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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, 23 Jan 2017 10:05:06 +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/2017/Jan/23/t1485162327q58tn32kera2lb9.htm/, Retrieved Wed, 15 May 2024 22:05:54 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=304191, Retrieved Wed, 15 May 2024 22:05:54 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact79

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [Vraag 8] [2017-01-23 09:05:06] [d5bfc1731fe289380efec318f4354749] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
10.24
10.89
9
12.25
13.69
7.29
12.96
12.25
14.44
11.56
13.69
12.25
7.84
14.44
18.49
10.89
12.96
12.96
10.89
7.84

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=304191&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=304191&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=304191&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
alpha0.0656064528120427
beta1
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.0656064528120427 \tabularnewline
beta & 1 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=304191&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]0.0656064528120427[/C][/ROW]
[ROW][C]beta[/C][C]1[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=304191&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=304191&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
alpha0.0656064528120427
beta1
gammaFALSE






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
3911.54-2.54
412.2511.85671921971480.393280780285174
513.6912.39168234347961.29831765652043
67.2913.0711997424257-5.78119974242575
712.9612.90697110910750.0530288908925236
812.2513.1289845587437-0.878984558743687
914.4413.23218485300041.20781514699958
1011.5613.5515331411275-1.99153314112748
1113.6913.53032611171320.159673888286752
1212.2513.6607277821804-1.41072778218041
137.8413.5955481238859-5.75554812388587
1414.4412.86771911846931.57228088153074
1518.4912.72379475237275.76620524762734
1610.8913.2333191597807-2.34331915978067
1712.9613.0570695789462-0.0970695789462486
1812.9613.0218200744903-0.0618200744902921
1910.8912.9848273691852-2.09482736918519
207.8412.6770220737883-4.83702207378832

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 9 & 11.54 & -2.54 \tabularnewline
4 & 12.25 & 11.8567192197148 & 0.393280780285174 \tabularnewline
5 & 13.69 & 12.3916823434796 & 1.29831765652043 \tabularnewline
6 & 7.29 & 13.0711997424257 & -5.78119974242575 \tabularnewline
7 & 12.96 & 12.9069711091075 & 0.0530288908925236 \tabularnewline
8 & 12.25 & 13.1289845587437 & -0.878984558743687 \tabularnewline
9 & 14.44 & 13.2321848530004 & 1.20781514699958 \tabularnewline
10 & 11.56 & 13.5515331411275 & -1.99153314112748 \tabularnewline
11 & 13.69 & 13.5303261117132 & 0.159673888286752 \tabularnewline
12 & 12.25 & 13.6607277821804 & -1.41072778218041 \tabularnewline
13 & 7.84 & 13.5955481238859 & -5.75554812388587 \tabularnewline
14 & 14.44 & 12.8677191184693 & 1.57228088153074 \tabularnewline
15 & 18.49 & 12.7237947523727 & 5.76620524762734 \tabularnewline
16 & 10.89 & 13.2333191597807 & -2.34331915978067 \tabularnewline
17 & 12.96 & 13.0570695789462 & -0.0970695789462486 \tabularnewline
18 & 12.96 & 13.0218200744903 & -0.0618200744902921 \tabularnewline
19 & 10.89 & 12.9848273691852 & -2.09482736918519 \tabularnewline
20 & 7.84 & 12.6770220737883 & -4.83702207378832 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=304191&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]9[/C][C]11.54[/C][C]-2.54[/C][/ROW]
[ROW][C]4[/C][C]12.25[/C][C]11.8567192197148[/C][C]0.393280780285174[/C][/ROW]
[ROW][C]5[/C][C]13.69[/C][C]12.3916823434796[/C][C]1.29831765652043[/C][/ROW]
[ROW][C]6[/C][C]7.29[/C][C]13.0711997424257[/C][C]-5.78119974242575[/C][/ROW]
[ROW][C]7[/C][C]12.96[/C][C]12.9069711091075[/C][C]0.0530288908925236[/C][/ROW]
[ROW][C]8[/C][C]12.25[/C][C]13.1289845587437[/C][C]-0.878984558743687[/C][/ROW]
[ROW][C]9[/C][C]14.44[/C][C]13.2321848530004[/C][C]1.20781514699958[/C][/ROW]
[ROW][C]10[/C][C]11.56[/C][C]13.5515331411275[/C][C]-1.99153314112748[/C][/ROW]
[ROW][C]11[/C][C]13.69[/C][C]13.5303261117132[/C][C]0.159673888286752[/C][/ROW]
[ROW][C]12[/C][C]12.25[/C][C]13.6607277821804[/C][C]-1.41072778218041[/C][/ROW]
[ROW][C]13[/C][C]7.84[/C][C]13.5955481238859[/C][C]-5.75554812388587[/C][/ROW]
[ROW][C]14[/C][C]14.44[/C][C]12.8677191184693[/C][C]1.57228088153074[/C][/ROW]
[ROW][C]15[/C][C]18.49[/C][C]12.7237947523727[/C][C]5.76620524762734[/C][/ROW]
[ROW][C]16[/C][C]10.89[/C][C]13.2333191597807[/C][C]-2.34331915978067[/C][/ROW]
[ROW][C]17[/C][C]12.96[/C][C]13.0570695789462[/C][C]-0.0970695789462486[/C][/ROW]
[ROW][C]18[/C][C]12.96[/C][C]13.0218200744903[/C][C]-0.0618200744902921[/C][/ROW]
[ROW][C]19[/C][C]10.89[/C][C]12.9848273691852[/C][C]-2.09482736918519[/C][/ROW]
[ROW][C]20[/C][C]7.84[/C][C]12.6770220737883[/C][C]-4.83702207378832[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=304191&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=304191&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
3911.54-2.54
412.2511.85671921971480.393280780285174
513.6912.39168234347961.29831765652043
67.2913.0711997424257-5.78119974242575
712.9612.90697110910750.0530288908925236
812.2513.1289845587437-0.878984558743687
914.4413.23218485300041.20781514699958
1011.5613.5515331411275-1.99153314112748
1113.6913.53032611171320.159673888286752
1212.2513.6607277821804-1.41072778218041
137.8413.5955481238859-5.75554812388587
1414.4412.86771911846931.57228088153074
1518.4912.72379475237275.76620524762734
1610.8913.2333191597807-2.34331915978067
1712.9613.0570695789462-0.0970695789462486
1812.9613.0218200744903-0.0618200744902921
1910.8912.9848273691852-2.09482736918519
207.8412.6770220737883-4.83702207378832






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
2111.87197125046776.3414096071976417.4025328937377
2211.38426028758185.8062924591402716.9622281160234
2310.8965493246965.2133632150698616.5797354343221
2410.40883836181014.5432560056306216.2744207179897
259.921127398924293.7813889957738816.0608658020747
269.433416436038452.9191317703830215.9477011016939
278.94570547315261.9537877091979615.9376232371072
288.457994510266760.88747787665363616.0285111438799
297.97028354738091-0.27451089841810216.2150779931799
307.48257258449507-1.5253106619430316.4904558309332
316.99486162160922-2.8576509506394616.8473741938579
326.50715065872337-4.2645749305502417.278876247997

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
21 & 11.8719712504677 & 6.34140960719764 & 17.4025328937377 \tabularnewline
22 & 11.3842602875818 & 5.80629245914027 & 16.9622281160234 \tabularnewline
23 & 10.896549324696 & 5.21336321506986 & 16.5797354343221 \tabularnewline
24 & 10.4088383618101 & 4.54325600563062 & 16.2744207179897 \tabularnewline
25 & 9.92112739892429 & 3.78138899577388 & 16.0608658020747 \tabularnewline
26 & 9.43341643603845 & 2.91913177038302 & 15.9477011016939 \tabularnewline
27 & 8.9457054731526 & 1.95378770919796 & 15.9376232371072 \tabularnewline
28 & 8.45799451026676 & 0.887477876653636 & 16.0285111438799 \tabularnewline
29 & 7.97028354738091 & -0.274510898418102 & 16.2150779931799 \tabularnewline
30 & 7.48257258449507 & -1.52531066194303 & 16.4904558309332 \tabularnewline
31 & 6.99486162160922 & -2.85765095063946 & 16.8473741938579 \tabularnewline
32 & 6.50715065872337 & -4.26457493055024 & 17.278876247997 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=304191&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]21[/C][C]11.8719712504677[/C][C]6.34140960719764[/C][C]17.4025328937377[/C][/ROW]
[ROW][C]22[/C][C]11.3842602875818[/C][C]5.80629245914027[/C][C]16.9622281160234[/C][/ROW]
[ROW][C]23[/C][C]10.896549324696[/C][C]5.21336321506986[/C][C]16.5797354343221[/C][/ROW]
[ROW][C]24[/C][C]10.4088383618101[/C][C]4.54325600563062[/C][C]16.2744207179897[/C][/ROW]
[ROW][C]25[/C][C]9.92112739892429[/C][C]3.78138899577388[/C][C]16.0608658020747[/C][/ROW]
[ROW][C]26[/C][C]9.43341643603845[/C][C]2.91913177038302[/C][C]15.9477011016939[/C][/ROW]
[ROW][C]27[/C][C]8.9457054731526[/C][C]1.95378770919796[/C][C]15.9376232371072[/C][/ROW]
[ROW][C]28[/C][C]8.45799451026676[/C][C]0.887477876653636[/C][C]16.0285111438799[/C][/ROW]
[ROW][C]29[/C][C]7.97028354738091[/C][C]-0.274510898418102[/C][C]16.2150779931799[/C][/ROW]
[ROW][C]30[/C][C]7.48257258449507[/C][C]-1.52531066194303[/C][C]16.4904558309332[/C][/ROW]
[ROW][C]31[/C][C]6.99486162160922[/C][C]-2.85765095063946[/C][C]16.8473741938579[/C][/ROW]
[ROW][C]32[/C][C]6.50715065872337[/C][C]-4.26457493055024[/C][C]17.278876247997[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=304191&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=304191&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
2111.87197125046776.3414096071976417.4025328937377
2211.38426028758185.8062924591402716.9622281160234
2310.8965493246965.2133632150698616.5797354343221
2410.40883836181014.5432560056306216.2744207179897
259.921127398924293.7813889957738816.0608658020747
269.433416436038452.9191317703830215.9477011016939
278.94570547315261.9537877091979615.9376232371072
288.457994510266760.88747787665363616.0285111438799
297.97028354738091-0.27451089841810216.2150779931799
307.48257258449507-1.5253106619430316.4904558309332
316.99486162160922-2.8576509506394616.8473741938579
326.50715065872337-4.2645749305502417.278876247997


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 1 ; par2 = 2 ; par3 = Exact Pearson Chi-Squared by Simulation ;
Parameters (R input):
par1 = 12 ; par2 = Double ; 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')