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Author's title

Author*The author of this computation has been verified*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationSun, 28 Nov 2010 09:23:28 +0000
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2010/Nov/28/t129093611169r0y8ocnim052n.htm/, Retrieved Thu, 02 May 2024 14:27:35 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=102459, Retrieved Thu, 02 May 2024 14:27:35 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact205
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-     [Multiple Regression] [Workshop 8 Regres...] [2010-11-27 09:22:21] [87d60b8864dc39f7ed759c345edfb471]
- RMP   [Spectral Analysis] [Workshop 8 Regres...] [2010-11-27 12:28:23] [87d60b8864dc39f7ed759c345edfb471]
- RMP     [Exponential Smoothing] [Workshop 8 Regres...] [2010-11-27 13:02:33] [87d60b8864dc39f7ed759c345edfb471]
-   P       [Exponential Smoothing] [Workshop 8 Regres...] [2010-11-27 13:15:31] [87d60b8864dc39f7ed759c345edfb471]
- R  D        [Exponential Smoothing] [ws 8 - exponentia...] [2010-11-28 09:11:22] [033eb2749a430605d9b2be7c4aac4a0c]
-   P           [Exponential Smoothing] [ws 8 - exponentia...] [2010-11-28 09:21:35] [033eb2749a430605d9b2be7c4aac4a0c]
-   P               [Exponential Smoothing] [ws 8 - exponentia...] [2010-11-28 09:23:28] [a948b7c78e10e31abd3f68e640bbd8ba] [Current]
Feedback Forum

Post a new message
Dataseries X:
444
454
469
471
443
437
444
451
457
460
454
439
441
446
459
456
433
424
430
428
424
419
409
397
397
413
413
390
385
397
398
406
412
409
404
412
418
434
431
406
416
424
427
401




Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time5 seconds
R Server'George Udny Yule' @ 72.249.76.132

\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 & 5 seconds \tabularnewline
R Server & 'George Udny Yule' @ 72.249.76.132 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=102459&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]5 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'George Udny Yule' @ 72.249.76.132[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=102459&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=102459&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 Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time5 seconds
R Server'George Udny Yule' @ 72.249.76.132







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.265744892612999
beta0.769328451910273
gamma1

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0.265744892612999 \tabularnewline
beta & 0.769328451910273 \tabularnewline
gamma & 1 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=102459&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]0.265744892612999[/C][/ROW]
[ROW][C]beta[/C][C]0.769328451910273[/C][/ROW]
[ROW][C]gamma[/C][C]1[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=102459&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=102459&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
ParameterValue
alpha0.265744892612999
beta0.769328451910273
gamma1







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13441446.912126068376-5.91212606837621
14446448.999290695025-2.99929069502508
15459460.039002805251-1.03900280525141
16456456.137232370713-0.137232370712638
17433432.947046336770.0529536632298573
18424423.8598940539970.140105946002848
19430420.6578793536839.34212064631686
20428428.811203915348-0.811203915347903
21424433.392154327235-9.39215432723483
22419431.064247665203-12.0642476652028
23409416.5597694432-7.55976944319997
24397392.6234420977694.3765579022309
25397385.57955653428911.4204434657109
26413392.63884104567620.3611589543239
27413414.329068164031-1.32906816403136
28390413.956284875514-23.9562848755143
29385382.6502046106062.34979538939410
30397372.78124859883124.2187514011687
31398386.20122878817811.7987712118219
32406391.52110700848214.4788929915182
33412400.95954314471811.0404568552824
34409413.371674972492-4.37167497249203
35404417.063778247386-13.0637782473863
36412412.148713774304-0.148713774303701
37418419.868715458689-1.86871545868939
38434438.038782486686-4.03878248668593
39431440.407783726959-9.40778372695905
40406422.711413717952-16.7114137179522
41416415.5646133240650.435386675935092
42424423.7715326986470.228467301352566
43427419.3193144120887.68068558791185
44401422.29333394394-21.2933339439398

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 441 & 446.912126068376 & -5.91212606837621 \tabularnewline
14 & 446 & 448.999290695025 & -2.99929069502508 \tabularnewline
15 & 459 & 460.039002805251 & -1.03900280525141 \tabularnewline
16 & 456 & 456.137232370713 & -0.137232370712638 \tabularnewline
17 & 433 & 432.94704633677 & 0.0529536632298573 \tabularnewline
18 & 424 & 423.859894053997 & 0.140105946002848 \tabularnewline
19 & 430 & 420.657879353683 & 9.34212064631686 \tabularnewline
20 & 428 & 428.811203915348 & -0.811203915347903 \tabularnewline
21 & 424 & 433.392154327235 & -9.39215432723483 \tabularnewline
22 & 419 & 431.064247665203 & -12.0642476652028 \tabularnewline
23 & 409 & 416.5597694432 & -7.55976944319997 \tabularnewline
24 & 397 & 392.623442097769 & 4.3765579022309 \tabularnewline
25 & 397 & 385.579556534289 & 11.4204434657109 \tabularnewline
26 & 413 & 392.638841045676 & 20.3611589543239 \tabularnewline
27 & 413 & 414.329068164031 & -1.32906816403136 \tabularnewline
28 & 390 & 413.956284875514 & -23.9562848755143 \tabularnewline
29 & 385 & 382.650204610606 & 2.34979538939410 \tabularnewline
30 & 397 & 372.781248598831 & 24.2187514011687 \tabularnewline
31 & 398 & 386.201228788178 & 11.7987712118219 \tabularnewline
32 & 406 & 391.521107008482 & 14.4788929915182 \tabularnewline
33 & 412 & 400.959543144718 & 11.0404568552824 \tabularnewline
34 & 409 & 413.371674972492 & -4.37167497249203 \tabularnewline
35 & 404 & 417.063778247386 & -13.0637782473863 \tabularnewline
36 & 412 & 412.148713774304 & -0.148713774303701 \tabularnewline
37 & 418 & 419.868715458689 & -1.86871545868939 \tabularnewline
38 & 434 & 438.038782486686 & -4.03878248668593 \tabularnewline
39 & 431 & 440.407783726959 & -9.40778372695905 \tabularnewline
40 & 406 & 422.711413717952 & -16.7114137179522 \tabularnewline
41 & 416 & 415.564613324065 & 0.435386675935092 \tabularnewline
42 & 424 & 423.771532698647 & 0.228467301352566 \tabularnewline
43 & 427 & 419.319314412088 & 7.68068558791185 \tabularnewline
44 & 401 & 422.29333394394 & -21.2933339439398 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=102459&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]13[/C][C]441[/C][C]446.912126068376[/C][C]-5.91212606837621[/C][/ROW]
[ROW][C]14[/C][C]446[/C][C]448.999290695025[/C][C]-2.99929069502508[/C][/ROW]
[ROW][C]15[/C][C]459[/C][C]460.039002805251[/C][C]-1.03900280525141[/C][/ROW]
[ROW][C]16[/C][C]456[/C][C]456.137232370713[/C][C]-0.137232370712638[/C][/ROW]
[ROW][C]17[/C][C]433[/C][C]432.94704633677[/C][C]0.0529536632298573[/C][/ROW]
[ROW][C]18[/C][C]424[/C][C]423.859894053997[/C][C]0.140105946002848[/C][/ROW]
[ROW][C]19[/C][C]430[/C][C]420.657879353683[/C][C]9.34212064631686[/C][/ROW]
[ROW][C]20[/C][C]428[/C][C]428.811203915348[/C][C]-0.811203915347903[/C][/ROW]
[ROW][C]21[/C][C]424[/C][C]433.392154327235[/C][C]-9.39215432723483[/C][/ROW]
[ROW][C]22[/C][C]419[/C][C]431.064247665203[/C][C]-12.0642476652028[/C][/ROW]
[ROW][C]23[/C][C]409[/C][C]416.5597694432[/C][C]-7.55976944319997[/C][/ROW]
[ROW][C]24[/C][C]397[/C][C]392.623442097769[/C][C]4.3765579022309[/C][/ROW]
[ROW][C]25[/C][C]397[/C][C]385.579556534289[/C][C]11.4204434657109[/C][/ROW]
[ROW][C]26[/C][C]413[/C][C]392.638841045676[/C][C]20.3611589543239[/C][/ROW]
[ROW][C]27[/C][C]413[/C][C]414.329068164031[/C][C]-1.32906816403136[/C][/ROW]
[ROW][C]28[/C][C]390[/C][C]413.956284875514[/C][C]-23.9562848755143[/C][/ROW]
[ROW][C]29[/C][C]385[/C][C]382.650204610606[/C][C]2.34979538939410[/C][/ROW]
[ROW][C]30[/C][C]397[/C][C]372.781248598831[/C][C]24.2187514011687[/C][/ROW]
[ROW][C]31[/C][C]398[/C][C]386.201228788178[/C][C]11.7987712118219[/C][/ROW]
[ROW][C]32[/C][C]406[/C][C]391.521107008482[/C][C]14.4788929915182[/C][/ROW]
[ROW][C]33[/C][C]412[/C][C]400.959543144718[/C][C]11.0404568552824[/C][/ROW]
[ROW][C]34[/C][C]409[/C][C]413.371674972492[/C][C]-4.37167497249203[/C][/ROW]
[ROW][C]35[/C][C]404[/C][C]417.063778247386[/C][C]-13.0637782473863[/C][/ROW]
[ROW][C]36[/C][C]412[/C][C]412.148713774304[/C][C]-0.148713774303701[/C][/ROW]
[ROW][C]37[/C][C]418[/C][C]419.868715458689[/C][C]-1.86871545868939[/C][/ROW]
[ROW][C]38[/C][C]434[/C][C]438.038782486686[/C][C]-4.03878248668593[/C][/ROW]
[ROW][C]39[/C][C]431[/C][C]440.407783726959[/C][C]-9.40778372695905[/C][/ROW]
[ROW][C]40[/C][C]406[/C][C]422.711413717952[/C][C]-16.7114137179522[/C][/ROW]
[ROW][C]41[/C][C]416[/C][C]415.564613324065[/C][C]0.435386675935092[/C][/ROW]
[ROW][C]42[/C][C]424[/C][C]423.771532698647[/C][C]0.228467301352566[/C][/ROW]
[ROW][C]43[/C][C]427[/C][C]419.319314412088[/C][C]7.68068558791185[/C][/ROW]
[ROW][C]44[/C][C]401[/C][C]422.29333394394[/C][C]-21.2933339439398[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=102459&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=102459&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
tObservedFittedResiduals
13441446.912126068376-5.91212606837621
14446448.999290695025-2.99929069502508
15459460.039002805251-1.03900280525141
16456456.137232370713-0.137232370712638
17433432.947046336770.0529536632298573
18424423.8598940539970.140105946002848
19430420.6578793536839.34212064631686
20428428.811203915348-0.811203915347903
21424433.392154327235-9.39215432723483
22419431.064247665203-12.0642476652028
23409416.5597694432-7.55976944319997
24397392.6234420977694.3765579022309
25397385.57955653428911.4204434657109
26413392.63884104567620.3611589543239
27413414.329068164031-1.32906816403136
28390413.956284875514-23.9562848755143
29385382.6502046106062.34979538939410
30397372.78124859883124.2187514011687
31398386.20122878817811.7987712118219
32406391.52110700848214.4788929915182
33412400.95954314471811.0404568552824
34409413.371674972492-4.37167497249203
35404417.063778247386-13.0637782473863
36412412.148713774304-0.148713774303701
37418419.868715458689-1.86871545868939
38434438.038782486686-4.03878248668593
39431440.407783726959-9.40778372695905
40406422.711413717952-16.7114137179522
41416415.5646133240650.435386675935092
42424423.7715326986470.228467301352566
43427419.3193144120887.68068558791185
44401422.29333394394-21.2933339439398







Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
45409.167945850895387.737362492628430.598529209163
46394.539680436358370.858365721759418.220995150956
47381.115064629995353.369193982166408.860935277825
48379.929161941626346.391886233272413.46643764998
49377.230744719644336.439272778955418.022216660332
50385.491061458579336.238526030081434.743596887077
51377.003872172532318.277847313356435.729897031707
52350.380960603284281.307533175343419.454388031224
53357.61794116767277.422041437929437.813840897411
54362.820896823557270.799862212444454.841931434671
55360.996754509321256.502751075469465.490757943172
56336.302031318540218.730125208514453.873937428567

\begin{tabular}{lllllllll}
\hline
Extrapolation Forecasts of Exponential Smoothing \tabularnewline
t & Forecast & 95% Lower Bound & 95% Upper Bound \tabularnewline
45 & 409.167945850895 & 387.737362492628 & 430.598529209163 \tabularnewline
46 & 394.539680436358 & 370.858365721759 & 418.220995150956 \tabularnewline
47 & 381.115064629995 & 353.369193982166 & 408.860935277825 \tabularnewline
48 & 379.929161941626 & 346.391886233272 & 413.46643764998 \tabularnewline
49 & 377.230744719644 & 336.439272778955 & 418.022216660332 \tabularnewline
50 & 385.491061458579 & 336.238526030081 & 434.743596887077 \tabularnewline
51 & 377.003872172532 & 318.277847313356 & 435.729897031707 \tabularnewline
52 & 350.380960603284 & 281.307533175343 & 419.454388031224 \tabularnewline
53 & 357.61794116767 & 277.422041437929 & 437.813840897411 \tabularnewline
54 & 362.820896823557 & 270.799862212444 & 454.841931434671 \tabularnewline
55 & 360.996754509321 & 256.502751075469 & 465.490757943172 \tabularnewline
56 & 336.302031318540 & 218.730125208514 & 453.873937428567 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=102459&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]45[/C][C]409.167945850895[/C][C]387.737362492628[/C][C]430.598529209163[/C][/ROW]
[ROW][C]46[/C][C]394.539680436358[/C][C]370.858365721759[/C][C]418.220995150956[/C][/ROW]
[ROW][C]47[/C][C]381.115064629995[/C][C]353.369193982166[/C][C]408.860935277825[/C][/ROW]
[ROW][C]48[/C][C]379.929161941626[/C][C]346.391886233272[/C][C]413.46643764998[/C][/ROW]
[ROW][C]49[/C][C]377.230744719644[/C][C]336.439272778955[/C][C]418.022216660332[/C][/ROW]
[ROW][C]50[/C][C]385.491061458579[/C][C]336.238526030081[/C][C]434.743596887077[/C][/ROW]
[ROW][C]51[/C][C]377.003872172532[/C][C]318.277847313356[/C][C]435.729897031707[/C][/ROW]
[ROW][C]52[/C][C]350.380960603284[/C][C]281.307533175343[/C][C]419.454388031224[/C][/ROW]
[ROW][C]53[/C][C]357.61794116767[/C][C]277.422041437929[/C][C]437.813840897411[/C][/ROW]
[ROW][C]54[/C][C]362.820896823557[/C][C]270.799862212444[/C][C]454.841931434671[/C][/ROW]
[ROW][C]55[/C][C]360.996754509321[/C][C]256.502751075469[/C][C]465.490757943172[/C][/ROW]
[ROW][C]56[/C][C]336.302031318540[/C][C]218.730125208514[/C][C]453.873937428567[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=102459&T=3

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=102459&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
tForecast95% Lower Bound95% Upper Bound
45409.167945850895387.737362492628430.598529209163
46394.539680436358370.858365721759418.220995150956
47381.115064629995353.369193982166408.860935277825
48379.929161941626346.391886233272413.46643764998
49377.230744719644336.439272778955418.022216660332
50385.491061458579336.238526030081434.743596887077
51377.003872172532318.277847313356435.729897031707
52350.380960603284281.307533175343419.454388031224
53357.61794116767277.422041437929437.813840897411
54362.820896823557270.799862212444454.841931434671
55360.996754509321256.502751075469465.490757943172
56336.302031318540218.730125208514453.873937428567



Parameters (Session):
par1 = 12 ; par2 = Triple ; par3 = additive ;
Parameters (R input):
par1 = 12 ; par2 = Triple ; 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')