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R Software Module

rwasp_exponentialsmoothing.wasp

Title produced by software

Exponential Smoothing

Date of computation

Mon, 14 Dec 2015 15:44:02 +0000

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Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2015/Dec/14/t1450109703649gndy4vs7xfmc.htm/, Retrieved Thu, 16 May 2024 23:44:31 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=286338, Retrieved Thu, 16 May 2024 23:44:31 +0000

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-       [Exponential Smoothing] [Double exponentia...] [2015-12-14 15:44:02] [ad3728658fedbe7404fc5a2c7dec5413] [Current]

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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	'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 & 1 seconds \tabularnewline
R Server & 'Sir Maurice George Kendall' @ kendall.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=286338&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]'Sir Maurice George Kendall' @ kendall.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=286338&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=286338&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	'Sir Maurice George Kendall' @ kendall.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.690366209464245
beta	0
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.690366209464245 \tabularnewline
beta & 0 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=286338&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]0.690366209464245[/C][/ROW]
[ROW][C]beta[/C][C]0[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=286338&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=286338&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	0.690366209464245
beta	0
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	657	671	-14
4	690	687.334873067501	2.66512693249945
5	721	715.174786645631	5.82521335436877
6	716	745.196317108407	-29.1963171084072
7	692	751.04016633596	-59.0401663359601
8	698	736.280830496465	-38.2808304964648
9	771	735.853038651477	35.1469613485228
10	856	786.117313131843	69.8826868681568
11	870	860.361958772189	9.63804122781062
12	929	893.015736761293	35.9842632387069
13	962	943.858056173763	18.1419438262371
14	1067	982.382641165396	84.6173588346045
15	976	1066.79960643892	-90.7996064389172

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 657 & 671 & -14 \tabularnewline
4 & 690 & 687.334873067501 & 2.66512693249945 \tabularnewline
5 & 721 & 715.174786645631 & 5.82521335436877 \tabularnewline
6 & 716 & 745.196317108407 & -29.1963171084072 \tabularnewline
7 & 692 & 751.04016633596 & -59.0401663359601 \tabularnewline
8 & 698 & 736.280830496465 & -38.2808304964648 \tabularnewline
9 & 771 & 735.853038651477 & 35.1469613485228 \tabularnewline
10 & 856 & 786.117313131843 & 69.8826868681568 \tabularnewline
11 & 870 & 860.361958772189 & 9.63804122781062 \tabularnewline
12 & 929 & 893.015736761293 & 35.9842632387069 \tabularnewline
13 & 962 & 943.858056173763 & 18.1419438262371 \tabularnewline
14 & 1067 & 982.382641165396 & 84.6173588346045 \tabularnewline
15 & 976 & 1066.79960643892 & -90.7996064389172 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=286338&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]657[/C][C]671[/C][C]-14[/C][/ROW]
[ROW][C]4[/C][C]690[/C][C]687.334873067501[/C][C]2.66512693249945[/C][/ROW]
[ROW][C]5[/C][C]721[/C][C]715.174786645631[/C][C]5.82521335436877[/C][/ROW]
[ROW][C]6[/C][C]716[/C][C]745.196317108407[/C][C]-29.1963171084072[/C][/ROW]
[ROW][C]7[/C][C]692[/C][C]751.04016633596[/C][C]-59.0401663359601[/C][/ROW]
[ROW][C]8[/C][C]698[/C][C]736.280830496465[/C][C]-38.2808304964648[/C][/ROW]
[ROW][C]9[/C][C]771[/C][C]735.853038651477[/C][C]35.1469613485228[/C][/ROW]
[ROW][C]10[/C][C]856[/C][C]786.117313131843[/C][C]69.8826868681568[/C][/ROW]
[ROW][C]11[/C][C]870[/C][C]860.361958772189[/C][C]9.63804122781062[/C][/ROW]
[ROW][C]12[/C][C]929[/C][C]893.015736761293[/C][C]35.9842632387069[/C][/ROW]
[ROW][C]13[/C][C]962[/C][C]943.858056173763[/C][C]18.1419438262371[/C][/ROW]
[ROW][C]14[/C][C]1067[/C][C]982.382641165396[/C][C]84.6173588346045[/C][/ROW]
[ROW][C]15[/C][C]976[/C][C]1066.79960643892[/C][C]-90.7996064389172[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=286338&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=286338&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	657	671	-14
4	690	687.334873067501	2.66512693249945
5	721	715.174786645631	5.82521335436877
6	716	745.196317108407	-29.1963171084072
7	692	751.04016633596	-59.0401663359601
8	698	736.280830496465	-38.2808304964648
9	771	735.853038651477	35.1469613485228
10	856	786.117313131843	69.8826868681568
11	870	860.361958772189	9.63804122781062
12	929	893.015736761293	35.9842632387069
13	962	943.858056173763	18.1419438262371
14	1067	982.382641165396	84.6173588346045
15	976	1066.79960643892	-90.7996064389172

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
16	1030.11462632084	933.416797412119	1126.81245522955
17	1056.11462632084	938.611623678309	1173.61762896336
18	1082.11462632084	946.972326709057	1217.25692593262
19	1108.11462632084	957.383320154167	1258.84593248751
20	1134.11462632084	969.26192791083	1298.96732473084
21	1160.11462632084	982.258230069786	1337.97102257189
22	1186.11462632084	996.142567318042	1376.08668532363
23	1212.11462632084	1010.75457885334	1413.47467378833
24	1238.11462632084	1025.97704306303	1450.25220957864
25	1264.11462632084	1041.72119173966	1486.50806090201
26	1290.11462632084	1057.91788805959	1522.31136458208
27	1316.11462632084	1074.51203781336	1557.71721482831

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
16 & 1030.11462632084 & 933.416797412119 & 1126.81245522955 \tabularnewline
17 & 1056.11462632084 & 938.611623678309 & 1173.61762896336 \tabularnewline
18 & 1082.11462632084 & 946.972326709057 & 1217.25692593262 \tabularnewline
19 & 1108.11462632084 & 957.383320154167 & 1258.84593248751 \tabularnewline
20 & 1134.11462632084 & 969.26192791083 & 1298.96732473084 \tabularnewline
21 & 1160.11462632084 & 982.258230069786 & 1337.97102257189 \tabularnewline
22 & 1186.11462632084 & 996.142567318042 & 1376.08668532363 \tabularnewline
23 & 1212.11462632084 & 1010.75457885334 & 1413.47467378833 \tabularnewline
24 & 1238.11462632084 & 1025.97704306303 & 1450.25220957864 \tabularnewline
25 & 1264.11462632084 & 1041.72119173966 & 1486.50806090201 \tabularnewline
26 & 1290.11462632084 & 1057.91788805959 & 1522.31136458208 \tabularnewline
27 & 1316.11462632084 & 1074.51203781336 & 1557.71721482831 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=286338&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]16[/C][C]1030.11462632084[/C][C]933.416797412119[/C][C]1126.81245522955[/C][/ROW]
[ROW][C]17[/C][C]1056.11462632084[/C][C]938.611623678309[/C][C]1173.61762896336[/C][/ROW]
[ROW][C]18[/C][C]1082.11462632084[/C][C]946.972326709057[/C][C]1217.25692593262[/C][/ROW]
[ROW][C]19[/C][C]1108.11462632084[/C][C]957.383320154167[/C][C]1258.84593248751[/C][/ROW]
[ROW][C]20[/C][C]1134.11462632084[/C][C]969.26192791083[/C][C]1298.96732473084[/C][/ROW]
[ROW][C]21[/C][C]1160.11462632084[/C][C]982.258230069786[/C][C]1337.97102257189[/C][/ROW]
[ROW][C]22[/C][C]1186.11462632084[/C][C]996.142567318042[/C][C]1376.08668532363[/C][/ROW]
[ROW][C]23[/C][C]1212.11462632084[/C][C]1010.75457885334[/C][C]1413.47467378833[/C][/ROW]
[ROW][C]24[/C][C]1238.11462632084[/C][C]1025.97704306303[/C][C]1450.25220957864[/C][/ROW]
[ROW][C]25[/C][C]1264.11462632084[/C][C]1041.72119173966[/C][C]1486.50806090201[/C][/ROW]
[ROW][C]26[/C][C]1290.11462632084[/C][C]1057.91788805959[/C][C]1522.31136458208[/C][/ROW]
[ROW][C]27[/C][C]1316.11462632084[/C][C]1074.51203781336[/C][C]1557.71721482831[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=286338&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=286338&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
16	1030.11462632084	933.416797412119	1126.81245522955
17	1056.11462632084	938.611623678309	1173.61762896336
18	1082.11462632084	946.972326709057	1217.25692593262
19	1108.11462632084	957.383320154167	1258.84593248751
20	1134.11462632084	969.26192791083	1298.96732473084
21	1160.11462632084	982.258230069786	1337.97102257189
22	1186.11462632084	996.142567318042	1376.08668532363
23	1212.11462632084	1010.75457885334	1413.47467378833
24	1238.11462632084	1025.97704306303	1450.25220957864
25	1264.11462632084	1041.72119173966	1486.50806090201
26	1290.11462632084	1057.91788805959	1522.31136458208
27	1316.11462632084	1074.51203781336	1557.71721482831

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):

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')

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Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code