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(921365,987921,1132614,1332224,1418133,1411549,1695920,1636173,1539653,1395314,1127575,1036076,989236,1008380,1207763,1368839,1469798,1498721,1761769,1653214,1599104,1421179,1163995,1037735,1015407,1039210,1258049,1469445,1552346,1549144,1785895,1662335,1629440,1467430,1202209,1076982,1039367,1063449,1335135,1491602,1591972,1641248,1898849,1798580,1762444,1622044,1368955,1262973,1195650,1269530,1479279,1607819,1712466,1721766,1949843,1821326,1757802,1590367,1260647,1149235)
> par10 = 'FALSE'
> par9 = '1'
> par8 = '0'
> par7 = '0'
> par6 = '0'
> par5 = '12'
> par4 = '1'
> par3 = '1'
> 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")

Coefficients:
         sma1
      -0.0905
s.e.   0.3015

sigma^2 estimated as 9.43e+08:  log likelihood = -270.33,  aic = 544.66
> (forecast <- predict(arima.out,par1))
$pred
Time Series:
Start = 37 
End = 60 
Frequency = 1 
 [1] 1052453 1076220 1292854 1500008 1584421 1583839 1823143 1701338 1666178
[10] 1503000 1238421 1113383 1088854 1112621 1329255 1536409 1620822 1620240
[19] 1859544 1737739 1702579 1539401 1274822 1149784

$se
Time Series:
Start = 37 
End = 60 
Frequency = 1 
 [1]  30710.55  43430.56  53191.06  61419.58  68669.06  75223.10  81250.17
 [8]  86860.04  92128.94  97112.40 101852.31 106381.24 121473.16 134886.26
[15] 147081.19 158339.67 168849.12 178741.71 188114.79 197042.50 205582.88
[22] 213782.36 221678.76 229303.39

> (lb <- forecast$pred - 1.96 * forecast$se)
Time Series:
Start = 37 
End = 60 
Frequency = 1 
 [1]  992260.8  991096.1 1188599.2 1379625.9 1449830.1 1436401.3 1663892.6
 [8] 1531092.0 1485605.2 1312660.1 1038790.5  904875.8  850767.1  848244.0
[15] 1040975.5 1226063.6 1289878.2 1269905.8 1490839.0 1351535.4 1299636.5
[22] 1120388.0  840331.7  700349.4
> (ub <- forecast$pred + 1.96 * forecast$se)
Time Series:
Start = 37 
End = 60 
Frequency = 1 
 [1] 1112646 1161344 1397108 1620391 1719013 1731276 1982393 1871583 1846751
[10] 1693341 1438052 1321890 1326942 1376998 1617534 1846755 1951767 1970573
[19] 2228249 2123942 2105521 1958415 1709312 1599219
> 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$pred))
 [1]  921365  987921 1132614 1332224 1418133 1411549 1695920 1636173 1539653
[10] 1395314 1127575 1036076  989236 1008380 1207763 1368839 1469798 1498721
[19] 1761769 1653214 1599104 1421179 1163995 1037735 1015407 1039210 1258049
[28] 1469445 1552346 1549144 1785895 1662335 1629440 1467430 1202209 1076982
[37] 1052453 1076220 1292854 1500008 1584421 1583839 1823143 1701338 1666178
[46] 1503000 1238421 1113383 1088854 1112621 1329255 1536409 1620822 1620240
[55] 1859544 1737739 1702579 1539401 1274822 1149784
> (perc.se <- (ub-forecast$pred)/1.96/forecast$pred)
Time Series:
Start = 37 
End = 60 
Frequency = 1 
 [1] 0.02917996 0.04035472 0.04114237 0.04094616 0.04334015 0.04749417
 [7] 0.04456599 0.05105397 0.05529358 0.06461236 0.08224368 0.09554775
[13] 0.11156051 0.12123289 0.11064937 0.10305826 0.10417496 0.11031808
[19] 0.10116178 0.11339018 0.12074793 0.13887369 0.17388996 0.19943170
> postscript(file="/var/www/html/rcomp/tmp/1ftkf1261313261.ps",horizontal=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> opar <- par(mar=c(4,4,2,2),las=1)
> ylim <- c( min(x[first:nx],lb), max(x[first:nx],ub))
> plot(x,ylim=ylim,type='n',xlim=c(first,lx))
> usr <- par('usr')
> rect(usr[1],usr[3],nx+1,usr[4],border=NA,col='lemonchiffon')
> rect(nx1,usr[3],usr[2],usr[4],border=NA,col='lavender')
> abline(h= (-3:3)*2 , col ='gray', lty =3)
> polygon( c(nx1:lx,lx:nx1), c(lb,rev(ub)), col = 'orange', lty=2,border=NA)
> lines(nx1:lx, lb , lty=2)
> lines(nx1:lx, ub , lty=2)
> lines(x, lwd=2)
> lines(nx1:lx, forecast$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/2acl61261313261.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/308b91261313262.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/4m1m81261313262.tab") 
> 
> try(system("convert tmp/1ftkf1261313261.ps tmp/1ftkf1261313261.png",intern=TRUE))
character(0)
> try(system("convert tmp/2acl61261313261.ps tmp/2acl61261313261.png",intern=TRUE))
character(0)
> 
> 
> proc.time()
   user  system elapsed 
  0.654   0.312   1.128