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Author's title

Author*Unverified author*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationThu, 11 Oct 2018 22:22:39 +0200
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/Oct/11/t1539289406lx7jtqyn856ijdj.htm/, Retrieved Fri, 03 May 2024 05:16:58 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=315574, Retrieved Fri, 03 May 2024 05:16:58 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact119
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [teste] [2018-10-11 20:22:39] [d41d8cd98f00b204e9800998ecf8427e] [Current]
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Dataseries X:
43.28
48.29
60.7
65.51
63.6
61.2
63.72
67.15
62.9
77.65
75.61
82.91
76.78
74.4
88.16
91.82
100.72
109.91
105.02
111.31
108.23
109.33
113.54
130.13
129.77
149.64
146.96
145.54
158.9
172.74
186.23
177.4
196.72
207.36
216.17
241.35
247.08
265.11
247.16
299.56
292.99
308.52




Summary of computational transaction
Raw Input view raw input (R code)
Raw Outputview raw output of R engine
Computing time2 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 time2 seconds \tabularnewline
R ServerBig Analytics Cloud Computing Center \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315574&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]2 seconds[/C][/ROW] [ROW]R Server[/C]Big Analytics Cloud Computing Center[/C][/ROW] [/TABLE] Source: https://freestatistics.org/blog/index.php?pk=315574&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315574&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 time2 seconds
R ServerBig Analytics Cloud Computing Center







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.937912320689768
betaFALSE
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.937912320689768 \tabularnewline
beta & FALSE \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315574&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]0.937912320689768[/C][/ROW]
[ROW][C]beta[/C][C]FALSE[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=315574&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315574&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
alpha0.937912320689768
betaFALSE
gammaFALSE







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
248.2943.285.01
360.747.978940726655712.7210592733443
465.5159.91017895135015.59982104864986
563.665.1623201065367-1.56232010653674
661.263.6970008297546-2.49700082975458
763.7261.35503298675522.36496701324481
867.1563.57316468650243.57683531349763
962.966.9279225961101-4.02792259611005
1077.6563.150084366433714.4999156335663
1175.6176.7497338881177-1.13973388811772
1282.9175.68076343214457.22923656785554
1376.7882.4611534783172-5.68115347831719
1474.477.132729635274-2.73272963527396
1588.1674.569668841236513.5903311587635
1691.8287.31620787729484.50379212270515
17100.7291.54036999900559.17963000099448
18109.91100.1500580763129.75994192368832
19105.02109.304027855756-4.28402785575555
20111.31105.2859853476646.02401465233577
21108.23110.935982910106-2.70598291010565
22109.33108.3980081991420.931991800858384
23113.54109.2721347919494.26786520805146
24130.13113.27501815362316.8549818463768
25129.77129.0835132923420.68648670765765
26149.64129.72737763344419.9126223665558
27146.96148.40367148828-1.44367148827951
28145.54147.049634212394-1.50963421239365
29158.9145.63372968485513.2662703151452
30172.74158.0763280630314.6636719369696
31186.23171.82956663926714.400433360733
32177.4185.33591051157-7.93591051157043
33196.72177.89272226687718.8272777331229
34207.36195.55105801782111.8089419821788
35216.17206.6268101972179.54318980278262
36241.35215.57748549192825.7725145080719
37247.08239.7498443842057.33015561579549
38265.11246.62488764883218.4851123511676
39247.16263.962302272327-16.8023022723271
40299.56248.20321595515851.3567840448422
41292.99296.371376461819-3.381376461819
42308.52293.19994181738915.3200581826114

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & 48.29 & 43.28 & 5.01 \tabularnewline
3 & 60.7 & 47.9789407266557 & 12.7210592733443 \tabularnewline
4 & 65.51 & 59.9101789513501 & 5.59982104864986 \tabularnewline
5 & 63.6 & 65.1623201065367 & -1.56232010653674 \tabularnewline
6 & 61.2 & 63.6970008297546 & -2.49700082975458 \tabularnewline
7 & 63.72 & 61.3550329867552 & 2.36496701324481 \tabularnewline
8 & 67.15 & 63.5731646865024 & 3.57683531349763 \tabularnewline
9 & 62.9 & 66.9279225961101 & -4.02792259611005 \tabularnewline
10 & 77.65 & 63.1500843664337 & 14.4999156335663 \tabularnewline
11 & 75.61 & 76.7497338881177 & -1.13973388811772 \tabularnewline
12 & 82.91 & 75.6807634321445 & 7.22923656785554 \tabularnewline
13 & 76.78 & 82.4611534783172 & -5.68115347831719 \tabularnewline
14 & 74.4 & 77.132729635274 & -2.73272963527396 \tabularnewline
15 & 88.16 & 74.5696688412365 & 13.5903311587635 \tabularnewline
16 & 91.82 & 87.3162078772948 & 4.50379212270515 \tabularnewline
17 & 100.72 & 91.5403699990055 & 9.17963000099448 \tabularnewline
18 & 109.91 & 100.150058076312 & 9.75994192368832 \tabularnewline
19 & 105.02 & 109.304027855756 & -4.28402785575555 \tabularnewline
20 & 111.31 & 105.285985347664 & 6.02401465233577 \tabularnewline
21 & 108.23 & 110.935982910106 & -2.70598291010565 \tabularnewline
22 & 109.33 & 108.398008199142 & 0.931991800858384 \tabularnewline
23 & 113.54 & 109.272134791949 & 4.26786520805146 \tabularnewline
24 & 130.13 & 113.275018153623 & 16.8549818463768 \tabularnewline
25 & 129.77 & 129.083513292342 & 0.68648670765765 \tabularnewline
26 & 149.64 & 129.727377633444 & 19.9126223665558 \tabularnewline
27 & 146.96 & 148.40367148828 & -1.44367148827951 \tabularnewline
28 & 145.54 & 147.049634212394 & -1.50963421239365 \tabularnewline
29 & 158.9 & 145.633729684855 & 13.2662703151452 \tabularnewline
30 & 172.74 & 158.07632806303 & 14.6636719369696 \tabularnewline
31 & 186.23 & 171.829566639267 & 14.400433360733 \tabularnewline
32 & 177.4 & 185.33591051157 & -7.93591051157043 \tabularnewline
33 & 196.72 & 177.892722266877 & 18.8272777331229 \tabularnewline
34 & 207.36 & 195.551058017821 & 11.8089419821788 \tabularnewline
35 & 216.17 & 206.626810197217 & 9.54318980278262 \tabularnewline
36 & 241.35 & 215.577485491928 & 25.7725145080719 \tabularnewline
37 & 247.08 & 239.749844384205 & 7.33015561579549 \tabularnewline
38 & 265.11 & 246.624887648832 & 18.4851123511676 \tabularnewline
39 & 247.16 & 263.962302272327 & -16.8023022723271 \tabularnewline
40 & 299.56 & 248.203215955158 & 51.3567840448422 \tabularnewline
41 & 292.99 & 296.371376461819 & -3.381376461819 \tabularnewline
42 & 308.52 & 293.199941817389 & 15.3200581826114 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315574&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]2[/C][C]48.29[/C][C]43.28[/C][C]5.01[/C][/ROW]
[ROW][C]3[/C][C]60.7[/C][C]47.9789407266557[/C][C]12.7210592733443[/C][/ROW]
[ROW][C]4[/C][C]65.51[/C][C]59.9101789513501[/C][C]5.59982104864986[/C][/ROW]
[ROW][C]5[/C][C]63.6[/C][C]65.1623201065367[/C][C]-1.56232010653674[/C][/ROW]
[ROW][C]6[/C][C]61.2[/C][C]63.6970008297546[/C][C]-2.49700082975458[/C][/ROW]
[ROW][C]7[/C][C]63.72[/C][C]61.3550329867552[/C][C]2.36496701324481[/C][/ROW]
[ROW][C]8[/C][C]67.15[/C][C]63.5731646865024[/C][C]3.57683531349763[/C][/ROW]
[ROW][C]9[/C][C]62.9[/C][C]66.9279225961101[/C][C]-4.02792259611005[/C][/ROW]
[ROW][C]10[/C][C]77.65[/C][C]63.1500843664337[/C][C]14.4999156335663[/C][/ROW]
[ROW][C]11[/C][C]75.61[/C][C]76.7497338881177[/C][C]-1.13973388811772[/C][/ROW]
[ROW][C]12[/C][C]82.91[/C][C]75.6807634321445[/C][C]7.22923656785554[/C][/ROW]
[ROW][C]13[/C][C]76.78[/C][C]82.4611534783172[/C][C]-5.68115347831719[/C][/ROW]
[ROW][C]14[/C][C]74.4[/C][C]77.132729635274[/C][C]-2.73272963527396[/C][/ROW]
[ROW][C]15[/C][C]88.16[/C][C]74.5696688412365[/C][C]13.5903311587635[/C][/ROW]
[ROW][C]16[/C][C]91.82[/C][C]87.3162078772948[/C][C]4.50379212270515[/C][/ROW]
[ROW][C]17[/C][C]100.72[/C][C]91.5403699990055[/C][C]9.17963000099448[/C][/ROW]
[ROW][C]18[/C][C]109.91[/C][C]100.150058076312[/C][C]9.75994192368832[/C][/ROW]
[ROW][C]19[/C][C]105.02[/C][C]109.304027855756[/C][C]-4.28402785575555[/C][/ROW]
[ROW][C]20[/C][C]111.31[/C][C]105.285985347664[/C][C]6.02401465233577[/C][/ROW]
[ROW][C]21[/C][C]108.23[/C][C]110.935982910106[/C][C]-2.70598291010565[/C][/ROW]
[ROW][C]22[/C][C]109.33[/C][C]108.398008199142[/C][C]0.931991800858384[/C][/ROW]
[ROW][C]23[/C][C]113.54[/C][C]109.272134791949[/C][C]4.26786520805146[/C][/ROW]
[ROW][C]24[/C][C]130.13[/C][C]113.275018153623[/C][C]16.8549818463768[/C][/ROW]
[ROW][C]25[/C][C]129.77[/C][C]129.083513292342[/C][C]0.68648670765765[/C][/ROW]
[ROW][C]26[/C][C]149.64[/C][C]129.727377633444[/C][C]19.9126223665558[/C][/ROW]
[ROW][C]27[/C][C]146.96[/C][C]148.40367148828[/C][C]-1.44367148827951[/C][/ROW]
[ROW][C]28[/C][C]145.54[/C][C]147.049634212394[/C][C]-1.50963421239365[/C][/ROW]
[ROW][C]29[/C][C]158.9[/C][C]145.633729684855[/C][C]13.2662703151452[/C][/ROW]
[ROW][C]30[/C][C]172.74[/C][C]158.07632806303[/C][C]14.6636719369696[/C][/ROW]
[ROW][C]31[/C][C]186.23[/C][C]171.829566639267[/C][C]14.400433360733[/C][/ROW]
[ROW][C]32[/C][C]177.4[/C][C]185.33591051157[/C][C]-7.93591051157043[/C][/ROW]
[ROW][C]33[/C][C]196.72[/C][C]177.892722266877[/C][C]18.8272777331229[/C][/ROW]
[ROW][C]34[/C][C]207.36[/C][C]195.551058017821[/C][C]11.8089419821788[/C][/ROW]
[ROW][C]35[/C][C]216.17[/C][C]206.626810197217[/C][C]9.54318980278262[/C][/ROW]
[ROW][C]36[/C][C]241.35[/C][C]215.577485491928[/C][C]25.7725145080719[/C][/ROW]
[ROW][C]37[/C][C]247.08[/C][C]239.749844384205[/C][C]7.33015561579549[/C][/ROW]
[ROW][C]38[/C][C]265.11[/C][C]246.624887648832[/C][C]18.4851123511676[/C][/ROW]
[ROW][C]39[/C][C]247.16[/C][C]263.962302272327[/C][C]-16.8023022723271[/C][/ROW]
[ROW][C]40[/C][C]299.56[/C][C]248.203215955158[/C][C]51.3567840448422[/C][/ROW]
[ROW][C]41[/C][C]292.99[/C][C]296.371376461819[/C][C]-3.381376461819[/C][/ROW]
[ROW][C]42[/C][C]308.52[/C][C]293.199941817389[/C][C]15.3200581826114[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=315574&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315574&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
248.2943.285.01
360.747.978940726655712.7210592733443
465.5159.91017895135015.59982104864986
563.665.1623201065367-1.56232010653674
661.263.6970008297546-2.49700082975458
763.7261.35503298675522.36496701324481
867.1563.57316468650243.57683531349763
962.966.9279225961101-4.02792259611005
1077.6563.150084366433714.4999156335663
1175.6176.7497338881177-1.13973388811772
1282.9175.68076343214457.22923656785554
1376.7882.4611534783172-5.68115347831719
1474.477.132729635274-2.73272963527396
1588.1674.569668841236513.5903311587635
1691.8287.31620787729484.50379212270515
17100.7291.54036999900559.17963000099448
18109.91100.1500580763129.75994192368832
19105.02109.304027855756-4.28402785575555
20111.31105.2859853476646.02401465233577
21108.23110.935982910106-2.70598291010565
22109.33108.3980081991420.931991800858384
23113.54109.2721347919494.26786520805146
24130.13113.27501815362316.8549818463768
25129.77129.0835132923420.68648670765765
26149.64129.72737763344419.9126223665558
27146.96148.40367148828-1.44367148827951
28145.54147.049634212394-1.50963421239365
29158.9145.63372968485513.2662703151452
30172.74158.0763280630314.6636719369696
31186.23171.82956663926714.400433360733
32177.4185.33591051157-7.93591051157043
33196.72177.89272226687718.8272777331229
34207.36195.55105801782111.8089419821788
35216.17206.6268101972179.54318980278262
36241.35215.57748549192825.7725145080719
37247.08239.7498443842057.33015561579549
38265.11246.62488764883218.4851123511676
39247.16263.962302272327-16.8023022723271
40299.56248.20321595515851.3567840448422
41292.99296.371376461819-3.381376461819
42308.52293.19994181738915.3200581826114







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
43307.568813140544285.111494727704330.026131553383
44307.568813140544276.779514103942338.358112177145
45307.568813140544270.264246127444344.873380153644

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
43 & 307.568813140544 & 285.111494727704 & 330.026131553383 \tabularnewline
44 & 307.568813140544 & 276.779514103942 & 338.358112177145 \tabularnewline
45 & 307.568813140544 & 270.264246127444 & 344.873380153644 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=315574&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]43[/C][C]307.568813140544[/C][C]285.111494727704[/C][C]330.026131553383[/C][/ROW]
[ROW][C]44[/C][C]307.568813140544[/C][C]276.779514103942[/C][C]338.358112177145[/C][/ROW]
[ROW][C]45[/C][C]307.568813140544[/C][C]270.264246127444[/C][C]344.873380153644[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=315574&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=315574&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
43307.568813140544285.111494727704330.026131553383
44307.568813140544276.779514103942338.358112177145
45307.568813140544270.264246127444344.873380153644



Parameters (Session):
Parameters (R input):
par1 = 1 ; par2 = Single ; par3 = additive ; par4 = 3 ;
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')