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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 computationWed, 12 Dec 2018 19:53:23 +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/2018/Dec/12/t1544641539c6j9ef5kyvhasey.htm/, Retrieved Mon, 06 May 2024 21:56:13 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=315866, Retrieved Mon, 06 May 2024 21:56:13 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact117

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...] [2018-12-12 18:53:23] [18ddfe38bfc2d98b48759e4537eb3733] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
0.386
0.374
0.393
0.425
0.406
0.344
0.327
0.288
0.269
0.256
0.286
0.298
0.329
0.318
0.381
0.381
0.47
0.443
0.386
0.342
0.319
0.307
0.284
0.326
0.309
0.359
0.376
0.416
0.437
0.548

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=315866&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=315866&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315866&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
alpha0.668776065119062
beta1
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.668776065119062 \tabularnewline
beta & 1 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315866&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]0.668776065119062[/C][/ROW]
[ROW][C]beta[/C][C]1[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=315866&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315866&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
alpha0.668776065119062
beta1
gamma1






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
50.4060.41686875-0.01086875
60.3440.352918730284474-0.00891873028447437
70.3270.331308203735694-0.00430820373569352
80.2880.332774863448245-0.0447748634482449
90.2690.2333840425743260.0356159574256735
100.2560.1861966617642570.0698033382357425
110.2860.2564370083886370.0295629916113633
120.2980.327480952534692-0.0294809525346924
130.3290.335502464750936-0.0065024647509358
140.3180.3138599449424050.00414005505759463
150.3810.3253326361821750.0556673638178254
160.3810.410210714182536-0.0292107141825358
170.470.4421376314623390.0278623685376609
180.4430.48609857589269-0.0430985758926897
190.3860.490550242602393-0.104550242602393
200.3420.3405192320904150.00148076790958479
210.3190.332755840509648-0.0137558405096485
220.3070.2184262890138070.0885737109861929
230.2840.2716889700739640.012311029926036
240.3260.294192015394850.0318079846051503
250.3090.381206153019886-0.0722061530198864
260.3590.3021324001669350.0568675998330652
270.3760.3281784513547560.0478215486452441
280.4160.423884208371587-0.00788420837158654
290.4370.466352265448919-0.0293522654489185
300.5480.503801217549940.0441987824500596

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
5 & 0.406 & 0.41686875 & -0.01086875 \tabularnewline
6 & 0.344 & 0.352918730284474 & -0.00891873028447437 \tabularnewline
7 & 0.327 & 0.331308203735694 & -0.00430820373569352 \tabularnewline
8 & 0.288 & 0.332774863448245 & -0.0447748634482449 \tabularnewline
9 & 0.269 & 0.233384042574326 & 0.0356159574256735 \tabularnewline
10 & 0.256 & 0.186196661764257 & 0.0698033382357425 \tabularnewline
11 & 0.286 & 0.256437008388637 & 0.0295629916113633 \tabularnewline
12 & 0.298 & 0.327480952534692 & -0.0294809525346924 \tabularnewline
13 & 0.329 & 0.335502464750936 & -0.0065024647509358 \tabularnewline
14 & 0.318 & 0.313859944942405 & 0.00414005505759463 \tabularnewline
15 & 0.381 & 0.325332636182175 & 0.0556673638178254 \tabularnewline
16 & 0.381 & 0.410210714182536 & -0.0292107141825358 \tabularnewline
17 & 0.47 & 0.442137631462339 & 0.0278623685376609 \tabularnewline
18 & 0.443 & 0.48609857589269 & -0.0430985758926897 \tabularnewline
19 & 0.386 & 0.490550242602393 & -0.104550242602393 \tabularnewline
20 & 0.342 & 0.340519232090415 & 0.00148076790958479 \tabularnewline
21 & 0.319 & 0.332755840509648 & -0.0137558405096485 \tabularnewline
22 & 0.307 & 0.218426289013807 & 0.0885737109861929 \tabularnewline
23 & 0.284 & 0.271688970073964 & 0.012311029926036 \tabularnewline
24 & 0.326 & 0.29419201539485 & 0.0318079846051503 \tabularnewline
25 & 0.309 & 0.381206153019886 & -0.0722061530198864 \tabularnewline
26 & 0.359 & 0.302132400166935 & 0.0568675998330652 \tabularnewline
27 & 0.376 & 0.328178451354756 & 0.0478215486452441 \tabularnewline
28 & 0.416 & 0.423884208371587 & -0.00788420837158654 \tabularnewline
29 & 0.437 & 0.466352265448919 & -0.0293522654489185 \tabularnewline
30 & 0.548 & 0.50380121754994 & 0.0441987824500596 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315866&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]5[/C][C]0.406[/C][C]0.41686875[/C][C]-0.01086875[/C][/ROW]
[ROW][C]6[/C][C]0.344[/C][C]0.352918730284474[/C][C]-0.00891873028447437[/C][/ROW]
[ROW][C]7[/C][C]0.327[/C][C]0.331308203735694[/C][C]-0.00430820373569352[/C][/ROW]
[ROW][C]8[/C][C]0.288[/C][C]0.332774863448245[/C][C]-0.0447748634482449[/C][/ROW]
[ROW][C]9[/C][C]0.269[/C][C]0.233384042574326[/C][C]0.0356159574256735[/C][/ROW]
[ROW][C]10[/C][C]0.256[/C][C]0.186196661764257[/C][C]0.0698033382357425[/C][/ROW]
[ROW][C]11[/C][C]0.286[/C][C]0.256437008388637[/C][C]0.0295629916113633[/C][/ROW]
[ROW][C]12[/C][C]0.298[/C][C]0.327480952534692[/C][C]-0.0294809525346924[/C][/ROW]
[ROW][C]13[/C][C]0.329[/C][C]0.335502464750936[/C][C]-0.0065024647509358[/C][/ROW]
[ROW][C]14[/C][C]0.318[/C][C]0.313859944942405[/C][C]0.00414005505759463[/C][/ROW]
[ROW][C]15[/C][C]0.381[/C][C]0.325332636182175[/C][C]0.0556673638178254[/C][/ROW]
[ROW][C]16[/C][C]0.381[/C][C]0.410210714182536[/C][C]-0.0292107141825358[/C][/ROW]
[ROW][C]17[/C][C]0.47[/C][C]0.442137631462339[/C][C]0.0278623685376609[/C][/ROW]
[ROW][C]18[/C][C]0.443[/C][C]0.48609857589269[/C][C]-0.0430985758926897[/C][/ROW]
[ROW][C]19[/C][C]0.386[/C][C]0.490550242602393[/C][C]-0.104550242602393[/C][/ROW]
[ROW][C]20[/C][C]0.342[/C][C]0.340519232090415[/C][C]0.00148076790958479[/C][/ROW]
[ROW][C]21[/C][C]0.319[/C][C]0.332755840509648[/C][C]-0.0137558405096485[/C][/ROW]
[ROW][C]22[/C][C]0.307[/C][C]0.218426289013807[/C][C]0.0885737109861929[/C][/ROW]
[ROW][C]23[/C][C]0.284[/C][C]0.271688970073964[/C][C]0.012311029926036[/C][/ROW]
[ROW][C]24[/C][C]0.326[/C][C]0.29419201539485[/C][C]0.0318079846051503[/C][/ROW]
[ROW][C]25[/C][C]0.309[/C][C]0.381206153019886[/C][C]-0.0722061530198864[/C][/ROW]
[ROW][C]26[/C][C]0.359[/C][C]0.302132400166935[/C][C]0.0568675998330652[/C][/ROW]
[ROW][C]27[/C][C]0.376[/C][C]0.328178451354756[/C][C]0.0478215486452441[/C][/ROW]
[ROW][C]28[/C][C]0.416[/C][C]0.423884208371587[/C][C]-0.00788420837158654[/C][/ROW]
[ROW][C]29[/C][C]0.437[/C][C]0.466352265448919[/C][C]-0.0293522654489185[/C][/ROW]
[ROW][C]30[/C][C]0.548[/C][C]0.50380121754994[/C][C]0.0441987824500596[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=315866&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315866&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
50.4060.41686875-0.01086875
60.3440.352918730284474-0.00891873028447437
70.3270.331308203735694-0.00430820373569352
80.2880.332774863448245-0.0447748634482449
90.2690.2333840425743260.0356159574256735
100.2560.1861966617642570.0698033382357425
110.2860.2564370083886370.0295629916113633
120.2980.327480952534692-0.0294809525346924
130.3290.335502464750936-0.0065024647509358
140.3180.3138599449424050.00414005505759463
150.3810.3253326361821750.0556673638178254
160.3810.410210714182536-0.0292107141825358
170.470.4421376314623390.0278623685376609
180.4430.48609857589269-0.0430985758926897
190.3860.490550242602393-0.104550242602393
200.3420.3405192320904150.00148076790958479
210.3190.332755840509648-0.0137558405096485
220.3070.2184262890138070.0885737109861929
230.2840.2716889700739640.012311029926036
240.3260.294192015394850.0318079846051503
250.3090.381206153019886-0.0722061530198864
260.3590.3021324001669350.0568675998330652
270.3760.3281784513547560.0478215486452441
280.4160.423884208371587-0.00788420837158654
290.4370.466352265448919-0.0293522654489185
300.5480.503801217549940.0441987824500596






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
310.5550165307293630.4672747062361720.642758355222554
320.6049455259501090.4584128515273850.751478200372833
330.6555046137601130.4264598196331550.884549407887071
340.7665046137601130.4385498831203661.09445934439986
350.7735211444894760.3135918188134941.23345047016546
360.8234501397102220.2442319107243741.40266836869607
370.8740092275202260.1639279512472161.58409050379324
380.9850092275202260.1337827186268411.83623573641361
390.992025758249589-0.02534087943695872.00939239593614
401.04195475347034-0.1325088680095652.21641837495024
411.09251384128034-0.2476368982445722.43266458080525
421.20351384128034-0.3103682853530782.71739596791376

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
31 & 0.555016530729363 & 0.467274706236172 & 0.642758355222554 \tabularnewline
32 & 0.604945525950109 & 0.458412851527385 & 0.751478200372833 \tabularnewline
33 & 0.655504613760113 & 0.426459819633155 & 0.884549407887071 \tabularnewline
34 & 0.766504613760113 & 0.438549883120366 & 1.09445934439986 \tabularnewline
35 & 0.773521144489476 & 0.313591818813494 & 1.23345047016546 \tabularnewline
36 & 0.823450139710222 & 0.244231910724374 & 1.40266836869607 \tabularnewline
37 & 0.874009227520226 & 0.163927951247216 & 1.58409050379324 \tabularnewline
38 & 0.985009227520226 & 0.133782718626841 & 1.83623573641361 \tabularnewline
39 & 0.992025758249589 & -0.0253408794369587 & 2.00939239593614 \tabularnewline
40 & 1.04195475347034 & -0.132508868009565 & 2.21641837495024 \tabularnewline
41 & 1.09251384128034 & -0.247636898244572 & 2.43266458080525 \tabularnewline
42 & 1.20351384128034 & -0.310368285353078 & 2.71739596791376 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315866&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]31[/C][C]0.555016530729363[/C][C]0.467274706236172[/C][C]0.642758355222554[/C][/ROW]
[ROW][C]32[/C][C]0.604945525950109[/C][C]0.458412851527385[/C][C]0.751478200372833[/C][/ROW]
[ROW][C]33[/C][C]0.655504613760113[/C][C]0.426459819633155[/C][C]0.884549407887071[/C][/ROW]
[ROW][C]34[/C][C]0.766504613760113[/C][C]0.438549883120366[/C][C]1.09445934439986[/C][/ROW]
[ROW][C]35[/C][C]0.773521144489476[/C][C]0.313591818813494[/C][C]1.23345047016546[/C][/ROW]
[ROW][C]36[/C][C]0.823450139710222[/C][C]0.244231910724374[/C][C]1.40266836869607[/C][/ROW]
[ROW][C]37[/C][C]0.874009227520226[/C][C]0.163927951247216[/C][C]1.58409050379324[/C][/ROW]
[ROW][C]38[/C][C]0.985009227520226[/C][C]0.133782718626841[/C][C]1.83623573641361[/C][/ROW]
[ROW][C]39[/C][C]0.992025758249589[/C][C]-0.0253408794369587[/C][C]2.00939239593614[/C][/ROW]
[ROW][C]40[/C][C]1.04195475347034[/C][C]-0.132508868009565[/C][C]2.21641837495024[/C][/ROW]
[ROW][C]41[/C][C]1.09251384128034[/C][C]-0.247636898244572[/C][C]2.43266458080525[/C][/ROW]
[ROW][C]42[/C][C]1.20351384128034[/C][C]-0.310368285353078[/C][C]2.71739596791376[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=315866&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315866&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
310.5550165307293630.4672747062361720.642758355222554
320.6049455259501090.4584128515273850.751478200372833
330.6555046137601130.4264598196331550.884549407887071
340.7665046137601130.4385498831203661.09445934439986
350.7735211444894760.3135918188134941.23345047016546
360.8234501397102220.2442319107243741.40266836869607
370.8740092275202260.1639279512472161.58409050379324
380.9850092275202260.1337827186268411.83623573641361
390.992025758249589-0.02534087943695872.00939239593614
401.04195475347034-0.1325088680095652.21641837495024
411.09251384128034-0.2476368982445722.43266458080525
421.20351384128034-0.3103682853530782.71739596791376


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
Parameters (R input):
par1 = 4 ; par2 = Triple ; par3 = additive ; par4 = 12 ;
R code (references can be found in the software module):
par4 <- '12'
par3 <- 'multiplicative'
par2 <- 'Triple'
par1 <- '4'
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')