Paper 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: Sun, 27 Dec 2009 06:08:29 -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/27/t1261919454pm1g9k6fmx6ykdr.htm/, Retrieved Sun, 27 Dec 2009 14:10:57 +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/27/t1261919454pm1g9k6fmx6ykdr.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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100.21 100.36 100.62 100.78 100.93 100.70 100.00 100.20 99.68 99.56 100.06 100.50 99.30 99.37 99.20 98.11 97.60 97.76 98.06 98.25 98.50 97.39 98.09 97.78 98.12 97.50 97.30 97.64 96.88 97.40 98.27 97.94 98.61 98.72 98.62 98.56 98.06 97.40 97.76 97.05 97.85 97.40 97.27 97.93 98.60 98.70 98.88 98.27 97.85 97.70 96.97 97.72 97.66 99.00 98.86 99.56 100.19 100.37 100.01 99.68 99.78 99.36 99.21 99.26 99.26 100.43 101.50 102.27 102.69 103.47 104.02 103.55 103.77 104.19 103.64 103.68 105.39 106.61 108.12 109.22 110.17 110.31 111.06 111.14 111.39 112.51 111.28 112.22 113.19 114.32 115.34 116.61 117.83 117.70 118.51 118.82 119.49 119.57 120.00 121.96 121.45 123.41 124.44 126.25 127.41 127.63 129.19 129.82 130.45 132.02 132.72 132.96 135.06 137.04 137.83 139.17 140.35 141.01 141.89 143.28 142.90 143.37 145.03 146.05 147.39 149.58 151.02 153.57 155.60 157.18 158.77 159.95 161.34 161.95 163. 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 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[190]) 178 258.37 - - - - - - - 179 258.22 - - - - - - - 180 258.59 - - - - - - - 181 257.45 - - - - - - - 182 257.45 - - - - - - - 183 256.73 - - - - - - - 184 258.82 - - - - - - - 185 257.99 - - - - - - - 186 262.85 - - - - - - - 187 262.58 - - - - - - - 188 261.55 - - - - - - - 189 261.25 - - - - - - - 190 259.78 - - - - - - - 191 256.26 259.784 254.0673 265.764 0.124 0.5005 0.6959 0.5005 192 254.29 260.1901 252.5713 268.2829 0.0765 0.8294 0.6508 0.5396 193 248.5 259.3232 249.7811 269.6234 0.0197 0.8309 0.6392 0.4654 194 241.88 259.5327 247.9076 272.3018 0.0034 0.9548 0.6254 0.4849 195 238.53 259.6815 246.1932 274.7333 0.0029 0.9898 0.6496 0.4949 196 232.24 259.8257 244.4804 277.2264 9e-04 0.9918 0.5451 0.5021 197 232.46 259.6751 242.5432 279.4111 0.0034 0.9968 0.5665 0.4958 198 225.79 260.8315 241.7694 283.1567 0.001 0.9936 0.4297 0.5368 199 221.63 260.9154 240.0704 285.7245 0.001 0.9972 0.4477 0.5357 200 219.62 261.1297 238.4918 288.5159 0.0015 0.9977 0.488 0.5385 201 215.94 261.1363 236.7462 291.1291 0.0016 0.9967 0.497 0.5353 202 211.81 260.7761 234.7087 293.3572 0.0016 0.9965 0.5239 0.5239 203 205.57 260.3818 232.2713 296.2333 0.0014 0.996 0.5891 0.5131 204 201.25 260.4862 230.3322 299.7248 0.0015 0.997 0.6215 0.5141 205 194.7 259.542 227.5666 301.9722 0.0014 0.9965 0.695 0.4956 206 187.94 259.3382 225.3716 305.3603 0.0012 0.997 0.7714 0.4925 207 185.61 259.1267 223.2009 308.8361 0.0019 0.9975 0.7916 0.4897 208 181.15 258.8937 221.0257 312.4203 0.0022 0.9964 0.8355 0.4871 209 186.5 258.4222 218.6963 315.784 0.007 0.9959 0.8125 0.4815 210 183.21 258.9807 217.1152 320.8486 0.0082 0.9892 0.8535 0.4899 211 182.61 258.6943 214.9468 324.7999 0.012 0.9874 0.8641 0.4872 212 187.09 258.51 212.8657 329.0722 0.0236 0.9825 0.86 0.4859 213 189.1 258.165 210.6931 333.2507 0.0357 0.9682 0.8648 0.4832 214 191.25 257.5367 208.3545 337.1122 0.0513 0.9541 0.87 0.478 215 190.74 256.8844 205.8339 341.6096 0.063 0.9355 0.8824 0.4733 216 190.79 256.6187 203.6251 346.8996 0.0765 0.9237 0.8853 0.4726 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 191 0.0117 -0.0136 0 12.4189 0 0 192 0.0159 -0.0227 0.0181 34.8117 23.6153 4.8596 193 0.0203 -0.0417 0.026 117.1421 54.7909 7.4021 194 0.0251 -0.068 0.0365 311.6195 118.998 10.9086 195 0.0296 -0.0815 0.0455 447.384 184.6752 13.5895 196 0.0342 -0.1062 0.0556 760.9716 280.7246 16.7548 197 0.0388 -0.1048 0.0626 740.6613 346.4299 18.6126 198 0.0437 -0.1343 0.0716 1227.904 456.6141 21.3685 199 0.0485 -0.1506 0.0804 1543.3417 577.3616 24.0284 200 0.0535 -0.159 0.0882 1723.0537 691.9309 26.3046 201 0.0586 -0.1731 0.0959 2042.7076 814.7287 28.5435 202 0.0637 -0.1878 0.1036 2397.6816 946.6415 30.7675 203 0.0702 -0.2105 0.1118 3004.333 1104.9254 33.2404 204 0.0769 -0.2274 0.1201 3508.9311 1276.6401 35.7301 205 0.0834 -0.2498 0.1287 4204.4845 1471.8298 38.3644 206 0.0905 -0.2753 0.1379 5097.7052 1698.447 41.2122 207 0.0979 -0.2837 0.1465 5404.6989 1916.4618 43.7774 208 0.1055 -0.3003 0.155 6044.0844 2145.7742 46.3225 209 0.1132 -0.2783 0.1615 5172.8002 2305.0913 48.0114 210 0.1219 -0.2926 0.1681 5741.1936 2476.8964 49.7684 211 0.1304 -0.2941 0.1741 5788.816 2634.6069 51.3284 212 0.1393 -0.2763 0.1787 5100.8132 2746.7072 52.409 213 0.1484 -0.2675 0.1826 4769.9725 2834.6752 53.2417 214 0.1576 -0.2574 0.1857 4393.9211 2899.6438 53.8483 215 0.1683 -0.2575 0.1886 4375.0769 2958.6611 54.3936 216 0.1795 -0.2565 0.1912 4333.4217 3011.5365 54.8775

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/27/t1261919454pm1g9k6fmx6ykdr/1430u1261919306.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/27/t1261919454pm1g9k6fmx6ykdr/1430u1261919306.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/27/t1261919454pm1g9k6fmx6ykdr/251oe1261919306.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/27/t1261919454pm1g9k6fmx6ykdr/251oe1261919306.ps (open in new window)

Parameters (Session):

par1 = Default ; par2 = -1.0 ; par3 = 1 ; par4 = 0 ; par5 = 12 ; par6 = White Noise ; par7 = 0.95 ;

Parameters (R input):

par1 = 24 ; par2 = -1.0 ; par3 = 2 ; par4 = 0 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 1 ; par9 = 1 ; par10 = FALSE ;

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

par1 <- 26 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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