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Author's title

Author

*The author of this computation has been verified*

R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Mon, 11 Jan 2016 09:57:41 +0000

Cite this page as follows

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2016/Jan/11/t1452506284pi6mjwf3vjpu84y.htm/, Retrieved Tue, 07 May 2024 12:14:44 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=289265, Retrieved Tue, 07 May 2024 12:14:44 +0000

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User-defined keywords

Estimated Impact

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

-       [Exponential Smoothing] [] [2016-01-11 09:57:41] [ea6a15f7ebbd64f3694cdf9efb8b4035] [Current]

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Dataseries X:

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Histogram

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0.7923
-2.468
-2.996
3.119
0.04315
0.731
2.45
2.119
-1.429
-1.644
-3.065
-1.461
1.141
1.329
0.3396
0.8429
2.225
-1.924
0.4999
-0.6433

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	1 seconds
R Server	'Gwilym Jenkins' @ jenkins.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 & 1 seconds \tabularnewline
R Server & 'Gwilym Jenkins' @ jenkins.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289265&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]1 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Gwilym Jenkins' @ jenkins.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=289265&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289265&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 Output	view raw output of R engine
Computing time	1 seconds
R Server	'Gwilym Jenkins' @ jenkins.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.878577320308712
beta	0.266400595402578
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.878577320308712 \tabularnewline
beta & 0.266400595402578 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289265&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]0.878577320308712[/C][/ROW]
[ROW][C]beta[/C][C]0.266400595402578[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=289265&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289265&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
Parameter	Value
alpha	0.878577320308712
beta	0.266400595402578
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	-2.996	-5.7283	2.7323
4	3.119	-5.94855875164344	9.06755875164344
5	0.04315	1.51949120904386	-1.47634120904386
6	0.731	-0.621633061857281	1.35263306185728
7	2.45	0.0393038324620748	2.41069616753793
8	2.119	2.1940629015337	-0.0750629015337032
9	-1.429	2.14732169234904	-3.57632169234904
10	-1.644	-1.81259676814963	0.168596768149627
11	-3.065	-2.44285413559407	-0.622145864405927
12	-1.461	-3.91345547649055	2.45245547649055
13	1.141	-2.10877597033397	3.24977597033397
14	1.329	1.15703274782027	0.171967252179725
15	0.3396	1.75899807086596	-1.41939807086596
16	0.8429	0.63061079623415	0.21228920376585
17	2.225	0.985473990592271	1.23952600940773
18	-1.924	2.53295957214099	-4.45695957214099
19	0.4999	-1.9675249655971	2.4674249655971
20	-0.6433	0.193107210204995	-0.836407210204995

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & -2.996 & -5.7283 & 2.7323 \tabularnewline
4 & 3.119 & -5.94855875164344 & 9.06755875164344 \tabularnewline
5 & 0.04315 & 1.51949120904386 & -1.47634120904386 \tabularnewline
6 & 0.731 & -0.621633061857281 & 1.35263306185728 \tabularnewline
7 & 2.45 & 0.0393038324620748 & 2.41069616753793 \tabularnewline
8 & 2.119 & 2.1940629015337 & -0.0750629015337032 \tabularnewline
9 & -1.429 & 2.14732169234904 & -3.57632169234904 \tabularnewline
10 & -1.644 & -1.81259676814963 & 0.168596768149627 \tabularnewline
11 & -3.065 & -2.44285413559407 & -0.622145864405927 \tabularnewline
12 & -1.461 & -3.91345547649055 & 2.45245547649055 \tabularnewline
13 & 1.141 & -2.10877597033397 & 3.24977597033397 \tabularnewline
14 & 1.329 & 1.15703274782027 & 0.171967252179725 \tabularnewline
15 & 0.3396 & 1.75899807086596 & -1.41939807086596 \tabularnewline
16 & 0.8429 & 0.63061079623415 & 0.21228920376585 \tabularnewline
17 & 2.225 & 0.985473990592271 & 1.23952600940773 \tabularnewline
18 & -1.924 & 2.53295957214099 & -4.45695957214099 \tabularnewline
19 & 0.4999 & -1.9675249655971 & 2.4674249655971 \tabularnewline
20 & -0.6433 & 0.193107210204995 & -0.836407210204995 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289265&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]-2.996[/C][C]-5.7283[/C][C]2.7323[/C][/ROW]
[ROW][C]4[/C][C]3.119[/C][C]-5.94855875164344[/C][C]9.06755875164344[/C][/ROW]
[ROW][C]5[/C][C]0.04315[/C][C]1.51949120904386[/C][C]-1.47634120904386[/C][/ROW]
[ROW][C]6[/C][C]0.731[/C][C]-0.621633061857281[/C][C]1.35263306185728[/C][/ROW]
[ROW][C]7[/C][C]2.45[/C][C]0.0393038324620748[/C][C]2.41069616753793[/C][/ROW]
[ROW][C]8[/C][C]2.119[/C][C]2.1940629015337[/C][C]-0.0750629015337032[/C][/ROW]
[ROW][C]9[/C][C]-1.429[/C][C]2.14732169234904[/C][C]-3.57632169234904[/C][/ROW]
[ROW][C]10[/C][C]-1.644[/C][C]-1.81259676814963[/C][C]0.168596768149627[/C][/ROW]
[ROW][C]11[/C][C]-3.065[/C][C]-2.44285413559407[/C][C]-0.622145864405927[/C][/ROW]
[ROW][C]12[/C][C]-1.461[/C][C]-3.91345547649055[/C][C]2.45245547649055[/C][/ROW]
[ROW][C]13[/C][C]1.141[/C][C]-2.10877597033397[/C][C]3.24977597033397[/C][/ROW]
[ROW][C]14[/C][C]1.329[/C][C]1.15703274782027[/C][C]0.171967252179725[/C][/ROW]
[ROW][C]15[/C][C]0.3396[/C][C]1.75899807086596[/C][C]-1.41939807086596[/C][/ROW]
[ROW][C]16[/C][C]0.8429[/C][C]0.63061079623415[/C][C]0.21228920376585[/C][/ROW]
[ROW][C]17[/C][C]2.225[/C][C]0.985473990592271[/C][C]1.23952600940773[/C][/ROW]
[ROW][C]18[/C][C]-1.924[/C][C]2.53295957214099[/C][C]-4.45695957214099[/C][/ROW]
[ROW][C]19[/C][C]0.4999[/C][C]-1.9675249655971[/C][C]2.4674249655971[/C][/ROW]
[ROW][C]20[/C][C]-0.6433[/C][C]0.193107210204995[/C][C]-0.836407210204995[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=289265&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289265&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
t	Observed	Fitted	Residuals
3	-2.996	-5.7283	2.7323
4	3.119	-5.94855875164344	9.06755875164344
5	0.04315	1.51949120904386	-1.47634120904386
6	0.731	-0.621633061857281	1.35263306185728
7	2.45	0.0393038324620748	2.41069616753793
8	2.119	2.1940629015337	-0.0750629015337032
9	-1.429	2.14732169234904	-3.57632169234904
10	-1.644	-1.81259676814963	0.168596768149627
11	-3.065	-2.44285413559407	-0.622145864405927
12	-1.461	-3.91345547649055	2.45245547649055
13	1.141	-2.10877597033397	3.24977597033397
14	1.329	1.15703274782027	0.171967252179725
15	0.3396	1.75899807086596	-1.41939807086596
16	0.8429	0.63061079623415	0.21228920376585
17	2.225	0.985473990592271	1.23952600940773
18	-1.924	2.53295957214099	-4.45695957214099
19	0.4999	-1.9675249655971	2.4674249655971
20	-0.6433	0.193107210204995	-0.836407210204995

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
21	-0.744696686495681	-6.56844278249977	5.0790494095084
22	-0.947652177767567	-9.6598427903154	7.76453843478026
23	-1.15060766903945	-12.8728497041045	10.5716343660256
24	-1.35356316031134	-16.2585290456635	13.5514027250408
25	-1.55651865158322	-19.8283364234666	16.7152991203002
26	-1.75947414285511	-23.5821875062251	20.0632392205149
27	-1.962429634127	-27.5160673374197	23.5912080691657
28	-2.16538512539888	-31.6246447725432	27.2938745217455
29	-2.36834061667077	-35.9022855989561	31.1656043656145
30	-2.57129610794265	-40.3434653797399	35.2008731638546
31	-2.77425159921454	-44.9429343566818	39.3944311582528
32	-2.97720709048642	-49.6957738317482	43.7413596507754

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
21 & -0.744696686495681 & -6.56844278249977 & 5.0790494095084 \tabularnewline
22 & -0.947652177767567 & -9.6598427903154 & 7.76453843478026 \tabularnewline
23 & -1.15060766903945 & -12.8728497041045 & 10.5716343660256 \tabularnewline
24 & -1.35356316031134 & -16.2585290456635 & 13.5514027250408 \tabularnewline
25 & -1.55651865158322 & -19.8283364234666 & 16.7152991203002 \tabularnewline
26 & -1.75947414285511 & -23.5821875062251 & 20.0632392205149 \tabularnewline
27 & -1.962429634127 & -27.5160673374197 & 23.5912080691657 \tabularnewline
28 & -2.16538512539888 & -31.6246447725432 & 27.2938745217455 \tabularnewline
29 & -2.36834061667077 & -35.9022855989561 & 31.1656043656145 \tabularnewline
30 & -2.57129610794265 & -40.3434653797399 & 35.2008731638546 \tabularnewline
31 & -2.77425159921454 & -44.9429343566818 & 39.3944311582528 \tabularnewline
32 & -2.97720709048642 & -49.6957738317482 & 43.7413596507754 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289265&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]-0.744696686495681[/C][C]-6.56844278249977[/C][C]5.0790494095084[/C][/ROW]
[ROW][C]22[/C][C]-0.947652177767567[/C][C]-9.6598427903154[/C][C]7.76453843478026[/C][/ROW]
[ROW][C]23[/C][C]-1.15060766903945[/C][C]-12.8728497041045[/C][C]10.5716343660256[/C][/ROW]
[ROW][C]24[/C][C]-1.35356316031134[/C][C]-16.2585290456635[/C][C]13.5514027250408[/C][/ROW]
[ROW][C]25[/C][C]-1.55651865158322[/C][C]-19.8283364234666[/C][C]16.7152991203002[/C][/ROW]
[ROW][C]26[/C][C]-1.75947414285511[/C][C]-23.5821875062251[/C][C]20.0632392205149[/C][/ROW]
[ROW][C]27[/C][C]-1.962429634127[/C][C]-27.5160673374197[/C][C]23.5912080691657[/C][/ROW]
[ROW][C]28[/C][C]-2.16538512539888[/C][C]-31.6246447725432[/C][C]27.2938745217455[/C][/ROW]
[ROW][C]29[/C][C]-2.36834061667077[/C][C]-35.9022855989561[/C][C]31.1656043656145[/C][/ROW]
[ROW][C]30[/C][C]-2.57129610794265[/C][C]-40.3434653797399[/C][C]35.2008731638546[/C][/ROW]
[ROW][C]31[/C][C]-2.77425159921454[/C][C]-44.9429343566818[/C][C]39.3944311582528[/C][/ROW]
[ROW][C]32[/C][C]-2.97720709048642[/C][C]-49.6957738317482[/C][C]43.7413596507754[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=289265&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289265&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
t	Forecast	95% Lower Bound	95% Upper Bound
21	-0.744696686495681	-6.56844278249977	5.0790494095084
22	-0.947652177767567	-9.6598427903154	7.76453843478026
23	-1.15060766903945	-12.8728497041045	10.5716343660256
24	-1.35356316031134	-16.2585290456635	13.5514027250408
25	-1.55651865158322	-19.8283364234666	16.7152991203002
26	-1.75947414285511	-23.5821875062251	20.0632392205149
27	-1.962429634127	-27.5160673374197	23.5912080691657
28	-2.16538512539888	-31.6246447725432	27.2938745217455
29	-2.36834061667077	-35.9022855989561	31.1656043656145
30	-2.57129610794265	-40.3434653797399	35.2008731638546
31	-2.77425159921454	-44.9429343566818	39.3944311582528
32	-2.97720709048642	-49.6957738317482	43.7413596507754

Figure 1

PNG link

Postscript link

PDF link

Figure 2

PNG link

Postscript link

PDF link

Figure 3

PNG link

Postscript link

PDF link

Parameters (Session):

Parameters (R input):

par1 = 12 ; par2 = Double ; par3 = additive ;

R code (references can be found in the software module):

par3 <- 'additive'
par2 <- 'Single'
par1 <- '12'
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')

Free Statistics

Description of Statistical Computation

Tree of Dependent Computations

Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code