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Author's title

Author*Unverified author*
R Software Modulerwasp_exponentialsmoothing.wasp
Title produced by softwareExponential Smoothing
Date of computationWed, 07 May 2008 11:09:02 -0600
Cite this page as followsStatistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?v=date/2008/May/07/t1210180177l5vr30tljt134cy.htm/, Retrieved Tue, 14 May 2024 12:10:02 +0000
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL https://freestatistics.org/blog/index.php?pk=11896, Retrieved Tue, 14 May 2024 12:10:02 +0000
QR Codes:

Original text written by user:
IsPrivate?No (this computation is public)
User-defined keywords
Estimated Impact153
Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)
-       [Exponential Smoothing] [exponential smoot...] [2008-05-07 17:09:02] [fe38921ec83f32fc5847db455ee6e655] [Current]
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Dataseries X:
56421
53152
53536
52408
41454
38271
35306
26414
31917
38030
27534
18387
50556
43901
48572
43899
37532
40357
35489
29027
34485
42598
30306
26451
47460
50104
61465
53726
39477
43895
31481
29896
33842
39120
33702
25094
51442
45594
52518
48564
41745
49585
32747
33379
35645
37034
35681
20972
58552
54955
65540
51570
51145
46641
35704
33253
35193
41668
34865
21210
56126
49231
59723
48103
47472
50497
40059
34149
36860
46356
36577




Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time7 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135

\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 & 7 seconds \tabularnewline
R Server & 'Gwilym Jenkins' @ 72.249.127.135 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=11896&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]7 seconds[/C][/ROW]
[ROW][C]R Server[/C][C]'Gwilym Jenkins' @ 72.249.127.135[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=11896&T=0

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=11896&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 time7 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135







Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0
beta0.00123082877145252
gamma0.371586577726775

\begin{tabular}{lllllllll}
\hline
Estimated Parameters of Exponential Smoothing \tabularnewline
Parameter & Value \tabularnewline
alpha & 0 \tabularnewline
beta & 0.00123082877145252 \tabularnewline
gamma & 0.371586577726775 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=11896&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[/C][/ROW]
[ROW][C]beta[/C][C]0.00123082877145252[/C][/ROW]
[ROW][C]gamma[/C][C]0.371586577726775[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=11896&T=1

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=11896&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
beta0.00123082877145252
gamma0.371586577726775







Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
135055650513.913472222342.0865277777484
144390143929.062602657-28.0626026570244
154857248570.83673309181.16326690820279
164389943787.1525301932111.847469806758
173753237300.9683272947231.031672705300
184035739861.1174577295495.882542270527
193548934972.7665881643516.233411835739
202902728567.9573852657459.042614734299
213448533416.93984903381068.06015096618
224259840769.92231280191828.07768719806
233030628174.07144323672131.92855676329
242645124277.26224033822173.73775966184
254746052769.341826265-5309.34182626495
265010446158.42448139103945.57551860905
276146550811.058552678610653.9414473214
285372646068.50311394357657.4968860565
293947739626.6061611191-149.606161119111
304389542285.17031978361609.82968021640
313148137404.3815601939-5923.38156019388
322989630978.3210247229-1082.32102472293
333384236053.606230555-2211.60623055504
343912043689.0010096239-4569.0010096239
353370231206.05704481972495.94295518028
362509427324.7835805437-2230.78358054374
375144253036.2512322789-1594.25123227890
384559449864.3369507308-4270.33695073079
395251857009.7097596075-4491.70975960753
404856451153.715741004-2589.71574100402
414174541810.8040849194-65.8040849193858
424958545123.15098659564461.84901340444
433274737443.1220428889-4696.12204288891
443337932815.9346243618563.065375638238
453564537471.5926052812-1826.59260528124
463703444231.0111260449-7197.01112604494
473568134373.30551095381307.69448904622
482097228735.6439094178-7763.64390941777
495855254683.6384380573868.36156194295
505495550517.32662268594437.67337731409
516554057580.44026711047959.55973288963
525157052431.2016967365-861.201696736542
535114544026.14173542117118.85826457889
544664149020.9037570375-2379.90375703746
553570437937.8956896019-2233.89568960192
563325335264.951725549-2011.95172554898
573519339032.6448754011-3839.64487540111
584166843796.4879570737-2128.48795707374
593486537099.016796068-2234.01679606799

\begin{tabular}{lllllllll}
\hline
Interpolation Forecasts of Exponential Smoothing \tabularnewline
t & Observed & Fitted & Residuals \tabularnewline
13 & 50556 & 50513.9134722223 & 42.0865277777484 \tabularnewline
14 & 43901 & 43929.062602657 & -28.0626026570244 \tabularnewline
15 & 48572 & 48570.8367330918 & 1.16326690820279 \tabularnewline
16 & 43899 & 43787.1525301932 & 111.847469806758 \tabularnewline
17 & 37532 & 37300.9683272947 & 231.031672705300 \tabularnewline
18 & 40357 & 39861.1174577295 & 495.882542270527 \tabularnewline
19 & 35489 & 34972.7665881643 & 516.233411835739 \tabularnewline
20 & 29027 & 28567.9573852657 & 459.042614734299 \tabularnewline
21 & 34485 & 33416.9398490338 & 1068.06015096618 \tabularnewline
22 & 42598 & 40769.9223128019 & 1828.07768719806 \tabularnewline
23 & 30306 & 28174.0714432367 & 2131.92855676329 \tabularnewline
24 & 26451 & 24277.2622403382 & 2173.73775966184 \tabularnewline
25 & 47460 & 52769.341826265 & -5309.34182626495 \tabularnewline
26 & 50104 & 46158.4244813910 & 3945.57551860905 \tabularnewline
27 & 61465 & 50811.0585526786 & 10653.9414473214 \tabularnewline
28 & 53726 & 46068.5031139435 & 7657.4968860565 \tabularnewline
29 & 39477 & 39626.6061611191 & -149.606161119111 \tabularnewline
30 & 43895 & 42285.1703197836 & 1609.82968021640 \tabularnewline
31 & 31481 & 37404.3815601939 & -5923.38156019388 \tabularnewline
32 & 29896 & 30978.3210247229 & -1082.32102472293 \tabularnewline
33 & 33842 & 36053.606230555 & -2211.60623055504 \tabularnewline
34 & 39120 & 43689.0010096239 & -4569.0010096239 \tabularnewline
35 & 33702 & 31206.0570448197 & 2495.94295518028 \tabularnewline
36 & 25094 & 27324.7835805437 & -2230.78358054374 \tabularnewline
37 & 51442 & 53036.2512322789 & -1594.25123227890 \tabularnewline
38 & 45594 & 49864.3369507308 & -4270.33695073079 \tabularnewline
39 & 52518 & 57009.7097596075 & -4491.70975960753 \tabularnewline
40 & 48564 & 51153.715741004 & -2589.71574100402 \tabularnewline
41 & 41745 & 41810.8040849194 & -65.8040849193858 \tabularnewline
42 & 49585 & 45123.1509865956 & 4461.84901340444 \tabularnewline
43 & 32747 & 37443.1220428889 & -4696.12204288891 \tabularnewline
44 & 33379 & 32815.9346243618 & 563.065375638238 \tabularnewline
45 & 35645 & 37471.5926052812 & -1826.59260528124 \tabularnewline
46 & 37034 & 44231.0111260449 & -7197.01112604494 \tabularnewline
47 & 35681 & 34373.3055109538 & 1307.69448904622 \tabularnewline
48 & 20972 & 28735.6439094178 & -7763.64390941777 \tabularnewline
49 & 58552 & 54683.638438057 & 3868.36156194295 \tabularnewline
50 & 54955 & 50517.3266226859 & 4437.67337731409 \tabularnewline
51 & 65540 & 57580.4402671104 & 7959.55973288963 \tabularnewline
52 & 51570 & 52431.2016967365 & -861.201696736542 \tabularnewline
53 & 51145 & 44026.1417354211 & 7118.85826457889 \tabularnewline
54 & 46641 & 49020.9037570375 & -2379.90375703746 \tabularnewline
55 & 35704 & 37937.8956896019 & -2233.89568960192 \tabularnewline
56 & 33253 & 35264.951725549 & -2011.95172554898 \tabularnewline
57 & 35193 & 39032.6448754011 & -3839.64487540111 \tabularnewline
58 & 41668 & 43796.4879570737 & -2128.48795707374 \tabularnewline
59 & 34865 & 37099.016796068 & -2234.01679606799 \tabularnewline
\hline
\end{tabular}
%Source: https://freestatistics.org/blog/index.php?pk=11896&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]50556[/C][C]50513.9134722223[/C][C]42.0865277777484[/C][/ROW]
[ROW][C]14[/C][C]43901[/C][C]43929.062602657[/C][C]-28.0626026570244[/C][/ROW]
[ROW][C]15[/C][C]48572[/C][C]48570.8367330918[/C][C]1.16326690820279[/C][/ROW]
[ROW][C]16[/C][C]43899[/C][C]43787.1525301932[/C][C]111.847469806758[/C][/ROW]
[ROW][C]17[/C][C]37532[/C][C]37300.9683272947[/C][C]231.031672705300[/C][/ROW]
[ROW][C]18[/C][C]40357[/C][C]39861.1174577295[/C][C]495.882542270527[/C][/ROW]
[ROW][C]19[/C][C]35489[/C][C]34972.7665881643[/C][C]516.233411835739[/C][/ROW]
[ROW][C]20[/C][C]29027[/C][C]28567.9573852657[/C][C]459.042614734299[/C][/ROW]
[ROW][C]21[/C][C]34485[/C][C]33416.9398490338[/C][C]1068.06015096618[/C][/ROW]
[ROW][C]22[/C][C]42598[/C][C]40769.9223128019[/C][C]1828.07768719806[/C][/ROW]
[ROW][C]23[/C][C]30306[/C][C]28174.0714432367[/C][C]2131.92855676329[/C][/ROW]
[ROW][C]24[/C][C]26451[/C][C]24277.2622403382[/C][C]2173.73775966184[/C][/ROW]
[ROW][C]25[/C][C]47460[/C][C]52769.341826265[/C][C]-5309.34182626495[/C][/ROW]
[ROW][C]26[/C][C]50104[/C][C]46158.4244813910[/C][C]3945.57551860905[/C][/ROW]
[ROW][C]27[/C][C]61465[/C][C]50811.0585526786[/C][C]10653.9414473214[/C][/ROW]
[ROW][C]28[/C][C]53726[/C][C]46068.5031139435[/C][C]7657.4968860565[/C][/ROW]
[ROW][C]29[/C][C]39477[/C][C]39626.6061611191[/C][C]-149.606161119111[/C][/ROW]
[ROW][C]30[/C][C]43895[/C][C]42285.1703197836[/C][C]1609.82968021640[/C][/ROW]
[ROW][C]31[/C][C]31481[/C][C]37404.3815601939[/C][C]-5923.38156019388[/C][/ROW]
[ROW][C]32[/C][C]29896[/C][C]30978.3210247229[/C][C]-1082.32102472293[/C][/ROW]
[ROW][C]33[/C][C]33842[/C][C]36053.606230555[/C][C]-2211.60623055504[/C][/ROW]
[ROW][C]34[/C][C]39120[/C][C]43689.0010096239[/C][C]-4569.0010096239[/C][/ROW]
[ROW][C]35[/C][C]33702[/C][C]31206.0570448197[/C][C]2495.94295518028[/C][/ROW]
[ROW][C]36[/C][C]25094[/C][C]27324.7835805437[/C][C]-2230.78358054374[/C][/ROW]
[ROW][C]37[/C][C]51442[/C][C]53036.2512322789[/C][C]-1594.25123227890[/C][/ROW]
[ROW][C]38[/C][C]45594[/C][C]49864.3369507308[/C][C]-4270.33695073079[/C][/ROW]
[ROW][C]39[/C][C]52518[/C][C]57009.7097596075[/C][C]-4491.70975960753[/C][/ROW]
[ROW][C]40[/C][C]48564[/C][C]51153.715741004[/C][C]-2589.71574100402[/C][/ROW]
[ROW][C]41[/C][C]41745[/C][C]41810.8040849194[/C][C]-65.8040849193858[/C][/ROW]
[ROW][C]42[/C][C]49585[/C][C]45123.1509865956[/C][C]4461.84901340444[/C][/ROW]
[ROW][C]43[/C][C]32747[/C][C]37443.1220428889[/C][C]-4696.12204288891[/C][/ROW]
[ROW][C]44[/C][C]33379[/C][C]32815.9346243618[/C][C]563.065375638238[/C][/ROW]
[ROW][C]45[/C][C]35645[/C][C]37471.5926052812[/C][C]-1826.59260528124[/C][/ROW]
[ROW][C]46[/C][C]37034[/C][C]44231.0111260449[/C][C]-7197.01112604494[/C][/ROW]
[ROW][C]47[/C][C]35681[/C][C]34373.3055109538[/C][C]1307.69448904622[/C][/ROW]
[ROW][C]48[/C][C]20972[/C][C]28735.6439094178[/C][C]-7763.64390941777[/C][/ROW]
[ROW][C]49[/C][C]58552[/C][C]54683.638438057[/C][C]3868.36156194295[/C][/ROW]
[ROW][C]50[/C][C]54955[/C][C]50517.3266226859[/C][C]4437.67337731409[/C][/ROW]
[ROW][C]51[/C][C]65540[/C][C]57580.4402671104[/C][C]7959.55973288963[/C][/ROW]
[ROW][C]52[/C][C]51570[/C][C]52431.2016967365[/C][C]-861.201696736542[/C][/ROW]
[ROW][C]53[/C][C]51145[/C][C]44026.1417354211[/C][C]7118.85826457889[/C][/ROW]
[ROW][C]54[/C][C]46641[/C][C]49020.9037570375[/C][C]-2379.90375703746[/C][/ROW]
[ROW][C]55[/C][C]35704[/C][C]37937.8956896019[/C][C]-2233.89568960192[/C][/ROW]
[ROW][C]56[/C][C]33253[/C][C]35264.951725549[/C][C]-2011.95172554898[/C][/ROW]
[ROW][C]57[/C][C]35193[/C][C]39032.6448754011[/C][C]-3839.64487540111[/C][/ROW]
[ROW][C]58[/C][C]41668[/C][C]43796.4879570737[/C][C]-2128.48795707374[/C][/ROW]
[ROW][C]59[/C][C]34865[/C][C]37099.016796068[/C][C]-2234.01679606799[/C][/ROW]
[/TABLE]
Source: https://freestatistics.org/blog/index.php?pk=11896&T=2

Globally Unique Identifier (entire table): ba.freestatistics.org/blog/index.php?pk=11896&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
135055650513.913472222342.0865277777484
144390143929.062602657-28.0626026570244
154857248570.83673309181.16326690820279
164389943787.1525301932111.847469806758
173753237300.9683272947231.031672705300
184035739861.1174577295495.882542270527
193548934972.7665881643516.233411835739
202902728567.9573852657459.042614734299
213448533416.93984903381068.06015096618
224259840769.92231280191828.07768719806
233030628174.07144323672131.92855676329
242645124277.26224033822173.73775966184
254746052769.341826265-5309.34182626495
265010446158.42448139103945.57551860905
276146550811.058552678610653.9414473214
285372646068.50311394357657.4968860565
293947739626.6061611191-149.606161119111
304389542285.17031978361609.82968021640
313148137404.3815601939-5923.38156019388
322989630978.3210247229-1082.32102472293
333384236053.606230555-2211.60623055504
343912043689.0010096239-4569.0010096239
353370231206.05704481972495.94295518028
362509427324.7835805437-2230.78358054374
375144253036.2512322789-1594.25123227890
384559449864.3369507308-4270.33695073079
395251857009.7097596075-4491.70975960753
404856451153.715741004-2589.71574100402
414174541810.8040849194-65.8040849193858
424958545123.15098659564461.84901340444
433274737443.1220428889-4696.12204288891
443337932815.9346243618563.065375638238
453564537471.5926052812-1826.59260528124
463703444231.0111260449-7197.01112604494
473568134373.30551095381307.69448904622
482097228735.6439094178-7763.64390941777
495855254683.6384380573868.36156194295
505495550517.32662268594437.67337731409
516554057580.44026711047959.55973288963
525157052431.2016967365-861.201696736542
535114544026.14173542117118.85826457889
544664149020.9037570375-2379.90375703746
553570437937.8956896019-2233.89568960192
563325335264.951725549-2011.95172554898
573519339032.6448754011-3839.64487540111
584166843796.4879570737-2128.48795707374
593486537099.016796068-2234.01679606799



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=0, beta=0)
if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)
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')