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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 09:58:55 +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/t1452506343hr22h02thhv6knh.htm/, Retrieved Wed, 08 May 2024 02:45:37 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=289266, Retrieved Wed, 08 May 2024 02:45:37 +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 09:58:55] [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=289266&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=289266&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289266&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	0.0540003928280715
beta	FALSE
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.0540003928280715 \tabularnewline
beta & FALSE \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289266&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]0.0540003928280715[/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=289266&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289266&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	0.0540003928280715
beta	FALSE
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
2	-2.468	0.7923	-3.2603
3	-2.996	0.616242519262638	-3.61224251926264
4	3.119	0.421180004232193	2.69781999576781
5	0.04315	0.566863343783081	-0.523713343783081
6	0.731	0.538582617489492	0.192417382510508
7	2.45	0.548973231732008	1.90102676826799
8	2.119	0.651629423995159	1.46737057600484
9	-1.429	0.730868011523774	-2.15986801152377
10	-1.644	0.614234290444705	-2.2582342904447
11	-3.065	0.492288751662869	-3.55728875166287
12	-1.461	0.300193761670194	-1.76119376167019
13	1.141	0.205088606693655	0.935911393306345
14	1.329	0.255628189584465	1.07337181041553
15	0.3396	0.313590688997482	0.0260093110025177
16	0.8429	0.314995202008806	0.527904797991194
17	2.225	0.343502268476154	1.88149773152385
18	-1.924	0.445103885083567	-2.36910388508357
19	0.4999	0.317171344638544	0.182728655361456
20	-0.6433	0.327038763809008	-0.970338763809008

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
2 & -2.468 & 0.7923 & -3.2603 \tabularnewline
3 & -2.996 & 0.616242519262638 & -3.61224251926264 \tabularnewline
4 & 3.119 & 0.421180004232193 & 2.69781999576781 \tabularnewline
5 & 0.04315 & 0.566863343783081 & -0.523713343783081 \tabularnewline
6 & 0.731 & 0.538582617489492 & 0.192417382510508 \tabularnewline
7 & 2.45 & 0.548973231732008 & 1.90102676826799 \tabularnewline
8 & 2.119 & 0.651629423995159 & 1.46737057600484 \tabularnewline
9 & -1.429 & 0.730868011523774 & -2.15986801152377 \tabularnewline
10 & -1.644 & 0.614234290444705 & -2.2582342904447 \tabularnewline
11 & -3.065 & 0.492288751662869 & -3.55728875166287 \tabularnewline
12 & -1.461 & 0.300193761670194 & -1.76119376167019 \tabularnewline
13 & 1.141 & 0.205088606693655 & 0.935911393306345 \tabularnewline
14 & 1.329 & 0.255628189584465 & 1.07337181041553 \tabularnewline
15 & 0.3396 & 0.313590688997482 & 0.0260093110025177 \tabularnewline
16 & 0.8429 & 0.314995202008806 & 0.527904797991194 \tabularnewline
17 & 2.225 & 0.343502268476154 & 1.88149773152385 \tabularnewline
18 & -1.924 & 0.445103885083567 & -2.36910388508357 \tabularnewline
19 & 0.4999 & 0.317171344638544 & 0.182728655361456 \tabularnewline
20 & -0.6433 & 0.327038763809008 & -0.970338763809008 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289266&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]-2.468[/C][C]0.7923[/C][C]-3.2603[/C][/ROW]
[ROW][C]3[/C][C]-2.996[/C][C]0.616242519262638[/C][C]-3.61224251926264[/C][/ROW]
[ROW][C]4[/C][C]3.119[/C][C]0.421180004232193[/C][C]2.69781999576781[/C][/ROW]
[ROW][C]5[/C][C]0.04315[/C][C]0.566863343783081[/C][C]-0.523713343783081[/C][/ROW]
[ROW][C]6[/C][C]0.731[/C][C]0.538582617489492[/C][C]0.192417382510508[/C][/ROW]
[ROW][C]7[/C][C]2.45[/C][C]0.548973231732008[/C][C]1.90102676826799[/C][/ROW]
[ROW][C]8[/C][C]2.119[/C][C]0.651629423995159[/C][C]1.46737057600484[/C][/ROW]
[ROW][C]9[/C][C]-1.429[/C][C]0.730868011523774[/C][C]-2.15986801152377[/C][/ROW]
[ROW][C]10[/C][C]-1.644[/C][C]0.614234290444705[/C][C]-2.2582342904447[/C][/ROW]
[ROW][C]11[/C][C]-3.065[/C][C]0.492288751662869[/C][C]-3.55728875166287[/C][/ROW]
[ROW][C]12[/C][C]-1.461[/C][C]0.300193761670194[/C][C]-1.76119376167019[/C][/ROW]
[ROW][C]13[/C][C]1.141[/C][C]0.205088606693655[/C][C]0.935911393306345[/C][/ROW]
[ROW][C]14[/C][C]1.329[/C][C]0.255628189584465[/C][C]1.07337181041553[/C][/ROW]
[ROW][C]15[/C][C]0.3396[/C][C]0.313590688997482[/C][C]0.0260093110025177[/C][/ROW]
[ROW][C]16[/C][C]0.8429[/C][C]0.314995202008806[/C][C]0.527904797991194[/C][/ROW]
[ROW][C]17[/C][C]2.225[/C][C]0.343502268476154[/C][C]1.88149773152385[/C][/ROW]
[ROW][C]18[/C][C]-1.924[/C][C]0.445103885083567[/C][C]-2.36910388508357[/C][/ROW]
[ROW][C]19[/C][C]0.4999[/C][C]0.317171344638544[/C][C]0.182728655361456[/C][/ROW]
[ROW][C]20[/C][C]-0.6433[/C][C]0.327038763809008[/C][C]-0.970338763809008[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=289266&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289266&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	-2.468	0.7923	-3.2603
3	-2.996	0.616242519262638	-3.61224251926264
4	3.119	0.421180004232193	2.69781999576781
5	0.04315	0.566863343783081	-0.523713343783081
6	0.731	0.538582617489492	0.192417382510508
7	2.45	0.548973231732008	1.90102676826799
8	2.119	0.651629423995159	1.46737057600484
9	-1.429	0.730868011523774	-2.15986801152377
10	-1.644	0.614234290444705	-2.2582342904447
11	-3.065	0.492288751662869	-3.55728875166287
12	-1.461	0.300193761670194	-1.76119376167019
13	1.141	0.205088606693655	0.935911393306345
14	1.329	0.255628189584465	1.07337181041553
15	0.3396	0.313590688997482	0.0260093110025177
16	0.8429	0.314995202008806	0.527904797991194
17	2.225	0.343502268476154	1.88149773152385
18	-1.924	0.445103885083567	-2.36910388508357
19	0.4999	0.317171344638544	0.182728655361456
20	-0.6433	0.327038763809008	-0.970338763809008

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
21	0.274640089387016	-3.5947008489117	4.14398102768573
22	0.274640089387016	-3.60033832329236	4.14961850206639
23	0.274640089387016	-3.60596760795473	4.15524778672876
24	0.274640089387016	-3.61158873848774	4.16086891726177
25	0.274640089387016	-3.61720175022332	4.16648192899736
26	0.274640089387016	-3.62280667823899	4.17208685701302
27	0.274640089387016	-3.62840355736038	4.17768373613441
28	0.274640089387016	-3.6339924221638	4.18327260093783
29	0.274640089387016	-3.6395733069787	4.18885348575273
30	0.274640089387016	-3.64514624589016	4.19442642466419
31	0.274640089387016	-3.65071127274128	4.19999145151531
32	0.274640089387016	-3.65626842113562	4.20554859990965

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
21 & 0.274640089387016 & -3.5947008489117 & 4.14398102768573 \tabularnewline
22 & 0.274640089387016 & -3.60033832329236 & 4.14961850206639 \tabularnewline
23 & 0.274640089387016 & -3.60596760795473 & 4.15524778672876 \tabularnewline
24 & 0.274640089387016 & -3.61158873848774 & 4.16086891726177 \tabularnewline
25 & 0.274640089387016 & -3.61720175022332 & 4.16648192899736 \tabularnewline
26 & 0.274640089387016 & -3.62280667823899 & 4.17208685701302 \tabularnewline
27 & 0.274640089387016 & -3.62840355736038 & 4.17768373613441 \tabularnewline
28 & 0.274640089387016 & -3.6339924221638 & 4.18327260093783 \tabularnewline
29 & 0.274640089387016 & -3.6395733069787 & 4.18885348575273 \tabularnewline
30 & 0.274640089387016 & -3.64514624589016 & 4.19442642466419 \tabularnewline
31 & 0.274640089387016 & -3.65071127274128 & 4.19999145151531 \tabularnewline
32 & 0.274640089387016 & -3.65626842113562 & 4.20554859990965 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=289266&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]0.274640089387016[/C][C]-3.5947008489117[/C][C]4.14398102768573[/C][/ROW]
[ROW][C]22[/C][C]0.274640089387016[/C][C]-3.60033832329236[/C][C]4.14961850206639[/C][/ROW]
[ROW][C]23[/C][C]0.274640089387016[/C][C]-3.60596760795473[/C][C]4.15524778672876[/C][/ROW]
[ROW][C]24[/C][C]0.274640089387016[/C][C]-3.61158873848774[/C][C]4.16086891726177[/C][/ROW]
[ROW][C]25[/C][C]0.274640089387016[/C][C]-3.61720175022332[/C][C]4.16648192899736[/C][/ROW]
[ROW][C]26[/C][C]0.274640089387016[/C][C]-3.62280667823899[/C][C]4.17208685701302[/C][/ROW]
[ROW][C]27[/C][C]0.274640089387016[/C][C]-3.62840355736038[/C][C]4.17768373613441[/C][/ROW]
[ROW][C]28[/C][C]0.274640089387016[/C][C]-3.6339924221638[/C][C]4.18327260093783[/C][/ROW]
[ROW][C]29[/C][C]0.274640089387016[/C][C]-3.6395733069787[/C][C]4.18885348575273[/C][/ROW]
[ROW][C]30[/C][C]0.274640089387016[/C][C]-3.64514624589016[/C][C]4.19442642466419[/C][/ROW]
[ROW][C]31[/C][C]0.274640089387016[/C][C]-3.65071127274128[/C][C]4.19999145151531[/C][/ROW]
[ROW][C]32[/C][C]0.274640089387016[/C][C]-3.65626842113562[/C][C]4.20554859990965[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=289266&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=289266&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	0.274640089387016	-3.5947008489117	4.14398102768573
22	0.274640089387016	-3.60033832329236	4.14961850206639
23	0.274640089387016	-3.60596760795473	4.15524778672876
24	0.274640089387016	-3.61158873848774	4.16086891726177
25	0.274640089387016	-3.61720175022332	4.16648192899736
26	0.274640089387016	-3.62280667823899	4.17208685701302
27	0.274640089387016	-3.62840355736038	4.17768373613441
28	0.274640089387016	-3.6339924221638	4.18327260093783
29	0.274640089387016	-3.6395733069787	4.18885348575273
30	0.274640089387016	-3.64514624589016	4.19442642466419
31	0.274640089387016	-3.65071127274128	4.19999145151531
32	0.274640089387016	-3.65626842113562	4.20554859990965

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 = 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