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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 computationFri, 23 Dec 2016 17:42:46 +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/23/t14825113825gk1rvlviybwhuw.htm/, Retrieved Fri, 01 Nov 2024 03:36:24 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=303005, Retrieved Fri, 01 Nov 2024 03:36:24 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact115
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2016-12-23 16:42:46] [2802fcbee976b89d2ab84425d3d65dcf] [Current]
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Dataseries X:
1550.61
1488.54
1200.03
1451.49
2576.19
2434.2
2586.21
1898.55
2958.18
3290.73
3408.39
3214.71
4205.43
4378.53
4279.68
4799.25
4902.84
5379.84
5527.05
6004.83
5827.71
6496.02
6858.99
6696.84
6831
7366.47
7881.03
7494.66
5813.55
6911.25
7252.59
7425.63
7603.5
6045.72
6064.35
5486.85
5808.27
6467.88




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=303005&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=303005&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=303005&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
alpha0.370572845174158
beta0.326592840518882
gamma0.175547325767585

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.370572845174158 \tabularnewline
beta & 0.326592840518882 \tabularnewline
gamma & 0.175547325767585 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=303005&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]0.370572845174158[/C][/ROW]
[ROW][C]beta[/C][C]0.326592840518882[/C][/ROW]
[ROW][C]gamma[/C][C]0.175547325767585[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=303005&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=303005&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
alpha0.370572845174158
beta0.326592840518882
gamma0.175547325767585







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
134205.432769.840336538461435.58966346154
144378.533613.67330683662764.856693163381
154279.684032.53998322859247.140016771408
164799.254945.42664380378-146.176643803778
174902.845254.67834187864-351.838341878641
185379.845807.01065804132-427.170658041318
195527.056150.52042025596-623.470420255955
206004.835368.20791684008636.622083159918
215827.716859.48833075396-1031.77833075396
226496.026861.47865008874-365.458650088737
236858.996882.6437905811-23.6537905811028
246696.846733.02479633339-36.1847963333885
2568317891.81599323692-1060.81599323692
267366.477443.80336341346-77.3333634134642
277881.037098.81197959459782.218020405405
287494.667836.72279541718-342.062795417185
295813.557697.15057385383-1883.60057385383
306911.257134.63649129076-223.386491290763
317252.597017.74328585021234.846714149794
327425.636282.382001045751143.24799895425
337603.57428.01990847284175.480091527163
346045.727748.08644408686-1702.36644408686
356064.356946.87067607801-882.520676078011
365486.856008.9220759024-522.072075902404
375808.276346.96331847518-538.693318475175
386467.885736.81753747166731.062462528338

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 4205.43 & 2769.84033653846 & 1435.58966346154 \tabularnewline
14 & 4378.53 & 3613.67330683662 & 764.856693163381 \tabularnewline
15 & 4279.68 & 4032.53998322859 & 247.140016771408 \tabularnewline
16 & 4799.25 & 4945.42664380378 & -146.176643803778 \tabularnewline
17 & 4902.84 & 5254.67834187864 & -351.838341878641 \tabularnewline
18 & 5379.84 & 5807.01065804132 & -427.170658041318 \tabularnewline
19 & 5527.05 & 6150.52042025596 & -623.470420255955 \tabularnewline
20 & 6004.83 & 5368.20791684008 & 636.622083159918 \tabularnewline
21 & 5827.71 & 6859.48833075396 & -1031.77833075396 \tabularnewline
22 & 6496.02 & 6861.47865008874 & -365.458650088737 \tabularnewline
23 & 6858.99 & 6882.6437905811 & -23.6537905811028 \tabularnewline
24 & 6696.84 & 6733.02479633339 & -36.1847963333885 \tabularnewline
25 & 6831 & 7891.81599323692 & -1060.81599323692 \tabularnewline
26 & 7366.47 & 7443.80336341346 & -77.3333634134642 \tabularnewline
27 & 7881.03 & 7098.81197959459 & 782.218020405405 \tabularnewline
28 & 7494.66 & 7836.72279541718 & -342.062795417185 \tabularnewline
29 & 5813.55 & 7697.15057385383 & -1883.60057385383 \tabularnewline
30 & 6911.25 & 7134.63649129076 & -223.386491290763 \tabularnewline
31 & 7252.59 & 7017.74328585021 & 234.846714149794 \tabularnewline
32 & 7425.63 & 6282.38200104575 & 1143.24799895425 \tabularnewline
33 & 7603.5 & 7428.01990847284 & 175.480091527163 \tabularnewline
34 & 6045.72 & 7748.08644408686 & -1702.36644408686 \tabularnewline
35 & 6064.35 & 6946.87067607801 & -882.520676078011 \tabularnewline
36 & 5486.85 & 6008.9220759024 & -522.072075902404 \tabularnewline
37 & 5808.27 & 6346.96331847518 & -538.693318475175 \tabularnewline
38 & 6467.88 & 5736.81753747166 & 731.062462528338 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=303005&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]4205.43[/C][C]2769.84033653846[/C][C]1435.58966346154[/C][/ROW]
[ROW][C]14[/C][C]4378.53[/C][C]3613.67330683662[/C][C]764.856693163381[/C][/ROW]
[ROW][C]15[/C][C]4279.68[/C][C]4032.53998322859[/C][C]247.140016771408[/C][/ROW]
[ROW][C]16[/C][C]4799.25[/C][C]4945.42664380378[/C][C]-146.176643803778[/C][/ROW]
[ROW][C]17[/C][C]4902.84[/C][C]5254.67834187864[/C][C]-351.838341878641[/C][/ROW]
[ROW][C]18[/C][C]5379.84[/C][C]5807.01065804132[/C][C]-427.170658041318[/C][/ROW]
[ROW][C]19[/C][C]5527.05[/C][C]6150.52042025596[/C][C]-623.470420255955[/C][/ROW]
[ROW][C]20[/C][C]6004.83[/C][C]5368.20791684008[/C][C]636.622083159918[/C][/ROW]
[ROW][C]21[/C][C]5827.71[/C][C]6859.48833075396[/C][C]-1031.77833075396[/C][/ROW]
[ROW][C]22[/C][C]6496.02[/C][C]6861.47865008874[/C][C]-365.458650088737[/C][/ROW]
[ROW][C]23[/C][C]6858.99[/C][C]6882.6437905811[/C][C]-23.6537905811028[/C][/ROW]
[ROW][C]24[/C][C]6696.84[/C][C]6733.02479633339[/C][C]-36.1847963333885[/C][/ROW]
[ROW][C]25[/C][C]6831[/C][C]7891.81599323692[/C][C]-1060.81599323692[/C][/ROW]
[ROW][C]26[/C][C]7366.47[/C][C]7443.80336341346[/C][C]-77.3333634134642[/C][/ROW]
[ROW][C]27[/C][C]7881.03[/C][C]7098.81197959459[/C][C]782.218020405405[/C][/ROW]
[ROW][C]28[/C][C]7494.66[/C][C]7836.72279541718[/C][C]-342.062795417185[/C][/ROW]
[ROW][C]29[/C][C]5813.55[/C][C]7697.15057385383[/C][C]-1883.60057385383[/C][/ROW]
[ROW][C]30[/C][C]6911.25[/C][C]7134.63649129076[/C][C]-223.386491290763[/C][/ROW]
[ROW][C]31[/C][C]7252.59[/C][C]7017.74328585021[/C][C]234.846714149794[/C][/ROW]
[ROW][C]32[/C][C]7425.63[/C][C]6282.38200104575[/C][C]1143.24799895425[/C][/ROW]
[ROW][C]33[/C][C]7603.5[/C][C]7428.01990847284[/C][C]175.480091527163[/C][/ROW]
[ROW][C]34[/C][C]6045.72[/C][C]7748.08644408686[/C][C]-1702.36644408686[/C][/ROW]
[ROW][C]35[/C][C]6064.35[/C][C]6946.87067607801[/C][C]-882.520676078011[/C][/ROW]
[ROW][C]36[/C][C]5486.85[/C][C]6008.9220759024[/C][C]-522.072075902404[/C][/ROW]
[ROW][C]37[/C][C]5808.27[/C][C]6346.96331847518[/C][C]-538.693318475175[/C][/ROW]
[ROW][C]38[/C][C]6467.88[/C][C]5736.81753747166[/C][C]731.062462528338[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=303005&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=303005&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
134205.432769.840336538461435.58966346154
144378.533613.67330683662764.856693163381
154279.684032.53998322859247.140016771408
164799.254945.42664380378-146.176643803778
174902.845254.67834187864-351.838341878641
185379.845807.01065804132-427.170658041318
195527.056150.52042025596-623.470420255955
206004.835368.20791684008636.622083159918
215827.716859.48833075396-1031.77833075396
226496.026861.47865008874-365.458650088737
236858.996882.6437905811-23.6537905811028
246696.846733.02479633339-36.1847963333885
2568317891.81599323692-1060.81599323692
267366.477443.80336341346-77.3333634134642
277881.037098.81197959459782.218020405405
287494.667836.72279541718-342.062795417185
295813.557697.15057385383-1883.60057385383
306911.257134.63649129076-223.386491290763
317252.597017.74328585021234.846714149794
327425.636282.382001045751143.24799895425
337603.57428.01990847284175.480091527163
346045.727748.08644408686-1702.36644408686
356064.356946.87067607801-882.520676078011
365486.856008.9220759024-522.072075902404
375808.276346.96331847518-538.693318475175
386467.885736.81753747166731.062462528338







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
395419.921635461843851.405744184346988.43752673934
405282.618515078423534.817481240297030.41954891655
414679.754443022552685.221266476876674.28761956824
424806.941201487142504.252162430767109.63024054352
434658.742272172991994.249186675227323.23535767076
443743.58480549446670.8539903075046816.31562068142
454027.12853428201505.6369633101887548.62010525382
463721.93320882422-284.2221004687887728.08851811723
473495.44535841495-1027.703919453898018.59463628379
482884.45968372556-2185.231851425777954.15121887689
493437.41186136607-2206.171332962569080.99505569469
503255.6697994767-2987.384638303629498.72423725702

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
39 & 5419.92163546184 & 3851.40574418434 & 6988.43752673934 \tabularnewline
40 & 5282.61851507842 & 3534.81748124029 & 7030.41954891655 \tabularnewline
41 & 4679.75444302255 & 2685.22126647687 & 6674.28761956824 \tabularnewline
42 & 4806.94120148714 & 2504.25216243076 & 7109.63024054352 \tabularnewline
43 & 4658.74227217299 & 1994.24918667522 & 7323.23535767076 \tabularnewline
44 & 3743.58480549446 & 670.853990307504 & 6816.31562068142 \tabularnewline
45 & 4027.12853428201 & 505.636963310188 & 7548.62010525382 \tabularnewline
46 & 3721.93320882422 & -284.222100468788 & 7728.08851811723 \tabularnewline
47 & 3495.44535841495 & -1027.70391945389 & 8018.59463628379 \tabularnewline
48 & 2884.45968372556 & -2185.23185142577 & 7954.15121887689 \tabularnewline
49 & 3437.41186136607 & -2206.17133296256 & 9080.99505569469 \tabularnewline
50 & 3255.6697994767 & -2987.38463830362 & 9498.72423725702 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=303005&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]39[/C][C]5419.92163546184[/C][C]3851.40574418434[/C][C]6988.43752673934[/C][/ROW]
[ROW][C]40[/C][C]5282.61851507842[/C][C]3534.81748124029[/C][C]7030.41954891655[/C][/ROW]
[ROW][C]41[/C][C]4679.75444302255[/C][C]2685.22126647687[/C][C]6674.28761956824[/C][/ROW]
[ROW][C]42[/C][C]4806.94120148714[/C][C]2504.25216243076[/C][C]7109.63024054352[/C][/ROW]
[ROW][C]43[/C][C]4658.74227217299[/C][C]1994.24918667522[/C][C]7323.23535767076[/C][/ROW]
[ROW][C]44[/C][C]3743.58480549446[/C][C]670.853990307504[/C][C]6816.31562068142[/C][/ROW]
[ROW][C]45[/C][C]4027.12853428201[/C][C]505.636963310188[/C][C]7548.62010525382[/C][/ROW]
[ROW][C]46[/C][C]3721.93320882422[/C][C]-284.222100468788[/C][C]7728.08851811723[/C][/ROW]
[ROW][C]47[/C][C]3495.44535841495[/C][C]-1027.70391945389[/C][C]8018.59463628379[/C][/ROW]
[ROW][C]48[/C][C]2884.45968372556[/C][C]-2185.23185142577[/C][C]7954.15121887689[/C][/ROW]
[ROW][C]49[/C][C]3437.41186136607[/C][C]-2206.17133296256[/C][C]9080.99505569469[/C][/ROW]
[ROW][C]50[/C][C]3255.6697994767[/C][C]-2987.38463830362[/C][C]9498.72423725702[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=303005&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=303005&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
395419.921635461843851.405744184346988.43752673934
405282.618515078423534.817481240297030.41954891655
414679.754443022552685.221266476876674.28761956824
424806.941201487142504.252162430767109.63024054352
434658.742272172991994.249186675227323.23535767076
443743.58480549446670.8539903075046816.31562068142
454027.12853428201505.6369633101887548.62010525382
463721.93320882422-284.2221004687887728.08851811723
473495.44535841495-1027.703919453898018.59463628379
482884.45968372556-2185.231851425777954.15121887689
493437.41186136607-2206.171332962569080.99505569469
503255.6697994767-2987.384638303629498.72423725702



Parameters (Session):
par1 = 12 ; par2 = Single ; 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')