ARIMA forecasting

*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: Fri, 11 Dec 2009 09:37:21 -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/11/t1260549491xptusslul5izcte.htm/, Retrieved Fri, 11 Dec 2009 17:38:14 +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/11/t1260549491xptusslul5izcte.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:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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562 561 555 544 537 543 594 611 613 611 594 595 591 589 584 573 567 569 621 629 628 612 595 597 593 590 580 574 573 573 620 626 620 588 566 557 561 549 532 526 511 499 555 565 542 527 510 514 517 508 493 490 469 478 528 534 518 506 502 516 528

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 '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[33]) 21 628 - - - - - - - 22 612 - - - - - - - 23 595 - - - - - - - 24 597 - - - - - - - 25 593 - - - - - - - 26 590 - - - - - - - 27 580 - - - - - - - 28 574 - - - - - - - 29 573 - - - - - - - 30 573 - - - - - - - 31 620 - - - - - - - 32 626 - - - - - - - 33 620 - - - - - - - 34 588 604.4992 596.9083 612.0901 0 0 0.0264 0 35 566 582.8089 570.6555 594.9624 0.0034 0.2012 0.0246 0 36 557 584.859 566.9219 602.7961 0.0012 0.9803 0.0923 1e-04 37 561 580.4244 559.2593 601.5895 0.036 0.985 0.1221 1e-04 38 549 575.5345 551.4921 599.577 0.0153 0.882 0.1191 1e-04 39 532 564.5376 537.4178 591.6574 0.0093 0.8693 0.1319 0 40 526 557.6209 527.7872 587.4546 0.0189 0.9538 0.1409 0 41 511 555.346 522.9695 587.7225 0.0036 0.9622 0.1426 0 42 499 554.2201 519.3174 589.1228 0.001 0.9924 0.1458 1e-04 43 555 600.1495 562.8209 637.4781 0.0089 1 0.1486 0.1486 44 565 604.999 565.3235 644.6746 0.0241 0.9932 0.1498 0.2293 45 542 597.869 555.8844 639.8536 0.0046 0.9375 0.1508 0.1508 46 527 581.2575 533.9718 628.5431 0.0123 0.9482 0.3899 0.0541 47 510 558.4404 505.4955 611.3852 0.0365 0.8778 0.3898 0.0113 48 514 559.3654 499.5452 619.1856 0.0686 0.9471 0.5309 0.0235 49 517 553.8115 488.5567 619.0662 0.1344 0.8841 0.4145 0.0234 50 508 547.7996 477.3921 618.207 0.1339 0.8044 0.4867 0.0222 51 493 535.6807 459.9748 611.3865 0.1346 0.7632 0.538 0.0145 52 490 527.6437 446.9552 608.3322 0.1803 0.8 0.5159 0.0124 53 469 524.2484 438.7528 609.744 0.1027 0.7838 0.6193 0.0141 54 478 522.0024 431.742 612.2627 0.1697 0.8751 0.6913 0.0167 55 528 566.8123 471.9017 661.7229 0.2114 0.9667 0.5964 0.136 56 534 570.5427 471.0779 670.0075 0.2357 0.7991 0.5435 0.1649 57 518 562.294 458.3326 666.2553 0.2018 0.7031 0.649 0.1383 58 506 544.5641 433.6353 655.4929 0.2478 0.6806 0.6218 0.0913 59 502 520.6292 402.3906 638.8677 0.3787 0.5958 0.5699 0.0498 60 516 520.4367 393.8322 647.0413 0.4726 0.6123 0.5397 0.0616 61 528 513.7657 379.8604 647.6711 0.4175 0.487 0.4811 0.06 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 34 0.0064 -0.0273 0 272.2239 0 0 35 0.0106 -0.0288 0.0281 282.5407 277.3823 16.6548 36 0.0156 -0.0476 0.0346 776.1219 443.6288 21.0625 37 0.0186 -0.0335 0.0343 377.3076 427.0485 20.6652 38 0.0213 -0.0461 0.0367 704.082 482.4552 21.9649 39 0.0245 -0.0576 0.0402 1058.6965 578.4954 24.0519 40 0.0273 -0.0567 0.0425 999.8824 638.6936 25.2724 41 0.0297 -0.0799 0.0472 1966.5669 804.6777 28.3668 42 0.0321 -0.0996 0.053 3049.2585 1054.0756 32.4665 43 0.0317 -0.0752 0.0552 2038.4756 1152.5156 33.9487 44 0.0335 -0.0661 0.0562 1599.9227 1193.189 34.5426 45 0.0358 -0.0934 0.0593 3121.3475 1353.8688 36.795 46 0.0415 -0.0933 0.0619 2943.8711 1476.1767 38.421 47 0.0484 -0.0867 0.0637 2346.4701 1538.3405 39.2217 48 0.0546 -0.0811 0.0649 2058.0192 1572.9858 39.6609 49 0.0601 -0.0665 0.065 1355.0831 1559.3669 39.4888 50 0.0656 -0.0727 0.0654 1584.0046 1560.8161 39.5072 51 0.0721 -0.0797 0.0662 1821.6395 1575.3063 39.6901 52 0.078 -0.0713 0.0665 1417.0475 1566.9769 39.5851 53 0.0832 -0.1054 0.0684 3052.386 1641.2474 40.5123 54 0.0882 -0.0843 0.0692 1936.207 1655.2931 40.6853 55 0.0854 -0.0685 0.0692 1506.3937 1648.5249 40.602 56 0.0889 -0.064 0.0689 1335.3718 1634.9096 40.434 57 0.0943 -0.0788 0.0693 1961.956 1648.5365 40.6022 58 0.1039 -0.0708 0.0694 1487.1916 1642.0827 40.5226 59 0.1159 -0.0358 0.0681 347.0463 1592.2736 39.9033 60 0.1241 -0.0085 0.0659 19.6846 1534.0296 39.1667 61 0.133 0.0277 0.0645 202.6141 1486.479 38.5549

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260549491xptusslul5izcte/14igv1260549439.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260549491xptusslul5izcte/14igv1260549439.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260549491xptusslul5izcte/25yww1260549439.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260549491xptusslul5izcte/25yww1260549439.ps (open in new window)

Parameters (Session):

par1 = 6 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 1 ; par7 = 1 ; par8 = 0 ; par9 = 0 ; par10 = FALSE ;

Parameters (R input):

par1 = 6 ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 1 ; par7 = 1 ; par8 = 0 ; par9 = 0 ; par10 = FALSE ;

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

par1 <- as.numeric(par1) #cut off periods par1 <- 28 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 par6 <- 3 par7 <- as.numeric(par7) #q par7 <- 3 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')

Copyright

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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