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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, 16 Dec 2016 12:58:09 +0100

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/Dec/16/t1481889576vg7focjw3uqbzk3.htm/, Retrieved Fri, 03 May 2024 02:26:12 +0000

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=300198, Retrieved Fri, 03 May 2024 02:26:12 +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] [n369..] [2016-12-16 11:58:09] [b7f10b15eba379294ac5bdad7f2e1205] [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	2 seconds
R Server	Big Analytics Cloud Computing Center

\begin{tabular}{lllllllll}
\hline
Summary of computational transaction \tabularnewline
Raw Input view raw input (R code)  \tabularnewline
Raw Outputview raw output of R engine  \tabularnewline
Computing time2 seconds \tabularnewline
R ServerBig Analytics Cloud Computing Center \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300198&T=0

[TABLE]
[ROW]

Summary of computational transaction[/C][/ROW] [ROW]

Raw Input[/C]

view raw input (R code) [/C][/ROW] [ROW]

Raw Output[/C]

view raw output of R engine [/C][/ROW] [ROW]

Computing time[/C]

2 seconds[/C][/ROW] [ROW]

R Server[/C]

Big Analytics Cloud Computing Center[/C][/ROW] [/TABLE] Source: https://freestatistics.org/blog/index.php?pk=300198&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300198&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	2 seconds
R Server	Big Analytics Cloud Computing Center

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	1
beta	0
gamma	FALSE

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 1 \tabularnewline
beta & 0 \tabularnewline
gamma & FALSE \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300198&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[/C][/ROW]
[ROW][C]gamma[/C][C]FALSE[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=300198&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300198&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	1
beta	0
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	1155	1315	-160
4	1160	1250	-90
5	1285	1255	30
6	1340	1380	-40
7	1470	1435	35
8	1275	1565	-290
9	1420	1370	50
10	1540	1515	25
11	1735	1635	100
12	1715	1830	-115
13	2035	1810	225
14	2005	2130	-125
15	1950	2100	-150
16	1975	2045	-70
17	2000	2070	-70
18	2075	2095	-20
19	2105	2170	-65
20	2330	2200	130
21	2425	2425	0
22	2625	2520	105
23	3050	2720	330
24	3295	3145	150
25	2870	3390	-520
26	2815	2965	-150
27	3150	2910	240
28	3495	3245	250
29	3490	3590	-100
30	3390	3585	-195
31	3530	3485	45
32	3945	3625	320
33	4370	4040	330
34	5405	4465	940
35	6435	5500	935
36	6115	6530	-415
37	5420	6210	-790
38	5995	5515	480
39	5375	6090	-715
40	4070	5470	-1400
41	4320	4165	155

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
3 & 1155 & 1315 & -160 \tabularnewline
4 & 1160 & 1250 & -90 \tabularnewline
5 & 1285 & 1255 & 30 \tabularnewline
6 & 1340 & 1380 & -40 \tabularnewline
7 & 1470 & 1435 & 35 \tabularnewline
8 & 1275 & 1565 & -290 \tabularnewline
9 & 1420 & 1370 & 50 \tabularnewline
10 & 1540 & 1515 & 25 \tabularnewline
11 & 1735 & 1635 & 100 \tabularnewline
12 & 1715 & 1830 & -115 \tabularnewline
13 & 2035 & 1810 & 225 \tabularnewline
14 & 2005 & 2130 & -125 \tabularnewline
15 & 1950 & 2100 & -150 \tabularnewline
16 & 1975 & 2045 & -70 \tabularnewline
17 & 2000 & 2070 & -70 \tabularnewline
18 & 2075 & 2095 & -20 \tabularnewline
19 & 2105 & 2170 & -65 \tabularnewline
20 & 2330 & 2200 & 130 \tabularnewline
21 & 2425 & 2425 & 0 \tabularnewline
22 & 2625 & 2520 & 105 \tabularnewline
23 & 3050 & 2720 & 330 \tabularnewline
24 & 3295 & 3145 & 150 \tabularnewline
25 & 2870 & 3390 & -520 \tabularnewline
26 & 2815 & 2965 & -150 \tabularnewline
27 & 3150 & 2910 & 240 \tabularnewline
28 & 3495 & 3245 & 250 \tabularnewline
29 & 3490 & 3590 & -100 \tabularnewline
30 & 3390 & 3585 & -195 \tabularnewline
31 & 3530 & 3485 & 45 \tabularnewline
32 & 3945 & 3625 & 320 \tabularnewline
33 & 4370 & 4040 & 330 \tabularnewline
34 & 5405 & 4465 & 940 \tabularnewline
35 & 6435 & 5500 & 935 \tabularnewline
36 & 6115 & 6530 & -415 \tabularnewline
37 & 5420 & 6210 & -790 \tabularnewline
38 & 5995 & 5515 & 480 \tabularnewline
39 & 5375 & 6090 & -715 \tabularnewline
40 & 4070 & 5470 & -1400 \tabularnewline
41 & 4320 & 4165 & 155 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300198&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]1155[/C][C]1315[/C][C]-160[/C][/ROW]
[ROW][C]4[/C][C]1160[/C][C]1250[/C][C]-90[/C][/ROW]
[ROW][C]5[/C][C]1285[/C][C]1255[/C][C]30[/C][/ROW]
[ROW][C]6[/C][C]1340[/C][C]1380[/C][C]-40[/C][/ROW]
[ROW][C]7[/C][C]1470[/C][C]1435[/C][C]35[/C][/ROW]
[ROW][C]8[/C][C]1275[/C][C]1565[/C][C]-290[/C][/ROW]
[ROW][C]9[/C][C]1420[/C][C]1370[/C][C]50[/C][/ROW]
[ROW][C]10[/C][C]1540[/C][C]1515[/C][C]25[/C][/ROW]
[ROW][C]11[/C][C]1735[/C][C]1635[/C][C]100[/C][/ROW]
[ROW][C]12[/C][C]1715[/C][C]1830[/C][C]-115[/C][/ROW]
[ROW][C]13[/C][C]2035[/C][C]1810[/C][C]225[/C][/ROW]
[ROW][C]14[/C][C]2005[/C][C]2130[/C][C]-125[/C][/ROW]
[ROW][C]15[/C][C]1950[/C][C]2100[/C][C]-150[/C][/ROW]
[ROW][C]16[/C][C]1975[/C][C]2045[/C][C]-70[/C][/ROW]
[ROW][C]17[/C][C]2000[/C][C]2070[/C][C]-70[/C][/ROW]
[ROW][C]18[/C][C]2075[/C][C]2095[/C][C]-20[/C][/ROW]
[ROW][C]19[/C][C]2105[/C][C]2170[/C][C]-65[/C][/ROW]
[ROW][C]20[/C][C]2330[/C][C]2200[/C][C]130[/C][/ROW]
[ROW][C]21[/C][C]2425[/C][C]2425[/C][C]0[/C][/ROW]
[ROW][C]22[/C][C]2625[/C][C]2520[/C][C]105[/C][/ROW]
[ROW][C]23[/C][C]3050[/C][C]2720[/C][C]330[/C][/ROW]
[ROW][C]24[/C][C]3295[/C][C]3145[/C][C]150[/C][/ROW]
[ROW][C]25[/C][C]2870[/C][C]3390[/C][C]-520[/C][/ROW]
[ROW][C]26[/C][C]2815[/C][C]2965[/C][C]-150[/C][/ROW]
[ROW][C]27[/C][C]3150[/C][C]2910[/C][C]240[/C][/ROW]
[ROW][C]28[/C][C]3495[/C][C]3245[/C][C]250[/C][/ROW]
[ROW][C]29[/C][C]3490[/C][C]3590[/C][C]-100[/C][/ROW]
[ROW][C]30[/C][C]3390[/C][C]3585[/C][C]-195[/C][/ROW]
[ROW][C]31[/C][C]3530[/C][C]3485[/C][C]45[/C][/ROW]
[ROW][C]32[/C][C]3945[/C][C]3625[/C][C]320[/C][/ROW]
[ROW][C]33[/C][C]4370[/C][C]4040[/C][C]330[/C][/ROW]
[ROW][C]34[/C][C]5405[/C][C]4465[/C][C]940[/C][/ROW]
[ROW][C]35[/C][C]6435[/C][C]5500[/C][C]935[/C][/ROW]
[ROW][C]36[/C][C]6115[/C][C]6530[/C][C]-415[/C][/ROW]
[ROW][C]37[/C][C]5420[/C][C]6210[/C][C]-790[/C][/ROW]
[ROW][C]38[/C][C]5995[/C][C]5515[/C][C]480[/C][/ROW]
[ROW][C]39[/C][C]5375[/C][C]6090[/C][C]-715[/C][/ROW]
[ROW][C]40[/C][C]4070[/C][C]5470[/C][C]-1400[/C][/ROW]
[ROW][C]41[/C][C]4320[/C][C]4165[/C][C]155[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=300198&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300198&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	1155	1315	-160
4	1160	1250	-90
5	1285	1255	30
6	1340	1380	-40
7	1470	1435	35
8	1275	1565	-290
9	1420	1370	50
10	1540	1515	25
11	1735	1635	100
12	1715	1830	-115
13	2035	1810	225
14	2005	2130	-125
15	1950	2100	-150
16	1975	2045	-70
17	2000	2070	-70
18	2075	2095	-20
19	2105	2170	-65
20	2330	2200	130
21	2425	2425	0
22	2625	2520	105
23	3050	2720	330
24	3295	3145	150
25	2870	3390	-520
26	2815	2965	-150
27	3150	2910	240
28	3495	3245	250
29	3490	3590	-100
30	3390	3585	-195
31	3530	3485	45
32	3945	3625	320
33	4370	4040	330
34	5405	4465	940
35	6435	5500	935
36	6115	6530	-415
37	5420	6210	-790
38	5995	5515	480
39	5375	6090	-715
40	4070	5470	-1400
41	4320	4165	155

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
42	4415	3613.47172908715	5216.52827091285
43	4510	3376.46784864959	5643.53215135041
44	4605	3216.71231107611	5993.28768892389
45	4700	3096.9434581743	6303.0565418257
46	4795	3002.728300351	6587.271699649
47	4890	2926.66472184824	6853.33527815176
48	4985	2864.35552637699	7105.64447362301
49	5080	2812.93569729918	7347.06430270082
50	5175	2770.41518726145	7579.58481273855
51	5270	2735.34505489891	7804.65494510109
52	5365	2706.63146651974	8023.36853348026
53	5460	2683.42462215223	8236.57537784777

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
42 & 4415 & 3613.47172908715 & 5216.52827091285 \tabularnewline
43 & 4510 & 3376.46784864959 & 5643.53215135041 \tabularnewline
44 & 4605 & 3216.71231107611 & 5993.28768892389 \tabularnewline
45 & 4700 & 3096.9434581743 & 6303.0565418257 \tabularnewline
46 & 4795 & 3002.728300351 & 6587.271699649 \tabularnewline
47 & 4890 & 2926.66472184824 & 6853.33527815176 \tabularnewline
48 & 4985 & 2864.35552637699 & 7105.64447362301 \tabularnewline
49 & 5080 & 2812.93569729918 & 7347.06430270082 \tabularnewline
50 & 5175 & 2770.41518726145 & 7579.58481273855 \tabularnewline
51 & 5270 & 2735.34505489891 & 7804.65494510109 \tabularnewline
52 & 5365 & 2706.63146651974 & 8023.36853348026 \tabularnewline
53 & 5460 & 2683.42462215223 & 8236.57537784777 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=300198&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]42[/C][C]4415[/C][C]3613.47172908715[/C][C]5216.52827091285[/C][/ROW]
[ROW][C]43[/C][C]4510[/C][C]3376.46784864959[/C][C]5643.53215135041[/C][/ROW]
[ROW][C]44[/C][C]4605[/C][C]3216.71231107611[/C][C]5993.28768892389[/C][/ROW]
[ROW][C]45[/C][C]4700[/C][C]3096.9434581743[/C][C]6303.0565418257[/C][/ROW]
[ROW][C]46[/C][C]4795[/C][C]3002.728300351[/C][C]6587.271699649[/C][/ROW]
[ROW][C]47[/C][C]4890[/C][C]2926.66472184824[/C][C]6853.33527815176[/C][/ROW]
[ROW][C]48[/C][C]4985[/C][C]2864.35552637699[/C][C]7105.64447362301[/C][/ROW]
[ROW][C]49[/C][C]5080[/C][C]2812.93569729918[/C][C]7347.06430270082[/C][/ROW]
[ROW][C]50[/C][C]5175[/C][C]2770.41518726145[/C][C]7579.58481273855[/C][/ROW]
[ROW][C]51[/C][C]5270[/C][C]2735.34505489891[/C][C]7804.65494510109[/C][/ROW]
[ROW][C]52[/C][C]5365[/C][C]2706.63146651974[/C][C]8023.36853348026[/C][/ROW]
[ROW][C]53[/C][C]5460[/C][C]2683.42462215223[/C][C]8236.57537784777[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=300198&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=300198&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
42	4415	3613.47172908715	5216.52827091285
43	4510	3376.46784864959	5643.53215135041
44	4605	3216.71231107611	5993.28768892389
45	4700	3096.9434581743	6303.0565418257
46	4795	3002.728300351	6587.271699649
47	4890	2926.66472184824	6853.33527815176
48	4985	2864.35552637699	7105.64447362301
49	5080	2812.93569729918	7347.06430270082
50	5175	2770.41518726145	7579.58481273855
51	5270	2735.34505489891	7804.65494510109
52	5365	2706.63146651974	8023.36853348026
53	5460	2683.42462215223	8236.57537784777

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 = 1 ; par2 = 0 ; par3 = 2 ; par4 = 1 ;

Parameters (R input):

par1 = 12 ; par2 = Double ; par3 = additive ; par4 = 12 ;

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

par4 <- '12'
par3 <- 'additive'
par2 <- 'Single'
par1 <- '12'
par1 <- as.numeric(par1)
par4 <- as.numeric(par4)
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$fitted[,'xhat']
bitmap(file='test1.png')
op <- par(mfrow=c(2,1))
plot(fit,ylab='Observed (black) / Fitted (red)',main='Interpolation Fit of Exponential Smoothing')
plot(myresid,ylab='Residuals',main='Interpolation Prediction Errors')
par(op)
dev.off()
bitmap(file='test2.png')
p <- predict(fit, par4, prediction.interval=TRUE)
np <- length(p[,1])
plot(fit,p,ylab='Observed (black) / Fitted (red)',main='Extrapolation Fit of Exponential Smoothing')
dev.off()
bitmap(file='test3.png')
op <- par(mfrow = c(2,2))
acf(as.numeric(myresid),lag.max = nx/2,main='Residual ACF')
spectrum(myresid,main='Residals Periodogram')
cpgram(myresid,main='Residal Cumulative Periodogram')
qqnorm(myresid,main='Residual Normal QQ Plot')
qqline(myresid)
par(op)
dev.off()
load(file='createtable')
a<-table.start()
a<-table.row.start(a)
a<-table.element(a,'Estimated Parameters of Exponential Smoothing',2,TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'Parameter',header=TRUE)
a<-table.element(a,'Value',header=TRUE)
a<-table.row.end(a)
a<-table.row.start(a)
a<-table.element(a,'alpha',header=TRUE)
a<-table.element(a,fit$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