFERENCES [] Cetral Statistical Office Trasport activity results, http://www.stat.gov.pl/cps/rde/xbcr/szczec/assets_sygala_trasport_wody pop.pdf [] Cetral Statistical Office Trasport activity results, http://www.stat.gov.pl/cps/rde/xbcr/gus/publ_til_trasport_wyii_dzialalosci_.pdf 6