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Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationFri, 16 Dec 2016 20:47:26 +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/16/t1481917663r5pgw76krknvg58.htm/, Retrieved Thu, 02 May 2024 19:48:24 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=300508, Retrieved Thu, 02 May 2024 19:48:24 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact51
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2016-12-16 19:47:26] [404ac5ee4f7301873f6a96ef36861981] [Current]
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Dataseries X:
2151.06
2332.26
2457.82
2533.53
2867.22
3186.52
3487.47
3733.52
4052.26
4482.15
5116.47
5671.98
6086.66
6229.16
6526.62
6735.6
6874.34
7206.03
7491.36




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=300508&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=300508&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300508&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
alpha1
beta0.870279173271583
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.870279173271583 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300508&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]0.870279173271583[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=300508&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300508&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
beta0.870279173271583
gammaFALSE







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
32457.822513.46-55.6400000000003
42533.532590.59766679917-57.0676667991693
52867.222616.64286491665250.57713508335
63186.523168.4049268777518.115073122251
73487.473503.47009773834-16.0000977383361
83733.523790.49554590635-56.9755459063522
94052.263986.9609149182865.299085081725
104482.154362.52934869859119.620651301409
115116.474896.52271021939219.947289780612
125671.985722.25825573299-50.2782557329856
136086.666234.01213690014-147.352136900145
146229.166520.45464101889-291.294641018886
156526.626409.44698165453117.173018345473
166735.66808.88021918996-73.2802191899609
176874.346954.08597061616-79.7459706161617
187206.037023.42471323659182.605286763411
197491.367514.03229123607-22.6722912360701

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 2457.82 & 2513.46 & -55.6400000000003 \tabularnewline
4 & 2533.53 & 2590.59766679917 & -57.0676667991693 \tabularnewline
5 & 2867.22 & 2616.64286491665 & 250.57713508335 \tabularnewline
6 & 3186.52 & 3168.40492687775 & 18.115073122251 \tabularnewline
7 & 3487.47 & 3503.47009773834 & -16.0000977383361 \tabularnewline
8 & 3733.52 & 3790.49554590635 & -56.9755459063522 \tabularnewline
9 & 4052.26 & 3986.96091491828 & 65.299085081725 \tabularnewline
10 & 4482.15 & 4362.52934869859 & 119.620651301409 \tabularnewline
11 & 5116.47 & 4896.52271021939 & 219.947289780612 \tabularnewline
12 & 5671.98 & 5722.25825573299 & -50.2782557329856 \tabularnewline
13 & 6086.66 & 6234.01213690014 & -147.352136900145 \tabularnewline
14 & 6229.16 & 6520.45464101889 & -291.294641018886 \tabularnewline
15 & 6526.62 & 6409.44698165453 & 117.173018345473 \tabularnewline
16 & 6735.6 & 6808.88021918996 & -73.2802191899609 \tabularnewline
17 & 6874.34 & 6954.08597061616 & -79.7459706161617 \tabularnewline
18 & 7206.03 & 7023.42471323659 & 182.605286763411 \tabularnewline
19 & 7491.36 & 7514.03229123607 & -22.6722912360701 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300508&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]3[/C][C]2457.82[/C][C]2513.46[/C][C]-55.6400000000003[/C][/ROW]
[ROW][C]4[/C][C]2533.53[/C][C]2590.59766679917[/C][C]-57.0676667991693[/C][/ROW]
[ROW][C]5[/C][C]2867.22[/C][C]2616.64286491665[/C][C]250.57713508335[/C][/ROW]
[ROW][C]6[/C][C]3186.52[/C][C]3168.40492687775[/C][C]18.115073122251[/C][/ROW]
[ROW][C]7[/C][C]3487.47[/C][C]3503.47009773834[/C][C]-16.0000977383361[/C][/ROW]
[ROW][C]8[/C][C]3733.52[/C][C]3790.49554590635[/C][C]-56.9755459063522[/C][/ROW]
[ROW][C]9[/C][C]4052.26[/C][C]3986.96091491828[/C][C]65.299085081725[/C][/ROW]
[ROW][C]10[/C][C]4482.15[/C][C]4362.52934869859[/C][C]119.620651301409[/C][/ROW]
[ROW][C]11[/C][C]5116.47[/C][C]4896.52271021939[/C][C]219.947289780612[/C][/ROW]
[ROW][C]12[/C][C]5671.98[/C][C]5722.25825573299[/C][C]-50.2782557329856[/C][/ROW]
[ROW][C]13[/C][C]6086.66[/C][C]6234.01213690014[/C][C]-147.352136900145[/C][/ROW]
[ROW][C]14[/C][C]6229.16[/C][C]6520.45464101889[/C][C]-291.294641018886[/C][/ROW]
[ROW][C]15[/C][C]6526.62[/C][C]6409.44698165453[/C][C]117.173018345473[/C][/ROW]
[ROW][C]16[/C][C]6735.6[/C][C]6808.88021918996[/C][C]-73.2802191899609[/C][/ROW]
[ROW][C]17[/C][C]6874.34[/C][C]6954.08597061616[/C][C]-79.7459706161617[/C][/ROW]
[ROW][C]18[/C][C]7206.03[/C][C]7023.42471323659[/C][C]182.605286763411[/C][/ROW]
[ROW][C]19[/C][C]7491.36[/C][C]7514.03229123607[/C][C]-22.6722912360701[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=300508&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300508&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
32457.822513.46-55.6400000000003
42533.532590.59766679917-57.0676667991693
52867.222616.64286491665250.57713508335
63186.523168.4049268777518.115073122251
73487.473503.47009773834-16.0000977383361
83733.523790.49554590635-56.9755459063522
94052.263986.9609149182865.299085081725
104482.154362.52934869859119.620651301409
115116.474896.52271021939219.947289780612
125671.985722.25825573299-50.2782557329856
136086.666234.01213690014-147.352136900145
146229.166520.45464101889-291.294641018886
156526.626409.44698165453117.173018345473
166735.66808.88021918996-73.2802191899609
176874.346954.08597061616-79.7459706161617
187206.037023.42471323659182.605286763411
197491.367514.03229123607-22.6722912360701







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
207779.631068362977507.943512435288051.31862429066
218067.902136725947491.697461209488644.1068122424
228356.173205088917414.682551504399297.66385867343
238644.444273451887284.7368980178510004.1516488859
248932.715341814857107.6066144183410757.8240692114
259220.986410177826887.5620249708611554.4107953848
269509.257478540796627.910886849312390.6040702323
279797.528546903766331.3058366031313263.7512572044
2810085.79961526675999.9329997251514171.6662308083
2910374.07068362975635.6333779458815112.5079893135
3010662.34175199275239.9845406163816084.698963369
3110950.61282035564814.3577381197417086.8679025916

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
20 & 7779.63106836297 & 7507.94351243528 & 8051.31862429066 \tabularnewline
21 & 8067.90213672594 & 7491.69746120948 & 8644.1068122424 \tabularnewline
22 & 8356.17320508891 & 7414.68255150439 & 9297.66385867343 \tabularnewline
23 & 8644.44427345188 & 7284.73689801785 & 10004.1516488859 \tabularnewline
24 & 8932.71534181485 & 7107.60661441834 & 10757.8240692114 \tabularnewline
25 & 9220.98641017782 & 6887.56202497086 & 11554.4107953848 \tabularnewline
26 & 9509.25747854079 & 6627.9108868493 & 12390.6040702323 \tabularnewline
27 & 9797.52854690376 & 6331.30583660313 & 13263.7512572044 \tabularnewline
28 & 10085.7996152667 & 5999.93299972515 & 14171.6662308083 \tabularnewline
29 & 10374.0706836297 & 5635.63337794588 & 15112.5079893135 \tabularnewline
30 & 10662.3417519927 & 5239.98454061638 & 16084.698963369 \tabularnewline
31 & 10950.6128203556 & 4814.35773811974 & 17086.8679025916 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300508&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]20[/C][C]7779.63106836297[/C][C]7507.94351243528[/C][C]8051.31862429066[/C][/ROW]
[ROW][C]21[/C][C]8067.90213672594[/C][C]7491.69746120948[/C][C]8644.1068122424[/C][/ROW]
[ROW][C]22[/C][C]8356.17320508891[/C][C]7414.68255150439[/C][C]9297.66385867343[/C][/ROW]
[ROW][C]23[/C][C]8644.44427345188[/C][C]7284.73689801785[/C][C]10004.1516488859[/C][/ROW]
[ROW][C]24[/C][C]8932.71534181485[/C][C]7107.60661441834[/C][C]10757.8240692114[/C][/ROW]
[ROW][C]25[/C][C]9220.98641017782[/C][C]6887.56202497086[/C][C]11554.4107953848[/C][/ROW]
[ROW][C]26[/C][C]9509.25747854079[/C][C]6627.9108868493[/C][C]12390.6040702323[/C][/ROW]
[ROW][C]27[/C][C]9797.52854690376[/C][C]6331.30583660313[/C][C]13263.7512572044[/C][/ROW]
[ROW][C]28[/C][C]10085.7996152667[/C][C]5999.93299972515[/C][C]14171.6662308083[/C][/ROW]
[ROW][C]29[/C][C]10374.0706836297[/C][C]5635.63337794588[/C][C]15112.5079893135[/C][/ROW]
[ROW][C]30[/C][C]10662.3417519927[/C][C]5239.98454061638[/C][C]16084.698963369[/C][/ROW]
[ROW][C]31[/C][C]10950.6128203556[/C][C]4814.35773811974[/C][C]17086.8679025916[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=300508&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300508&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
207779.631068362977507.943512435288051.31862429066
218067.902136725947491.697461209488644.1068122424
228356.173205088917414.682551504399297.66385867343
238644.444273451887284.7368980178510004.1516488859
248932.715341814857107.6066144183410757.8240692114
259220.986410177826887.5620249708611554.4107953848
269509.257478540796627.910886849312390.6040702323
279797.528546903766331.3058366031313263.7512572044
2810085.79961526675999.9329997251514171.6662308083
2910374.07068362975635.6333779458815112.5079893135
3010662.34175199275239.9845406163816084.698963369
3110950.61282035564814.3577381197417086.8679025916



Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = additive ; par4 = 12 ;
Parameters (R input):
par1 = 12 ; par2 = Double ; par3 = additive ; par4 = 12 ;
R code (references can be found in the software module):
par4 <- '12'
par3 <- 'additive'
par2 <- 'Triple'
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')