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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 computationSun, 10 Jan 2016 11:31:00 +0000
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/Jan/10/t1452426348cygpyu1efa0zeam.htm/, Retrieved Sat, 04 May 2024 21:29:48 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=287762, Retrieved Sat, 04 May 2024 21:29:48 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact89

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 7] [2016-01-10 11:31:00] [c582792ff1d3c5b3510908504e5a21b2] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
19.6427
13.7242
14.8027
8.42832
6.80835
8.82499
8.76937
7.69536
7.65933
1.86731
3.72965
3.26542
-7.7517
0.957393
-3.04433
-12.5655
-6.9441
-6.8168
-8.56987
-4.46034
-11.229
-3.03305
-9.87928
-4.30931
-2.82241
-2.62636
-4.92814
-2.61029
-6.89788
-7.6867

Tables (Output of Computation)





Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 seconds
R Server'George Udny Yule' @ yule.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 & 'George Udny Yule' @ yule.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=287762&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]'George Udny Yule' @ yule.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=287762&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=287762&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 Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time2 seconds
R Server'George Udny Yule' @ yule.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.364865249555852
beta1
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.364865249555852 \tabularnewline
beta & 1 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=287762&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]0.364865249555852[/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=287762&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=287762&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
alpha0.364865249555852
beta1
gammaFALSE






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
314.80277.80576.997
48.428326.99312430228461.4351956977154
56.808354.674892526243732.13345747375627
68.824993.38985670095225.4351332990478
78.769375.292578917134593.47679108286541
87.695367.74933035799709-0.0539703579970858
97.659338.89813773647788-1.23880773647788
101.867319.16264123525445-7.29533123525445
113.729654.55551692446531-0.825866924465307
123.265422.007545182429971.25787481757003
13-7.75170.678813200335009-8.43051320033501
140.957393-7.260876196430948.21826919643094
15-3.04433-6.127442607926853.08311260792685
16-12.5655-5.74272855849566-6.82277144150434
17-6.9441-11.46171956949854.51761956949854
18-6.8168-11.39467359258744.57787359258742
19-8.56987-9.635336025710681.06546602571068
20-4.46034-8.768802394900074.30846239490007
21-11.229-5.14700387761924-6.08199612238076
22-3.03305-7.535431633235994.50238163323599
23-9.87928-4.41922515944988-5.46005484055012
24-4.30931-6.930149827842032.62083982784203
25-2.82241-5.536383468585222.71397346858522
26-2.62636-3.118401273353290.492041273353292
27-4.92814-1.33159616103702-3.59654383896298
28-2.61029-2.34882754139815-0.261462458601846
29-6.89788-2.244602186832-4.653277813168
30-7.6867-5.4406170081671-2.2460829918329

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 14.8027 & 7.8057 & 6.997 \tabularnewline
4 & 8.42832 & 6.9931243022846 & 1.4351956977154 \tabularnewline
5 & 6.80835 & 4.67489252624373 & 2.13345747375627 \tabularnewline
6 & 8.82499 & 3.3898567009522 & 5.4351332990478 \tabularnewline
7 & 8.76937 & 5.29257891713459 & 3.47679108286541 \tabularnewline
8 & 7.69536 & 7.74933035799709 & -0.0539703579970858 \tabularnewline
9 & 7.65933 & 8.89813773647788 & -1.23880773647788 \tabularnewline
10 & 1.86731 & 9.16264123525445 & -7.29533123525445 \tabularnewline
11 & 3.72965 & 4.55551692446531 & -0.825866924465307 \tabularnewline
12 & 3.26542 & 2.00754518242997 & 1.25787481757003 \tabularnewline
13 & -7.7517 & 0.678813200335009 & -8.43051320033501 \tabularnewline
14 & 0.957393 & -7.26087619643094 & 8.21826919643094 \tabularnewline
15 & -3.04433 & -6.12744260792685 & 3.08311260792685 \tabularnewline
16 & -12.5655 & -5.74272855849566 & -6.82277144150434 \tabularnewline
17 & -6.9441 & -11.4617195694985 & 4.51761956949854 \tabularnewline
18 & -6.8168 & -11.3946735925874 & 4.57787359258742 \tabularnewline
19 & -8.56987 & -9.63533602571068 & 1.06546602571068 \tabularnewline
20 & -4.46034 & -8.76880239490007 & 4.30846239490007 \tabularnewline
21 & -11.229 & -5.14700387761924 & -6.08199612238076 \tabularnewline
22 & -3.03305 & -7.53543163323599 & 4.50238163323599 \tabularnewline
23 & -9.87928 & -4.41922515944988 & -5.46005484055012 \tabularnewline
24 & -4.30931 & -6.93014982784203 & 2.62083982784203 \tabularnewline
25 & -2.82241 & -5.53638346858522 & 2.71397346858522 \tabularnewline
26 & -2.62636 & -3.11840127335329 & 0.492041273353292 \tabularnewline
27 & -4.92814 & -1.33159616103702 & -3.59654383896298 \tabularnewline
28 & -2.61029 & -2.34882754139815 & -0.261462458601846 \tabularnewline
29 & -6.89788 & -2.244602186832 & -4.653277813168 \tabularnewline
30 & -7.6867 & -5.4406170081671 & -2.2460829918329 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=287762&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]14.8027[/C][C]7.8057[/C][C]6.997[/C][/ROW]
[ROW][C]4[/C][C]8.42832[/C][C]6.9931243022846[/C][C]1.4351956977154[/C][/ROW]
[ROW][C]5[/C][C]6.80835[/C][C]4.67489252624373[/C][C]2.13345747375627[/C][/ROW]
[ROW][C]6[/C][C]8.82499[/C][C]3.3898567009522[/C][C]5.4351332990478[/C][/ROW]
[ROW][C]7[/C][C]8.76937[/C][C]5.29257891713459[/C][C]3.47679108286541[/C][/ROW]
[ROW][C]8[/C][C]7.69536[/C][C]7.74933035799709[/C][C]-0.0539703579970858[/C][/ROW]
[ROW][C]9[/C][C]7.65933[/C][C]8.89813773647788[/C][C]-1.23880773647788[/C][/ROW]
[ROW][C]10[/C][C]1.86731[/C][C]9.16264123525445[/C][C]-7.29533123525445[/C][/ROW]
[ROW][C]11[/C][C]3.72965[/C][C]4.55551692446531[/C][C]-0.825866924465307[/C][/ROW]
[ROW][C]12[/C][C]3.26542[/C][C]2.00754518242997[/C][C]1.25787481757003[/C][/ROW]
[ROW][C]13[/C][C]-7.7517[/C][C]0.678813200335009[/C][C]-8.43051320033501[/C][/ROW]
[ROW][C]14[/C][C]0.957393[/C][C]-7.26087619643094[/C][C]8.21826919643094[/C][/ROW]
[ROW][C]15[/C][C]-3.04433[/C][C]-6.12744260792685[/C][C]3.08311260792685[/C][/ROW]
[ROW][C]16[/C][C]-12.5655[/C][C]-5.74272855849566[/C][C]-6.82277144150434[/C][/ROW]
[ROW][C]17[/C][C]-6.9441[/C][C]-11.4617195694985[/C][C]4.51761956949854[/C][/ROW]
[ROW][C]18[/C][C]-6.8168[/C][C]-11.3946735925874[/C][C]4.57787359258742[/C][/ROW]
[ROW][C]19[/C][C]-8.56987[/C][C]-9.63533602571068[/C][C]1.06546602571068[/C][/ROW]
[ROW][C]20[/C][C]-4.46034[/C][C]-8.76880239490007[/C][C]4.30846239490007[/C][/ROW]
[ROW][C]21[/C][C]-11.229[/C][C]-5.14700387761924[/C][C]-6.08199612238076[/C][/ROW]
[ROW][C]22[/C][C]-3.03305[/C][C]-7.53543163323599[/C][C]4.50238163323599[/C][/ROW]
[ROW][C]23[/C][C]-9.87928[/C][C]-4.41922515944988[/C][C]-5.46005484055012[/C][/ROW]
[ROW][C]24[/C][C]-4.30931[/C][C]-6.93014982784203[/C][C]2.62083982784203[/C][/ROW]
[ROW][C]25[/C][C]-2.82241[/C][C]-5.53638346858522[/C][C]2.71397346858522[/C][/ROW]
[ROW][C]26[/C][C]-2.62636[/C][C]-3.11840127335329[/C][C]0.492041273353292[/C][/ROW]
[ROW][C]27[/C][C]-4.92814[/C][C]-1.33159616103702[/C][C]-3.59654383896298[/C][/ROW]
[ROW][C]28[/C][C]-2.61029[/C][C]-2.34882754139815[/C][C]-0.261462458601846[/C][/ROW]
[ROW][C]29[/C][C]-6.89788[/C][C]-2.244602186832[/C][C]-4.653277813168[/C][/ROW]
[ROW][C]30[/C][C]-7.6867[/C][C]-5.4406170081671[/C][C]-2.2460829918329[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=287762&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=287762&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
314.80277.80576.997
48.428326.99312430228461.4351956977154
56.808354.674892526243732.13345747375627
68.824993.38985670095225.4351332990478
78.769375.292578917134593.47679108286541
87.695367.74933035799709-0.0539703579970858
97.659338.89813773647788-1.23880773647788
101.867319.16264123525445-7.29533123525445
113.729654.55551692446531-0.825866924465307
123.265422.007545182429971.25787481757003
13-7.75170.678813200335009-8.43051320033501
140.957393-7.260876196430948.21826919643094
15-3.04433-6.127442607926853.08311260792685
16-12.5655-5.74272855849566-6.82277144150434
17-6.9441-11.46171956949854.51761956949854
18-6.8168-11.39467359258744.57787359258742
19-8.56987-9.635336025710681.06546602571068
20-4.46034-8.768802394900074.30846239490007
21-11.229-5.14700387761924-6.08199612238076
22-3.03305-7.535431633235994.50238163323599
23-9.87928-4.41922515944988-5.46005484055012
24-4.30931-6.930149827842032.62083982784203
25-2.82241-5.536383468585222.71397346858522
26-2.62636-3.118401273353290.492041273353292
27-4.92814-1.33159616103702-3.59654383896298
28-2.61029-2.34882754139815-0.261462458601846
29-6.89788-2.244602186832-4.653277813168
30-7.6867-5.4406170081671-2.2460829918329






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
31-8.57784772162448-17.40557836058750.249882917338542
32-10.8955608037436-21.82380145368110.0326798461939237
33-13.2132738858627-27.80080549706691.37425772534151
34-15.5309869679818-34.99342618775393.93145225179028
35-17.8487000501009-43.11026447243437.41286437223245
36-20.1664131322201-51.972468786504611.6396425220644
37-22.4841262143392-61.470931235743316.502678807065
38-24.8018392964583-71.534406241563621.930727648647
39-27.1195523785774-82.112820389518227.8737156323634
40-29.4372654606965-93.168732674661334.2942017532683
41-31.7549785428156-104.67275950999641.1628024243648
42-34.0726916249347-116.60097125111348.4555880012435

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
31 & -8.57784772162448 & -17.4055783605875 & 0.249882917338542 \tabularnewline
32 & -10.8955608037436 & -21.8238014536811 & 0.0326798461939237 \tabularnewline
33 & -13.2132738858627 & -27.8008054970669 & 1.37425772534151 \tabularnewline
34 & -15.5309869679818 & -34.9934261877539 & 3.93145225179028 \tabularnewline
35 & -17.8487000501009 & -43.1102644724343 & 7.41286437223245 \tabularnewline
36 & -20.1664131322201 & -51.9724687865046 & 11.6396425220644 \tabularnewline
37 & -22.4841262143392 & -61.4709312357433 & 16.502678807065 \tabularnewline
38 & -24.8018392964583 & -71.5344062415636 & 21.930727648647 \tabularnewline
39 & -27.1195523785774 & -82.1128203895182 & 27.8737156323634 \tabularnewline
40 & -29.4372654606965 & -93.1687326746613 & 34.2942017532683 \tabularnewline
41 & -31.7549785428156 & -104.672759509996 & 41.1628024243648 \tabularnewline
42 & -34.0726916249347 & -116.600971251113 & 48.4555880012435 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=287762&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]-8.57784772162448[/C][C]-17.4055783605875[/C][C]0.249882917338542[/C][/ROW]
[ROW][C]32[/C][C]-10.8955608037436[/C][C]-21.8238014536811[/C][C]0.0326798461939237[/C][/ROW]
[ROW][C]33[/C][C]-13.2132738858627[/C][C]-27.8008054970669[/C][C]1.37425772534151[/C][/ROW]
[ROW][C]34[/C][C]-15.5309869679818[/C][C]-34.9934261877539[/C][C]3.93145225179028[/C][/ROW]
[ROW][C]35[/C][C]-17.8487000501009[/C][C]-43.1102644724343[/C][C]7.41286437223245[/C][/ROW]
[ROW][C]36[/C][C]-20.1664131322201[/C][C]-51.9724687865046[/C][C]11.6396425220644[/C][/ROW]
[ROW][C]37[/C][C]-22.4841262143392[/C][C]-61.4709312357433[/C][C]16.502678807065[/C][/ROW]
[ROW][C]38[/C][C]-24.8018392964583[/C][C]-71.5344062415636[/C][C]21.930727648647[/C][/ROW]
[ROW][C]39[/C][C]-27.1195523785774[/C][C]-82.1128203895182[/C][C]27.8737156323634[/C][/ROW]
[ROW][C]40[/C][C]-29.4372654606965[/C][C]-93.1687326746613[/C][C]34.2942017532683[/C][/ROW]
[ROW][C]41[/C][C]-31.7549785428156[/C][C]-104.672759509996[/C][C]41.1628024243648[/C][/ROW]
[ROW][C]42[/C][C]-34.0726916249347[/C][C]-116.600971251113[/C][C]48.4555880012435[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=287762&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=287762&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
31-8.57784772162448-17.40557836058750.249882917338542
32-10.8955608037436-21.82380145368110.0326798461939237
33-13.2132738858627-27.80080549706691.37425772534151
34-15.5309869679818-34.99342618775393.93145225179028
35-17.8487000501009-43.11026447243437.41286437223245
36-20.1664131322201-51.972468786504611.6396425220644
37-22.4841262143392-61.470931235743316.502678807065
38-24.8018392964583-71.534406241563621.930727648647
39-27.1195523785774-82.112820389518227.8737156323634
40-29.4372654606965-93.168732674661334.2942017532683
41-31.7549785428156-104.67275950999641.1628024243648
42-34.0726916249347-116.60097125111348.4555880012435


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ; par4 = 0 ; par5 = 0 ;
Parameters (R input):
par1 = 12 ; par2 = Double ; par3 = additive ;
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')