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 13:42:17 -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/t12605642026u8lqeaduwoh006.htm/, Retrieved Fri, 11 Dec 2009 21:43:25 +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/t12605642026u8lqeaduwoh006.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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11.40 11.00 9.20 7.10 9.30 9.30 11.50 13.00 13.20 13.10 13.90 11.00 11.30 10.80 11.20 12.90 13.90 14.50 14.50 13.30 12.00 11.50 11.00 12.10 13.00 14.00 15.10 14.50 14.20 13.30 12.70 11.80 11.40 10.50 9.60 10.80 10.70 11.90 12.00 11.10 10.90 10.40 10.70 12.10 12.80 13.90 13.50 12.00 12.00 11.50 12.50 13.10 12.70 12.80 12.50 13.00 13.20 12.80 12.40 12.00 11.80 11.10 8.50 6.30

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[36]) 24 12.1 - - - - - - - 25 13 - - - - - - - 26 14 - - - - - - - 27 15.1 - - - - - - - 28 14.5 - - - - - - - 29 14.2 - - - - - - - 30 13.3 - - - - - - - 31 12.7 - - - - - - - 32 11.8 - - - - - - - 33 11.4 - - - - - - - 34 10.5 - - - - - - - 35 9.6 - - - - - - - 36 10.8 - - - - - - - 37 10.7 10.6528 9.0349 12.2707 0.4772 0.4293 0.0022 0.4293 38 11.9 11.7535 9.7142 13.7929 0.444 0.8444 0.0154 0.8203 39 12 12.5863 9.7969 15.3756 0.3402 0.6852 0.0387 0.8953 40 11.1 13.0653 9.7668 16.3638 0.1214 0.7366 0.197 0.9109 41 10.9 12.9903 9.4487 16.532 0.1237 0.8522 0.2516 0.8873 42 10.4 12.4614 8.863 16.0598 0.1308 0.8025 0.3239 0.8172 43 10.7 11.7338 8.1341 15.3335 0.2868 0.7661 0.2994 0.6944 44 12.1 11.1171 7.5148 14.7193 0.2964 0.5898 0.3551 0.5685 45 12.8 10.8431 7.2409 14.4454 0.1435 0.247 0.3809 0.5094 46 13.9 10.9837 7.3573 14.61 0.0575 0.1631 0.6031 0.5395 47 13.5 11.4387 7.6992 15.1781 0.14 0.0985 0.8324 0.6311 48 12 11.9945 8.0405 15.9485 0.4989 0.2278 0.7231 0.7231 49 12 12.4206 8.226 16.6152 0.4221 0.5779 0.7893 0.7756 50 11.5 12.5623 8.1881 16.9365 0.317 0.5995 0.6167 0.7851 51 12.5 12.3941 7.9285 16.8596 0.4815 0.6526 0.5687 0.7579 52 13.1 12.0156 7.5189 16.5124 0.3182 0.4164 0.6551 0.7019 53 12.7 11.5991 7.0936 16.1046 0.316 0.2569 0.6195 0.6359 54 12.8 11.3133 6.8015 15.8252 0.2592 0.2735 0.6542 0.5882 55 12.5 11.2573 6.7279 15.7868 0.2954 0.2522 0.5953 0.5784 56 13 11.4286 6.8492 16.008 0.2506 0.3233 0.3869 0.6061 57 13.2 11.7351 7.0592 16.4111 0.2696 0.298 0.3277 0.6525 58 12.8 12.0412 7.2367 16.8457 0.3784 0.3182 0.2241 0.6937 59 12.4 12.2257 7.2968 17.1547 0.4724 0.4097 0.3062 0.7146 60 12 12.2289 7.2086 17.2491 0.4644 0.4734 0.5356 0.7115 61 11.8 12.0688 6.9949 17.1427 0.4587 0.5106 0.5106 0.688 62 11.1 11.8262 6.7235 16.929 0.3901 0.504 0.5499 0.6533 63 8.5 11.606 6.4838 16.7281 0.1173 0.5768 0.3661 0.6211 64 6.3 11.4927 6.3486 16.6368 0.0239 0.8729 0.2701 0.6041 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 37 0.0775 0.0044 0 0.0022 0 0 38 0.0885 0.0125 0.0084 0.0215 0.0118 0.1088 39 0.1131 -0.0466 0.0212 0.3437 0.1225 0.3499 40 0.1288 -0.1504 0.0535 3.8625 1.0575 1.0283 41 0.1391 -0.1609 0.075 4.3695 1.7199 1.3114 42 0.1473 -0.1654 0.09 4.2494 2.1415 1.4634 43 0.1565 -0.0881 0.0898 1.0687 1.9882 1.41 44 0.1653 0.0884 0.0896 0.9662 1.8605 1.364 45 0.1695 0.1805 0.0997 3.8293 2.0792 1.4419 46 0.1684 0.2655 0.1163 8.505 2.7218 1.6498 47 0.1668 0.1802 0.1221 4.2491 2.8606 1.6913 48 0.1682 5e-04 0.1119 0 2.6223 1.6193 49 0.1723 -0.0339 0.1059 0.1769 2.4341 1.5602 50 0.1777 -0.0846 0.1044 1.1286 2.3409 1.53 51 0.1838 0.0085 0.098 0.0112 2.1856 1.4784 52 0.1909 0.0902 0.0975 1.1759 2.1225 1.4569 53 0.1982 0.0949 0.0974 1.212 2.0689 1.4384 54 0.2035 0.1314 0.0993 2.2102 2.0768 1.4411 55 0.2053 0.1104 0.0999 1.5442 2.0487 1.4313 56 0.2044 0.1375 0.1017 2.4692 2.0698 1.4387 57 0.2033 0.1248 0.1028 2.1458 2.0734 1.4399 58 0.2036 0.063 0.101 0.5758 2.0053 1.4161 59 0.2057 0.0143 0.0973 0.0304 1.9194 1.3854 60 0.2095 -0.0187 0.094 0.0524 1.8417 1.3571 61 0.2145 -0.0223 0.0911 0.0722 1.7709 1.3307 62 0.2201 -0.0614 0.09 0.5274 1.7231 1.3127 63 0.2252 -0.2676 0.0966 9.6469 2.0165 1.42 64 0.2284 -0.4518 0.1092 26.9643 2.9075 1.7051

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/11/t12605642026u8lqeaduwoh006/1z9un1260564135.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t12605642026u8lqeaduwoh006/1z9un1260564135.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t12605642026u8lqeaduwoh006/2k38w1260564135.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t12605642026u8lqeaduwoh006/2k38w1260564135.ps (open in new window)

Parameters (Session):

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

Parameters (R input):

par1 = 24 ; par2 = 1 ; par3 = 1 ; par4 = 0 ; par5 = 12 ; par6 = 3 ; 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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