Paper: ARIMA: Forecast

*The author of this computation has been verified*

R Software Module: /rwasp_arimaforecasting.wasp (opens new window with default values)

Title produced by software: ARIMA Forecasting

Date of computation: Tue, 22 Dec 2009 11:20:51 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/22/t1261506141h62uj61n9lwld9s.htm/, Retrieved Tue, 22 Dec 2009 19:22:26 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Dec/22/t1261506141h62uj61n9lwld9s.htm/}, year = {2009}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

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Dataseries X:

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90.2 90 88.8 85.8 84.2 80 77.8 76.8 86.4 89.2 86.2 84.6 83.2 83.2 82.6 79.8 77.2 74.8 73 73 83.6 85.6 84.8 84.2 83.4 84.6 84.6 83.8 81.2 79.6 78 78.2 88.8 92 91 91.2 90.4 91.8 92.2 90.2 88.6 87.8 86 87.2 97.6 101.2 100.4 100.2 100.2 103 104.2 104 102.4 101.8 101 102.2 114 118.4 118.8 117.2 117.2 118.4 118.8 117.2 114.4 112.6 111 110.8 120.2 124.4 123.4 121.2 119 119.8 120 118.4 115 113.4 111 111 121.6 126.2 125.8 124.8 122 123.2 124.2 120.8 116.8 114.8 111 109 119.8 124 121.6 118 115.8 116 115.8 114.4 112 110.2 107.4 108.2 117.6 121.4 119.8 115.6 112.6 113.2 112.2 110.8 108 105.2 102.4 101 110.8 116.8 113.8 108 104.4 105.2 105.4 103.2 100.6 97.8 95.8 95 104.8 110.4 106.4 102.2 98.4 98.4 98.6 96.2 92.4 91.4 88.4 87.8 97.6 104.2 100.2 97 92.8 92 93.4 92 89.6 88.6 87.2 86.2 96.8 102 102.6 100.6 94.2 94.2 95.2 95 94 92.2 91 91.2 103.4 105 104.6 103. etc...

Output produced by software:

Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Univariate ARIMA Extrapolation Forecast time Y[t] F[t] 95% LB 95% UB p-value (H0: Y[t] = F[t]) P(F[t]>Y[t-1]) P(F[t]>Y[t-s]) P(F[t]>Y[229]) 217 113.2 - - - - - - - 218 111.4 - - - - - - - 219 112.2 - - - - - - - 220 109.8 - - - - - - - 221 106.4 - - - - - - - 222 105.2 - - - - - - - 223 102.2 - - - - - - - 224 99.8 - - - - - - - 225 111 - - - - - - - 226 113 - - - - - - - 227 108.4 - - - - - - - 228 105.4 - - - - - - - 229 102 - - - - - - - 230 102.8 100.9071 98.7558 103.0584 0.0423 0.1597 0 0.1597 231 103.4 100.9975 97.8076 104.1874 0.0699 0.134 0 0.269 232 101.6 98.7162 94.6315 102.801 0.0832 0.0123 0 0.0576 233 98.6 95.5317 90.6132 100.4501 0.1107 0.0078 0 0.005 234 98 93.6661 87.9457 99.3864 0.0688 0.0455 0 0.0021 235 93.8 90.8982 84.3944 97.402 0.1909 0.0162 3e-04 4e-04 236 95.6 89.2991 82.0232 96.5749 0.0448 0.1127 0.0023 3e-04 237 105.6 99.7794 91.7388 107.82 0.078 0.8458 0.0031 0.2941 238 106.8 102.1549 93.3547 110.955 0.1504 0.2214 0.0079 0.5138 239 103.6 98.8719 89.3159 108.428 0.1661 0.052 0.0253 0.2606 240 101.2 95.7353 85.4264 106.0442 0.1494 0.0674 0.0331 0.1168 241 100.4 92.2384 81.1792 103.2977 0.074 0.0561 0.0418 0.0418 242 103.2 91.5696 79.33 103.8091 0.0313 0.0787 0.0361 0.0474 243 105.6 91.4775 78.069 104.886 0.0195 0.0433 0.0407 0.062 244 106.6 89.361 74.792 103.9299 0.0102 0.0145 0.0498 0.0445 245 107.2 86.3778 70.655 102.1005 0.0047 0.0059 0.0638 0.0257 246 107.4 84.335 67.4641 101.2059 0.0037 0.0039 0.0562 0.0201 247 104.8 81.7673 63.7533 99.7813 0.0061 0.0026 0.0952 0.0139 248 107.2 80.6059 61.4536 99.7581 0.0032 0.0066 0.0625 0.0143 249 117.4 90.8743 70.5884 111.1602 0.0052 0.0574 0.0774 0.1412 250 119.4 93.4994 72.0844 114.9143 0.0089 0.0144 0.1117 0.2183 251 116.2 90.8625 68.3233 113.4017 0.0138 0.0065 0.134 0.1664 252 112.8 87.7511 64.0925 111.4097 0.019 0.0092 0.1326 0.1189 253 111.6 84.2931 59.5202 109.066 0.0154 0.0121 0.1013 0.0806 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 230 0.0109 0.0188 0 3.5831 0 0 231 0.0161 0.0238 0.0213 5.772 4.6775 2.1628 232 0.0211 0.0292 0.0239 8.3161 5.8904 2.427 233 0.0263 0.0321 0.026 9.4146 6.7715 2.6022 234 0.0312 0.0463 0.03 18.7831 9.1738 3.0288 235 0.0365 0.0319 0.0303 8.4204 9.0482 3.008 236 0.0416 0.0706 0.0361 39.7017 13.4273 3.6643 237 0.0411 0.0583 0.0389 33.8794 15.9838 3.998 238 0.044 0.0455 0.0396 21.5773 16.6053 4.075 239 0.0493 0.0478 0.0404 22.3546 17.1802 4.1449 240 0.0549 0.0571 0.0419 29.863 18.3332 4.2817 241 0.0612 0.0885 0.0458 66.6111 22.3564 4.7283 242 0.0682 0.127 0.0521 135.2672 31.0418 5.5715 243 0.0748 0.1544 0.0594 199.4448 43.0706 6.5628 244 0.0832 0.1929 0.0683 297.1848 60.0116 7.7467 245 0.0929 0.2411 0.0791 433.5654 83.3587 9.1301 246 0.1021 0.2735 0.0905 531.9926 109.7489 10.4761 247 0.1124 0.2817 0.1011 530.5055 133.1243 11.5379 248 0.1212 0.3299 0.1132 707.2482 163.3413 12.7805 249 0.1139 0.2919 0.1221 703.6123 190.3549 13.7969 250 0.1169 0.277 0.1295 670.8422 213.2352 14.6026 251 0.1266 0.2789 0.1363 641.9907 232.7241 15.2553 252 0.1376 0.2855 0.1428 627.4488 249.8861 15.8078 253 0.1499 0.324 0.1503 745.666 270.5435 16.4482

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/22/t1261506141h62uj61n9lwld9s/1xy301261506046.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/22/t1261506141h62uj61n9lwld9s/1xy301261506046.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/22/t1261506141h62uj61n9lwld9s/2mct31261506046.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/22/t1261506141h62uj61n9lwld9s/2mct31261506046.ps (open in new window)

Parameters (Session):

par1 = 24 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 1 ; par7 = 1 ; par8 = 1 ; par9 = 1 ; par10 = FALSE ;

Parameters (R input):

par1 = 24 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 1 ; par7 = 1 ; par8 = 1 ; par9 = 1 ; par10 = FALSE ;

R code (references can be found in the software module):

par1 <- as.numeric(par1) #cut off periods par2 <- as.numeric(par2) #lambda par3 <- as.numeric(par3) #degree of non-seasonal differencing par4 <- as.numeric(par4) #degree of seasonal differencing par5 <- as.numeric(par5) #seasonal period par6 <- as.numeric(par6) #p par7 <- as.numeric(par7) #q par8 <- as.numeric(par8) #P par9 <- as.numeric(par9) #Q if (par10 == 'TRUE') par10 <- TRUE if (par10 == 'FALSE') par10 <- FALSE if (par2 == 0) x <- log(x) if (par2 != 0) x <- x^par2 lx <- length(x) first <- lx - 2*par1 nx <- lx - par1 nx1 <- nx + 1 fx <- lx - nx if (fx < 1) { fx <- par5 nx1 <- lx + fx - 1 first <- lx - 2*fx } first <- 1 if (fx < 3) fx <- round(lx/10,0) (arima.out <- arima(x[1:nx], order=c(par6,par3,par7), seasonal=list(order=c(par8,par4,par9), period=par5), include.mean=par10, method='ML')) (forecast <- predict(arima.out,par1)) (lb <- forecast$pred - 1.96 * forecast$se) (ub <- forecast$pred + 1.96 * forecast$se) if (par2 == 0) { x <- exp(x) forecast$pred <- exp(forecast$pred) lb <- exp(lb) ub <- exp(ub) } if (par2 != 0) { x <- x^(1/par2) forecast$pred <- forecast$pred^(1/par2) lb <- lb^(1/par2) ub <- ub^(1/par2) } if (par2 < 0) { olb <- lb lb <- ub ub <- olb } (actandfor <- c(x[1:nx], forecast$pred)) (perc.se <- (ub-forecast$pred)/1.96/forecast$pred) bitmap(file='test1.png') opar <- par(mar=c(4,4,2,2),las=1) ylim <- c( min(x[first:nx],lb), max(x[first:nx],ub)) plot(x,ylim=ylim,type='n',xlim=c(first,lx)) usr <- par('usr') rect(usr[1],usr[3],nx+1,usr[4],border=NA,col='lemonchiffon') rect(nx1,usr[3],usr[2],usr[4],border=NA,col='lavender') abline(h= (-3:3)*2 , col ='gray', lty =3) polygon( c(nx1:lx,lx:nx1), c(lb,rev(ub)), col = 'orange', lty=2,border=NA) lines(nx1:lx, lb , lty=2) lines(nx1:lx, ub , lty=2) lines(x, lwd=2) lines(nx1:lx, forecast$pred , lwd=2 , col ='white') box() par(opar) dev.off() prob.dec <- array(NA, dim=fx) prob.sdec <- array(NA, dim=fx) prob.ldec <- array(NA, dim=fx) prob.pval <- array(NA, dim=fx) perf.pe <- array(0, dim=fx) perf.mape <- array(0, dim=fx) perf.mape1 <- array(0, dim=fx) perf.se <- array(0, dim=fx) perf.mse <- array(0, dim=fx) perf.mse1 <- array(0, dim=fx) perf.rmse <- array(0, dim=fx) for (i in 1:fx) { locSD <- (ub[i] - forecast$pred[i]) / 1.96 perf.pe[i] = (x[nx+i] - forecast$pred[i]) / forecast$pred[i] perf.se[i] = (x[nx+i] - forecast$pred[i])^2 prob.dec[i] = pnorm((x[nx+i-1] - forecast$pred[i]) / locSD) prob.sdec[i] = pnorm((x[nx+i-par5] - forecast$pred[i]) / locSD) prob.ldec[i] = pnorm((x[nx] - forecast$pred[i]) / locSD) prob.pval[i] = pnorm(abs(x[nx+i] - forecast$pred[i]) / locSD) } perf.mape[1] = abs(perf.pe[1]) perf.mse[1] = abs(perf.se[1]) for (i in 2:fx) { perf.mape[i] = perf.mape[i-1] + abs(perf.pe[i]) perf.mape1[i] = perf.mape[i] / i perf.mse[i] = perf.mse[i-1] + perf.se[i] perf.mse1[i] = perf.mse[i] / i } perf.rmse = sqrt(perf.mse1) bitmap(file='test2.png') plot(forecast$pred, pch=19, type='b',main='ARIMA Extrapolation Forecast', ylab='Forecast and 95% CI', xlab='time',ylim=c(min(lb),max(ub))) dum <- forecast$pred dum[1:par1] <- x[(nx+1):lx] lines(dum, lty=1) lines(ub,lty=3) lines(lb,lty=3) dev.off() load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Univariate ARIMA Extrapolation Forecast',9,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'time',1,header=TRUE) a<-table.element(a,'Y[t]',1,header=TRUE) a<-table.element(a,'F[t]',1,header=TRUE) a<-table.element(a,'95% LB',1,header=TRUE) a<-table.element(a,'95% UB',1,header=TRUE) a<-table.element(a,'p-value<br />(H0: Y[t] = F[t])',1,header=TRUE) a<-table.element(a,'P(F[t]>Y[t-1])',1,header=TRUE) a<-table.element(a,'P(F[t]>Y[t-s])',1,header=TRUE) mylab <- paste('P(F[t]>Y[',nx,sep='') mylab <- paste(mylab,'])',sep='') a<-table.element(a,mylab,1,header=TRUE) a<-table.row.end(a) for (i in (nx-par5):nx) { a<-table.row.start(a) a<-table.element(a,i,header=TRUE) a<-table.element(a,x[i]) a<-table.element(a,'-') a<-table.element(a,'-') a<-table.element(a,'-') a<-table.element(a,'-') a<-table.element(a,'-') a<-table.element(a,'-') a<-table.element(a,'-') a<-table.row.end(a) } for (i in 1:fx) { a<-table.row.start(a) a<-table.element(a,nx+i,header=TRUE) a<-table.element(a,round(x[nx+i],4)) a<-table.element(a,round(forecast$pred[i],4)) a<-table.element(a,round(lb[i],4)) a<-table.element(a,round(ub[i],4)) a<-table.element(a,round((1-prob.pval[i]),4)) a<-table.element(a,round((1-prob.dec[i]),4)) a<-table.element(a,round((1-prob.sdec[i]),4)) a<-table.element(a,round((1-prob.ldec[i]),4)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Univariate ARIMA Extrapolation Forecast Performance',7,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'time',1,header=TRUE) a<-table.element(a,'% S.E.',1,header=TRUE) a<-table.element(a,'PE',1,header=TRUE) a<-table.element(a,'MAPE',1,header=TRUE) a<-table.element(a,'Sq.E',1,header=TRUE) a<-table.element(a,'MSE',1,header=TRUE) a<-table.element(a,'RMSE',1,header=TRUE) a<-table.row.end(a) for (i in 1:fx) { a<-table.row.start(a) a<-table.element(a,nx+i,header=TRUE) a<-table.element(a,round(perc.se[i],4)) a<-table.element(a,round(perf.pe[i],4)) a<-table.element(a,round(perf.mape1[i],4)) a<-table.element(a,round(perf.se[i],4)) a<-table.element(a,round(perf.mse1[i],4)) a<-table.element(a,round(perf.rmse[i],4)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable1.tab')

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This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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