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Paper: Exponential Smoothing

*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: Wed, 29 Dec 2010 17:48:54 +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/29/t1293644810n5qrdptjx2q8rnv.htm/, Retrieved Wed, 29 Dec 2010 18:46:54 +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/29/t1293644810n5qrdptjx2q8rnv.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 «
31245 30951 30872 30752 30967 30781 30681 31356 31434 31594 31949 32396 32441 32447 32288 32418 32346 32091 31855 31683 31615 31840 31536 31383 31638 31626 31720 31472 31372 31419 31341 31171 31036 30532 30666 30571 30173 30032 29874 30018 29911 29963 30050 29901 29544 29451 29293 29334 29389 29563
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'George Udny Yule' @ 72.249.76.132


Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.953617426395245
beta0.348663695525690
gammaFALSE


Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
33087230657215
43075230639.513478516112.486521484017
53096730561.6691607835405.330839216465
63078130897.8554582621-116.855458262118
73068130697.2223237869-16.2223237869039
83135630587.1609107732768.8390892268
93143431481.3804162387-47.3804162386623
103159431581.485178785512.5148212144813
113194931742.8681586578206.131841342241
123239632157.4248449895238.575155010458
133244132682.2443176054-241.244317605357
143244732669.2878279993-222.287827999317
153228832600.4997025092-312.499702509169
163241832341.780380330476.2196196695586
173234632479.0929744682-133.092974468225
183209132372.5491118175-281.549111817545
193185532030.8221253145-175.822125314469
203168331731.4588248743-48.4588248742584
213161531537.439226471577.5607735284975
223184031489.3824325117350.617567488269
233153631818.2948133802-282.294813380224
243138331449.7902146135-66.7902146135239
253163831264.5873596225373.412640377519
263162631623.32625028152.67374971845493
273172031629.411073958290.5889260418298
283147231749.4534147274-277.453414727373
293137231426.2731870665-54.2731870665048
303141931297.8761253727121.123874627276
313134131377.0134504400-36.0134504400448
323117131294.3277078892-123.327707889170
333103631087.3721192354-51.3721192353878
343053230931.9538266867-399.953826686688
353066630311.1405852349354.859414765102
363057130528.118241552642.8817584473582
373017330461.8464001842-288.846400184200
383003229983.193753328948.8062466710981
392987429842.760230219131.2397697809029
403001829695.9619751629322.03802483714
412991129933.5489985127-22.5489985127133
422996329835.0344749239127.965525076135
433005029922.6007086549127.399291345148
442990130051.9861886736-150.98618867364
452954429865.6967576668-321.696757666832
462945129409.653226906741.3467730933189
472929329313.5617956373-20.5617956373280
482933429151.5966464339182.403353566056
492938929243.8302154382145.1697845618
502956329348.8249636843214.175036315726


Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
5129590.835760716229141.973793977530039.6977274549
5229628.605510818528897.349317233330359.8617044038
5329666.375260920928635.560455249630697.1900665922
5429704.145011023328351.534724262331056.7552977843
5529741.914761125728044.796972843831439.0325494075
5629779.684511228127715.928552833831843.4404696224
5729817.454261330427365.772983669832269.1355389911
5829855.224011432826995.204260550332715.2437623154
5929892.993761535226605.053492264633180.9340308058
6029930.763511637626196.087796444233665.439226831
6129968.533261740025769.007774331434168.0587491486
6230006.303011842325324.451564671934688.1544590128
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293644810n5qrdptjx2q8rnv/1a4zu1293644931.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293644810n5qrdptjx2q8rnv/1a4zu1293644931.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293644810n5qrdptjx2q8rnv/23vgx1293644931.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293644810n5qrdptjx2q8rnv/23vgx1293644931.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293644810n5qrdptjx2q8rnv/33vgx1293644931.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293644810n5qrdptjx2q8rnv/33vgx1293644931.ps (open in new window)


 
Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = additive ;
 
Parameters (R input):
par1 = 12 ; par2 = Double ; 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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