R version 3.3.0 (2016-05-03) -- "Supposedly Educational"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> x <- c(2341
+ ,2115
+ ,2402
+ ,2180
+ ,2453
+ ,2507
+ ,2679
+ ,2622
+ ,2618
+ ,2648
+ ,2523
+ ,2473
+ ,2513
+ ,2466
+ ,2544
+ ,2537
+ ,2564
+ ,2582
+ ,2716
+ ,2904
+ ,2851
+ ,2932
+ ,2772
+ ,2811
+ ,2935
+ ,2783
+ ,3003
+ ,2995
+ ,3127
+ ,2985
+ ,3287
+ ,3236
+ ,3252
+ ,3228
+ ,2856
+ ,3176
+ ,3362
+ ,3036
+ ,3330
+ ,3251
+ ,3318
+ ,3238
+ ,3597
+ ,3708
+ ,3902
+ ,3745
+ ,3426
+ ,3526
+ ,3483
+ ,3458
+ ,3824
+ ,3696
+ ,3518
+ ,3814
+ ,3996
+ ,4136
+ ,4037
+ ,3915
+ ,3760
+ ,3955
+ ,4160
+ ,4115
+ ,4202
+ ,4018
+ ,4233
+ ,4029
+ ,4401
+ ,4645
+ ,4491
+ ,4379
+ ,4394
+ ,4472
+ ,4614
+ ,4160
+ ,4328
+ ,4202
+ ,4635
+ ,4542
+ ,4920
+ ,4774
+ ,4698
+ ,4916
+ ,4703
+ ,4616
+ ,4873
+ ,4375
+ ,4801
+ ,4427
+ ,4684
+ ,4648
+ ,5225
+ ,5174
+ ,5181
+ ,5266
+ ,4839
+ ,5032
+ ,5221
+ ,4658
+ ,5014
+ ,4980
+ ,4952
+ ,4946
+ ,5365
+ ,5456
+ ,5397
+ ,5436
+ ,4995
+ ,5019
+ ,5249
+ ,4799
+ ,5137
+ ,4979
+ ,4951
+ ,5265
+ ,5612
+ ,5572
+ ,5403
+ ,5373
+ ,5252
+ ,5437
+ ,5296
+ ,5011
+ ,5294
+ ,5335
+ ,5398
+ ,5396
+ ,5724
+ ,5898
+ ,5718
+ ,5625
+ ,5380
+ ,5488
+ ,5678
+ ,5224
+ ,5596
+ ,5184
+ ,5620
+ ,5531
+ ,5816
+ ,6086
+ ,6175
+ ,6112
+ ,5813
+ ,5740
+ ,5821
+ ,5294
+ ,5881
+ ,5589
+ ,5845
+ ,5706
+ ,6355
+ ,6404
+ ,6426
+ ,6375
+ ,5869
+ ,5994
+ ,6105
+ ,5792
+ ,6011
+ ,5968
+ ,6255
+ ,6208
+ ,6897
+ ,6814
+ ,6897
+ ,6596
+ ,6188
+ ,6406
+ ,6548
+ ,5842
+ ,6555
+ ,6424
+ ,6596
+ ,6645
+ ,7203
+ ,7128
+ ,7133
+ ,6778
+ ,6593
+ ,6591
+ ,6120
+ ,5612
+ ,6070
+ ,5983
+ ,6145
+ ,6303
+ ,6588
+ ,6640
+ ,6719
+ ,6575
+ ,6487
+ ,6510
+ ,6365
+ ,5844
+ ,5974
+ ,5880
+ ,6279
+ ,6342
+ ,6598
+ ,6801
+ ,6529
+ ,6369
+ ,6028
+ ,6187
+ ,6164
+ ,5866
+ ,6198
+ ,5898
+ ,6462
+ ,6063
+ ,6496
+ ,6678
+ ,6554
+ ,6513
+ ,6210
+ ,5928
+ ,6268
+ ,5582
+ ,5869
+ ,5764
+ ,6082
+ ,6062
+ ,6810
+ ,6727
+ ,6537
+ ,6175
+ ,6014
+ ,6109)
> 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: 0.3977449
 beta : 0.02385914
 gamma: 0.2723045

Coefficients:
            [,1]
a   6178.7183573
b      1.7765375
s1     0.9913508
s2     0.9135356
s3     0.9746619
s4     0.9522800
s5     0.9997545
s6     0.9862356
s7     1.0621995
s8     1.0668308
s9     1.0491120
s10    1.0225647
s11    0.9798662
s12    0.9836193
> myresid <- x - fit$fitted[,'xhat']
> postscript(file="/var/wessaorg/rcomp/tmp/1yo5b1463658704.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> op <- par(mfrow=c(2,1))
> plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
> plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
> par(op)
> dev.off()
null device 
          1 
> postscript(file="/var/wessaorg/rcomp/tmp/2x0fw1463658704.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> p <- predict(fit, par1, prediction.interval=TRUE)
> np <- length(p[,1])
> plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
> dev.off()
null device 
          1 
> postscript(file="/var/wessaorg/rcomp/tmp/3b6ld1463658704.ps",horizontal=F,onefile=F,pagecentre=F,paper="special",width=8.3333333333333,height=5.5555555555556) 
> op <- par(mfrow = c(2,2))
> acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
> spectrum(myresid,main='Residals Periodogram')
> cpgram(myresid,main='Residal Cumulative Periodogram')
> qqnorm(myresid,main='Residual Normal QQ Plot')
> qqline(myresid)
> par(op)
> dev.off()
null device 
          1 
> 
> #Note: the /var/wessaorg/rcomp/createtable file can be downloaded at http://www.wessa.net/cretab
> load(file="/var/wessaorg/rcomp/createtable")
> 
> a<-table.start()
> a<-table.row.start(a)
> a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'Parameter',header=TRUE)
> a<-table.element(a,'Value',header=TRUE)
> a<-table.row.end(a)
> a<-table.row.start(a)
> a<-table.element(a,'alpha',header=TRUE)
> a<-table.element(a,fit$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/wessaorg/rcomp/tmp/4l0xu1463658704.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/wessaorg/rcomp/tmp/57cbp1463658704.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/wessaorg/rcomp/tmp/6abno1463658704.tab") 
> 
> try(system("convert tmp/1yo5b1463658704.ps tmp/1yo5b1463658704.png",intern=TRUE))
character(0)
> try(system("convert tmp/2x0fw1463658704.ps tmp/2x0fw1463658704.png",intern=TRUE))
character(0)
> try(system("convert tmp/3b6ld1463658704.ps tmp/3b6ld1463658704.png",intern=TRUE))
character(0)
> 
> 
> proc.time()
   user  system elapsed 
  1.745   0.232   1.974