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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 computationSat, 15 Nov 2014 10:45:02 +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/2014/Nov/15/t1416048328h3vhfkn9ug1f1gw.htm/, Retrieved Sun, 19 May 2024 15:36:11 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=254904, Retrieved Sun, 19 May 2024 15:36:11 +0000
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Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact101

Tree of Dependent Computations

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [] [2014-11-15 10:45:02] [91c7a3a259bd23f54bd28f7298631f67] [Current]
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Dataset

Dataseries X:
Download CSV
Histogram
Boxplots 
26,6
25,0
24,2
24,2
24,8
24,5
24,3
21,6
22,4
23,5
23,4
23,4
23,0
22,0
21,7
22,2
22,8
22,2
19,9
16,1
15,8
16,8
18,4
19,3
18,6

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'Sir Maurice George Kendall' @ kendall.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 & 'Sir Maurice George Kendall' @ kendall.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=254904&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]'Sir Maurice George Kendall' @ kendall.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=254904&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=254904&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'Sir Maurice George Kendall' @ kendall.wessa.net






Estimated Parameters of Exponential Smoothing
ParameterValue
alpha1
beta0.882309582362307
gammaFALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0.882309582362307 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=254904&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]1[/C][/ROW]
[ROW][C]beta[/C][C]0.882309582362307[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=254904&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=254904&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
alpha1
beta0.882309582362307
gammaFALSE






Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
324.223.40.800000000000001
424.223.30584766588980.894152334110156
524.824.09476683836690.705233161633142
624.525.3170008146754-0.817000814675445
724.324.29615316708950.00384683291051147
821.624.0995472646282-2.49954726462818
922.419.19417276147923.20582723852075
1023.522.82270485342420.677295146575805
1123.424.5202888513355-1.12028885133551
1223.423.4318472627885-0.0318472627885278
132323.4037481176582-0.403748117658196
142222.6475172845876-0.647517284587629
1521.721.07620657965070.623793420349255
1622.221.32658549183940.87341450816055
1722.822.59720748176380.202792518236237
1822.223.376133263835-1.17613326383498
1919.921.7384196150183-1.83841961501832
2016.117.8163643722848-1.71636437228483
2115.812.50199963979273.29800036020734
2216.815.11185696023791.68814303976206
2318.417.60132174061820.798678259381759
2419.319.9060032220952-0.606003222095211
2518.620.2713207722982-1.67132077229817

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 24.2 & 23.4 & 0.800000000000001 \tabularnewline
4 & 24.2 & 23.3058476658898 & 0.894152334110156 \tabularnewline
5 & 24.8 & 24.0947668383669 & 0.705233161633142 \tabularnewline
6 & 24.5 & 25.3170008146754 & -0.817000814675445 \tabularnewline
7 & 24.3 & 24.2961531670895 & 0.00384683291051147 \tabularnewline
8 & 21.6 & 24.0995472646282 & -2.49954726462818 \tabularnewline
9 & 22.4 & 19.1941727614792 & 3.20582723852075 \tabularnewline
10 & 23.5 & 22.8227048534242 & 0.677295146575805 \tabularnewline
11 & 23.4 & 24.5202888513355 & -1.12028885133551 \tabularnewline
12 & 23.4 & 23.4318472627885 & -0.0318472627885278 \tabularnewline
13 & 23 & 23.4037481176582 & -0.403748117658196 \tabularnewline
14 & 22 & 22.6475172845876 & -0.647517284587629 \tabularnewline
15 & 21.7 & 21.0762065796507 & 0.623793420349255 \tabularnewline
16 & 22.2 & 21.3265854918394 & 0.87341450816055 \tabularnewline
17 & 22.8 & 22.5972074817638 & 0.202792518236237 \tabularnewline
18 & 22.2 & 23.376133263835 & -1.17613326383498 \tabularnewline
19 & 19.9 & 21.7384196150183 & -1.83841961501832 \tabularnewline
20 & 16.1 & 17.8163643722848 & -1.71636437228483 \tabularnewline
21 & 15.8 & 12.5019996397927 & 3.29800036020734 \tabularnewline
22 & 16.8 & 15.1118569602379 & 1.68814303976206 \tabularnewline
23 & 18.4 & 17.6013217406182 & 0.798678259381759 \tabularnewline
24 & 19.3 & 19.9060032220952 & -0.606003222095211 \tabularnewline
25 & 18.6 & 20.2713207722982 & -1.67132077229817 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=254904&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]24.2[/C][C]23.4[/C][C]0.800000000000001[/C][/ROW]
[ROW][C]4[/C][C]24.2[/C][C]23.3058476658898[/C][C]0.894152334110156[/C][/ROW]
[ROW][C]5[/C][C]24.8[/C][C]24.0947668383669[/C][C]0.705233161633142[/C][/ROW]
[ROW][C]6[/C][C]24.5[/C][C]25.3170008146754[/C][C]-0.817000814675445[/C][/ROW]
[ROW][C]7[/C][C]24.3[/C][C]24.2961531670895[/C][C]0.00384683291051147[/C][/ROW]
[ROW][C]8[/C][C]21.6[/C][C]24.0995472646282[/C][C]-2.49954726462818[/C][/ROW]
[ROW][C]9[/C][C]22.4[/C][C]19.1941727614792[/C][C]3.20582723852075[/C][/ROW]
[ROW][C]10[/C][C]23.5[/C][C]22.8227048534242[/C][C]0.677295146575805[/C][/ROW]
[ROW][C]11[/C][C]23.4[/C][C]24.5202888513355[/C][C]-1.12028885133551[/C][/ROW]
[ROW][C]12[/C][C]23.4[/C][C]23.4318472627885[/C][C]-0.0318472627885278[/C][/ROW]
[ROW][C]13[/C][C]23[/C][C]23.4037481176582[/C][C]-0.403748117658196[/C][/ROW]
[ROW][C]14[/C][C]22[/C][C]22.6475172845876[/C][C]-0.647517284587629[/C][/ROW]
[ROW][C]15[/C][C]21.7[/C][C]21.0762065796507[/C][C]0.623793420349255[/C][/ROW]
[ROW][C]16[/C][C]22.2[/C][C]21.3265854918394[/C][C]0.87341450816055[/C][/ROW]
[ROW][C]17[/C][C]22.8[/C][C]22.5972074817638[/C][C]0.202792518236237[/C][/ROW]
[ROW][C]18[/C][C]22.2[/C][C]23.376133263835[/C][C]-1.17613326383498[/C][/ROW]
[ROW][C]19[/C][C]19.9[/C][C]21.7384196150183[/C][C]-1.83841961501832[/C][/ROW]
[ROW][C]20[/C][C]16.1[/C][C]17.8163643722848[/C][C]-1.71636437228483[/C][/ROW]
[ROW][C]21[/C][C]15.8[/C][C]12.5019996397927[/C][C]3.29800036020734[/C][/ROW]
[ROW][C]22[/C][C]16.8[/C][C]15.1118569602379[/C][C]1.68814303976206[/C][/ROW]
[ROW][C]23[/C][C]18.4[/C][C]17.6013217406182[/C][C]0.798678259381759[/C][/ROW]
[ROW][C]24[/C][C]19.3[/C][C]19.9060032220952[/C][C]-0.606003222095211[/C][/ROW]
[ROW][C]25[/C][C]18.6[/C][C]20.2713207722982[/C][C]-1.67132077229817[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=254904&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=254904&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
324.223.40.800000000000001
424.223.30584766588980.894152334110156
524.824.09476683836690.705233161633142
624.525.3170008146754-0.817000814675445
724.324.29615316708950.00384683291051147
821.624.0995472646282-2.49954726462818
922.419.19417276147923.20582723852075
1023.522.82270485342420.677295146575805
1123.424.5202888513355-1.12028885133551
1223.423.4318472627885-0.0318472627885278
132323.4037481176582-0.403748117658196
142222.6475172845876-0.647517284587629
1521.721.07620657965070.623793420349255
1622.221.32658549183940.87341450816055
1722.822.59720748176380.202792518236237
1822.223.376133263835-1.17613326383498
1919.921.7384196150183-1.83841961501832
2016.117.8163643722848-1.71636437228483
2115.812.50199963979273.29800036020734
2216.815.11185696023791.68814303976206
2318.417.60132174061820.798678259381759
2419.319.9060032220952-0.606003222095211
2518.620.2713207722982-1.67132077229817






Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
2618.096698439698315.208085710695520.9853111687011
2717.593396879396611.436456324902223.7503374338911
2817.0900953190957.0063092659812827.1738813722087
2916.58679375879332.0039340699996931.1696534475869
3016.0834921984916-3.5080959017747835.675080298758
3115.5801906381899-9.4834396777090840.643820954089
3215.0768890778883-15.886262543876746.0400406996533
3314.5735875175866-22.687855640137151.8350306753103
3414.0702859572849-29.864574561764858.0051464763346
3513.5669843969832-37.396516924387564.530485718354
3613.0636828366816-45.266634162251371.3939998356145
3712.5603812763799-53.460111392949578.5808739457093

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
26 & 18.0966984396983 & 15.2080857106955 & 20.9853111687011 \tabularnewline
27 & 17.5933968793966 & 11.4364563249022 & 23.7503374338911 \tabularnewline
28 & 17.090095319095 & 7.00630926598128 & 27.1738813722087 \tabularnewline
29 & 16.5867937587933 & 2.00393406999969 & 31.1696534475869 \tabularnewline
30 & 16.0834921984916 & -3.50809590177478 & 35.675080298758 \tabularnewline
31 & 15.5801906381899 & -9.48343967770908 & 40.643820954089 \tabularnewline
32 & 15.0768890778883 & -15.8862625438767 & 46.0400406996533 \tabularnewline
33 & 14.5735875175866 & -22.6878556401371 & 51.8350306753103 \tabularnewline
34 & 14.0702859572849 & -29.8645745617648 & 58.0051464763346 \tabularnewline
35 & 13.5669843969832 & -37.3965169243875 & 64.530485718354 \tabularnewline
36 & 13.0636828366816 & -45.2666341622513 & 71.3939998356145 \tabularnewline
37 & 12.5603812763799 & -53.4601113929495 & 78.5808739457093 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=254904&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]26[/C][C]18.0966984396983[/C][C]15.2080857106955[/C][C]20.9853111687011[/C][/ROW]
[ROW][C]27[/C][C]17.5933968793966[/C][C]11.4364563249022[/C][C]23.7503374338911[/C][/ROW]
[ROW][C]28[/C][C]17.090095319095[/C][C]7.00630926598128[/C][C]27.1738813722087[/C][/ROW]
[ROW][C]29[/C][C]16.5867937587933[/C][C]2.00393406999969[/C][C]31.1696534475869[/C][/ROW]
[ROW][C]30[/C][C]16.0834921984916[/C][C]-3.50809590177478[/C][C]35.675080298758[/C][/ROW]
[ROW][C]31[/C][C]15.5801906381899[/C][C]-9.48343967770908[/C][C]40.643820954089[/C][/ROW]
[ROW][C]32[/C][C]15.0768890778883[/C][C]-15.8862625438767[/C][C]46.0400406996533[/C][/ROW]
[ROW][C]33[/C][C]14.5735875175866[/C][C]-22.6878556401371[/C][C]51.8350306753103[/C][/ROW]
[ROW][C]34[/C][C]14.0702859572849[/C][C]-29.8645745617648[/C][C]58.0051464763346[/C][/ROW]
[ROW][C]35[/C][C]13.5669843969832[/C][C]-37.3965169243875[/C][C]64.530485718354[/C][/ROW]
[ROW][C]36[/C][C]13.0636828366816[/C][C]-45.2666341622513[/C][C]71.3939998356145[/C][/ROW]
[ROW][C]37[/C][C]12.5603812763799[/C][C]-53.4601113929495[/C][C]78.5808739457093[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=254904&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=254904&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
2618.096698439698315.208085710695520.9853111687011
2717.593396879396611.436456324902223.7503374338911
2817.0900953190957.0063092659812827.1738813722087
2916.58679375879332.0039340699996931.1696534475869
3016.0834921984916-3.5080959017747835.675080298758
3115.5801906381899-9.4834396777090840.643820954089
3215.0768890778883-15.886262543876746.0400406996533
3314.5735875175866-22.687855640137151.8350306753103
3414.0702859572849-29.864574561764858.0051464763346
3513.5669843969832-37.396516924387564.530485718354
3613.0636828366816-45.266634162251371.3939998356145
3712.5603812763799-53.460111392949578.5808739457093


Figures (Output of Computation)

Input Parameters & R Code

Parameters (Session):
par1 = 12 ; par2 = Double ; par3 = additive ;
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')