R version 2.9.0 (2009-04-17)
Copyright (C) 2009 The R Foundation for Statistical Computing
ISBN 3-900051-07-0

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> x <- c(100.01,103.84,104.48,95.43,104.80,108.64,105.65,108.42,115.35,113.64,115.24,100.33,101.29,104.48,99.26,100.11,103.52,101.18,96.39,97.56,96.39,85.10,79.77,79.13,80.84,82.75,92.55,96.60,96.92,95.32,98.52,100.22,104.91,103.10,97.13,103.42,111.72,118.11,111.62,100.22,102.03,105.76,107.68,110.77,105.44,112.26,114.07,117.90,124.72,126.42,134.73,135.79,143.36,140.37,144.74,151.98,150.92,163.38,154.43,146.66,157.95,162.10,180.42,179.57,171.58,185.43,190.64,203.00,202.36,193.41,186.17,192.24,209.60,206.41,209.82,230.37,235.80,232.07,244.64,242.19,217.48,209.39,211.73,221.00,203.11,214.71,224.19,238.04,238.36,246.24,259.87,249.97,266.48,282.98,306.31,301.73,314.62,332.62,355.51,370.32,408.13,433.58,440.51,386.29,342.84,254.97,203.42,170.09,174.03,167.85,177.01,188.19,211.20,240.91,230.26,251.25,241.66)
> par10 = 'FALSE'
> par9 = '0'
> par8 = '0'
> par7 = '0'
> par6 = '0'
> par5 = '12'
> par4 = '0'
> par3 = '2'
> par2 = '1'
> par1 = '24'
> #'GNU S' R Code compiled by R2WASP v. 1.0.44 ()
> #Author: Prof. Dr. P. Wessa
> #To cite this work: Wessa P., (2009), ARIMA Forecasting (v1.0.5) in Free Statistics Software (v$_version), Office for Research Development and Education, URL http://www.wessa.net/rwasp_arimaforecasting.wasp/
> #Source of accompanying publication: 
> #Technical description: 
> 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'))

Call:
arima(x = x[1:nx], order = c(par6, par3, par7), seasonal = list(order = c(par8, 
    par4, par9), period = par5), include.mean = par10, method = "ML")


sigma^2 estimated as 125.5:  log likelihood = -348.99,  aic = 699.98
> (forecast <- predict(arima.out,par1))
$pred
Time Series:
Start = 94 
End = 117 
Frequency = 1 
 [1] 282.99 299.50 316.01 332.52 349.03 365.54 382.05 398.56 415.07 431.58
[11] 448.09 464.60 481.11 497.62 514.13 530.64 547.15 563.66 580.17 596.68
[21] 613.19 629.70 646.21 662.72

$se
Time Series:
Start = 94 
End = 117 
Frequency = 1 
 [1]  11.20214  25.04874  41.91456  61.35664  83.07728 106.86159 132.54549
 [8] 159.99854 189.11386 219.80183 251.98588 285.59961 320.58477 356.88966
[15] 394.46800 433.27810 473.28207 514.44533 556.73614 600.12523 644.58550
[22] 690.09174 736.62047 784.14970

> (lb <- forecast$pred - 1.96 * forecast$se)
Time Series:
Start = 94 
End = 117 
Frequency = 1 
 [1]  261.0338085  250.4044632  233.8574538  212.2609863  186.1985256
 [6]  156.0912819  122.2608384   84.9628594   44.4068259    0.7684132
[11]  -45.8023154  -95.1752451 -147.2361542 -201.8837251 -259.0272862
[16] -318.5850714 -380.4828519 -444.6528464 -511.0328387 -579.5654563
[21] -650.1975732 -722.8798105 -797.5661159 -874.2134066
> (ub <- forecast$pred + 1.96 * forecast$se)
Time Series:
Start = 94 
End = 117 
Frequency = 1 
 [1]  304.9462  348.5955  398.1625  452.7790  511.8615  574.9887  641.8392
 [8]  712.1571  785.7332  862.3916  941.9823 1024.3752 1109.4562 1197.1237
[15] 1287.2873 1379.8651 1474.7829 1571.9728 1671.3728 1772.9255 1876.5776
[22] 1982.2798 2089.9861 2199.6534
> 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$pred))
  [1] 100.01 103.84 104.48  95.43 104.80 108.64 105.65 108.42 115.35 113.64
 [11] 115.24 100.33 101.29 104.48  99.26 100.11 103.52 101.18  96.39  97.56
 [21]  96.39  85.10  79.77  79.13  80.84  82.75  92.55  96.60  96.92  95.32
 [31]  98.52 100.22 104.91 103.10  97.13 103.42 111.72 118.11 111.62 100.22
 [41] 102.03 105.76 107.68 110.77 105.44 112.26 114.07 117.90 124.72 126.42
 [51] 134.73 135.79 143.36 140.37 144.74 151.98 150.92 163.38 154.43 146.66
 [61] 157.95 162.10 180.42 179.57 171.58 185.43 190.64 203.00 202.36 193.41
 [71] 186.17 192.24 209.60 206.41 209.82 230.37 235.80 232.07 244.64 242.19
 [81] 217.48 209.39 211.73 221.00 203.11 214.71 224.19 238.04 238.36 246.24
 [91] 259.87 249.97 266.48 282.99 299.50 316.01 332.52 349.03 365.54 382.05
[101] 398.56 415.07 431.58 448.09 464.60 481.11 497.62 514.13 530.64 547.15
[111] 563.66 580.17 596.68 613.19 629.70 646.21 662.72
> (perc.se <- (ub-forecast$pred)/1.96/forecast$pred)
Time Series:
Start = 94 
End = 117 
Frequency = 1 
 [1] 0.03958493 0.08363520 0.13263683 0.18452015 0.23802333 0.29233898
 [7] 0.34693231 0.40144154 0.45561921 0.50929568 0.56235550 0.61472151
[13] 0.66634402 0.71719315 0.76725342 0.81651986 0.86499510 0.91268731
[19] 0.95960864 1.00577400 1.05120027 1.09590557 1.13990880 1.18322926
> postscript(file="/var/www/html/rcomp/tmp/1o8wm1260610787.ps",horizontal=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> opar <- par(mar=c(4,4,2,2),las=1)
> ylim <- c( min(x[first:nx],lb), max(x[first:nx],ub))
> plot(x,ylim=ylim,type='n',xlim=c(first,lx))
> usr <- par('usr')
> rect(usr[1],usr[3],nx+1,usr[4],border=NA,col='lemonchiffon')
> rect(nx1,usr[3],usr[2],usr[4],border=NA,col='lavender')
> abline(h= (-3:3)*2 , col ='gray', lty =3)
> polygon( c(nx1:lx,lx:nx1), c(lb,rev(ub)), col = 'orange', lty=2,border=NA)
> lines(nx1:lx, lb , lty=2)
> lines(nx1:lx, ub , lty=2)
> lines(x, lwd=2)
> lines(nx1:lx, forecast$pred , lwd=2 , col ='white')
> box()
> par(opar)
> dev.off()
null device 
          1 
> 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)
> postscript(file="/var/www/html/rcomp/tmp/2y1yj1260610787.ps",horizontal=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> 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()
null device 
          1 
> 
> #Note: the /var/www/html/rcomp/createtable file can be downloaded at http://www.wessa.net/cretab
> load(file="/var/www/html/rcomp/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="/var/www/html/rcomp/tmp/36ep41260610787.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="/var/www/html/rcomp/tmp/4hb6q1260610787.tab") 
> 
> system("convert tmp/1o8wm1260610787.ps tmp/1o8wm1260610787.png")
> system("convert tmp/2y1yj1260610787.ps tmp/2y1yj1260610787.png")
> 
> 
> proc.time()
   user  system elapsed 
  0.669   0.335   1.152