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Author's title

Author*Unverified author*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationWed, 16 Jan 2013 03:14:35 -0500
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2013/Jan/16/t1358324091byeo0vanfxskoun.htm/, Retrieved Fri, 03 May 2024 19:55:49 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=205630, Retrieved Fri, 03 May 2024 19:55:49 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact109
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2013-01-16 08:14:35] [7d095200a4be23015976b6928166d958] [Current]
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Dataseries X:
1103
1125
1150
1187
1248
1172
1092
5255
5297
5461
5729
5795
5411
5503
5583
5763
5840
5907
5836
5756
5916
6056
6469
6603
6247
6342
6431




Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Sir Ronald Aylmer Fisher' @ fisher.wessa.net

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input & view raw input (R code)  \tabularnewline
Raw Output & view raw output of R engine  \tabularnewline
Computing time & 3 seconds \tabularnewline
R Server & 'Sir Ronald Aylmer Fisher' @ fisher.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=205630&T=0

[TABLE]
[ROW][C]Summary of computational transaction[/C][/ROW]
[ROW][C]Raw Input[/C][C]view raw input (R code) [/C][/ROW]
[ROW][C]Raw Output[/C][C]view raw output of R engine [/C][/ROW]
[ROW][C]Computing time[/C][C]3 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Sir Ronald Aylmer Fisher' @ fisher.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=205630&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=205630&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 Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Sir Ronald Aylmer Fisher' @ fisher.wessa.net







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.975226252579178
beta0.00822673416033337
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.975226252579178 \tabularnewline
beta & 0.00822673416033337 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=205630&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]0.975226252579178[/C][/ROW]
[ROW][C]beta[/C][C]0.00822673416033337[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=205630&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=205630&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
alpha0.975226252579178
beta0.00822673416033337
gammaFALSE







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
3115011473
411871171.9497475391215.0502524608839
512481208.7719647070239.2280352929833
611721269.4877140904-97.4877140903964
710921196.09253870823-104.092538708227
852551115.421038112644139.57896188736
952975206.5009325640990.4990674359078
1054615349.53788257899111.462117421014
1157295513.91280171888215.08719828112
1257955781.0712490567713.9287509432343
1354115852.16644697645-441.166446976454
1455035475.9014142075527.098585792447
1555835556.5181445387726.481855461233
1657635636.74588525675126.254114743249
1758405815.2870800676524.7129199323463
1859075895.000905936411.9990940636017
1958365962.41214290448-126.412142904475
2057565893.82691251894-137.826912518944
2159165813.00392386308102.996076136922
2260565967.8641659821288.135834017884
2364696108.93941724051360.060582759487
2466036518.091562014284.908437985805
2562476659.58932596147-412.589325961471
2663426312.6040358127129.3959641872907
2764316396.8902455493234.1097544506774

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 1150 & 1147 & 3 \tabularnewline
4 & 1187 & 1171.94974753912 & 15.0502524608839 \tabularnewline
5 & 1248 & 1208.77196470702 & 39.2280352929833 \tabularnewline
6 & 1172 & 1269.4877140904 & -97.4877140903964 \tabularnewline
7 & 1092 & 1196.09253870823 & -104.092538708227 \tabularnewline
8 & 5255 & 1115.42103811264 & 4139.57896188736 \tabularnewline
9 & 5297 & 5206.50093256409 & 90.4990674359078 \tabularnewline
10 & 5461 & 5349.53788257899 & 111.462117421014 \tabularnewline
11 & 5729 & 5513.91280171888 & 215.08719828112 \tabularnewline
12 & 5795 & 5781.07124905677 & 13.9287509432343 \tabularnewline
13 & 5411 & 5852.16644697645 & -441.166446976454 \tabularnewline
14 & 5503 & 5475.90141420755 & 27.098585792447 \tabularnewline
15 & 5583 & 5556.51814453877 & 26.481855461233 \tabularnewline
16 & 5763 & 5636.74588525675 & 126.254114743249 \tabularnewline
17 & 5840 & 5815.28708006765 & 24.7129199323463 \tabularnewline
18 & 5907 & 5895.0009059364 & 11.9990940636017 \tabularnewline
19 & 5836 & 5962.41214290448 & -126.412142904475 \tabularnewline
20 & 5756 & 5893.82691251894 & -137.826912518944 \tabularnewline
21 & 5916 & 5813.00392386308 & 102.996076136922 \tabularnewline
22 & 6056 & 5967.86416598212 & 88.135834017884 \tabularnewline
23 & 6469 & 6108.93941724051 & 360.060582759487 \tabularnewline
24 & 6603 & 6518.0915620142 & 84.908437985805 \tabularnewline
25 & 6247 & 6659.58932596147 & -412.589325961471 \tabularnewline
26 & 6342 & 6312.60403581271 & 29.3959641872907 \tabularnewline
27 & 6431 & 6396.89024554932 & 34.1097544506774 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=205630&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]1150[/C][C]1147[/C][C]3[/C][/ROW]
[ROW][C]4[/C][C]1187[/C][C]1171.94974753912[/C][C]15.0502524608839[/C][/ROW]
[ROW][C]5[/C][C]1248[/C][C]1208.77196470702[/C][C]39.2280352929833[/C][/ROW]
[ROW][C]6[/C][C]1172[/C][C]1269.4877140904[/C][C]-97.4877140903964[/C][/ROW]
[ROW][C]7[/C][C]1092[/C][C]1196.09253870823[/C][C]-104.092538708227[/C][/ROW]
[ROW][C]8[/C][C]5255[/C][C]1115.42103811264[/C][C]4139.57896188736[/C][/ROW]
[ROW][C]9[/C][C]5297[/C][C]5206.50093256409[/C][C]90.4990674359078[/C][/ROW]
[ROW][C]10[/C][C]5461[/C][C]5349.53788257899[/C][C]111.462117421014[/C][/ROW]
[ROW][C]11[/C][C]5729[/C][C]5513.91280171888[/C][C]215.08719828112[/C][/ROW]
[ROW][C]12[/C][C]5795[/C][C]5781.07124905677[/C][C]13.9287509432343[/C][/ROW]
[ROW][C]13[/C][C]5411[/C][C]5852.16644697645[/C][C]-441.166446976454[/C][/ROW]
[ROW][C]14[/C][C]5503[/C][C]5475.90141420755[/C][C]27.098585792447[/C][/ROW]
[ROW][C]15[/C][C]5583[/C][C]5556.51814453877[/C][C]26.481855461233[/C][/ROW]
[ROW][C]16[/C][C]5763[/C][C]5636.74588525675[/C][C]126.254114743249[/C][/ROW]
[ROW][C]17[/C][C]5840[/C][C]5815.28708006765[/C][C]24.7129199323463[/C][/ROW]
[ROW][C]18[/C][C]5907[/C][C]5895.0009059364[/C][C]11.9990940636017[/C][/ROW]
[ROW][C]19[/C][C]5836[/C][C]5962.41214290448[/C][C]-126.412142904475[/C][/ROW]
[ROW][C]20[/C][C]5756[/C][C]5893.82691251894[/C][C]-137.826912518944[/C][/ROW]
[ROW][C]21[/C][C]5916[/C][C]5813.00392386308[/C][C]102.996076136922[/C][/ROW]
[ROW][C]22[/C][C]6056[/C][C]5967.86416598212[/C][C]88.135834017884[/C][/ROW]
[ROW][C]23[/C][C]6469[/C][C]6108.93941724051[/C][C]360.060582759487[/C][/ROW]
[ROW][C]24[/C][C]6603[/C][C]6518.0915620142[/C][C]84.908437985805[/C][/ROW]
[ROW][C]25[/C][C]6247[/C][C]6659.58932596147[/C][C]-412.589325961471[/C][/ROW]
[ROW][C]26[/C][C]6342[/C][C]6312.60403581271[/C][C]29.3959641872907[/C][/ROW]
[ROW][C]27[/C][C]6431[/C][C]6396.89024554932[/C][C]34.1097544506774[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=205630&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=205630&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
3115011473
411871171.9497475391215.0502524608839
512481208.7719647070239.2280352929833
611721269.4877140904-97.4877140903964
710921196.09253870823-104.092538708227
852551115.421038112644139.57896188736
952975206.5009325640990.4990674359078
1054615349.53788257899111.462117421014
1157295513.91280171888215.08719828112
1257955781.0712490567713.9287509432343
1354115852.16644697645-441.166446976454
1455035475.9014142075527.098585792447
1555835556.5181445387726.481855461233
1657635636.74588525675126.254114743249
1758405815.2870800676524.7129199323463
1859075895.000905936411.9990940636017
1958365962.41214290448-126.412142904475
2057565893.82691251894-137.826912518944
2159165813.00392386308102.996076136922
2260565967.8641659821288.135834017884
2364696108.93941724051360.060582759487
2466036518.091562014284.908437985805
2562476659.58932596147-412.589325961471
2663426312.6040358127129.3959641872907
2764316396.8902455493234.1097544506774







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
286486.047127374194832.29096961358139.80328513488
296541.939281189734222.680284419328861.19827796014
306597.831435005273757.701145112069437.96172489847
316653.723588820813367.784633889619939.66254375201
326709.615742636353025.5312392902710393.7002459824
336765.507896451892716.8113681667810814.204424737
346821.400050267422433.2179956585311209.5821048763
356877.292204082962169.2750838588211585.3093243071
366933.18435789851921.1841548117311945.1845609853
376989.076511714041686.1851950412212291.9678283869
387044.968665529581462.1998073170212627.7375237421
397100.860819345121247.6178261923812954.1038124979

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
28 & 6486.04712737419 & 4832.2909696135 & 8139.80328513488 \tabularnewline
29 & 6541.93928118973 & 4222.68028441932 & 8861.19827796014 \tabularnewline
30 & 6597.83143500527 & 3757.70114511206 & 9437.96172489847 \tabularnewline
31 & 6653.72358882081 & 3367.78463388961 & 9939.66254375201 \tabularnewline
32 & 6709.61574263635 & 3025.53123929027 & 10393.7002459824 \tabularnewline
33 & 6765.50789645189 & 2716.81136816678 & 10814.204424737 \tabularnewline
34 & 6821.40005026742 & 2433.21799565853 & 11209.5821048763 \tabularnewline
35 & 6877.29220408296 & 2169.27508385882 & 11585.3093243071 \tabularnewline
36 & 6933.1843578985 & 1921.18415481173 & 11945.1845609853 \tabularnewline
37 & 6989.07651171404 & 1686.18519504122 & 12291.9678283869 \tabularnewline
38 & 7044.96866552958 & 1462.19980731702 & 12627.7375237421 \tabularnewline
39 & 7100.86081934512 & 1247.61782619238 & 12954.1038124979 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=205630&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]28[/C][C]6486.04712737419[/C][C]4832.2909696135[/C][C]8139.80328513488[/C][/ROW]
[ROW][C]29[/C][C]6541.93928118973[/C][C]4222.68028441932[/C][C]8861.19827796014[/C][/ROW]
[ROW][C]30[/C][C]6597.83143500527[/C][C]3757.70114511206[/C][C]9437.96172489847[/C][/ROW]
[ROW][C]31[/C][C]6653.72358882081[/C][C]3367.78463388961[/C][C]9939.66254375201[/C][/ROW]
[ROW][C]32[/C][C]6709.61574263635[/C][C]3025.53123929027[/C][C]10393.7002459824[/C][/ROW]
[ROW][C]33[/C][C]6765.50789645189[/C][C]2716.81136816678[/C][C]10814.204424737[/C][/ROW]
[ROW][C]34[/C][C]6821.40005026742[/C][C]2433.21799565853[/C][C]11209.5821048763[/C][/ROW]
[ROW][C]35[/C][C]6877.29220408296[/C][C]2169.27508385882[/C][C]11585.3093243071[/C][/ROW]
[ROW][C]36[/C][C]6933.1843578985[/C][C]1921.18415481173[/C][C]11945.1845609853[/C][/ROW]
[ROW][C]37[/C][C]6989.07651171404[/C][C]1686.18519504122[/C][C]12291.9678283869[/C][/ROW]
[ROW][C]38[/C][C]7044.96866552958[/C][C]1462.19980731702[/C][C]12627.7375237421[/C][/ROW]
[ROW][C]39[/C][C]7100.86081934512[/C][C]1247.61782619238[/C][C]12954.1038124979[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=205630&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=205630&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
286486.047127374194832.29096961358139.80328513488
296541.939281189734222.680284419328861.19827796014
306597.831435005273757.701145112069437.96172489847
316653.723588820813367.784633889619939.66254375201
326709.615742636353025.5312392902710393.7002459824
336765.507896451892716.8113681667810814.204424737
346821.400050267422433.2179956585311209.5821048763
356877.292204082962169.2750838588211585.3093243071
366933.18435789851921.1841548117311945.1845609853
376989.076511714041686.1851950412212291.9678283869
387044.968665529581462.1998073170212627.7375237421
397100.860819345121247.6178261923812954.1038124979



Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = additive ;
Parameters (R input):
par1 = 12 ; par2 = Double ; par3 = additive ;
R code (references can be found in the software module):
par1 <- as.numeric(par1)
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$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par1, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$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')