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Author's title

Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationWed, 21 Dec 2016 10:50:33 +0100
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2016/Dec/21/t1482314124kg5ar5tfl160y9f.htm/, Retrieved Mon, 06 May 2024 17:23:19 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=301963, Retrieved Mon, 06 May 2024 17:23:19 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact53
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2016-12-21 09:50:33] [1a4fa2544711480e714211476e711237] [Current]
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Dataseries X:
142.2
162.3
143.4
257.1
235.8
188.1
190.2
298.2
363.6
325.2
321
391.8
481.5
416.7
603
499.2
551.4
613.5
776.1
956.4
1351.2
1593.6
1488.6
1361.7
1774.8
1893
1716.9
1453.8
1869.6
2110.8
2106.9
2845.2
3255
4645.8
4773
6724.2
9043.8
9135.3
11113.8
14501.4
19131.3




Summary of computational transaction
Raw Input view raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R ServerBig Analytics Cloud Computing Center

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input view raw input (R code)  \tabularnewline
Raw Outputview raw output of R engine  \tabularnewline
Computing time1 seconds \tabularnewline
R ServerBig Analytics Cloud Computing Center \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301963&T=0

[TABLE]
[ROW]
Summary of computational transaction[/C][/ROW] [ROW]Raw Input[/C] view raw input (R code) [/C][/ROW] [ROW]Raw Output[/C]view raw output of R engine [/C][/ROW] [ROW]Computing time[/C]1 seconds[/C][/ROW] [ROW]R Server[/C]Big Analytics Cloud Computing Center[/C][/ROW] [/TABLE] Source: https://freestatistics.org/blog/index.php?pk=301963&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301963&T=0

As an alternative you can also use a QR Code:  

The GUIDs for individual cells are displayed in the table below:

Summary of computational transaction
Raw Input view raw input (R code)
Raw Outputview raw output of R engine
Computing time1 seconds
R ServerBig Analytics Cloud Computing Center







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha1
beta0.688967277470754
gamma0.0346590184648955

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.688967277470754 \tabularnewline
gamma & 0.0346590184648955 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301963&T=1

[TABLE]
[ROW][C]Estimated Parameters of Exponential Smoothing[/C][/ROW]
[ROW][C]Parameter[/C][C]Value[/C][/ROW]
[ROW][C]alpha[/C][C]1[/C][/ROW]
[ROW][C]beta[/C][C]0.688967277470754[/C][/ROW]
[ROW][C]gamma[/C][C]0.0346590184648955[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=301963&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301963&T=1

As an alternative you can also use a QR Code:  

The GUIDs for individual cells are displayed in the table below:

Estimated Parameters of Exponential Smoothing
ParameterValue
alpha1
beta0.688967277470754
gamma0.0346590184648955







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13481.5298.600961538462182.899038461538
14416.7541.762414119326-125.062414119326
15603625.161003149614-22.161003149614
16499.2480.66779714360418.5322028563963
17551.4538.13587849111113.2641215088894
18613.5621.811924175131-8.31192417513137
19776.1866.797780405648-90.6977804056479
20956.4888.52247756692867.8775224330716
211351.21067.9753693991283.224630600899
221593.61554.6203720568638.9796279431368
231488.61864.07606019767-375.476060197668
241361.71567.74784124784-206.047841247836
251774.81306.52512103459468.274878965412
2618931878.0511895033314.9488104966651
271716.92240.91293077265-524.012930772648
281453.81388.2601684987565.5398315012535
291869.61318.81496777406550.785032225943
302110.82136.42533189841-25.6253318984063
312106.92548.58281674608-441.682816746078
322845.22161.99030898692683.209691013082
3332553323.38692974584-68.3869297458364
344645.83582.783072944261063.01692705574
3547735746.16695108318-973.166951083183
366724.25270.249266270891453.95073372911
379043.88230.81124486484812.988755135158
389135.310946.3338941046-1811.03389410465
3911113.810024.45330267611089.34669732389
4014501.412437.95253095312063.44746904689
4119131.317395.70031590631735.59968409373

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 481.5 & 298.600961538462 & 182.899038461538 \tabularnewline
14 & 416.7 & 541.762414119326 & -125.062414119326 \tabularnewline
15 & 603 & 625.161003149614 & -22.161003149614 \tabularnewline
16 & 499.2 & 480.667797143604 & 18.5322028563963 \tabularnewline
17 & 551.4 & 538.135878491111 & 13.2641215088894 \tabularnewline
18 & 613.5 & 621.811924175131 & -8.31192417513137 \tabularnewline
19 & 776.1 & 866.797780405648 & -90.6977804056479 \tabularnewline
20 & 956.4 & 888.522477566928 & 67.8775224330716 \tabularnewline
21 & 1351.2 & 1067.9753693991 & 283.224630600899 \tabularnewline
22 & 1593.6 & 1554.62037205686 & 38.9796279431368 \tabularnewline
23 & 1488.6 & 1864.07606019767 & -375.476060197668 \tabularnewline
24 & 1361.7 & 1567.74784124784 & -206.047841247836 \tabularnewline
25 & 1774.8 & 1306.52512103459 & 468.274878965412 \tabularnewline
26 & 1893 & 1878.05118950333 & 14.9488104966651 \tabularnewline
27 & 1716.9 & 2240.91293077265 & -524.012930772648 \tabularnewline
28 & 1453.8 & 1388.26016849875 & 65.5398315012535 \tabularnewline
29 & 1869.6 & 1318.81496777406 & 550.785032225943 \tabularnewline
30 & 2110.8 & 2136.42533189841 & -25.6253318984063 \tabularnewline
31 & 2106.9 & 2548.58281674608 & -441.682816746078 \tabularnewline
32 & 2845.2 & 2161.99030898692 & 683.209691013082 \tabularnewline
33 & 3255 & 3323.38692974584 & -68.3869297458364 \tabularnewline
34 & 4645.8 & 3582.78307294426 & 1063.01692705574 \tabularnewline
35 & 4773 & 5746.16695108318 & -973.166951083183 \tabularnewline
36 & 6724.2 & 5270.24926627089 & 1453.95073372911 \tabularnewline
37 & 9043.8 & 8230.81124486484 & 812.988755135158 \tabularnewline
38 & 9135.3 & 10946.3338941046 & -1811.03389410465 \tabularnewline
39 & 11113.8 & 10024.4533026761 & 1089.34669732389 \tabularnewline
40 & 14501.4 & 12437.9525309531 & 2063.44746904689 \tabularnewline
41 & 19131.3 & 17395.7003159063 & 1735.59968409373 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301963&T=2

[TABLE]
[ROW][C]Interpolation Forecasts of Exponential Smoothing[/C][/ROW]
[ROW][C]t[/C][C]Observed[/C][C]Fitted[/C][C]Residuals[/C][/ROW]
[ROW][C]13[/C][C]481.5[/C][C]298.600961538462[/C][C]182.899038461538[/C][/ROW]
[ROW][C]14[/C][C]416.7[/C][C]541.762414119326[/C][C]-125.062414119326[/C][/ROW]
[ROW][C]15[/C][C]603[/C][C]625.161003149614[/C][C]-22.161003149614[/C][/ROW]
[ROW][C]16[/C][C]499.2[/C][C]480.667797143604[/C][C]18.5322028563963[/C][/ROW]
[ROW][C]17[/C][C]551.4[/C][C]538.135878491111[/C][C]13.2641215088894[/C][/ROW]
[ROW][C]18[/C][C]613.5[/C][C]621.811924175131[/C][C]-8.31192417513137[/C][/ROW]
[ROW][C]19[/C][C]776.1[/C][C]866.797780405648[/C][C]-90.6977804056479[/C][/ROW]
[ROW][C]20[/C][C]956.4[/C][C]888.522477566928[/C][C]67.8775224330716[/C][/ROW]
[ROW][C]21[/C][C]1351.2[/C][C]1067.9753693991[/C][C]283.224630600899[/C][/ROW]
[ROW][C]22[/C][C]1593.6[/C][C]1554.62037205686[/C][C]38.9796279431368[/C][/ROW]
[ROW][C]23[/C][C]1488.6[/C][C]1864.07606019767[/C][C]-375.476060197668[/C][/ROW]
[ROW][C]24[/C][C]1361.7[/C][C]1567.74784124784[/C][C]-206.047841247836[/C][/ROW]
[ROW][C]25[/C][C]1774.8[/C][C]1306.52512103459[/C][C]468.274878965412[/C][/ROW]
[ROW][C]26[/C][C]1893[/C][C]1878.05118950333[/C][C]14.9488104966651[/C][/ROW]
[ROW][C]27[/C][C]1716.9[/C][C]2240.91293077265[/C][C]-524.012930772648[/C][/ROW]
[ROW][C]28[/C][C]1453.8[/C][C]1388.26016849875[/C][C]65.5398315012535[/C][/ROW]
[ROW][C]29[/C][C]1869.6[/C][C]1318.81496777406[/C][C]550.785032225943[/C][/ROW]
[ROW][C]30[/C][C]2110.8[/C][C]2136.42533189841[/C][C]-25.6253318984063[/C][/ROW]
[ROW][C]31[/C][C]2106.9[/C][C]2548.58281674608[/C][C]-441.682816746078[/C][/ROW]
[ROW][C]32[/C][C]2845.2[/C][C]2161.99030898692[/C][C]683.209691013082[/C][/ROW]
[ROW][C]33[/C][C]3255[/C][C]3323.38692974584[/C][C]-68.3869297458364[/C][/ROW]
[ROW][C]34[/C][C]4645.8[/C][C]3582.78307294426[/C][C]1063.01692705574[/C][/ROW]
[ROW][C]35[/C][C]4773[/C][C]5746.16695108318[/C][C]-973.166951083183[/C][/ROW]
[ROW][C]36[/C][C]6724.2[/C][C]5270.24926627089[/C][C]1453.95073372911[/C][/ROW]
[ROW][C]37[/C][C]9043.8[/C][C]8230.81124486484[/C][C]812.988755135158[/C][/ROW]
[ROW][C]38[/C][C]9135.3[/C][C]10946.3338941046[/C][C]-1811.03389410465[/C][/ROW]
[ROW][C]39[/C][C]11113.8[/C][C]10024.4533026761[/C][C]1089.34669732389[/C][/ROW]
[ROW][C]40[/C][C]14501.4[/C][C]12437.9525309531[/C][C]2063.44746904689[/C][/ROW]
[ROW][C]41[/C][C]19131.3[/C][C]17395.7003159063[/C][C]1735.59968409373[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=301963&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301963&T=2

As an alternative you can also use a QR Code:  

The GUIDs for individual cells are displayed in the table below:

Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13481.5298.600961538462182.899038461538
14416.7541.762414119326-125.062414119326
15603625.161003149614-22.161003149614
16499.2480.66779714360418.5322028563963
17551.4538.13587849111113.2641215088894
18613.5621.811924175131-8.31192417513137
19776.1866.797780405648-90.6977804056479
20956.4888.52247756692867.8775224330716
211351.21067.9753693991283.224630600899
221593.61554.6203720568638.9796279431368
231488.61864.07606019767-375.476060197668
241361.71567.74784124784-206.047841247836
251774.81306.52512103459468.274878965412
2618931878.0511895033314.9488104966651
271716.92240.91293077265-524.012930772648
281453.81388.2601684987565.5398315012535
291869.61318.81496777406550.785032225943
302110.82136.42533189841-25.6253318984063
312106.92548.58281674608-441.682816746078
322845.22161.99030898692683.209691013082
3332553323.38692974584-68.3869297458364
344645.83582.783072944261063.01692705574
3547735746.16695108318-973.166951083183
366724.25270.249266270891453.95073372911
379043.88230.81124486484812.988755135158
389135.310946.3338941046-1811.03389410465
3911113.810024.45330267611089.34669732389
4014501.412437.95253095312063.44746904689
4119131.317395.70031590631735.59968409373







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
4223243.709205035421708.836212085324778.5821979856
4327544.730910070924532.071789647130557.3900304946
4431767.365115106327034.780625050936499.9496051616
4535942.386820141729267.353158446342617.420481837
4640014.121025177131194.325868542748833.9161818115
4744126.055230212632975.628827713555276.4816327116
4848305.35193524834651.698292951761959.0055775443
4952492.286140283436173.706256449168810.8660241177
5056515.020345318837378.93139739775651.1092932407
5160772.117050354338673.693429692282870.5406710164
5264713.688755389739514.802973418889912.5745373606
5368803.760460425140372.141249274397235.3796715759

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
42 & 23243.7092050354 & 21708.8362120853 & 24778.5821979856 \tabularnewline
43 & 27544.7309100709 & 24532.0717896471 & 30557.3900304946 \tabularnewline
44 & 31767.3651151063 & 27034.7806250509 & 36499.9496051616 \tabularnewline
45 & 35942.3868201417 & 29267.3531584463 & 42617.420481837 \tabularnewline
46 & 40014.1210251771 & 31194.3258685427 & 48833.9161818115 \tabularnewline
47 & 44126.0552302126 & 32975.6288277135 & 55276.4816327116 \tabularnewline
48 & 48305.351935248 & 34651.6982929517 & 61959.0055775443 \tabularnewline
49 & 52492.2861402834 & 36173.7062564491 & 68810.8660241177 \tabularnewline
50 & 56515.0203453188 & 37378.931397397 & 75651.1092932407 \tabularnewline
51 & 60772.1170503543 & 38673.6934296922 & 82870.5406710164 \tabularnewline
52 & 64713.6887553897 & 39514.8029734188 & 89912.5745373606 \tabularnewline
53 & 68803.7604604251 & 40372.1412492743 & 97235.3796715759 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=301963&T=3

[TABLE]
[ROW][C]Extrapolation Forecasts of Exponential Smoothing[/C][/ROW]
[ROW][C]t[/C][C]Forecast[/C][C]95% Lower Bound[/C][C]95% Upper Bound[/C][/ROW]
[ROW][C]42[/C][C]23243.7092050354[/C][C]21708.8362120853[/C][C]24778.5821979856[/C][/ROW]
[ROW][C]43[/C][C]27544.7309100709[/C][C]24532.0717896471[/C][C]30557.3900304946[/C][/ROW]
[ROW][C]44[/C][C]31767.3651151063[/C][C]27034.7806250509[/C][C]36499.9496051616[/C][/ROW]
[ROW][C]45[/C][C]35942.3868201417[/C][C]29267.3531584463[/C][C]42617.420481837[/C][/ROW]
[ROW][C]46[/C][C]40014.1210251771[/C][C]31194.3258685427[/C][C]48833.9161818115[/C][/ROW]
[ROW][C]47[/C][C]44126.0552302126[/C][C]32975.6288277135[/C][C]55276.4816327116[/C][/ROW]
[ROW][C]48[/C][C]48305.351935248[/C][C]34651.6982929517[/C][C]61959.0055775443[/C][/ROW]
[ROW][C]49[/C][C]52492.2861402834[/C][C]36173.7062564491[/C][C]68810.8660241177[/C][/ROW]
[ROW][C]50[/C][C]56515.0203453188[/C][C]37378.931397397[/C][C]75651.1092932407[/C][/ROW]
[ROW][C]51[/C][C]60772.1170503543[/C][C]38673.6934296922[/C][C]82870.5406710164[/C][/ROW]
[ROW][C]52[/C][C]64713.6887553897[/C][C]39514.8029734188[/C][C]89912.5745373606[/C][/ROW]
[ROW][C]53[/C][C]68803.7604604251[/C][C]40372.1412492743[/C][C]97235.3796715759[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=301963&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=301963&T=3

As an alternative you can also use a QR Code:  

The GUIDs for individual cells are displayed in the table below:

Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
4223243.709205035421708.836212085324778.5821979856
4327544.730910070924532.071789647130557.3900304946
4431767.365115106327034.780625050936499.9496051616
4535942.386820141729267.353158446342617.420481837
4640014.121025177131194.325868542748833.9161818115
4744126.055230212632975.628827713555276.4816327116
4848305.35193524834651.698292951761959.0055775443
4952492.286140283436173.706256449168810.8660241177
5056515.020345318837378.93139739775651.1092932407
5160772.117050354338673.693429692282870.5406710164
5264713.688755389739514.802973418889912.5745373606
5368803.760460425140372.141249274397235.3796715759



Parameters (Session):
par1 = 12 ; par2 = Triple ; par3 = additive ; par4 = 12 ;
Parameters (R input):
par1 = 12 ; par2 = Triple ; par3 = additive ; par4 = 12 ;
R code (references can be found in the software module):
par1 <- as.numeric(par1)
par4 <- as.numeric(par4)
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$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par4, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$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')