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Holt-Winters model

*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: Tue, 30 Nov 2010 19:51:51 +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/Nov/30/t1291146639u81jk3ro71ieadw.htm/, Retrieved Tue, 30 Nov 2010 20:50:43 +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/Nov/30/t1291146639u81jk3ro71ieadw.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 «

0,96 0,95 0,95 0,96 0,96 0,96 0,95 0,96 0,96 0,96 0,95 0,95 0,96 0,96 0,96 0,96 0,96 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,96 0,96 0,96 0,97 0,97 0,97 0,96 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,95 0,94 0,94 0,94 0,93 0,93 0,93 0,93 0,92 0,93

Output produced by software:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	2 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.839082605513062
beta	0
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	0.96	0.960080128205128	-8.01282051284824e-05
14	0.96	0.96018771919682	-0.000187719196819036
15	0.96	0.960621699125539	-0.00062169912553911
16	0.96	0.960691534044928	-0.000691534044928499
17	0.96	0.960286105031534	-0.000286105031534145
18	0.95	0.949804197784383	0.000195802215617435
19	0.95	0.94805998385912	0.00194001614087991
20	0.95	0.959029309498839	-0.00902930949883884
21	0.95	0.950377798133395	-0.00037779813339478
22	0.95	0.94940228613276	0.000597713867240279
23	0.95	0.939661975949994	0.0103380240500064
24	0.95	0.948511257280555	0.00148874271944510
25	0.95	0.959922366553357	-0.00992236655335677
26	0.95	0.951754193285682	-0.00175419328568227
27	0.95	0.95080393713506	-0.000803937135060995
28	0.96	0.950709621657325	0.0092903783426751
29	0.96	0.95874508227861	0.00125491772139119
30	0.96	0.949633767676733	0.0103662323232674
31	0.97	0.956704059105667	0.0132959408943334
32	0.97	0.97543678837430	-0.00543678837430095
33	0.97	0.971191877661696	-0.00119187766169615
34	0.96	0.969690262538792	-0.00969026253879213
35	0.95	0.9528848756439	-0.00288487564390072
36	0.95	0.949215048552065	0.000784951447935223
37	0.95	0.958199372838646	-0.0081993728386458
38	0.95	0.952791334786346	-0.00279133478634563
39	0.95	0.951123743986915	-0.00112374398691517
40	0.95	0.95238543508847	-0.00238543508847089
41	0.95	0.949330878367785	0.000669121632214997
42	0.95	0.941194201463188	0.00880579853681185
43	0.95	0.947426601114714	0.00257339888528552
44	0.95	0.954147809911136	-0.00414780991113584
45	0.95	0.951667538577556	-0.00166753857755586
46	0.95	0.948399266702262	0.00160073329773802
47	0.94	0.942163063140325	-0.00216306314032488
48	0.94	0.939689435378517	0.000310564621482934
49	0.94	0.946829975875315	-0.00682997587531498
50	0.93	0.94344122238765	-0.0134412223876503
51	0.93	0.93310584051781	-0.0031058405178106
52	0.93	0.932501360853135	-0.00250136085313446
53	0.93	0.929841064148594	0.000158935851406139
54	0.92	0.92258563207701	-0.00258563207700990
55	0.93	0.918256778935244	0.0117432210647556

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
56	0.931590666610784	0.920364779892464	0.942816553329105
57	0.932989869225233	0.918335637728013	0.947644100722454
58	0.931646721759036	0.914226260544254	0.949067182973817
59	0.923461710414709	0.903657709882057	0.94326571094736
60	0.923201121042935	0.901271132322946	0.945131109762924
61	0.928932034995985	0.90506468546948	0.952799384522492
62	0.930210330898296	0.904551485608961	0.95586917618763
63	0.932816387652288	0.905483215320055	0.960149559984521
64	0.934915236034265	0.906004541163592	0.963825930904937
65	0.934781875725957	0.904375392107053	0.965188359344862
66	0.926951434626033	0.895119371949727	0.958783497302339
67	0.927097902097902	0.893901423634218	0.960294380561586

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146639u81jk3ro71ieadw/1df5e1291146708.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146639u81jk3ro71ieadw/1df5e1291146708.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146639u81jk3ro71ieadw/26o4h1291146708.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146639u81jk3ro71ieadw/26o4h1291146708.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146639u81jk3ro71ieadw/36o4h1291146708.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Nov/30/t1291146639u81jk3ro71ieadw/36o4h1291146708.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Triple ; par3 = additive ;

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=0, beta=0)

if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)

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')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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