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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

Fri, 22 Jan 2016 09:21:40 +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/22/t1453454623efc76g8d4n5oko3.htm/, Retrieved Tue, 07 May 2024 18:02:21 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=290667, Retrieved Tue, 07 May 2024 18:02:21 +0000

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Estimated Impact

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

-       [Exponential Smoothing] [vraag5] [2016-01-22 09:21:40] [18c782d74737b748f6df91564ed160d2] [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	'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 & 1 seconds \tabularnewline
R Server & 'George Udny Yule' @ yule.wessa.net \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=290667&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]'George Udny Yule' @ yule.wessa.net[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=290667&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=290667&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	'George Udny Yule' @ yule.wessa.net

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.127158571611458
beta	FALSE
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.127158571611458 \tabularnewline
beta & FALSE \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=290667&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]0.127158571611458[/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=290667&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=290667&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	0.127158571611458
beta	FALSE
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
2	10.89	10.24	0.65
3	9	10.3226530715474	-1.32265307154745
4	12.25	10.154466396232	2.09553360376803
5	13.69	10.4209314560509	3.26906854394908
6	7.29	10.8366215425994	-3.54662154259944
7	12.96	10.3856382131961	2.57436178680394
8	12.25	10.7129903808172	1.53700961918283
9	14.44	10.9084343285455	3.53156567145446
10	11.56	11.3575031748797	0.202496825120255
11	13.69	11.3832523819179	2.30674761808211
12	12.25	11.6765751141013	0.573424885898653
13	7.84	11.7494910035187	-3.90949100351868
14	14.44	11.2523657117834	3.1876342882166
15	18.49	11.6577007346927	6.83229926530727
16	10.89	12.5264861500912	-1.63648615009122
17	12.96	12.3183929087837	0.641607091216315
18	12.96	12.3999787500385	0.560021249961466
19	10.89	12.4711902522557	-1.5811902522557
20	7.84	12.2701283583329	-4.4301283583329

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & 10.89 & 10.24 & 0.65 \tabularnewline
3 & 9 & 10.3226530715474 & -1.32265307154745 \tabularnewline
4 & 12.25 & 10.154466396232 & 2.09553360376803 \tabularnewline
5 & 13.69 & 10.4209314560509 & 3.26906854394908 \tabularnewline
6 & 7.29 & 10.8366215425994 & -3.54662154259944 \tabularnewline
7 & 12.96 & 10.3856382131961 & 2.57436178680394 \tabularnewline
8 & 12.25 & 10.7129903808172 & 1.53700961918283 \tabularnewline
9 & 14.44 & 10.9084343285455 & 3.53156567145446 \tabularnewline
10 & 11.56 & 11.3575031748797 & 0.202496825120255 \tabularnewline
11 & 13.69 & 11.3832523819179 & 2.30674761808211 \tabularnewline
12 & 12.25 & 11.6765751141013 & 0.573424885898653 \tabularnewline
13 & 7.84 & 11.7494910035187 & -3.90949100351868 \tabularnewline
14 & 14.44 & 11.2523657117834 & 3.1876342882166 \tabularnewline
15 & 18.49 & 11.6577007346927 & 6.83229926530727 \tabularnewline
16 & 10.89 & 12.5264861500912 & -1.63648615009122 \tabularnewline
17 & 12.96 & 12.3183929087837 & 0.641607091216315 \tabularnewline
18 & 12.96 & 12.3999787500385 & 0.560021249961466 \tabularnewline
19 & 10.89 & 12.4711902522557 & -1.5811902522557 \tabularnewline
20 & 7.84 & 12.2701283583329 & -4.4301283583329 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=290667&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]10.89[/C][C]10.24[/C][C]0.65[/C][/ROW]
[ROW][C]3[/C][C]9[/C][C]10.3226530715474[/C][C]-1.32265307154745[/C][/ROW]
[ROW][C]4[/C][C]12.25[/C][C]10.154466396232[/C][C]2.09553360376803[/C][/ROW]
[ROW][C]5[/C][C]13.69[/C][C]10.4209314560509[/C][C]3.26906854394908[/C][/ROW]
[ROW][C]6[/C][C]7.29[/C][C]10.8366215425994[/C][C]-3.54662154259944[/C][/ROW]
[ROW][C]7[/C][C]12.96[/C][C]10.3856382131961[/C][C]2.57436178680394[/C][/ROW]
[ROW][C]8[/C][C]12.25[/C][C]10.7129903808172[/C][C]1.53700961918283[/C][/ROW]
[ROW][C]9[/C][C]14.44[/C][C]10.9084343285455[/C][C]3.53156567145446[/C][/ROW]
[ROW][C]10[/C][C]11.56[/C][C]11.3575031748797[/C][C]0.202496825120255[/C][/ROW]
[ROW][C]11[/C][C]13.69[/C][C]11.3832523819179[/C][C]2.30674761808211[/C][/ROW]
[ROW][C]12[/C][C]12.25[/C][C]11.6765751141013[/C][C]0.573424885898653[/C][/ROW]
[ROW][C]13[/C][C]7.84[/C][C]11.7494910035187[/C][C]-3.90949100351868[/C][/ROW]
[ROW][C]14[/C][C]14.44[/C][C]11.2523657117834[/C][C]3.1876342882166[/C][/ROW]
[ROW][C]15[/C][C]18.49[/C][C]11.6577007346927[/C][C]6.83229926530727[/C][/ROW]
[ROW][C]16[/C][C]10.89[/C][C]12.5264861500912[/C][C]-1.63648615009122[/C][/ROW]
[ROW][C]17[/C][C]12.96[/C][C]12.3183929087837[/C][C]0.641607091216315[/C][/ROW]
[ROW][C]18[/C][C]12.96[/C][C]12.3999787500385[/C][C]0.560021249961466[/C][/ROW]
[ROW][C]19[/C][C]10.89[/C][C]12.4711902522557[/C][C]-1.5811902522557[/C][/ROW]
[ROW][C]20[/C][C]7.84[/C][C]12.2701283583329[/C][C]-4.4301283583329[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=290667&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=290667&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
2	10.89	10.24	0.65
3	9	10.3226530715474	-1.32265307154745
4	12.25	10.154466396232	2.09553360376803
5	13.69	10.4209314560509	3.26906854394908
6	7.29	10.8366215425994	-3.54662154259944
7	12.96	10.3856382131961	2.57436178680394
8	12.25	10.7129903808172	1.53700961918283
9	14.44	10.9084343285455	3.53156567145446
10	11.56	11.3575031748797	0.202496825120255
11	13.69	11.3832523819179	2.30674761808211
12	12.25	11.6765751141013	0.573424885898653
13	7.84	11.7494910035187	-3.90949100351868
14	14.44	11.2523657117834	3.1876342882166
15	18.49	11.6577007346927	6.83229926530727
16	10.89	12.5264861500912	-1.63648615009122
17	12.96	12.3183929087837	0.641607091216315
18	12.96	12.3999787500385	0.560021249961466
19	10.89	12.4711902522557	-1.5811902522557
20	7.84	12.2701283583329	-4.4301283583329

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
21	11.7067995642319	6.09369101675273	17.319908111711
22	11.7067995642319	6.04849296478213	17.3651061636816
23	11.7067995642319	6.00365310100917	17.4099460274546
24	11.7067995642319	5.95916304227237	17.4544360861914
25	11.7067995642319	5.91501472739998	17.4985844010638
26	11.7067995642319	5.8712004001572	17.5423987283065
27	11.7067995642319	5.82771259333732	17.5858865351264
28	11.7067995642319	5.78454411390425	17.6290550145595
29	11.7067995642319	5.74168802910293	17.6719110993608
30	11.7067995642319	5.69913765346147	17.7144614750023
31	11.7067995642319	5.65688653661613	17.7567125918476
32	11.7067995642319	5.61492845189637	17.7986706765674

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
21 & 11.7067995642319 & 6.09369101675273 & 17.319908111711 \tabularnewline
22 & 11.7067995642319 & 6.04849296478213 & 17.3651061636816 \tabularnewline
23 & 11.7067995642319 & 6.00365310100917 & 17.4099460274546 \tabularnewline
24 & 11.7067995642319 & 5.95916304227237 & 17.4544360861914 \tabularnewline
25 & 11.7067995642319 & 5.91501472739998 & 17.4985844010638 \tabularnewline
26 & 11.7067995642319 & 5.8712004001572 & 17.5423987283065 \tabularnewline
27 & 11.7067995642319 & 5.82771259333732 & 17.5858865351264 \tabularnewline
28 & 11.7067995642319 & 5.78454411390425 & 17.6290550145595 \tabularnewline
29 & 11.7067995642319 & 5.74168802910293 & 17.6719110993608 \tabularnewline
30 & 11.7067995642319 & 5.69913765346147 & 17.7144614750023 \tabularnewline
31 & 11.7067995642319 & 5.65688653661613 & 17.7567125918476 \tabularnewline
32 & 11.7067995642319 & 5.61492845189637 & 17.7986706765674 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=290667&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]11.7067995642319[/C][C]6.09369101675273[/C][C]17.319908111711[/C][/ROW]
[ROW][C]22[/C][C]11.7067995642319[/C][C]6.04849296478213[/C][C]17.3651061636816[/C][/ROW]
[ROW][C]23[/C][C]11.7067995642319[/C][C]6.00365310100917[/C][C]17.4099460274546[/C][/ROW]
[ROW][C]24[/C][C]11.7067995642319[/C][C]5.95916304227237[/C][C]17.4544360861914[/C][/ROW]
[ROW][C]25[/C][C]11.7067995642319[/C][C]5.91501472739998[/C][C]17.4985844010638[/C][/ROW]
[ROW][C]26[/C][C]11.7067995642319[/C][C]5.8712004001572[/C][C]17.5423987283065[/C][/ROW]
[ROW][C]27[/C][C]11.7067995642319[/C][C]5.82771259333732[/C][C]17.5858865351264[/C][/ROW]
[ROW][C]28[/C][C]11.7067995642319[/C][C]5.78454411390425[/C][C]17.6290550145595[/C][/ROW]
[ROW][C]29[/C][C]11.7067995642319[/C][C]5.74168802910293[/C][C]17.6719110993608[/C][/ROW]
[ROW][C]30[/C][C]11.7067995642319[/C][C]5.69913765346147[/C][C]17.7144614750023[/C][/ROW]
[ROW][C]31[/C][C]11.7067995642319[/C][C]5.65688653661613[/C][C]17.7567125918476[/C][/ROW]
[ROW][C]32[/C][C]11.7067995642319[/C][C]5.61492845189637[/C][C]17.7986706765674[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=290667&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=290667&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	11.7067995642319	6.09369101675273	17.319908111711
22	11.7067995642319	6.04849296478213	17.3651061636816
23	11.7067995642319	6.00365310100917	17.4099460274546
24	11.7067995642319	5.95916304227237	17.4544360861914
25	11.7067995642319	5.91501472739998	17.4985844010638
26	11.7067995642319	5.8712004001572	17.5423987283065
27	11.7067995642319	5.82771259333732	17.5858865351264
28	11.7067995642319	5.78454411390425	17.6290550145595
29	11.7067995642319	5.74168802910293	17.6719110993608
30	11.7067995642319	5.69913765346147	17.7144614750023
31	11.7067995642319	5.65688653661613	17.7567125918476
32	11.7067995642319	5.61492845189637	17.7986706765674

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 = Single ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Single ; 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')

Free Statistics

Description of Statistical Computation

Tree of Dependent Computations

Dataset

Tables (Output of Computation)

Figures (Output of Computation)

Input Parameters & R Code