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Double 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: Tue, 21 Dec 2010 20:32:31 +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/21/t1292963451xfpovn40kgg4d7t.htm/, Retrieved Tue, 21 Dec 2010 21:30:51 +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/21/t1292963451xfpovn40kgg4d7t.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 «
27951 29781 32914 33488 35652 36488 35387 35676 34844 32447 31068 29010 29812 30951 32974 32936 34012 32946 31948 30599 27691 25073 23406 22248 22896 25317 26558 26471 27543 26198 24725 25005 23462 20780 19815 19761 21454 23899 24939 23580 24562 24696 23785 23812 21917 19713 19282 18788 21453 24482 27474 27264 27349 30632 29429 30084 26290 24379 23335 21346 21106 24514 28353 30805 31348 34556 33855 34787 32529 29998 29257 28155 30466 35704 39327 39351 42234 43630 43722 43121 37985 37135 34646 33026 35087 38846 42013 43908 42868 44423 44167 43636 44382 42142 43452 36912 42413 45344 44873 47510 49554 47369 45998 48140 48441 44928 40454 38661 37246 36843 36424 37594 38144 38737 34560 36080 33508 35462 33374 32110 35533 35532 37903 36763 40399 44164 44496 43110 43880 43930 44327
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 seconds
R Server'Sir Ronald Aylmer Fisher' @ 193.190.124.24


Estimated Parameters of Exponential Smoothing
ParameterValue
alpha1
beta-6.64073830647371e-18
gammaFALSE


Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
332914297813133
43348832914574
535652334882164
63648835652836
73538736488-1101
83567635387289
93484435676-832
103244734844-2397
113106832447-1379
122901031068-2058
132981229010802
1430951298121139
1532974309512023
163293632974-38
1734012329361076
183294634012-1066
193194832946-998
203059931948-1349
212769130599-2908
222507327691-2618
232340625073-1667
242224823406-1158
252289622248648
2625317228962421
2726558253171241
282647126558-87
2927543264711072
302619827543-1345
312472526198-1473
322500524725280
332346225005-1543
342078023462-2682
351981520780-965
361976119815-54
3721454197611693
3823899214542445
3924939238991040
402358024939-1359
412456223580982
422469624562134
432378524696-911
44238122378527
452191723812-1895
461971321917-2204
471928219713-431
481878819282-494
4921453187882665
5024482214533029
5127474244822992
522726427474-210
53273492726485
5430632273493283
552942930632-1203
563008429429655
572629030084-3794
582437926290-1911
592333524379-1044
602134623335-1989
612110621346-240
6224514211063408
6328353245143839
6430805283532452
653134830805543
6634556313483208
673385534556-701
683478733855932
693252934787-2258
702999832529-2531
712925729998-741
722815529257-1102
7330466281552311
7435704304665238
7539327357043623
76393513932724
7742234393512883
7843630422341396
79437224363092
804312143722-601
813798543121-5136
823713537985-850
833464637135-2489
843302634646-1620
8535087330262061
8638846350873759
8742013388463167
8843908420131895
894286843908-1040
9044423428681555
914416744423-256
924363644167-531
934438243636746
944214244382-2240
9543452421421310
963691243452-6540
9742413369125501
9845344424132931
994487345344-471
10047510448732637
10149554475102044
1024736949554-2185
1034599847369-1371
10448140459982142
1054844148140301
1064492848441-3513
1074045444928-4474
1083866140454-1793
1093724638661-1415
1103684337246-403
1113642436843-419
11237594364241170
1133814437594550
1143873738144593
1153456038737-4177
11636080345601520
1173350836080-2572
11835462335081954
1193337435462-2088
1203211033374-1264
12135533321103423
1223553235533-1
12337903355322371
1243676337903-1140
12540399367633636
12644164403993765
1274449644164332
1284311044496-1386
1294388043110770
130439304388050
1314432743930397


Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
1324432740126.895802299648527.1041977004
1334432738387.155680231950266.8443197681
1344432737052.206132499651601.7938675004
1354432735926.791604599252727.2083954008
1364432734935.281501359753718.7184986403
1374432734038.887849112354615.1121508877
1384432733214.568812326355439.4311876737
1394432732447.311360463856206.6886395362
1404432731726.687406898856927.3125931012
1414432731045.104325232657608.8956747674
1424432730396.830295831058257.169704169
1434432729777.412264999158876.5877350009
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/21/t1292963451xfpovn40kgg4d7t/1jnsu1292963547.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/21/t1292963451xfpovn40kgg4d7t/1jnsu1292963547.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/21/t1292963451xfpovn40kgg4d7t/2jnsu1292963547.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/21/t1292963451xfpovn40kgg4d7t/2jnsu1292963547.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/21/t1292963451xfpovn40kgg4d7t/3cw9x1292963547.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/21/t1292963451xfpovn40kgg4d7t/3cw9x1292963547.ps (open in new window)


 
Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = multiplicative ;
 
Parameters (R input):
par1 = 12 ; par2 = Double ; par3 = multiplicative ;
 
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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Software written by Ed van Stee & Patrick Wessa


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