Paper ARIMA Forecasting 33

*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: Wed, 30 Dec 2009 07:04:28 -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/30/t1262181997a6uwhj4kqm04x0i.htm/, Retrieved Wed, 30 Dec 2009 15:06:39 +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/30/t1262181997a6uwhj4kqm04x0i.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 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[183]) 171 247.94 - - - - - - - 172 248.8 - - - - - - - 173 250.18 - - - - - - - 174 251.55 - - - - - - - 175 254.4 - - - - - - - 176 255.72 - - - - - - - 177 257.69 - - - - - - - 178 258.37 - - - - - - - 179 258.22 - - - - - - - 180 258.59 - - - - - - - 181 257.45 - - - - - - - 182 257.45 - - - - - - - 183 256.73 - - - - - - - 184 258.82 257.589 251.9262 263.5123 0.3419 0.6119 0.9982 0.6119 185 257.99 258.4618 250.8861 266.5092 0.4543 0.4652 0.9782 0.6634 186 262.85 259.6428 250.0186 270.0376 0.2727 0.6223 0.9365 0.7086 187 262.58 261.1406 249.2966 274.1661 0.4143 0.3985 0.8448 0.7466 188 261.55 263.0623 249.157 278.6115 0.4244 0.5242 0.8226 0.7876 189 261.25 264.4204 248.4822 282.5432 0.3658 0.6219 0.7667 0.7972 190 259.78 265.6029 247.6625 286.3454 0.2911 0.6596 0.7528 0.7991 191 256.26 266.1472 246.3129 289.4557 0.2029 0.7038 0.7475 0.7858 192 254.29 267.4502 245.6076 293.557 0.1616 0.7996 0.747 0.7895 193 248.5 267.3428 243.718 296.0394 0.0991 0.8137 0.7504 0.7657 194 241.88 268.167 242.6084 299.7449 0.0514 0.8889 0.747 0.7611 195 238.53 269.0059 241.5004 303.5823 0.042 0.9379 0.7567 0.7567 196 232.24 269.5212 239.7159 307.7908 0.0281 0.9438 0.7082 0.7438 197 232.46 270.398 238.3032 312.4832 0.0386 0.9622 0.7183 0.7378 198 225.79 271.4541 236.9882 317.651 0.0263 0.951 0.6425 0.7339 199 221.63 272.61 235.7039 323.2189 0.0242 0.9651 0.6512 0.7307 200 219.62 274.1984 234.7364 329.6097 0.0268 0.9685 0.6727 0.7317 201 215.94 275.2603 233.3557 335.5088 0.0268 0.9649 0.6757 0.7267 202 211.81 276.1564 231.8445 341.4092 0.0266 0.9648 0.6886 0.7202 203 205.57 276.5105 229.952 346.709 0.0238 0.9646 0.7141 0.7096 204 201.25 277.507 228.5003 353.2739 0.0243 0.9686 0.7259 0.7045 205 194.7 277.2944 226.2341 358.1213 0.0226 0.9674 0.7575 0.691 206 187.94 277.8788 224.5089 364.5356 0.021 0.97 0.7922 0.6838 207 185.61 278.473 222.7905 371.2637 0.0249 0.9721 0.8006 0.677 208 181.15 278.7901 220.6785 378.4469 0.0274 0.9666 0.82 0.6678 209 186.5 279.4132 218.8078 386.4531 0.0444 0.964 0.805 0.6611 210 183.21 280.1863 217.0118 395.2471 0.0493 0.9447 0.8229 0.6553 211 182.61 281.041 215.246 404.7679 0.0595 0.9394 0.8267 0.6499 212 187.09 282.2573 213.6942 415.6018 0.0809 0.9285 0.8214 0.6463 213 189.1 283.0258 211.8782 426.112 0.0991 0.9056 0.8209 0.6407 214 191.25 283.6526 209.9872 436.9327 0.1187 0.8867 0.8209 0.6347 215 190.74 283.8232 207.8598 447.2864 0.1322 0.8665 0.826 0.6274 216 190.79 284.5306 206.0321 459.6627 0.1471 0.8531 0.8243 0.6221 Univariate ARIMA Extrapolation Forecast Performance time % S.E. PE MAPE Sq.E MSE RMSE 184 0.0117 0.0048 0 1.5154 0 0 185 0.0159 -0.0018 0.0033 0.2226 0.869 0.9322 186 0.0204 0.0124 0.0063 10.286 4.008 2.002 187 0.0254 0.0055 0.0061 2.0719 3.524 1.8772 188 0.0302 -0.0057 0.006 2.287 3.2766 1.8101 189 0.035 -0.012 0.007 10.0513 4.4057 2.099 190 0.0398 -0.0219 0.0092 33.9058 8.62 2.936 191 0.0447 -0.0371 0.0127 97.7572 19.7621 4.4455 192 0.0498 -0.0492 0.0167 173.1908 36.8098 6.0671 193 0.0548 -0.0705 0.0221 355.0515 68.6339 8.2846 194 0.0601 -0.098 0.029 691.006 125.2132 11.1899 195 0.0656 -0.1133 0.036 928.7828 192.1773 13.8628 196 0.0724 -0.1383 0.0439 1389.8897 284.3091 16.8615 197 0.0794 -0.1403 0.0508 1439.2881 366.8076 19.1522 198 0.0868 -0.1682 0.0586 2085.2094 481.3677 21.9401 199 0.0947 -0.187 0.0666 2598.9581 613.7171 24.7733 200 0.1031 -0.199 0.0744 2978.805 752.8399 27.4379 201 0.1117 -0.2155 0.0823 3518.8925 906.5095 30.1083 202 0.1206 -0.233 0.0902 4140.4643 1076.7177 32.8134 203 0.1295 -0.2566 0.0985 5032.5593 1274.5097 35.7003 204 0.1393 -0.2748 0.1069 5815.129 1490.7297 38.61 205 0.1487 -0.2979 0.1156 6821.8412 1733.0529 41.63 206 0.1591 -0.3237 0.1246 8088.9865 2009.3979 44.8263 207 0.17 -0.3335 0.1333 8623.5299 2284.9867 47.8015 208 0.1824 -0.3502 0.142 9533.5849 2574.9306 50.7438 209 0.1955 -0.3325 0.1493 8632.8714 2807.9284 52.9899 210 0.2095 -0.3461 0.1566 9404.4057 3052.2423 55.2471 211 0.2246 -0.3502 0.1635 9688.6688 3289.2576 57.352 212 0.241 -0.3372 0.1695 9056.8161 3488.1389 59.0605 213 0.2579 -0.3319 0.1749 8822.061 3665.9363 60.547 214 0.2757 -0.3258 0.1798 8538.2494 3823.1077 61.8313 215 0.2938 -0.328 0.1844 8664.4902 3974.4009 63.0428 216 0.314 -0.3295 0.1888 8787.3029 4120.2464 64.1891

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/30/t1262181997a6uwhj4kqm04x0i/1bwiy1262181865.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/30/t1262181997a6uwhj4kqm04x0i/1bwiy1262181865.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/30/t1262181997a6uwhj4kqm04x0i/21ep91262181865.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/30/t1262181997a6uwhj4kqm04x0i/21ep91262181865.ps (open in new window)

Parameters (Session):

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

Parameters (R input):

par1 = 0 ; 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 <- 33 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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