R version 3.0.1 (2013-05-16) -- "Good Sport"
Copyright (C) 2013 The R Foundation for Statistical Computing
Platform: i686-pc-linux-gnu (32-bit)

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> x <- c(865911,858030,850038,833499,997113,988455,865911,784437,792318,792318,801087,816849,865911,850038,874569,914862,1144188,1144188,1095237,1046175,1086468,1135530,1144188,1168719,1242312,1193250,1193250,1266843,1470861,1487400,1446330,1348206,1421799,1421799,1429680,1470861,1503273,1519812,1519812,1568874,1757130,1806081,1813962,1691418,1757130,1732599,1683537,1789542,1813962,1772892,1781550,1838493,2051280,2157174,2157174,2108223,2181705,2108223,2067042,2222886,2247306,2189586,2336661,2394381,2565987,2679873,2664111,2655342,2721054,2713062,2615049,2762124,2811186,2762124,2966142,3064266,3292704,3382836,3358416,3309354,3350424,3399486,3235761,3366297,3448548,3415359,3628035,3701517,4012317,4069260,3995778,4036848,4061379,4085910,3930066,4077141,4158615,4077141,4314459,4387941,4706622,4755684,4771446,4853697,4853697,4886109,4739034,4812627,4861578,4771446,5033184,5082246,5408808,5466528,5548002,5621595,5629476,5638134,5491059,5638134)
> par3 = 'multiplicative'
> par2 = 'Triple'
> par1 = '12'
> par3 <- 'multiplicative'
> par2 <- 'Triple'
> par1 <- '12'
> #'GNU S' R Code compiled by R2WASP v. 1.2.327 ()
> #Author: root
> #To cite this work: Wessa P., (2013), Exponential Smoothing (v1.0.5) in Free Statistics Software (v$_version), Office for Research Development and Education, URL http://www.wessa.net/rwasp_exponentialsmoothing.wasp/
> #Source of accompanying publication: 
> #
> 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: 0.2142297
 beta : 0.1078698
 gamma: 1

Coefficients:
            [,1]
a   5.386044e+06
b   6.209673e+04
s1  1.023105e+00
s2  9.929738e-01
s3  1.036649e+00
s4  1.037611e+00
s5  1.096477e+00
s6  1.100006e+00
s7  1.103986e+00
s8  1.105610e+00
s9  1.092857e+00
s10 1.080747e+00
s11 1.037061e+00
s12 1.046804e+00
> myresid <- x - fit$fitted[,'xhat']
> postscript(file="/var/fisher/rcomp/tmp/1p26p1377072796.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> op <- par(mfrow=c(2,1))
> plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
> plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
> par(op)
> dev.off()
null device 
          1 
> postscript(file="/var/fisher/rcomp/tmp/21bgq1377072796.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> p <- predict(fit, par1, prediction.interval=TRUE)
> np <- length(p[,1])
> plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
> dev.off()
null device 
          1 
> postscript(file="/var/fisher/rcomp/tmp/3ay1m1377072796.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> op <- par(mfrow = c(2,2))
> acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
> spectrum(myresid,main='Residals Periodogram')
> cpgram(myresid,main='Residal Cumulative Periodogram')
> qqnorm(myresid,main='Residual Normal QQ Plot')
> qqline(myresid)
> par(op)
> dev.off()
null device 
          1 
> 
> #Note: the /var/fisher/rcomp/createtable file can be downloaded at http://www.wessa.net/cretab
> load(file="/var/fisher/rcomp/createtable")
> 
> a<-table.start()
> a<-table.row.start(a)
> a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'Parameter',header=TRUE)
> a<-table.element(a,'Value',header=TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'alpha',header=TRUE)
> a<-table.element(a,fit$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/fisher/rcomp/tmp/4cyiq1377072796.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/fisher/rcomp/tmp/5a4et1377072796.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/fisher/rcomp/tmp/6aam21377072796.tab") 
> 
> try(system("convert tmp/1p26p1377072796.ps tmp/1p26p1377072796.png",intern=TRUE))
character(0)
> try(system("convert tmp/21bgq1377072796.ps tmp/21bgq1377072796.png",intern=TRUE))
character(0)
> try(system("convert tmp/3ay1m1377072796.ps tmp/3ay1m1377072796.png",intern=TRUE))
character(0)
> 
> 
> proc.time()
   user  system elapsed 
  1.970   0.366   2.319