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Exponential Smoothing Paper

*The author of this computation has been verified*
R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)
Title produced by software: Exponential Smoothing
Date of computation: Fri, 24 Dec 2010 14:38:01 +0000
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo.htm/, Retrieved Fri, 24 Dec 2010 15:36:16 +0100
 
BibTeX entries for LaTeX users:
@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo.htm/},
    year = {2010},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2010},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}
 
Original text written by user:
 
IsPrivate?
No (this computation is public)
 
User-defined keywords:
 
Dataseries X:
» Textbox « » Textfile « » CSV «
44164 40399 36763 37903 35532 35533 32110 33374 35462 33508 36080 34560 38737 38144 37594 36424 36843 37246 38661 40454 44928 48441 48140 45998 47369 49554 47510 44873 45344 42413 36912 43452 42142 44382 43636 44167 44423 42868 43908 42013 38846 35087 33026 34646 37135 37985 43121 43722 43630 42234 39351 39327 35704 30466 28155 29257 29998 32529 34787 33855 34556 31348 30805 28353 24514 21106 21346 23335 24379 26290 30084 29429 30632 27349 27264 27474 24482 21453 18788 19282 19713 21917 23812 23785 24696 24562 23580 24939 23899 21454 19761 19815 20780 23462 25005 24725 26198 27543 26471 26558 25317 22896
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time5 seconds
R Server'George Udny Yule' @ 72.249.76.132


Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.855314864777257
beta0.0252228018917085
gamma1


Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
133873738442.7943376068294.205662393157
143814437937.7174870782206.282512921782
153759437283.4305493924310.569450607611
163642435777.8334969757646.166503024266
173684336054.1342913292788.865708670812
183724636599.1313824386646.86861756137
193866136911.42309705381749.57690294621
204045440489.9969874451-35.9969874451199
214492843103.81640847171824.18359152826
224844143273.69650702945167.30349297063
234814050920.4732504005-2780.4732504005
244599847484.4140376552-1486.41403765516
254736950644.3496980102-3275.34969801022
264955447154.60818547822399.39181452175
274751048519.6721805846-1009.67218058464
284487346033.3897680229-1160.38976802291
294534444846.1699821568497.83001784324
304241345176.4238107366-2763.42381073661
313691242712.5442437365-5800.54424373655
324345239393.31624904314058.68375095692
334214245685.1287516123-3543.12875161225
344438241538.78649826452843.21350173555
354363645788.4906863834-2152.49068638339
364416742831.01448433761335.98551566244
374442347961.2759558062-3538.27595580615
384286844877.1461433188-2009.14614331885
394390841692.61958200562215.38041799444
404201341726.8798484619286.120151538118
413884641831.9212902576-2985.92129025756
423508738450.5794353890-3363.57943538904
433302634760.9677878567-1734.96778785666
443464636160.2960914411-1514.2960914411
453713536280.0830782005854.916921799471
463798536608.84067513561376.15932486445
474312138638.67529826094482.32470173906
484372241761.64956401741960.35043598259
494363046635.0397405614-3005.03974056143
504223444154.0741689972-1920.07416899723
513935141584.717002687-2233.71700268699
523932737366.23911777131960.76088222873
533570438298.1139580916-2594.11395809161
543046635073.6058965703-4607.60589657035
552815530405.1145719043-2250.11457190428
562925731234.1633517367-1977.16335173673
572999831129.2626216296-1131.26262162965
583252929620.19826459132908.80173540870
593478733228.97595257141558.02404742860
603385533241.4082660266613.5917339734
613455635970.9713958884-1414.97139588836
623134834767.7903853884-3419.7903853884
633080530598.767235971206.232764029002
642835328855.1744048542-502.174404854162
652451426749.3885697929-2235.38856979291
662110623276.0674204115-2170.0674204115
672134620821.8051312095524.194868790517
682333523911.3775826198-576.377582619753
692437925005.3223520738-626.322352073832
702629024401.91481585491888.0851841451
713008426809.43729239993274.56270760011
722942928057.65324022131371.34675977875
733063231062.4278767054-430.427876705362
742734930353.1093545012-3004.10935450122
752726427015.058976107248.941023893000
762747425157.22348805342316.77651194663
772448225224.2967310677-742.29673106769
782145323082.2401301385-1629.24013013846
791878821536.7925738898-2748.79257388979
801928221653.5015583878-2371.50155838779
811971321151.9046658262-1438.90466582622
822191720146.83142281221770.16857718777
832381222581.10752467911230.89247532091
842378521688.89265965242096.10734034764
852469624951.4291364799-255.429136479881
862456223921.7448591370640.255140862955
872358024152.3917831075-572.391783107512
882493921854.47436454583084.52563545419
892389922115.4065781821783.59342181799
902145422039.7401379344-585.740137934441
911976121281.6292132008-1520.62921320078
921981522586.6868569339-2771.68685693386
932078021952.3989437535-1172.39894375353
942346221719.98715337511742.01284662486
952500524131.9585396449873.041460355107
962472523130.93457042231594.06542957765
972619825685.0864539043512.913546095737
982754325519.99677655962023.00322344036
992647126865.5348163776-394.5348163776
1002655825360.33779335211197.66220664792
1012531723889.97108212151427.02891787852
1022289623229.6190197624-333.619019762369


Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
10322620.422246850218403.82861411526837.0158795853
10425146.428179214619538.306187035230754.5501713939
10527275.334199248120507.515428865634043.1529696305
10628653.793168097220852.945295524436454.64104067
10729598.915075610920845.792665597138352.0374856247
10828085.499959104718436.059878540237734.9400396691
10929215.420648882918710.497843795339720.3434539705
11028914.673882208917585.118307487640244.2294569301
11128221.040203776316090.725995701440351.3544118511
11227333.088216054914420.805973271640245.3704588382
11324895.117780104311215.830185664538574.4053745441
11422752.26977784478317.9806825137537186.5588731757
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo/11eho1293201476.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo/11eho1293201476.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo/2tngr1293201476.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo/2tngr1293201476.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo/3tngr1293201476.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/24/t1293201372vf9uoe46njhcimo/3tngr1293201476.ps (open in new window)


 
Parameters (Session):
par1 = additive ; par2 = 12 ;
 
Parameters (R input):
par1 = 12 ; par2 = Triple ; par3 = additive ;
 
R code (references can be found in the software module):
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
myresid <- x - fit$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par1, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$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='mytable.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='mytable1.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='mytable2.tab')
 




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