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(21790,13253,37702,30364,32609,30212,29965,28352,25814,22414,20506,28806,22228,13971,36845,35338,35022,34777,26887,23970,22780,17351,21382,24561,17409,11514,31514,27071,29462,26105,22397,23843,21705,18089,20764,25316,17704,15548,28029,29383,36438,32034,22679,24319,18004,17537,20366,22782,19169,13807,29743,25591,29096,26482,22405,27044,17970,18730,19684,19785,18479,10698,31956,29506,34506,27165,26736,23691,18157,17328,18205,20995,17382,9367,31124,26551,30651,25859,25100,25778,20418,18688,20424,24776,19814,12738,31566,30111,30019,31934,25826,26835,20205,17789,20520,22518,15572,11509,25447,24090,27786,26195,20516,22759,19028,16971,20036,22485,18730)
> par10 = 'FALSE'
> par9 = '1'
> par8 = '0'
> par7 = '1'
> par6 = '3'
> par5 = '12'
> par4 = '1'
> par3 = '1'
> par2 = '-0.5'
> par1 = '12'
> #'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")

Coefficients:
         ar1     ar2     ar3      ma1     sma1
      0.0818  0.2949  0.2103  -0.9620  -0.9999
s.e.  0.1292  0.1262  0.1194   0.1001   0.5382

sigma^2 estimated as 9.843e-08:  log likelihood = 544.57,  aic = -1077.14
> (forecast <- predict(arima.out,par1))
$pred
Time Series:
Start = 98 
End = 109 
Frequency = 1 
 [1] 0.009221732 0.005991522 0.006251908 0.005901877 0.006204901 0.006632362
 [7] 0.006580747 0.007304062 0.007663750 0.007324210 0.006813933 0.007649058

$se
Time Series:
Start = 98 
End = 109 
Frequency = 1 
 [1] 0.0003348739 0.0003369950 0.0003565264 0.0003723374 0.0003780915
 [6] 0.0003856731 0.0003907859 0.0003947120 0.0003984879 0.0004015648
[11] 0.0004039485 0.0004066558

> (lb <- forecast$pred - 1.96 * forecast$se)
Time Series:
Start = 98 
End = 109 
Frequency = 1 
 [1] 0.008565379 0.005331012 0.005553116 0.005172096 0.005463842 0.005876442
 [7] 0.005814807 0.006530426 0.006882714 0.006537143 0.006022194 0.006852013
> (ub <- forecast$pred + 1.96 * forecast$se)
Time Series:
Start = 98 
End = 109 
Frequency = 1 
 [1] 0.009878085 0.006652032 0.006950699 0.006631659 0.006945961 0.007388281
 [7] 0.007346688 0.008077697 0.008444787 0.008111277 0.007605672 0.008446104
> 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$pred))
  [1] 21790.00 13253.00 37702.00 30364.00 32609.00 30212.00 29965.00 28352.00
  [9] 25814.00 22414.00 20506.00 28806.00 22228.00 13971.00 36845.00 35338.00
 [17] 35022.00 34777.00 26887.00 23970.00 22780.00 17351.00 21382.00 24561.00
 [25] 17409.00 11514.00 31514.00 27071.00 29462.00 26105.00 22397.00 23843.00
 [33] 21705.00 18089.00 20764.00 25316.00 17704.00 15548.00 28029.00 29383.00
 [41] 36438.00 32034.00 22679.00 24319.00 18004.00 17537.00 20366.00 22782.00
 [49] 19169.00 13807.00 29743.00 25591.00 29096.00 26482.00 22405.00 27044.00
 [57] 17970.00 18730.00 19684.00 19785.00 18479.00 10698.00 31956.00 29506.00
 [65] 34506.00 27165.00 26736.00 23691.00 18157.00 17328.00 18205.00 20995.00
 [73] 17382.00  9367.00 31124.00 26551.00 30651.00 25859.00 25100.00 25778.00
 [81] 20418.00 18688.00 20424.00 24776.00 19814.00 12738.00 31566.00 30111.00
 [89] 30019.00 31934.00 25826.00 26835.00 20205.00 17789.00 20520.00 22518.00
 [97] 15572.00 11759.12 27856.45 25584.38 28709.10 25973.49 22733.36 23091.36
[105] 18744.38 17026.18 18641.39 21537.94 17091.65
> (perc.se <- (ub-forecast$pred)/1.96/forecast$pred)
Time Series:
Start = 98 
End = 109 
Frequency = 1 
 [1] 0.08118833 0.13426035 0.13648505 0.15413701 0.14778308 0.13970316
 [7] 0.14326308 0.12804433 0.12236386 0.13025226 0.14297184 0.12560032
> postscript(file="/var/www/html/rcomp/tmp/1hg3f1260467605.ps",horizontal=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> opar <- par(mar=c(4,4,2,2),las=1)
> ylim <- c( min(x[first:nx],lb), max(x[first:nx],ub))
> plot(x,ylim=ylim,type='n',xlim=c(first,lx))
> usr <- par('usr')
> rect(usr[1],usr[3],nx+1,usr[4],border=NA,col='lemonchiffon')
> rect(nx1,usr[3],usr[2],usr[4],border=NA,col='lavender')
> abline(h= (-3:3)*2 , col ='gray', lty =3)
> polygon( c(nx1:lx,lx:nx1), c(lb,rev(ub)), col = 'orange', lty=2,border=NA)
> lines(nx1:lx, lb , lty=2)
> lines(nx1:lx, ub , lty=2)
> lines(x, lwd=2)
> lines(nx1:lx, forecast$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/2kxiq1260467605.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/3hg081260467605.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/449wh1260467605.tab") 
> 
> system("convert tmp/1hg3f1260467605.ps tmp/1hg3f1260467605.png")
> system("convert tmp/2kxiq1260467605.ps tmp/2kxiq1260467605.png")
> 
> 
> proc.time()
   user  system elapsed 
  1.260   0.351   1.502