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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, 07 Dec 2016 11:37:07 +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/07/t14811071015xry8xhoc970e0k.htm/, Retrieved Tue, 07 May 2024 06:20:46 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=297987, Retrieved Tue, 07 May 2024 06:20:46 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact80
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [Voorbeeld Exponen...] [2016-12-07 10:37:07] [fc6d28d208bad0c833791fcb11cb4db1] [Current]
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Dataseries X:
2157.07
2267.88
2375.38
2803.62
2367.53
2439.08
2533.76
2956.28
2484.24
2588.49
2668.42
3085.62
2595.93
2686.64
2779.43
3221.13
2752.7
2886.58
2958.05
3444.61
2939.78
3088.73
3161.34
3672.39
3092.36
3228.05
3311.16
3801.93
3246.26
3309.22
3458.64
4005.04
3477.65
3524.42
3699.5
4247.68
3697.6
3746.72
3950.67
4566.86
3967.9
4059.35
4215.38
4856.13




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=297987&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=297987&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=297987&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
alpha0.449104645448336
beta0.124988500961914
gamma0.741727782869552

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.449104645448336 \tabularnewline
beta & 0.124988500961914 \tabularnewline
gamma & 0.741727782869552 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=297987&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]0.449104645448336[/C][/ROW]
[ROW][C]beta[/C][C]0.124988500961914[/C][/ROW]
[ROW][C]gamma[/C][C]0.741727782869552[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=297987&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=297987&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
alpha0.449104645448336
beta0.124988500961914
gamma0.741727782869552







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
132595.932387.17917200855208.750827991451
142686.642582.33343545803104.306564541967
152779.432737.21424333742.2157566630008
163221.133214.993224497826.13677550218472
172752.72765.22594272255-12.5259427225533
182886.582905.07861313341-18.4986131334063
192958.053011.7580484123-53.7080484123021
203444.613426.9553900184517.6546099815514
212939.782982.08345443382-42.3034544338202
223088.733084.251540649924.47845935007763
233161.343184.14765732034-22.8076573203439
243672.393606.5296085826465.8603914173605
253092.363249.20832050802-156.848320508023
263228.053233.78589044528-5.73589044527489
273311.163303.990930525617.16906947438656
283801.933739.43662553862.4933744619984
293246.263298.66569394575-52.4056939457491
303309.223407.24129201517-98.0212920151694
313458.643448.4294292557310.2105707442711
324005.043909.6904556669295.3495443330776
333477.653467.770918258799.87908174120685
343524.423607.97831512743-83.5583151274304
353699.53647.7335746750351.7664253249673
364247.684134.57056145363113.109438546371
373697.63704.85171022574-7.25171022573932
383746.723824.14247496676-77.4224749667587
393950.673869.1838609526581.486139047352
404566.864366.54160983693200.318390163067
413967.93954.3852459602213.5147540397843
424059.354091.29326631579-31.9432663157913
434215.384227.45803336696-12.0780333669645
444856.134733.32261446669122.807385533309

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 2595.93 & 2387.17917200855 & 208.750827991451 \tabularnewline
14 & 2686.64 & 2582.33343545803 & 104.306564541967 \tabularnewline
15 & 2779.43 & 2737.214243337 & 42.2157566630008 \tabularnewline
16 & 3221.13 & 3214.99322449782 & 6.13677550218472 \tabularnewline
17 & 2752.7 & 2765.22594272255 & -12.5259427225533 \tabularnewline
18 & 2886.58 & 2905.07861313341 & -18.4986131334063 \tabularnewline
19 & 2958.05 & 3011.7580484123 & -53.7080484123021 \tabularnewline
20 & 3444.61 & 3426.95539001845 & 17.6546099815514 \tabularnewline
21 & 2939.78 & 2982.08345443382 & -42.3034544338202 \tabularnewline
22 & 3088.73 & 3084.25154064992 & 4.47845935007763 \tabularnewline
23 & 3161.34 & 3184.14765732034 & -22.8076573203439 \tabularnewline
24 & 3672.39 & 3606.52960858264 & 65.8603914173605 \tabularnewline
25 & 3092.36 & 3249.20832050802 & -156.848320508023 \tabularnewline
26 & 3228.05 & 3233.78589044528 & -5.73589044527489 \tabularnewline
27 & 3311.16 & 3303.99093052561 & 7.16906947438656 \tabularnewline
28 & 3801.93 & 3739.436625538 & 62.4933744619984 \tabularnewline
29 & 3246.26 & 3298.66569394575 & -52.4056939457491 \tabularnewline
30 & 3309.22 & 3407.24129201517 & -98.0212920151694 \tabularnewline
31 & 3458.64 & 3448.42942925573 & 10.2105707442711 \tabularnewline
32 & 4005.04 & 3909.69045566692 & 95.3495443330776 \tabularnewline
33 & 3477.65 & 3467.77091825879 & 9.87908174120685 \tabularnewline
34 & 3524.42 & 3607.97831512743 & -83.5583151274304 \tabularnewline
35 & 3699.5 & 3647.73357467503 & 51.7664253249673 \tabularnewline
36 & 4247.68 & 4134.57056145363 & 113.109438546371 \tabularnewline
37 & 3697.6 & 3704.85171022574 & -7.25171022573932 \tabularnewline
38 & 3746.72 & 3824.14247496676 & -77.4224749667587 \tabularnewline
39 & 3950.67 & 3869.18386095265 & 81.486139047352 \tabularnewline
40 & 4566.86 & 4366.54160983693 & 200.318390163067 \tabularnewline
41 & 3967.9 & 3954.38524596022 & 13.5147540397843 \tabularnewline
42 & 4059.35 & 4091.29326631579 & -31.9432663157913 \tabularnewline
43 & 4215.38 & 4227.45803336696 & -12.0780333669645 \tabularnewline
44 & 4856.13 & 4733.32261446669 & 122.807385533309 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=297987&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]2595.93[/C][C]2387.17917200855[/C][C]208.750827991451[/C][/ROW]
[ROW][C]14[/C][C]2686.64[/C][C]2582.33343545803[/C][C]104.306564541967[/C][/ROW]
[ROW][C]15[/C][C]2779.43[/C][C]2737.214243337[/C][C]42.2157566630008[/C][/ROW]
[ROW][C]16[/C][C]3221.13[/C][C]3214.99322449782[/C][C]6.13677550218472[/C][/ROW]
[ROW][C]17[/C][C]2752.7[/C][C]2765.22594272255[/C][C]-12.5259427225533[/C][/ROW]
[ROW][C]18[/C][C]2886.58[/C][C]2905.07861313341[/C][C]-18.4986131334063[/C][/ROW]
[ROW][C]19[/C][C]2958.05[/C][C]3011.7580484123[/C][C]-53.7080484123021[/C][/ROW]
[ROW][C]20[/C][C]3444.61[/C][C]3426.95539001845[/C][C]17.6546099815514[/C][/ROW]
[ROW][C]21[/C][C]2939.78[/C][C]2982.08345443382[/C][C]-42.3034544338202[/C][/ROW]
[ROW][C]22[/C][C]3088.73[/C][C]3084.25154064992[/C][C]4.47845935007763[/C][/ROW]
[ROW][C]23[/C][C]3161.34[/C][C]3184.14765732034[/C][C]-22.8076573203439[/C][/ROW]
[ROW][C]24[/C][C]3672.39[/C][C]3606.52960858264[/C][C]65.8603914173605[/C][/ROW]
[ROW][C]25[/C][C]3092.36[/C][C]3249.20832050802[/C][C]-156.848320508023[/C][/ROW]
[ROW][C]26[/C][C]3228.05[/C][C]3233.78589044528[/C][C]-5.73589044527489[/C][/ROW]
[ROW][C]27[/C][C]3311.16[/C][C]3303.99093052561[/C][C]7.16906947438656[/C][/ROW]
[ROW][C]28[/C][C]3801.93[/C][C]3739.436625538[/C][C]62.4933744619984[/C][/ROW]
[ROW][C]29[/C][C]3246.26[/C][C]3298.66569394575[/C][C]-52.4056939457491[/C][/ROW]
[ROW][C]30[/C][C]3309.22[/C][C]3407.24129201517[/C][C]-98.0212920151694[/C][/ROW]
[ROW][C]31[/C][C]3458.64[/C][C]3448.42942925573[/C][C]10.2105707442711[/C][/ROW]
[ROW][C]32[/C][C]4005.04[/C][C]3909.69045566692[/C][C]95.3495443330776[/C][/ROW]
[ROW][C]33[/C][C]3477.65[/C][C]3467.77091825879[/C][C]9.87908174120685[/C][/ROW]
[ROW][C]34[/C][C]3524.42[/C][C]3607.97831512743[/C][C]-83.5583151274304[/C][/ROW]
[ROW][C]35[/C][C]3699.5[/C][C]3647.73357467503[/C][C]51.7664253249673[/C][/ROW]
[ROW][C]36[/C][C]4247.68[/C][C]4134.57056145363[/C][C]113.109438546371[/C][/ROW]
[ROW][C]37[/C][C]3697.6[/C][C]3704.85171022574[/C][C]-7.25171022573932[/C][/ROW]
[ROW][C]38[/C][C]3746.72[/C][C]3824.14247496676[/C][C]-77.4224749667587[/C][/ROW]
[ROW][C]39[/C][C]3950.67[/C][C]3869.18386095265[/C][C]81.486139047352[/C][/ROW]
[ROW][C]40[/C][C]4566.86[/C][C]4366.54160983693[/C][C]200.318390163067[/C][/ROW]
[ROW][C]41[/C][C]3967.9[/C][C]3954.38524596022[/C][C]13.5147540397843[/C][/ROW]
[ROW][C]42[/C][C]4059.35[/C][C]4091.29326631579[/C][C]-31.9432663157913[/C][/ROW]
[ROW][C]43[/C][C]4215.38[/C][C]4227.45803336696[/C][C]-12.0780333669645[/C][/ROW]
[ROW][C]44[/C][C]4856.13[/C][C]4733.32261446669[/C][C]122.807385533309[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=297987&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=297987&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
132595.932387.17917200855208.750827991451
142686.642582.33343545803104.306564541967
152779.432737.21424333742.2157566630008
163221.133214.993224497826.13677550218472
172752.72765.22594272255-12.5259427225533
182886.582905.07861313341-18.4986131334063
192958.053011.7580484123-53.7080484123021
203444.613426.9553900184517.6546099815514
212939.782982.08345443382-42.3034544338202
223088.733084.251540649924.47845935007763
233161.343184.14765732034-22.8076573203439
243672.393606.5296085826465.8603914173605
253092.363249.20832050802-156.848320508023
263228.053233.78589044528-5.73589044527489
273311.163303.990930525617.16906947438656
283801.933739.43662553862.4933744619984
293246.263298.66569394575-52.4056939457491
303309.223407.24129201517-98.0212920151694
313458.643448.4294292557310.2105707442711
324005.043909.6904556669295.3495443330776
333477.653467.770918258799.87908174120685
343524.423607.97831512743-83.5583151274304
353699.53647.7335746750351.7664253249673
364247.684134.57056145363113.109438546371
373697.63704.85171022574-7.25171022573932
383746.723824.14247496676-77.4224749667587
393950.673869.1838609526581.486139047352
404566.864366.54160983693200.318390163067
413967.93954.3852459602213.5147540397843
424059.354091.29326631579-31.9432663157913
434215.384227.45803336696-12.0780333669645
444856.134733.32261446669122.807385533309







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
454290.175809455534132.989916672924447.36170223813
464408.577807522734232.468919477114584.68669556834
474566.656710358894369.678149838564763.63527087921
485077.906572081124858.312071251285297.50107291096
494564.45506563454320.664958666134808.24517260287
504674.983498996734405.554716153594944.41228183987
514840.727649933584544.328924501745137.12637536542
525366.471733805525041.86377975585691.07968785523
534793.20191185774439.221787351595147.1820363638
544909.888084908434525.436891759665294.33927805719
555074.731502764164658.764678036935490.69832749138
565648.030044617175199.549645979036096.51044325532

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
45 & 4290.17580945553 & 4132.98991667292 & 4447.36170223813 \tabularnewline
46 & 4408.57780752273 & 4232.46891947711 & 4584.68669556834 \tabularnewline
47 & 4566.65671035889 & 4369.67814983856 & 4763.63527087921 \tabularnewline
48 & 5077.90657208112 & 4858.31207125128 & 5297.50107291096 \tabularnewline
49 & 4564.4550656345 & 4320.66495866613 & 4808.24517260287 \tabularnewline
50 & 4674.98349899673 & 4405.55471615359 & 4944.41228183987 \tabularnewline
51 & 4840.72764993358 & 4544.32892450174 & 5137.12637536542 \tabularnewline
52 & 5366.47173380552 & 5041.8637797558 & 5691.07968785523 \tabularnewline
53 & 4793.2019118577 & 4439.22178735159 & 5147.1820363638 \tabularnewline
54 & 4909.88808490843 & 4525.43689175966 & 5294.33927805719 \tabularnewline
55 & 5074.73150276416 & 4658.76467803693 & 5490.69832749138 \tabularnewline
56 & 5648.03004461717 & 5199.54964597903 & 6096.51044325532 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=297987&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]45[/C][C]4290.17580945553[/C][C]4132.98991667292[/C][C]4447.36170223813[/C][/ROW]
[ROW][C]46[/C][C]4408.57780752273[/C][C]4232.46891947711[/C][C]4584.68669556834[/C][/ROW]
[ROW][C]47[/C][C]4566.65671035889[/C][C]4369.67814983856[/C][C]4763.63527087921[/C][/ROW]
[ROW][C]48[/C][C]5077.90657208112[/C][C]4858.31207125128[/C][C]5297.50107291096[/C][/ROW]
[ROW][C]49[/C][C]4564.4550656345[/C][C]4320.66495866613[/C][C]4808.24517260287[/C][/ROW]
[ROW][C]50[/C][C]4674.98349899673[/C][C]4405.55471615359[/C][C]4944.41228183987[/C][/ROW]
[ROW][C]51[/C][C]4840.72764993358[/C][C]4544.32892450174[/C][C]5137.12637536542[/C][/ROW]
[ROW][C]52[/C][C]5366.47173380552[/C][C]5041.8637797558[/C][C]5691.07968785523[/C][/ROW]
[ROW][C]53[/C][C]4793.2019118577[/C][C]4439.22178735159[/C][C]5147.1820363638[/C][/ROW]
[ROW][C]54[/C][C]4909.88808490843[/C][C]4525.43689175966[/C][C]5294.33927805719[/C][/ROW]
[ROW][C]55[/C][C]5074.73150276416[/C][C]4658.76467803693[/C][C]5490.69832749138[/C][/ROW]
[ROW][C]56[/C][C]5648.03004461717[/C][C]5199.54964597903[/C][C]6096.51044325532[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=297987&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=297987&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
454290.175809455534132.989916672924447.36170223813
464408.577807522734232.468919477114584.68669556834
474566.656710358894369.678149838564763.63527087921
485077.906572081124858.312071251285297.50107291096
494564.45506563454320.664958666134808.24517260287
504674.983498996734405.554716153594944.41228183987
514840.727649933584544.328924501745137.12637536542
525366.471733805525041.86377975585691.07968785523
534793.20191185774439.221787351595147.1820363638
544909.888084908434525.436891759665294.33927805719
555074.731502764164658.764678036935490.69832749138
565648.030044617175199.549645979036096.51044325532



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):
par4 <- '12'
par3 <- 'additive'
par2 <- 'Double'
par1 <- '12'
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')