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Author's title

Author*Unverified author*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationThu, 23 Aug 2012 22:06:30 -0400
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2012/Aug/23/t1345774114dv7tqzadbp2s9wv.htm/, Retrieved Thu, 13 Aug 2026 17:15:05 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=169491, Retrieved Thu, 13 Aug 2026 17:15:05 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywordstaller de mod
Estimated Impact470
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [taller de mod] [2012-08-24 02:06:30] [d41d8cd98f00b204e9800998ecf8427e] [Current]
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Dataseries X:
96
319
405
830
874
1719
2797
2235
1471
735
383
422
144
364
692
656
1223
2199
3530
2973
2099
1244
525
209
356
540
770
646
1355
2379
3946
3503
2723
1243
499
322
238
479
630
921




Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 seconds
R Server'Gwilym Jenkins' @ jenkins.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 & 2 seconds \tabularnewline
R Server & 'Gwilym Jenkins' @ jenkins.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=169491&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]2 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Gwilym Jenkins' @ jenkins.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=169491&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=169491&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 time2 seconds
R Server'Gwilym Jenkins' @ jenkins.wessa.net







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.381725117441034
beta0
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.381725117441034 \tabularnewline
beta & 0 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=169491&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]0.381725117441034[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=169491&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=169491&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
alpha0.381725117441034
beta0
gamma1







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13144124.87011610682219.1298838931782
14364327.24854075822436.751459241776
15692637.95184761230954.0481523876913
16656618.56548494226637.4345150577341
1712231186.6907624517236.3092375482788
1821992219.01584767519-20.0158476751881
1935303739.44773283779-209.447732837787
2029732986.17548381301-13.17548381301
2120991983.477077109115.522922891002
2212441032.1249956928211.875004307197
23525588.167426623316-63.167426623316
24209614.815206205526-405.815206205526
25356168.026750642675187.973249357325
26540573.972700688958-33.9727006889584
277701022.60203208303-252.602032083028
28646852.246205441435-206.246205441435
2913551419.81286189455-64.8128618945457
3023792509.28283310371-130.282833103709
3139464024.03696956858-78.0369695685763
3235033360.0973527098142.902647290198
3327232351.08953149971371.910468500289
3412431365.22805597315-122.228055973145
35499579.578334321198-80.5783343211978
36322291.80547519899730.1945248010026
37238358.964572094279-120.964572094279
38479486.692315931337-7.69231593133651
39630763.297241624734-133.297241624734
40921659.614485010454261.385514989546

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 144 & 124.870116106822 & 19.1298838931782 \tabularnewline
14 & 364 & 327.248540758224 & 36.751459241776 \tabularnewline
15 & 692 & 637.951847612309 & 54.0481523876913 \tabularnewline
16 & 656 & 618.565484942266 & 37.4345150577341 \tabularnewline
17 & 1223 & 1186.69076245172 & 36.3092375482788 \tabularnewline
18 & 2199 & 2219.01584767519 & -20.0158476751881 \tabularnewline
19 & 3530 & 3739.44773283779 & -209.447732837787 \tabularnewline
20 & 2973 & 2986.17548381301 & -13.17548381301 \tabularnewline
21 & 2099 & 1983.477077109 & 115.522922891002 \tabularnewline
22 & 1244 & 1032.1249956928 & 211.875004307197 \tabularnewline
23 & 525 & 588.167426623316 & -63.167426623316 \tabularnewline
24 & 209 & 614.815206205526 & -405.815206205526 \tabularnewline
25 & 356 & 168.026750642675 & 187.973249357325 \tabularnewline
26 & 540 & 573.972700688958 & -33.9727006889584 \tabularnewline
27 & 770 & 1022.60203208303 & -252.602032083028 \tabularnewline
28 & 646 & 852.246205441435 & -206.246205441435 \tabularnewline
29 & 1355 & 1419.81286189455 & -64.8128618945457 \tabularnewline
30 & 2379 & 2509.28283310371 & -130.282833103709 \tabularnewline
31 & 3946 & 4024.03696956858 & -78.0369695685763 \tabularnewline
32 & 3503 & 3360.0973527098 & 142.902647290198 \tabularnewline
33 & 2723 & 2351.08953149971 & 371.910468500289 \tabularnewline
34 & 1243 & 1365.22805597315 & -122.228055973145 \tabularnewline
35 & 499 & 579.578334321198 & -80.5783343211978 \tabularnewline
36 & 322 & 291.805475198997 & 30.1945248010026 \tabularnewline
37 & 238 & 358.964572094279 & -120.964572094279 \tabularnewline
38 & 479 & 486.692315931337 & -7.69231593133651 \tabularnewline
39 & 630 & 763.297241624734 & -133.297241624734 \tabularnewline
40 & 921 & 659.614485010454 & 261.385514989546 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=169491&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]144[/C][C]124.870116106822[/C][C]19.1298838931782[/C][/ROW]
[ROW][C]14[/C][C]364[/C][C]327.248540758224[/C][C]36.751459241776[/C][/ROW]
[ROW][C]15[/C][C]692[/C][C]637.951847612309[/C][C]54.0481523876913[/C][/ROW]
[ROW][C]16[/C][C]656[/C][C]618.565484942266[/C][C]37.4345150577341[/C][/ROW]
[ROW][C]17[/C][C]1223[/C][C]1186.69076245172[/C][C]36.3092375482788[/C][/ROW]
[ROW][C]18[/C][C]2199[/C][C]2219.01584767519[/C][C]-20.0158476751881[/C][/ROW]
[ROW][C]19[/C][C]3530[/C][C]3739.44773283779[/C][C]-209.447732837787[/C][/ROW]
[ROW][C]20[/C][C]2973[/C][C]2986.17548381301[/C][C]-13.17548381301[/C][/ROW]
[ROW][C]21[/C][C]2099[/C][C]1983.477077109[/C][C]115.522922891002[/C][/ROW]
[ROW][C]22[/C][C]1244[/C][C]1032.1249956928[/C][C]211.875004307197[/C][/ROW]
[ROW][C]23[/C][C]525[/C][C]588.167426623316[/C][C]-63.167426623316[/C][/ROW]
[ROW][C]24[/C][C]209[/C][C]614.815206205526[/C][C]-405.815206205526[/C][/ROW]
[ROW][C]25[/C][C]356[/C][C]168.026750642675[/C][C]187.973249357325[/C][/ROW]
[ROW][C]26[/C][C]540[/C][C]573.972700688958[/C][C]-33.9727006889584[/C][/ROW]
[ROW][C]27[/C][C]770[/C][C]1022.60203208303[/C][C]-252.602032083028[/C][/ROW]
[ROW][C]28[/C][C]646[/C][C]852.246205441435[/C][C]-206.246205441435[/C][/ROW]
[ROW][C]29[/C][C]1355[/C][C]1419.81286189455[/C][C]-64.8128618945457[/C][/ROW]
[ROW][C]30[/C][C]2379[/C][C]2509.28283310371[/C][C]-130.282833103709[/C][/ROW]
[ROW][C]31[/C][C]3946[/C][C]4024.03696956858[/C][C]-78.0369695685763[/C][/ROW]
[ROW][C]32[/C][C]3503[/C][C]3360.0973527098[/C][C]142.902647290198[/C][/ROW]
[ROW][C]33[/C][C]2723[/C][C]2351.08953149971[/C][C]371.910468500289[/C][/ROW]
[ROW][C]34[/C][C]1243[/C][C]1365.22805597315[/C][C]-122.228055973145[/C][/ROW]
[ROW][C]35[/C][C]499[/C][C]579.578334321198[/C][C]-80.5783343211978[/C][/ROW]
[ROW][C]36[/C][C]322[/C][C]291.805475198997[/C][C]30.1945248010026[/C][/ROW]
[ROW][C]37[/C][C]238[/C][C]358.964572094279[/C][C]-120.964572094279[/C][/ROW]
[ROW][C]38[/C][C]479[/C][C]486.692315931337[/C][C]-7.69231593133651[/C][/ROW]
[ROW][C]39[/C][C]630[/C][C]763.297241624734[/C][C]-133.297241624734[/C][/ROW]
[ROW][C]40[/C][C]921[/C][C]659.614485010454[/C][C]261.385514989546[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=169491&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=169491&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
13144124.87011610682219.1298838931782
14364327.24854075822436.751459241776
15692637.95184761230954.0481523876913
16656618.56548494226637.4345150577341
1712231186.6907624517236.3092375482788
1821992219.01584767519-20.0158476751881
1935303739.44773283779-209.447732837787
2029732986.17548381301-13.17548381301
2120991983.477077109115.522922891002
2212441032.1249956928211.875004307197
23525588.167426623316-63.167426623316
24209614.815206205526-405.815206205526
25356168.026750642675187.973249357325
26540573.972700688958-33.9727006889584
277701022.60203208303-252.602032083028
28646852.246205441435-206.246205441435
2913551419.81286189455-64.8128618945457
3023792509.28283310371-130.282833103709
3139464024.03696956858-78.0369695685763
3235033360.0973527098142.902647290198
3327232351.08953149971371.910468500289
3412431365.22805597315-122.228055973145
35499579.578334321198-80.5783343211978
36322291.80547519899730.1945248010026
37238358.964572094279-120.964572094279
38479486.692315931337-7.69231593133651
39630763.297241624734-133.297241624734
40921659.614485010454261.385514989546







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
411615.733514164091294.816368623821936.65065970437
422883.847912452542497.792315417423269.90350948765
434800.514488947414284.690329068955316.33864882588
444177.332867092423694.56374605524660.10198812964
453051.953298700072630.367700275153473.538897125
461439.457087625041089.784829526391789.1293457237
47608.622180939511278.60460978557938.639752093452
48376.43533125194143.6744119665115709.196250537371
49318.425673776321-26.1548590491932663.006206601835
50641.256918598439168.2663987228591114.24743847402
51898.808643467706297.7657966968571499.85149023855
521133.76303770925484.6493461374831782.87672928101

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
41 & 1615.73351416409 & 1294.81636862382 & 1936.65065970437 \tabularnewline
42 & 2883.84791245254 & 2497.79231541742 & 3269.90350948765 \tabularnewline
43 & 4800.51448894741 & 4284.69032906895 & 5316.33864882588 \tabularnewline
44 & 4177.33286709242 & 3694.5637460552 & 4660.10198812964 \tabularnewline
45 & 3051.95329870007 & 2630.36770027515 & 3473.538897125 \tabularnewline
46 & 1439.45708762504 & 1089.78482952639 & 1789.1293457237 \tabularnewline
47 & 608.622180939511 & 278.60460978557 & 938.639752093452 \tabularnewline
48 & 376.435331251941 & 43.6744119665115 & 709.196250537371 \tabularnewline
49 & 318.425673776321 & -26.1548590491932 & 663.006206601835 \tabularnewline
50 & 641.256918598439 & 168.266398722859 & 1114.24743847402 \tabularnewline
51 & 898.808643467706 & 297.765796696857 & 1499.85149023855 \tabularnewline
52 & 1133.76303770925 & 484.649346137483 & 1782.87672928101 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=169491&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]41[/C][C]1615.73351416409[/C][C]1294.81636862382[/C][C]1936.65065970437[/C][/ROW]
[ROW][C]42[/C][C]2883.84791245254[/C][C]2497.79231541742[/C][C]3269.90350948765[/C][/ROW]
[ROW][C]43[/C][C]4800.51448894741[/C][C]4284.69032906895[/C][C]5316.33864882588[/C][/ROW]
[ROW][C]44[/C][C]4177.33286709242[/C][C]3694.5637460552[/C][C]4660.10198812964[/C][/ROW]
[ROW][C]45[/C][C]3051.95329870007[/C][C]2630.36770027515[/C][C]3473.538897125[/C][/ROW]
[ROW][C]46[/C][C]1439.45708762504[/C][C]1089.78482952639[/C][C]1789.1293457237[/C][/ROW]
[ROW][C]47[/C][C]608.622180939511[/C][C]278.60460978557[/C][C]938.639752093452[/C][/ROW]
[ROW][C]48[/C][C]376.435331251941[/C][C]43.6744119665115[/C][C]709.196250537371[/C][/ROW]
[ROW][C]49[/C][C]318.425673776321[/C][C]-26.1548590491932[/C][C]663.006206601835[/C][/ROW]
[ROW][C]50[/C][C]641.256918598439[/C][C]168.266398722859[/C][C]1114.24743847402[/C][/ROW]
[ROW][C]51[/C][C]898.808643467706[/C][C]297.765796696857[/C][C]1499.85149023855[/C][/ROW]
[ROW][C]52[/C][C]1133.76303770925[/C][C]484.649346137483[/C][C]1782.87672928101[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=169491&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=169491&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
411615.733514164091294.816368623821936.65065970437
422883.847912452542497.792315417423269.90350948765
434800.514488947414284.690329068955316.33864882588
444177.332867092423694.56374605524660.10198812964
453051.953298700072630.367700275153473.538897125
461439.457087625041089.784829526391789.1293457237
47608.622180939511278.60460978557938.639752093452
48376.43533125194143.6744119665115709.196250537371
49318.425673776321-26.1548590491932663.006206601835
50641.256918598439168.2663987228591114.24743847402
51898.808643467706297.7657966968571499.85149023855
521133.76303770925484.6493461374831782.87672928101



Parameters (Session):
par1 = 12 ; par2 = Triple ; par3 = multiplicative ;
Parameters (R input):
par1 = 12 ; par2 = Triple ; par3 = multiplicative ;
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')