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Description of Statistical Computation

Author's title

Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationMon, 12 Dec 2016 17:51:07 +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/12/t14815621907kbr9p2fp31rix7.htm/, Retrieved Fri, 04 Apr 2025 14:23:16 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=298935, Retrieved Fri, 04 Apr 2025 14:23:16 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact110

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [Exponential smoot...] [2016-12-12 16:51:07] [cf891dd8f76140334edd84f5996aae27] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
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

Tables (Output of Computation)





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=298935&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=298935&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298935&T=0

As an alternative you can also use a QR Code:  
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
alpha0.626552721558841
beta0
gamma0

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.626552721558841 \tabularnewline
beta & 0 \tabularnewline
gamma & 0 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298935&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]0.626552721558841[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]0[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=298935&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298935&T=1

As an alternative you can also use a QR Code:  
QR Code


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

Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.626552721558841
beta0
gamma0






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
133707.62755.23322649573952.366773504271
144513.64183.24950998408330.350490015918
155634.25528.50229819889105.697701801108
164344.44359.08785388526-14.6878538852625
1740604174.79551201997-114.795512019967
184530.354655.91169450149-125.561694501488
195348.755913.00521271508-564.25521271508
204504.94258.60077982839246.29922017161
214281.354177.68059950511103.669400494895
224423.455079.28698415459-655.836984154595
235197.95973.44132646063-775.54132646063
244883.93972.74250397908911.157496020925
254155.254465.2660858401-310.016085840098
264415.755102.332953157-686.582953157004
275384.055810.42332490762-426.373324907616
285153.84307.63833074703846.161669252972
294564.14662.71360045662-98.6136004566233
3055455153.96860366818391.031396331818
317585.46734.73492888188850.665071118124
326252.25966.85265061969285.347349380312
335785.655910.3981819239-124.748181923896
346664.956668.888908657-3.93890865700087
358639.857971.49176434502668.358235654976
366841.356875.47214216462-34.1221421646214

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 3707.6 & 2755.23322649573 & 952.366773504271 \tabularnewline
14 & 4513.6 & 4183.24950998408 & 330.350490015918 \tabularnewline
15 & 5634.2 & 5528.50229819889 & 105.697701801108 \tabularnewline
16 & 4344.4 & 4359.08785388526 & -14.6878538852625 \tabularnewline
17 & 4060 & 4174.79551201997 & -114.795512019967 \tabularnewline
18 & 4530.35 & 4655.91169450149 & -125.561694501488 \tabularnewline
19 & 5348.75 & 5913.00521271508 & -564.25521271508 \tabularnewline
20 & 4504.9 & 4258.60077982839 & 246.29922017161 \tabularnewline
21 & 4281.35 & 4177.68059950511 & 103.669400494895 \tabularnewline
22 & 4423.45 & 5079.28698415459 & -655.836984154595 \tabularnewline
23 & 5197.9 & 5973.44132646063 & -775.54132646063 \tabularnewline
24 & 4883.9 & 3972.74250397908 & 911.157496020925 \tabularnewline
25 & 4155.25 & 4465.2660858401 & -310.016085840098 \tabularnewline
26 & 4415.75 & 5102.332953157 & -686.582953157004 \tabularnewline
27 & 5384.05 & 5810.42332490762 & -426.373324907616 \tabularnewline
28 & 5153.8 & 4307.63833074703 & 846.161669252972 \tabularnewline
29 & 4564.1 & 4662.71360045662 & -98.6136004566233 \tabularnewline
30 & 5545 & 5153.96860366818 & 391.031396331818 \tabularnewline
31 & 7585.4 & 6734.73492888188 & 850.665071118124 \tabularnewline
32 & 6252.2 & 5966.85265061969 & 285.347349380312 \tabularnewline
33 & 5785.65 & 5910.3981819239 & -124.748181923896 \tabularnewline
34 & 6664.95 & 6668.888908657 & -3.93890865700087 \tabularnewline
35 & 8639.85 & 7971.49176434502 & 668.358235654976 \tabularnewline
36 & 6841.35 & 6875.47214216462 & -34.1221421646214 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298935&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]2755.23322649573[/C][C]952.366773504271[/C][/ROW]
[ROW][C]14[/C][C]4513.6[/C][C]4183.24950998408[/C][C]330.350490015918[/C][/ROW]
[ROW][C]15[/C][C]5634.2[/C][C]5528.50229819889[/C][C]105.697701801108[/C][/ROW]
[ROW][C]16[/C][C]4344.4[/C][C]4359.08785388526[/C][C]-14.6878538852625[/C][/ROW]
[ROW][C]17[/C][C]4060[/C][C]4174.79551201997[/C][C]-114.795512019967[/C][/ROW]
[ROW][C]18[/C][C]4530.35[/C][C]4655.91169450149[/C][C]-125.561694501488[/C][/ROW]
[ROW][C]19[/C][C]5348.75[/C][C]5913.00521271508[/C][C]-564.25521271508[/C][/ROW]
[ROW][C]20[/C][C]4504.9[/C][C]4258.60077982839[/C][C]246.29922017161[/C][/ROW]
[ROW][C]21[/C][C]4281.35[/C][C]4177.68059950511[/C][C]103.669400494895[/C][/ROW]
[ROW][C]22[/C][C]4423.45[/C][C]5079.28698415459[/C][C]-655.836984154595[/C][/ROW]
[ROW][C]23[/C][C]5197.9[/C][C]5973.44132646063[/C][C]-775.54132646063[/C][/ROW]
[ROW][C]24[/C][C]4883.9[/C][C]3972.74250397908[/C][C]911.157496020925[/C][/ROW]
[ROW][C]25[/C][C]4155.25[/C][C]4465.2660858401[/C][C]-310.016085840098[/C][/ROW]
[ROW][C]26[/C][C]4415.75[/C][C]5102.332953157[/C][C]-686.582953157004[/C][/ROW]
[ROW][C]27[/C][C]5384.05[/C][C]5810.42332490762[/C][C]-426.373324907616[/C][/ROW]
[ROW][C]28[/C][C]5153.8[/C][C]4307.63833074703[/C][C]846.161669252972[/C][/ROW]
[ROW][C]29[/C][C]4564.1[/C][C]4662.71360045662[/C][C]-98.6136004566233[/C][/ROW]
[ROW][C]30[/C][C]5545[/C][C]5153.96860366818[/C][C]391.031396331818[/C][/ROW]
[ROW][C]31[/C][C]7585.4[/C][C]6734.73492888188[/C][C]850.665071118124[/C][/ROW]
[ROW][C]32[/C][C]6252.2[/C][C]5966.85265061969[/C][C]285.347349380312[/C][/ROW]
[ROW][C]33[/C][C]5785.65[/C][C]5910.3981819239[/C][C]-124.748181923896[/C][/ROW]
[ROW][C]34[/C][C]6664.95[/C][C]6668.888908657[/C][C]-3.93890865700087[/C][/ROW]
[ROW][C]35[/C][C]8639.85[/C][C]7971.49176434502[/C][C]668.358235654976[/C][/ROW]
[ROW][C]36[/C][C]6841.35[/C][C]6875.47214216462[/C][C]-34.1221421646214[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=298935&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298935&T=2

As an alternative you can also use a QR Code:  
QR Code


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

Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
133707.62755.23322649573952.366773504271
144513.64183.24950998408330.350490015918
155634.25528.50229819889105.697701801108
164344.44359.08785388526-14.6878538852625
1740604174.79551201997-114.795512019967
184530.354655.91169450149-125.561694501488
195348.755913.00521271508-564.25521271508
204504.94258.60077982839246.29922017161
214281.354177.68059950511103.669400494895
224423.455079.28698415459-655.836984154595
235197.95973.44132646063-775.54132646063
244883.93972.74250397908911.157496020925
254155.254465.2660858401-310.016085840098
264415.755102.332953157-686.582953157004
275384.055810.42332490762-426.373324907616
285153.84307.63833074703846.161669252972
294564.14662.71360045662-98.6136004566233
3055455153.96860366818391.031396331818
317585.46734.73492888188850.665071118124
326252.25966.85265061969285.347349380312
335785.655910.3981819239-124.748181923896
346664.956668.888908657-3.93890865700087
358639.857971.49176434502668.358235654976
366841.356875.47214216462-34.1221421646214






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
376775.728194086335768.349793541737783.10659463094
387607.036483713376418.258127999088795.81483942766
398745.307273340417399.3590142510310091.2555324298
407509.667646300786023.074408970318996.26088363126
417334.578019261165719.541737808748949.61430071358
427887.619642221536153.628602008639621.61068243443
439223.384181848577378.0908001150311068.6775635821
447922.515388142275972.261455496449872.76932078811
457687.275761102655637.428659748139737.12286245717
468523.927800729696379.1070911918310668.7485102675
479828.998590356737593.2350489254212064.762131788
488314.217296650435991.0682806972110637.3663126037

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
37 & 6775.72819408633 & 5768.34979354173 & 7783.10659463094 \tabularnewline
38 & 7607.03648371337 & 6418.25812799908 & 8795.81483942766 \tabularnewline
39 & 8745.30727334041 & 7399.35901425103 & 10091.2555324298 \tabularnewline
40 & 7509.66764630078 & 6023.07440897031 & 8996.26088363126 \tabularnewline
41 & 7334.57801926116 & 5719.54173780874 & 8949.61430071358 \tabularnewline
42 & 7887.61964222153 & 6153.62860200863 & 9621.61068243443 \tabularnewline
43 & 9223.38418184857 & 7378.09080011503 & 11068.6775635821 \tabularnewline
44 & 7922.51538814227 & 5972.26145549644 & 9872.76932078811 \tabularnewline
45 & 7687.27576110265 & 5637.42865974813 & 9737.12286245717 \tabularnewline
46 & 8523.92780072969 & 6379.10709119183 & 10668.7485102675 \tabularnewline
47 & 9828.99859035673 & 7593.23504892542 & 12064.762131788 \tabularnewline
48 & 8314.21729665043 & 5991.06828069721 & 10637.3663126037 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=298935&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]6775.72819408633[/C][C]5768.34979354173[/C][C]7783.10659463094[/C][/ROW]
[ROW][C]38[/C][C]7607.03648371337[/C][C]6418.25812799908[/C][C]8795.81483942766[/C][/ROW]
[ROW][C]39[/C][C]8745.30727334041[/C][C]7399.35901425103[/C][C]10091.2555324298[/C][/ROW]
[ROW][C]40[/C][C]7509.66764630078[/C][C]6023.07440897031[/C][C]8996.26088363126[/C][/ROW]
[ROW][C]41[/C][C]7334.57801926116[/C][C]5719.54173780874[/C][C]8949.61430071358[/C][/ROW]
[ROW][C]42[/C][C]7887.61964222153[/C][C]6153.62860200863[/C][C]9621.61068243443[/C][/ROW]
[ROW][C]43[/C][C]9223.38418184857[/C][C]7378.09080011503[/C][C]11068.6775635821[/C][/ROW]
[ROW][C]44[/C][C]7922.51538814227[/C][C]5972.26145549644[/C][C]9872.76932078811[/C][/ROW]
[ROW][C]45[/C][C]7687.27576110265[/C][C]5637.42865974813[/C][C]9737.12286245717[/C][/ROW]
[ROW][C]46[/C][C]8523.92780072969[/C][C]6379.10709119183[/C][C]10668.7485102675[/C][/ROW]
[ROW][C]47[/C][C]9828.99859035673[/C][C]7593.23504892542[/C][C]12064.762131788[/C][/ROW]
[ROW][C]48[/C][C]8314.21729665043[/C][C]5991.06828069721[/C][C]10637.3663126037[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=298935&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=298935&T=3

As an alternative you can also use a QR Code:  
QR Code


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

Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
376775.728194086335768.349793541737783.10659463094
387607.036483713376418.258127999088795.81483942766
398745.307273340417399.3590142510310091.2555324298
407509.667646300786023.074408970318996.26088363126
417334.578019261165719.541737808748949.61430071358
427887.619642221536153.628602008639621.61068243443
439223.384181848577378.0908001150311068.6775635821
447922.515388142275972.261455496449872.76932078811
457687.275761102655637.428659748139737.12286245717
468523.927800729696379.1070911918310668.7485102675
479828.998590356737593.2350489254212064.762131788
488314.217296650435991.0682806972110637.3663126037


Figures (Output of Computation)

Input Parameters & R Code

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')