R version 2.12.1 (2010-12-16)
Copyright (C) 2010 The R Foundation for Statistical Computing
ISBN 3-900051-07-0
Platform: x86_64-redhat-linux-gnu (64-bit)

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> x <- c(132,131.4,132.7,130.9,126,109.7,68.3,70.6,75.3,74.1,74.9,74,74.2,76,76.2,74.9,74.1,76.5,57.8,59.2,57.3,57.5,60.4,59.9,59.9,60,60.2,65.4,62.4,78.8,65.6,64.4,67.4,65.3,66.7,66.8,69.4,71.7,77.1,81.1,82.1,92.1,77.1,78.2,77.7,77.3,78.5,78.8,78.7,79.8,82.2,84,81.7,77.6,64.3,72.6,73.8,73.8,70.1,70,72.3,72.1,73.3,79.1,77,76.1,66.4,72.7,73.2,70.7,73.6,74.2,72.6,73.6,79.1,79.6,78,85.4,82,91.9,89.4,92.1,93.8,93.6,95.6,99.9,103.7,99.2,93.7,93.5,80.7,91.8,105.8,111.3,110.3,109.4,111.4,111.6,111.8,106.6,104.3,105.5,98.5,108.5,106,101.8,101.3,92.4,88.9,84.9,86.4,90.7,86.8,90.6,88.3,95.4,93.6,91.3,91.3,89.3)
> par3 = 'multiplicative'
> par2 = 'Triple'
> par1 = '12'
> #'GNU S' R Code compiled by R2WASP v. 1.0.44 ()
> #Author: Prof. Dr. P. Wessa
> #To cite this work: Wessa P., (2010), Exponential Smoothing (v1.0.4) in Free Statistics Software (v$_version), Office for Research Development and Education, URL http://www.wessa.net/rwasp_exponentialsmoothing.wasp/
> #Source of accompanying publication: 
> #Technical description: 
> par1 <- as.numeric(par1)
> if (par2 == 'Single') K <- 1
> if (par2 == 'Double') K <- 2
> if (par2 == 'Triple') K <- par1
> nx <- length(x)
> nxmK <- nx - K
> x <- ts(x, frequency = par1)
> if (par2 == 'Single') fit <- HoltWinters(x, gamma=F, beta=F)
> if (par2 == 'Double') fit <- HoltWinters(x, gamma=F)
> if (par2 == 'Triple') fit <- HoltWinters(x, seasonal=par3)
> fit
Holt-Winters exponential smoothing with trend and multiplicative seasonal component.

Call:
 HoltWinters(x = x, seasonal = par3) 

Smoothing parameters:
 alpha:  0.4227806 
 beta :  0.05094075 
 gamma:  1 

Coefficients:
          [,1]
a   82.4979048
b   -0.2261623
s1   1.0459649
s2   1.0269828
s3   1.0568967
s4   1.0773002
s5   1.0263078
s6   1.0632580
s7   1.0140297
s8   1.1002694
s9   1.1033810
s10  1.1021890
s11  1.1187136
s12  1.0824517
> myresid <- x - fit$fitted[,'xhat']
> postscript(file="/var/www/wessaorg/rcomp/tmp/1hbda1295191756.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> op <- par(mfrow=c(2,1))
> plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
> plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
> par(op)
> dev.off()
null device 
          1 
> postscript(file="/var/www/wessaorg/rcomp/tmp/29bvw1295191756.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> p <- predict(fit, par1, prediction.interval=TRUE)
> np <- length(p[,1])
> plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
> dev.off()
null device 
          1 
> postscript(file="/var/www/wessaorg/rcomp/tmp/3e5u31295191756.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> op <- par(mfrow = c(2,2))
> acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
> spectrum(myresid,main='Residals Periodogram')
> cpgram(myresid,main='Residal Cumulative Periodogram')
> qqnorm(myresid,main='Residual Normal QQ Plot')
> qqline(myresid)
> par(op)
> dev.off()
null device 
          1 
> 
> #Note: the /var/www/wessaorg/rcomp/createtable file can be downloaded at http://www.wessa.net/cretab
> load(file="/var/www/wessaorg/rcomp/createtable")
> 
> a<-table.start()
> a<-table.row.start(a)
> a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'Parameter',header=TRUE)
> a<-table.element(a,'Value',header=TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'alpha',header=TRUE)
> a<-table.element(a,fit$alpha)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'beta',header=TRUE)
> a<-table.element(a,fit$beta)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'gamma',header=TRUE)
> a<-table.element(a,fit$gamma)
> a<-table.row.end(a)
> a<-table.end(a)
> table.save(a,file="/var/www/wessaorg/rcomp/tmp/43n671295191756.tab") 
> a<-table.start()
> a<-table.row.start(a)
> a<-table.element(a,'Interpolation Forecasts of Exponential Smoothing',4,TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'t',header=TRUE)
> a<-table.element(a,'Observed',header=TRUE)
> a<-table.element(a,'Fitted',header=TRUE)
> a<-table.element(a,'Residuals',header=TRUE)
> a<-table.row.end(a)
> for (i in 1:nxmK) {
+ a<-table.row.start(a)
+ a<-table.element(a,i+K,header=TRUE)
+ a<-table.element(a,x[i+K])
+ a<-table.element(a,fit$fitted[i,'xhat'])
+ a<-table.element(a,myresid[i])
+ a<-table.row.end(a)
+ }
> a<-table.end(a)
> table.save(a,file="/var/www/wessaorg/rcomp/tmp/5eurq1295191756.tab") 
> a<-table.start()
> a<-table.row.start(a)
> a<-table.element(a,'Extrapolation Forecasts of Exponential Smoothing',4,TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'t',header=TRUE)
> a<-table.element(a,'Forecast',header=TRUE)
> a<-table.element(a,'95% Lower Bound',header=TRUE)
> a<-table.element(a,'95% Upper Bound',header=TRUE)
> a<-table.row.end(a)
> for (i in 1:np) {
+ a<-table.row.start(a)
+ a<-table.element(a,nx+i,header=TRUE)
+ a<-table.element(a,p[i,'fit'])
+ a<-table.element(a,p[i,'lwr'])
+ a<-table.element(a,p[i,'upr'])
+ a<-table.row.end(a)
+ }
> a<-table.end(a)
> table.save(a,file="/var/www/wessaorg/rcomp/tmp/6xd7g1295191756.tab") 
> 
> try(system("convert tmp/1hbda1295191756.ps tmp/1hbda1295191756.png",intern=TRUE))
character(0)
> try(system("convert tmp/29bvw1295191756.ps tmp/29bvw1295191756.png",intern=TRUE))
character(0)
> try(system("convert tmp/3e5u31295191756.ps tmp/3e5u31295191756.png",intern=TRUE))
character(0)
> 
> 
> proc.time()
   user  system elapsed 
  1.230   0.150   1.452