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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 21:01:20 +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/t1482350590kzomiwintjgoyco.htm/, Retrieved Mon, 06 May 2024 19:46:42 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=302485, Retrieved Mon, 06 May 2024 19:46:42 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact41
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2016-12-21 20:01:20] [9412b5b3b31fe4708efb1e5c8c74b28f] [Current]
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Dataseries X:
588,55
930,75
3228,65
2268,55
2414,5
3305,25
4342,05
3198,75
3091,35
3993,05
5331,5
3814,65
3707,6
4513,6
5634,2
4344,4
4060
4530,35
5348,75
4504,9
4281,35
4423,45
5197,9
4883,9
4155,25
4415,75
5384,05
5153,8
4564,1
5545
7585,4
6252,2
5785,65
6664,95
8639,85
6841,35




Summary of computational transaction
Raw Input view raw input (R code)
Raw Outputview raw output of R engine
Computing time2 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 time2 seconds \tabularnewline
R ServerBig Analytics Cloud Computing Center \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302485&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]2 seconds[/C][/ROW] [ROW]R Server[/C]Big Analytics Cloud Computing Center[/C][/ROW] [/TABLE] Source: https://freestatistics.org/blog/index.php?pk=302485&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302485&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 time2 seconds
R ServerBig Analytics Cloud Computing Center







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.724832086297686
beta0
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.724832086297686 \tabularnewline
beta & 0 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302485&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]0.724832086297686[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=302485&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302485&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
alpha0.724832086297686
beta0
gamma1







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
133707.62883.91267160657823.687328393426
144513.64265.84685315706247.753146842944
155634.25568.3251724130965.8748275869139
164344.44378.77154086858-34.3715408685848
1740604163.04688756575-103.046887565748
184530.354639.61239258057-109.262392580565
195348.756464.14980232309-1115.39980232309
204504.93938.41663044408566.483369555923
214281.354017.90186617015263.448133829849
224423.455296.23181993748-872.781819937482
235197.96143.1466273643-945.246627364305
244883.93900.75522787443983.144772125569
254155.254780.66666062085-625.416660620849
264415.755034.55233456814-618.802334568141
275384.055673.54005205182-289.490052051822
285153.84240.45666835497913.343331645029
294564.14651.8122534347-87.7122534347027
3055455193.63494363594351.365056364065
317585.47321.21987259998264.180127400023
326252.25684.37279601451567.827203985487
335785.655495.29823705153290.351762948471
346664.956656.678975272968.27102472703609
358639.858737.17070377842-97.3207037784214
366841.356809.0819201125532.2680798874544

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 3707.6 & 2883.91267160657 & 823.687328393426 \tabularnewline
14 & 4513.6 & 4265.84685315706 & 247.753146842944 \tabularnewline
15 & 5634.2 & 5568.32517241309 & 65.8748275869139 \tabularnewline
16 & 4344.4 & 4378.77154086858 & -34.3715408685848 \tabularnewline
17 & 4060 & 4163.04688756575 & -103.046887565748 \tabularnewline
18 & 4530.35 & 4639.61239258057 & -109.262392580565 \tabularnewline
19 & 5348.75 & 6464.14980232309 & -1115.39980232309 \tabularnewline
20 & 4504.9 & 3938.41663044408 & 566.483369555923 \tabularnewline
21 & 4281.35 & 4017.90186617015 & 263.448133829849 \tabularnewline
22 & 4423.45 & 5296.23181993748 & -872.781819937482 \tabularnewline
23 & 5197.9 & 6143.1466273643 & -945.246627364305 \tabularnewline
24 & 4883.9 & 3900.75522787443 & 983.144772125569 \tabularnewline
25 & 4155.25 & 4780.66666062085 & -625.416660620849 \tabularnewline
26 & 4415.75 & 5034.55233456814 & -618.802334568141 \tabularnewline
27 & 5384.05 & 5673.54005205182 & -289.490052051822 \tabularnewline
28 & 5153.8 & 4240.45666835497 & 913.343331645029 \tabularnewline
29 & 4564.1 & 4651.8122534347 & -87.7122534347027 \tabularnewline
30 & 5545 & 5193.63494363594 & 351.365056364065 \tabularnewline
31 & 7585.4 & 7321.21987259998 & 264.180127400023 \tabularnewline
32 & 6252.2 & 5684.37279601451 & 567.827203985487 \tabularnewline
33 & 5785.65 & 5495.29823705153 & 290.351762948471 \tabularnewline
34 & 6664.95 & 6656.67897527296 & 8.27102472703609 \tabularnewline
35 & 8639.85 & 8737.17070377842 & -97.3207037784214 \tabularnewline
36 & 6841.35 & 6809.08192011255 & 32.2680798874544 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302485&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]3707.6[/C][C]2883.91267160657[/C][C]823.687328393426[/C][/ROW]
[ROW][C]14[/C][C]4513.6[/C][C]4265.84685315706[/C][C]247.753146842944[/C][/ROW]
[ROW][C]15[/C][C]5634.2[/C][C]5568.32517241309[/C][C]65.8748275869139[/C][/ROW]
[ROW][C]16[/C][C]4344.4[/C][C]4378.77154086858[/C][C]-34.3715408685848[/C][/ROW]
[ROW][C]17[/C][C]4060[/C][C]4163.04688756575[/C][C]-103.046887565748[/C][/ROW]
[ROW][C]18[/C][C]4530.35[/C][C]4639.61239258057[/C][C]-109.262392580565[/C][/ROW]
[ROW][C]19[/C][C]5348.75[/C][C]6464.14980232309[/C][C]-1115.39980232309[/C][/ROW]
[ROW][C]20[/C][C]4504.9[/C][C]3938.41663044408[/C][C]566.483369555923[/C][/ROW]
[ROW][C]21[/C][C]4281.35[/C][C]4017.90186617015[/C][C]263.448133829849[/C][/ROW]
[ROW][C]22[/C][C]4423.45[/C][C]5296.23181993748[/C][C]-872.781819937482[/C][/ROW]
[ROW][C]23[/C][C]5197.9[/C][C]6143.1466273643[/C][C]-945.246627364305[/C][/ROW]
[ROW][C]24[/C][C]4883.9[/C][C]3900.75522787443[/C][C]983.144772125569[/C][/ROW]
[ROW][C]25[/C][C]4155.25[/C][C]4780.66666062085[/C][C]-625.416660620849[/C][/ROW]
[ROW][C]26[/C][C]4415.75[/C][C]5034.55233456814[/C][C]-618.802334568141[/C][/ROW]
[ROW][C]27[/C][C]5384.05[/C][C]5673.54005205182[/C][C]-289.490052051822[/C][/ROW]
[ROW][C]28[/C][C]5153.8[/C][C]4240.45666835497[/C][C]913.343331645029[/C][/ROW]
[ROW][C]29[/C][C]4564.1[/C][C]4651.8122534347[/C][C]-87.7122534347027[/C][/ROW]
[ROW][C]30[/C][C]5545[/C][C]5193.63494363594[/C][C]351.365056364065[/C][/ROW]
[ROW][C]31[/C][C]7585.4[/C][C]7321.21987259998[/C][C]264.180127400023[/C][/ROW]
[ROW][C]32[/C][C]6252.2[/C][C]5684.37279601451[/C][C]567.827203985487[/C][/ROW]
[ROW][C]33[/C][C]5785.65[/C][C]5495.29823705153[/C][C]290.351762948471[/C][/ROW]
[ROW][C]34[/C][C]6664.95[/C][C]6656.67897527296[/C][C]8.27102472703609[/C][/ROW]
[ROW][C]35[/C][C]8639.85[/C][C]8737.17070377842[/C][C]-97.3207037784214[/C][/ROW]
[ROW][C]36[/C][C]6841.35[/C][C]6809.08192011255[/C][C]32.2680798874544[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=302485&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302485&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
133707.62883.91267160657823.687328393426
144513.64265.84685315706247.753146842944
155634.25568.3251724130965.8748275869139
164344.44378.77154086858-34.3715408685848
1740604163.04688756575-103.046887565748
184530.354639.61239258057-109.262392580565
195348.756464.14980232309-1115.39980232309
204504.93938.41663044408566.483369555923
214281.354017.90186617015263.448133829849
224423.455296.23181993748-872.781819937482
235197.96143.1466273643-945.246627364305
244883.93900.75522787443983.144772125569
254155.254780.66666062085-625.416660620849
264415.755034.55233456814-618.802334568141
275384.055673.54005205182-289.490052051822
285153.84240.45666835497913.343331645029
294564.14651.8122534347-87.7122534347027
3055455193.63494363594351.365056364065
317585.47321.21987259998264.180127400023
326252.25684.37279601451567.827203985487
335785.655495.29823705153290.351762948471
346664.956656.678975272968.27102472703609
358639.858737.17070377842-97.3207037784214
366841.356809.0819201125532.2680798874544







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
376380.248115435665274.17122185777486.32500901362
387379.492832627865945.727115182738813.258550073
399244.545064925537383.0455932611811106.0445365899
407569.240127488335809.380157889529329.10009708713
416747.294548328694969.140733508228525.44836314917
427757.871161590865628.821422872129886.9209003096
4310280.6885159097459.6997969110513101.677234907
447860.490034227045534.5517055228410186.4283629312
456979.557059055254761.187191609469197.92692650104
468009.414521274665406.1706638101410612.6583787392
4710437.4922403567053.5585509773313821.4259297346
488212.719256892735654.5491090554410770.88940473

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
37 & 6380.24811543566 & 5274.1712218577 & 7486.32500901362 \tabularnewline
38 & 7379.49283262786 & 5945.72711518273 & 8813.258550073 \tabularnewline
39 & 9244.54506492553 & 7383.04559326118 & 11106.0445365899 \tabularnewline
40 & 7569.24012748833 & 5809.38015788952 & 9329.10009708713 \tabularnewline
41 & 6747.29454832869 & 4969.14073350822 & 8525.44836314917 \tabularnewline
42 & 7757.87116159086 & 5628.82142287212 & 9886.9209003096 \tabularnewline
43 & 10280.688515909 & 7459.69979691105 & 13101.677234907 \tabularnewline
44 & 7860.49003422704 & 5534.55170552284 & 10186.4283629312 \tabularnewline
45 & 6979.55705905525 & 4761.18719160946 & 9197.92692650104 \tabularnewline
46 & 8009.41452127466 & 5406.17066381014 & 10612.6583787392 \tabularnewline
47 & 10437.492240356 & 7053.55855097733 & 13821.4259297346 \tabularnewline
48 & 8212.71925689273 & 5654.54910905544 & 10770.88940473 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302485&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]37[/C][C]6380.24811543566[/C][C]5274.1712218577[/C][C]7486.32500901362[/C][/ROW]
[ROW][C]38[/C][C]7379.49283262786[/C][C]5945.72711518273[/C][C]8813.258550073[/C][/ROW]
[ROW][C]39[/C][C]9244.54506492553[/C][C]7383.04559326118[/C][C]11106.0445365899[/C][/ROW]
[ROW][C]40[/C][C]7569.24012748833[/C][C]5809.38015788952[/C][C]9329.10009708713[/C][/ROW]
[ROW][C]41[/C][C]6747.29454832869[/C][C]4969.14073350822[/C][C]8525.44836314917[/C][/ROW]
[ROW][C]42[/C][C]7757.87116159086[/C][C]5628.82142287212[/C][C]9886.9209003096[/C][/ROW]
[ROW][C]43[/C][C]10280.688515909[/C][C]7459.69979691105[/C][C]13101.677234907[/C][/ROW]
[ROW][C]44[/C][C]7860.49003422704[/C][C]5534.55170552284[/C][C]10186.4283629312[/C][/ROW]
[ROW][C]45[/C][C]6979.55705905525[/C][C]4761.18719160946[/C][C]9197.92692650104[/C][/ROW]
[ROW][C]46[/C][C]8009.41452127466[/C][C]5406.17066381014[/C][C]10612.6583787392[/C][/ROW]
[ROW][C]47[/C][C]10437.492240356[/C][C]7053.55855097733[/C][C]13821.4259297346[/C][/ROW]
[ROW][C]48[/C][C]8212.71925689273[/C][C]5654.54910905544[/C][C]10770.88940473[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=302485&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302485&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
376380.248115435665274.17122185777486.32500901362
387379.492832627865945.727115182738813.258550073
399244.545064925537383.0455932611811106.0445365899
407569.240127488335809.380157889529329.10009708713
416747.294548328694969.140733508228525.44836314917
427757.871161590865628.821422872129886.9209003096
4310280.6885159097459.6997969110513101.677234907
447860.490034227045534.5517055228410186.4283629312
456979.557059055254761.187191609469197.92692650104
468009.414521274665406.1706638101410612.6583787392
4710437.4922403567053.5585509773313821.4259297346
488212.719256892735654.5491090554410770.88940473



Parameters (Session):
Parameters (R input):
par1 = 12 ; par2 = Triple ; par3 = multiplicative ; 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')