Free Statistics

of Irreproducible Research!

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 17:00:44 +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/t1482508865qsji5u88r4qnnm6.htm/, Retrieved Tue, 07 May 2024 23:13:34 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=302984, Retrieved Tue, 07 May 2024 23:13:34 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact70
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [Exponential Smoot...] [2016-12-23 16:00:44] [e6ca2edb1c884165f9811941c39250b2] [Current]
Feedback Forum

Post a new message
Dataseries X:
2436.4
2823.8
2702
2631.4
2525.9
2845.8
2776.9
2767.3
2605.9
2985.2
3050.8
2953.9
2867.8
3452.6
3510.9
3376.6
3469.5
3958.6
4081.2
3845.4
3936
4469.3
4383.7
4485.5
4474.2
4956.8
5034.8
4886.3
4759.4
5403.1
5412.4
5197.5
5322.6
6063.5
6271.8
5986.3




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=302984&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=302984&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302984&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
alpha0.67653765820993
beta0.0716084620482283
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.67653765820993 \tabularnewline
beta & 0.0716084620482283 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302984&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]0.67653765820993[/C][/ROW]
[ROW][C]beta[/C][C]0.0716084620482283[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=302984&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302984&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
alpha0.67653765820993
beta0.0716084620482283
gamma1







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
132867.82458.19298908215409.607010917849
143452.63313.23040962281139.369590377194
153510.93487.1076760832723.7923239167308
163376.63373.528165086293.07183491371461
173469.53472.60867195851-3.10867195850824
183958.63961.98416418637-3.38416418636689
194081.23878.54534510966202.654654890339
203845.44067.77893583953-222.378935839532
2139363720.87086309938215.12913690062
224469.34465.301960110383.99803988961776
234383.74593.08310025483-209.383100254833
244485.54307.04532134884178.454678651164
254474.24488.14872781362-13.9487278136157
264956.85192.25521089253-235.455210892525
275034.85036.47089313219-1.6708931321873
284886.34787.0687343656199.2312656343911
294759.44942.60030032948-183.200300329479
305403.15444.58093192695-41.4809319269471
315412.45339.8806986380672.5193013619355
325197.55218.12539325852-20.6253932585187
335322.65082.93913192265239.660868077348
346063.55904.18899079994159.311009200063
356271.86043.02792791434228.772072085656
365986.36144.34445370325-158.044453703254

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 2867.8 & 2458.19298908215 & 409.607010917849 \tabularnewline
14 & 3452.6 & 3313.23040962281 & 139.369590377194 \tabularnewline
15 & 3510.9 & 3487.10767608327 & 23.7923239167308 \tabularnewline
16 & 3376.6 & 3373.52816508629 & 3.07183491371461 \tabularnewline
17 & 3469.5 & 3472.60867195851 & -3.10867195850824 \tabularnewline
18 & 3958.6 & 3961.98416418637 & -3.38416418636689 \tabularnewline
19 & 4081.2 & 3878.54534510966 & 202.654654890339 \tabularnewline
20 & 3845.4 & 4067.77893583953 & -222.378935839532 \tabularnewline
21 & 3936 & 3720.87086309938 & 215.12913690062 \tabularnewline
22 & 4469.3 & 4465.30196011038 & 3.99803988961776 \tabularnewline
23 & 4383.7 & 4593.08310025483 & -209.383100254833 \tabularnewline
24 & 4485.5 & 4307.04532134884 & 178.454678651164 \tabularnewline
25 & 4474.2 & 4488.14872781362 & -13.9487278136157 \tabularnewline
26 & 4956.8 & 5192.25521089253 & -235.455210892525 \tabularnewline
27 & 5034.8 & 5036.47089313219 & -1.6708931321873 \tabularnewline
28 & 4886.3 & 4787.06873436561 & 99.2312656343911 \tabularnewline
29 & 4759.4 & 4942.60030032948 & -183.200300329479 \tabularnewline
30 & 5403.1 & 5444.58093192695 & -41.4809319269471 \tabularnewline
31 & 5412.4 & 5339.88069863806 & 72.5193013619355 \tabularnewline
32 & 5197.5 & 5218.12539325852 & -20.6253932585187 \tabularnewline
33 & 5322.6 & 5082.93913192265 & 239.660868077348 \tabularnewline
34 & 6063.5 & 5904.18899079994 & 159.311009200063 \tabularnewline
35 & 6271.8 & 6043.02792791434 & 228.772072085656 \tabularnewline
36 & 5986.3 & 6144.34445370325 & -158.044453703254 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302984&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]2867.8[/C][C]2458.19298908215[/C][C]409.607010917849[/C][/ROW]
[ROW][C]14[/C][C]3452.6[/C][C]3313.23040962281[/C][C]139.369590377194[/C][/ROW]
[ROW][C]15[/C][C]3510.9[/C][C]3487.10767608327[/C][C]23.7923239167308[/C][/ROW]
[ROW][C]16[/C][C]3376.6[/C][C]3373.52816508629[/C][C]3.07183491371461[/C][/ROW]
[ROW][C]17[/C][C]3469.5[/C][C]3472.60867195851[/C][C]-3.10867195850824[/C][/ROW]
[ROW][C]18[/C][C]3958.6[/C][C]3961.98416418637[/C][C]-3.38416418636689[/C][/ROW]
[ROW][C]19[/C][C]4081.2[/C][C]3878.54534510966[/C][C]202.654654890339[/C][/ROW]
[ROW][C]20[/C][C]3845.4[/C][C]4067.77893583953[/C][C]-222.378935839532[/C][/ROW]
[ROW][C]21[/C][C]3936[/C][C]3720.87086309938[/C][C]215.12913690062[/C][/ROW]
[ROW][C]22[/C][C]4469.3[/C][C]4465.30196011038[/C][C]3.99803988961776[/C][/ROW]
[ROW][C]23[/C][C]4383.7[/C][C]4593.08310025483[/C][C]-209.383100254833[/C][/ROW]
[ROW][C]24[/C][C]4485.5[/C][C]4307.04532134884[/C][C]178.454678651164[/C][/ROW]
[ROW][C]25[/C][C]4474.2[/C][C]4488.14872781362[/C][C]-13.9487278136157[/C][/ROW]
[ROW][C]26[/C][C]4956.8[/C][C]5192.25521089253[/C][C]-235.455210892525[/C][/ROW]
[ROW][C]27[/C][C]5034.8[/C][C]5036.47089313219[/C][C]-1.6708931321873[/C][/ROW]
[ROW][C]28[/C][C]4886.3[/C][C]4787.06873436561[/C][C]99.2312656343911[/C][/ROW]
[ROW][C]29[/C][C]4759.4[/C][C]4942.60030032948[/C][C]-183.200300329479[/C][/ROW]
[ROW][C]30[/C][C]5403.1[/C][C]5444.58093192695[/C][C]-41.4809319269471[/C][/ROW]
[ROW][C]31[/C][C]5412.4[/C][C]5339.88069863806[/C][C]72.5193013619355[/C][/ROW]
[ROW][C]32[/C][C]5197.5[/C][C]5218.12539325852[/C][C]-20.6253932585187[/C][/ROW]
[ROW][C]33[/C][C]5322.6[/C][C]5082.93913192265[/C][C]239.660868077348[/C][/ROW]
[ROW][C]34[/C][C]6063.5[/C][C]5904.18899079994[/C][C]159.311009200063[/C][/ROW]
[ROW][C]35[/C][C]6271.8[/C][C]6043.02792791434[/C][C]228.772072085656[/C][/ROW]
[ROW][C]36[/C][C]5986.3[/C][C]6144.34445370325[/C][C]-158.044453703254[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=302984&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302984&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
132867.82458.19298908215409.607010917849
143452.63313.23040962281139.369590377194
153510.93487.1076760832723.7923239167308
163376.63373.528165086293.07183491371461
173469.53472.60867195851-3.10867195850824
183958.63961.98416418637-3.38416418636689
194081.23878.54534510966202.654654890339
203845.44067.77893583953-222.378935839532
2139363720.87086309938215.12913690062
224469.34465.301960110383.99803988961776
234383.74593.08310025483-209.383100254833
244485.54307.04532134884178.454678651164
254474.24488.14872781362-13.9487278136157
264956.85192.25521089253-235.455210892525
275034.85036.47089313219-1.6708931321873
284886.34787.0687343656199.2312656343911
294759.44942.60030032948-183.200300329479
305403.15444.58093192695-41.4809319269471
315412.45339.8806986380672.5193013619355
325197.55218.12539325852-20.6253932585187
335322.65082.93913192265239.660868077348
346063.55904.18899079994159.311009200063
356271.86043.02792791434228.772072085656
365986.36144.34445370325-158.044453703254







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
375999.966637179415674.453657942526325.4796164163
386820.344112688336402.142348314527238.54587706213
396902.990681357646415.433761942147390.54760077315
406582.899180037336049.214823468417116.58353660626
416549.941318681945957.662827441577142.21980992232
427453.968883922926734.270270523348173.6674973225
437381.331481967026610.608117372988152.05484656107
447087.969275707146288.558449480677887.38010193362
457016.773815440136169.110534740167864.4370961401
467819.516676025596826.419701820448812.61365023073
477849.724603194646797.637891646418901.81131474287
487582.126571775096560.1104840358604.14265951519

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
37 & 5999.96663717941 & 5674.45365794252 & 6325.4796164163 \tabularnewline
38 & 6820.34411268833 & 6402.14234831452 & 7238.54587706213 \tabularnewline
39 & 6902.99068135764 & 6415.43376194214 & 7390.54760077315 \tabularnewline
40 & 6582.89918003733 & 6049.21482346841 & 7116.58353660626 \tabularnewline
41 & 6549.94131868194 & 5957.66282744157 & 7142.21980992232 \tabularnewline
42 & 7453.96888392292 & 6734.27027052334 & 8173.6674973225 \tabularnewline
43 & 7381.33148196702 & 6610.60811737298 & 8152.05484656107 \tabularnewline
44 & 7087.96927570714 & 6288.55844948067 & 7887.38010193362 \tabularnewline
45 & 7016.77381544013 & 6169.11053474016 & 7864.4370961401 \tabularnewline
46 & 7819.51667602559 & 6826.41970182044 & 8812.61365023073 \tabularnewline
47 & 7849.72460319464 & 6797.63789164641 & 8901.81131474287 \tabularnewline
48 & 7582.12657177509 & 6560.110484035 & 8604.14265951519 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=302984&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]5999.96663717941[/C][C]5674.45365794252[/C][C]6325.4796164163[/C][/ROW]
[ROW][C]38[/C][C]6820.34411268833[/C][C]6402.14234831452[/C][C]7238.54587706213[/C][/ROW]
[ROW][C]39[/C][C]6902.99068135764[/C][C]6415.43376194214[/C][C]7390.54760077315[/C][/ROW]
[ROW][C]40[/C][C]6582.89918003733[/C][C]6049.21482346841[/C][C]7116.58353660626[/C][/ROW]
[ROW][C]41[/C][C]6549.94131868194[/C][C]5957.66282744157[/C][C]7142.21980992232[/C][/ROW]
[ROW][C]42[/C][C]7453.96888392292[/C][C]6734.27027052334[/C][C]8173.6674973225[/C][/ROW]
[ROW][C]43[/C][C]7381.33148196702[/C][C]6610.60811737298[/C][C]8152.05484656107[/C][/ROW]
[ROW][C]44[/C][C]7087.96927570714[/C][C]6288.55844948067[/C][C]7887.38010193362[/C][/ROW]
[ROW][C]45[/C][C]7016.77381544013[/C][C]6169.11053474016[/C][C]7864.4370961401[/C][/ROW]
[ROW][C]46[/C][C]7819.51667602559[/C][C]6826.41970182044[/C][C]8812.61365023073[/C][/ROW]
[ROW][C]47[/C][C]7849.72460319464[/C][C]6797.63789164641[/C][C]8901.81131474287[/C][/ROW]
[ROW][C]48[/C][C]7582.12657177509[/C][C]6560.110484035[/C][C]8604.14265951519[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=302984&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=302984&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
375999.966637179415674.453657942526325.4796164163
386820.344112688336402.142348314527238.54587706213
396902.990681357646415.433761942147390.54760077315
406582.899180037336049.214823468417116.58353660626
416549.941318681945957.662827441577142.21980992232
427453.968883922926734.270270523348173.6674973225
437381.331481967026610.608117372988152.05484656107
447087.969275707146288.558449480677887.38010193362
457016.773815440136169.110534740167864.4370961401
467819.516676025596826.419701820448812.61365023073
477849.724603194646797.637891646418901.81131474287
487582.126571775096560.1104840358604.14265951519



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