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(13.193,15.234,14.718,16.961,13.945,15.876,16.226,18.316,16.748,17.904,17.209,18.950,17.225,18.710,17.236,18.687,17.580,19.568,17.381,19.580,17.260,18.661,15.658,18.674,15.908,17.475,17.725,19.562,16.368,19.555,17.743,19.867,15.703,19.324,18.162,19.074,15.323,19.704,18.375,18.352,13.927,17.795,16.761,18.902,16.239,19.158,18.279,15.698,16.239,18.431,18.414,19.801,14.995,18.706,18.232,19.409,16.263,19.017,20.298,19.891,15.203,17.845,17.502,18.532,15.737,17.770,17.224,17.601,14.940,18.507,17.635,19.392,15.699,17.661,18.243,19.643,15.770,17.344,17.229,17.322,16.152,17.919,16.918,18.114,16.308,17.759,16.021,17.952,15.954,17.762,16.610,17.751,15.458,18.106,15.990,15.349,13.185,15.409,16.007,16.633,14.800,15.974,15.693)
> par10 = 'FALSE'
> par9 = '1'
> par8 = '1'
> par7 = '1'
> par6 = '2'
> par5 = '4'
> par4 = '1'
> par3 = '0'
> par2 = '1'
> 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      ma1    sar1     sma1
      1.1917  -0.2418  -0.7811  0.2711  -1.0000
s.e.  0.3423   0.2100   0.2932  0.1186   0.1477

sigma^2 estimated as 0.8625:  log likelihood = -121.76,  aic = 255.51
> (forecast <- predict(arima.out,par1))
$pred
Time Series:
Start = 92 
End = 103 
Frequency = 1 
 [1] 18.10769 15.54409 17.76636 16.87974 18.19334 15.45809 17.78700 16.96999
 [9] 18.23224 15.44933 17.80614 17.00708

$se
Time Series:
Start = 92 
End = 103 
Frequency = 1 
 [1] 0.9479443 1.0233257 1.0492687 1.0649854 1.1590867 1.1885809 1.2067494
 [8] 1.2205463 1.2538243 1.2688178 1.2804923 1.2898772

> (lb <- forecast$pred - 1.96 * forecast$se)
Time Series:
Start = 92 
End = 103 
Frequency = 1 
 [1] 16.24972 13.53838 15.70979 14.79237 15.92153 13.12848 15.42177 14.57772
 [9] 15.77474 12.96244 15.29637 14.47892
> (ub <- forecast$pred + 1.96 * forecast$se)
Time Series:
Start = 92 
End = 103 
Frequency = 1 
 [1] 19.96566 17.54981 19.82292 18.96711 20.46515 17.78771 20.15223 19.36226
 [9] 20.68973 17.93621 20.31590 19.53524
> 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$pred))
  [1] 13.19300 15.23400 14.71800 16.96100 13.94500 15.87600 16.22600 18.31600
  [9] 16.74800 17.90400 17.20900 18.95000 17.22500 18.71000 17.23600 18.68700
 [17] 17.58000 19.56800 17.38100 19.58000 17.26000 18.66100 15.65800 18.67400
 [25] 15.90800 17.47500 17.72500 19.56200 16.36800 19.55500 17.74300 19.86700
 [33] 15.70300 19.32400 18.16200 19.07400 15.32300 19.70400 18.37500 18.35200
 [41] 13.92700 17.79500 16.76100 18.90200 16.23900 19.15800 18.27900 15.69800
 [49] 16.23900 18.43100 18.41400 19.80100 14.99500 18.70600 18.23200 19.40900
 [57] 16.26300 19.01700 20.29800 19.89100 15.20300 17.84500 17.50200 18.53200
 [65] 15.73700 17.77000 17.22400 17.60100 14.94000 18.50700 17.63500 19.39200
 [73] 15.69900 17.66100 18.24300 19.64300 15.77000 17.34400 17.22900 17.32200
 [81] 16.15200 17.91900 16.91800 18.11400 16.30800 17.75900 16.02100 17.95200
 [89] 15.95400 17.76200 16.61000 18.10769 15.54409 17.76636 16.87974 18.19334
 [97] 15.45809 17.78700 16.96999 18.23224 15.44933 17.80614 17.00708
> (perc.se <- (ub-forecast$pred)/1.96/forecast$pred)
Time Series:
Start = 92 
End = 103 
Frequency = 1 
 [1] 0.05235036 0.06583373 0.05905930 0.06309251 0.06370941 0.07689052
 [7] 0.06784447 0.07192383 0.06876963 0.08212771 0.07191297 0.07584355
> postscript(file="/var/www/html/rcomp/tmp/15q1d1293197997.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> opar <- par(mar=c(4,4,2,2),las=1)
> ylim <- c( min(x[first:nx],lb), max(x[first:nx],ub))
> plot(x,ylim=ylim,type='n',xlim=c(first,lx))
> usr <- par('usr')
> rect(usr[1],usr[3],nx+1,usr[4],border=NA,col='lemonchiffon')
> rect(nx1,usr[3],usr[2],usr[4],border=NA,col='lavender')
> abline(h= (-3:3)*2 , col ='gray', lty =3)
> polygon( c(nx1:lx,lx:nx1), c(lb,rev(ub)), col = 'orange', lty=2,border=NA)
> lines(nx1:lx, lb , lty=2)
> lines(nx1:lx, ub , lty=2)
> lines(x, lwd=2)
> lines(nx1:lx, forecast$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/2j0z41293197997.ps",horizontal=F,onefile=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/381ey1293197997.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/4tjv41293197997.tab") 
> 
> try(system("convert tmp/15q1d1293197997.ps tmp/15q1d1293197997.png",intern=TRUE))
character(0)
> try(system("convert tmp/2j0z41293197997.ps tmp/2j0z41293197997.png",intern=TRUE))
character(0)
> 
> 
> proc.time()
   user  system elapsed 
  0.825   0.354   1.920