Paper Box Jenkins 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: Sun, 19 Dec 2010 15:30:52 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/19/t1292772545b8ymgem4zc3h7tc.htm/, Retrieved Sun, 19 Dec 2010 16:29:07 +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/2010/Dec/19/t1292772545b8ymgem4zc3h7tc.htm/}, year = {2010}, } @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 = {2010}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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36700 35600 80900 174000 169422 153452 173570 193036 174652 105367 95963 82896 121747 120196 103983 81103 70944 57248 47830 60095 60931 82955 99559 77911 70753 69287 88426 91756 96933 174484 232595 266197 290435 304296 322310 415555 490042 545109 545720 505944 477930 466106 424476 383018 364696 391116 435721 511435 553997 555252 544897 540562 505282 507626 474427 469740 491480 538974 576612

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 2 seconds R Server 'George Udny Yule' @ 72.249.76.132 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[35]) 34 304296 - - - - - - - 35 322310 - - - - - - - 36 415555 334738.5428 278652.7006 390824.385 0.0024 0.668 0.668 0.668 37 490042 344827.498 246220.5683 443434.4277 0.0019 0.0799 0.0799 0.6728 38 545109 353936.467 218345.6624 489527.2716 0.0029 0.0246 0.0246 0.6762 39 545720 362634.9481 194437.7759 530832.1202 0.0164 0.0167 0.0167 0.6808 40 505944 371161.4874 173596.049 568726.9259 0.0906 0.0417 0.0417 0.686 41 477930 379616.0054 155056.9551 604175.0557 0.1954 0.1351 0.1351 0.6915 42 466106 388040.3556 138252.7383 637827.973 0.2701 0.2403 0.2403 0.697 43 424476 396452.0695 122776.1503 670127.9887 0.4205 0.3089 0.3089 0.7023 44 383018 404858.4904 108335.8075 701381.1732 0.4426 0.4484 0.4484 0.7073 45 364696 413262.6941 94720.7578 731804.6305 0.3825 0.5738 0.5738 0.7121 46 391116 421665.9692 81775.5139 761556.4245 0.4301 0.6287 0.6287 0.7167 47 435721 430068.8553 69383.1501 790754.5605 0.4877 0.5838 0.5838 0.7209 48 511435 438471.5784 57453.964 819489.1928 0.3537 0.5056 0.5056 0.7249 49 553997 446874.2333 45917.8318 847830.6348 0.3003 0.3762 0.3762 0.7287 50 555252 455276.8596 34718.987 875834.7322 0.3206 0.3227 0.3227 0.7323 51 544897 463679.4739 23812.395 903546.5529 0.3587 0.3416 0.3416 0.7356 52 540562 472082.0832 13161.1886 931002.9779 0.385 0.3779 0.3779 0.7388 53 505282 480484.6905 2734.8208 958234.5601 0.4595 0.4027 0.4027 0.7418 54 507626 488887.2968 -7492.291 985266.8846 0.4705 0.4742 0.4742 0.7446 55 474427 497289.9028 -17541.7766 1012121.5821 0.4653 0.4843 0.4843 0.7473 56 469740 505692.5086 -27432.079 1038817.0962 0.4474 0.5458 0.5458 0.7499 57 491480 514095.1143 -37179.0448 1065369.2734 0.468 0.5627 0.5627 0.7523 58 538974 522497.72 -46796.383 1091791.823 0.4774 0.5425 0.5425 0.7547 59 576612 530900.3257 -56296.0277 1118096.6792 0.4394 0.4893 0.4893 0.7569 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 36 0.0855 0.2414 0 6531299753.6303 0 0 37 0.1459 0.4211 0.3313 21087251602.302 13809275677.9662 117512.8745 38 0.1955 0.5401 0.4009 36546937362.9658 21388496239.6327 146248.0641 39 0.2366 0.5049 0.4269 33520136244.3275 24421406240.8064 156273.4982 40 0.2716 0.3631 0.4141 18166325692.6411 23170390131.1734 152218.2319 41 0.3018 0.259 0.3883 9665641535.1529 20919598698.5033 144636.0906 42 0.3284 0.2012 0.3616 6094244828.1875 18801691002.7439 137119.2583 43 0.3522 0.0707 0.3252 785340679.8858 16549647212.3866 128645.4321 44 0.3737 -0.0539 0.2951 477007019.2278 14763798302.0356 121506.3714 45 0.3933 -0.1175 0.2773 2358723777.2755 13523290849.5596 116289.6851 46 0.4113 -0.0725 0.2587 933300617.6311 12378746283.0207 111259.8143 47 0.4279 0.0131 0.2382 31946740.0182 11349846321.1038 106535.6575 48 0.4434 0.1664 0.2327 5323660889.5146 10886293595.5969 104337.4027 49 0.4578 0.2397 0.2332 11475287144.3927 10928364563.3681 104538.8185 50 0.4713 0.2196 0.2323 9995028696.475 10866142172.2419 104240.7894 51 0.484 0.1752 0.2287 6596286540.0417 10599276195.2293 102952.7862 52 0.496 0.1451 0.2238 4689498998.4385 10251642242.4769 101250.3938 53 0.5073 0.0516 0.2142 614906560.503 9716268037.9228 98571.1319 54 0.518 0.0383 0.205 351138998.0199 9223366509.5069 96038.3596 55 0.5282 -0.046 0.197 522712322.2259 8788333800.1428 93746.1135 56 0.5379 -0.0711 0.191 1292582871.7124 8431393279.7414 91822.6186 57 0.5471 -0.044 0.1843 511443394.9177 8071395557.704 89840.9459 58 0.5559 0.0315 0.1777 271467802.0241 7732268263.9787 87933.3171 59 0.5643 0.0861 0.1739 2089557165.1721 7497155301.5285 86586.1149

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/19/t1292772545b8ymgem4zc3h7tc/1zbjd1292772650.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/19/t1292772545b8ymgem4zc3h7tc/1zbjd1292772650.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/19/t1292772545b8ymgem4zc3h7tc/26uy71292772650.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/19/t1292772545b8ymgem4zc3h7tc/26uy71292772650.ps (open in new window)

Parameters (Session):

par1 = 24 ; par2 = 1 ; par3 = 2 ; par4 = 0 ; par5 = 1 ; par6 = 1 ; par7 = 0 ; par8 = 0 ; par9 = 1 ; par10 = FALSE ;

Parameters (R input):

par1 = 24 ; par2 = 1 ; par3 = 2 ; par4 = 0 ; par5 = 1 ; par6 = 1 ; par7 = 0 ; par8 = 0 ; 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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