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ES

*The author of this computation has been verified*
R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)
Title produced by software: Exponential Smoothing
Date of computation: Wed, 29 Dec 2010 17:20:44 +0000
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k.htm/, Retrieved Wed, 29 Dec 2010 18:18:26 +0100
 
BibTeX entries for LaTeX users:
@Manual{KEY,
    author = {{YOUR NAME}},
    publisher = {Office for Research Development and Education},
    title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k.htm/},
    year = {2010},
}
@Manual{R,
    title = {R: A Language and Environment for Statistical Computing},
    author = {{R Development Core Team}},
    organization = {R Foundation for Statistical Computing},
    address = {Vienna, Austria},
    year = {2010},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}
 
Original text written by user:
 
IsPrivate?
No (this computation is public)
 
User-defined keywords:
 
Dataseries X:
» Textbox « » Textfile « » CSV «
347553 353245 347966 364343 341713 361162 354400 345183 341807 348712 349011 416259 360289 367557 364611 378688 351763 377765 373212 365819 364686 363333 362536 428803 375361 377205 381266 390516 366117 393928 387784 385656 385320 389053 390595 462440 402532 409070 421329 428841 404864 432633 416886 414893 417914 417518 423137 506710 436264 443712 452849 453009 437876 460041 450426 447873 444955 452043 480877 546696 483668 489627 490007 496590 480846 497677 492795 488495 486769 494455 498054 558648 506424 512528 512866 529324 508431 523026 521355 514103 513885 522150 527384 654047 543115 543200 571201 568892 537223 553186 550954 543433 557195 565522 571691 633972 575265 572364 586744 600389 578540 610778 596577 590558 597294 602384 614190 690042 639497 649304 678762 691885 667973 682032 672651 671865 671463 675917 680952 754718 694413 699390 710573 714217 702996 712370 708445 707083 etc...
 
Output produced by software:


Summary of computational transaction
Raw Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Sir Ronald Aylmer Fisher' @ 193.190.124.24


Estimated Parameters of Exponential Smoothing
ParameterValue
alpha0.543297682285428
beta0.0389250695236751
gamma0.426603189321503


Interpolation Forecasts of Exponential Smoothing
tObservedFittedResiduals
13360289353148.6153846167140.38461538439
14367557364169.4453822023387.55461779761
15364611362839.5528732441771.44712675619
16378688377942.720296868745.279703132284
17351763351891.884438275-128.884438274836
18377765378376.933259166-611.93325916573
19373212370894.8100051642317.18999483594
20365819363485.0763286092333.92367139086
21364686361811.9166042812874.08339571871
22363333370772.630234647-7439.63023464713
23362536367641.386263898-5105.38626389753
24428803432525.280100561-3722.28010056051
25375361375889.97645441-528.976454409771
26377205381926.026619807-4721.026619807
27381266375617.5465814025648.45341859845
28390516392450.797358555-1934.79735855520
29366117364540.533563731576.46643627004
30393928391661.0131299352266.98687006492
31387784386177.5969479741606.40305202571
32385656378233.8363609017422.16363909881
33385320379386.8326414705933.16735852952
34389053388021.2923573241031.70764267631
35390595390147.620480725447.379519274633
36462440458635.5514811983804.4485188016
37402532407188.590917175-4656.5909171754
38409070410555.022804861-1485.02280486142
39421329408483.02853501912845.9714649811
40428841428359.516484422481.483515577856
41404864403107.5057094601756.49429053953
42432633431125.5271182111507.47288178880
43416886425749.903207132-8863.90320713172
44414893413678.449455911214.55054409028
45417914411465.195266966448.80473304004
46417518419732.183037035-2214.18303703493
47423137420219.8755896422917.12441035832
48506710490994.5975195415715.4024804597
49436264444913.176625733-8649.17662573297
50443712447186.745805042-3474.74580504175
51452849447242.1594537965606.84054620378
52453009461039.873187615-8030.87318761466
53437876431494.7217215076381.27827849274
54460041462157.85679372-2116.85679371987
55450426452896.830153274-2470.83015327365
56447873446501.8593812381371.14061876165
57444955445636.334350919-681.334350919293
58452043448433.7918507573609.20814924350
59480877453300.28131581727576.7186841834
60546696540702.7252611065993.27473889396
61483668485123.449724558-1455.44972455798
62489627492996.746056515-3369.74605651473
63490007495564.052061195-5557.05206119537
64496590501088.78581606-4498.78581606009
65480846476794.6163618864051.38363811432
66497677505011.033457708-7334.03345770802
67492795493211.03532353-416.035323530377
68488495489088.891498397-593.891498397395
69486769487122.255663244-353.255663243646
70494455491307.1966327413147.80336725916
71498054500956.164059429-2902.16405942885
72558648567313.372628912-8665.3726289121
73506424501727.8361205494696.16387945064
74512528512109.408059064418.591940935992
75512866515927.95224783-3061.95224783046
76529324522686.414000946637.58599906013
77508431506015.918960322415.0810396797
78523026530997.978109583-7971.97810958303
79521355520058.6029642261296.39703577431
80514103516727.760843013-2624.76084301283
81513885513557.286026758327.713973242266
82522150518661.3553464643488.6446535358
83527384527191.035417446192.964582554414
84654047594046.67945803660000.3205419643
85543115569762.136179214-26647.1361792145
86543200563010.56290679-19810.5629067896
87571201555461.73252013815739.2674798615
88568892575023.422288848-6131.42228884797
89537223551021.644902338-13798.6449023378
90553186565256.984138224-12070.9841382240
91550954553895.584300676-2941.58430067613
92543433547407.862624716-3974.86262471648
93557195543960.1291360413234.8708639605
94565522556846.4482343928675.55176560825
95571691567815.7306975143875.26930248609
96633972648665.809642751-14693.8096427508
97575265565680.4532471929584.54675280815
98572364579473.583798736-7109.58379873598
99586744585548.1024094361195.89759056352
100600389592436.5096328047952.49036719592
101578540574379.7020907254160.29790927458
102610778598875.51137996911902.4886200312
103596577602991.336129799-6414.33612979855
104590558595015.926079282-4457.92607928207
105597294595248.8773317292045.12266827142
106602384601521.045491262862.954508738127
107614190607498.8082147226691.19178527757
108690042686508.7846291153533.21537088545
109639497618789.5996266420707.4003733596
110649304636241.74982433213062.2501756676
111678762656188.86109015722573.1389098426
112691885677755.06874930914129.9312506909
113667973664193.4583520173779.5416479829
114682032691860.61082862-9828.61082862038
115672651682011.524357447-9360.5243574474
116671865674164.543944143-2299.54394414276
117671463678230.695972898-6767.69597289804
118675917680691.748424216-4774.74842421641
119680952685830.044821743-4878.04482174281
120754718758782.511361205-4064.5113612049
121694413690964.1811644373448.81883556303
122699390697867.9157960071522.08420399297
123710573713471.88562168-2898.88562167948
124714217719089.117000886-4872.11700088589
125702996692320.22249679410675.7775032058
126712370720361.697395478-7991.69739547826
127708445710919.545061452-2474.5450614522
128707083707652.766481269-569.766481269151
129700632711288.120454612-10656.1204546120
130706309711442.607072392-5133.60707239178
131709523715775.940863703-6252.9408637028
132769096787520.98444293-18424.9844429301
133715100712441.8293831542658.17061684572
134713872717601.258888248-3729.25888824789
135714032728440.430020106-14408.4300201064
136732269726126.288255986142.71174402058
137711137707310.053442463826.94655753963
138715284726787.861547874-11503.8615478739
139716888715232.4778734131655.52212658734
140716426713388.0155938543037.98440614576
141714726715901.968654268-1175.96865426842
142718016721367.087478926-3351.08747892606
143725932725572.608782551359.391217448749
144779564797800.295941077-18236.2959410772
145732144726196.956633495947.04336650996
146730816731233.965292309-417.965292308596
147746719741196.7608012575522.23919874278
148760065753541.6261111366523.37388886407
149734516734315.800349932200.199650067603
150740167748594.373905543-8427.37390554289
151740976741097.431748395-121.431748395436
152735764738342.463185092-2578.4631850922
153734711736650.792528223-1939.79252822255
154737916740927.781013858-3011.78101385839
155739132745698.370634877-6566.37063487747
156792705810051.640878883-17346.6408788832
157747488743173.463076474314.53692352993
158746616745579.0696989151036.93030108477
159749781757016.042393475-7235.0423934753
160760911761881.569992392-970.569992391625
161739543736450.4845419193092.51545808092
162745626749778.827978661-4152.82797866117
163746246745472.183924967773.816075033392
164744769741993.5239432772775.47605672327
165741388742716.920852126-1328.92085212586
166744469746511.709082406-2042.70908240601
167745566750531.51482205-4965.5148220507
168798367813103.318478794-14736.3184787936
169752440751367.9066899751072.09331002494
170756627750809.074280945817.92571905942
171758941762768.776122426-3827.77612242557
172771287770314.890802406972.109197593527
173749858746380.8715675133477.12843248714
174758370758164.692489126205.307510874001
175755407756935.973327917-1528.97332791693
176752063752297.797429844-234.797429843806
177756298750223.9958702056074.00412979478
178755892757696.203087381-1804.20308738085
179758486761075.675133339-2589.67513333866
180812777822884.397568516-10107.3975685160
181762561766691.505883886-4130.50588388648
182763579764068.397744732-489.397744732443
183764615770426.330881655-5811.33088165522
184773312777492.265712633-4180.26571263326
185755697750800.3817893164896.61821068358
186762909762301.330492911607.669507089071
187760337760545.2050157-208.205015699728
188759270756496.5585416662773.44145833421
189754929756969.70604817-2040.70604817057
190766116758010.1164451458105.88355485466
191765945766342.081841088-397.081841088133
192814783827645.121297608-12862.1212976075
193768494770829.588788023-2335.58878802252
194769222769638.525604513-416.525604513357
195768977774748.186461629-5771.18646162876
196783341781903.5743742841437.42562571564
197764061759900.888042834160.11195717007
198767394770019.163890115-2625.16389011464
199763910766132.410520931-2222.41052093077
200761677761312.491189817364.508810182917
201759173759230.107902647-57.1079026468797
202762486763058.19860539-572.198605389567
203762690764568.349685376-1878.34968537570
204809542822156.32813409-12614.3281340905
205769041767049.833022451991.16697754967
206770889768198.400987792690.59901220945
207773527773633.634884097-106.634884096798
208789890785071.5441731934818.45582680719
209768325765308.2531361453016.74686385528
210772712773331.208432919-619.208432919113
211772944770503.023329112440.97667089070
212769637768709.612119379927.387880621012
213768546766851.6890263341694.31097366649
214775013771568.7980767443444.20192325604
215776635775129.3424038931505.65759610722


Extrapolation Forecasts of Exponential Smoothing
tForecast95% Lower Bound95% Upper Bound
216832658.481422083816588.031774783848728.931069384
217787712.02151017769258.282717093806165.760303248
218788334.056511121767619.262237866809048.850784376
219792124.599371005769229.686498364815019.512243645
220804944.349061166779924.467347024829964.230775309
221782474.620978812755367.683804466809581.558153157
222788348.838239279759180.658214508817517.01826405
223786665.029632085755452.603669029817877.45559514
224783410.670503283750164.385400157816656.955606409
225781338.83011411746064.001319434816613.658908784
226785581.035491782748279.014647542822883.056336022
227786924.506512076747593.499463834826255.513560317
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k/197io1293643239.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k/197io1293643239.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k/297io1293643239.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k/297io1293643239.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k/32yhr1293643239.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293643106x5s8m62tvmbz39k/32yhr1293643239.ps (open in new window)


 
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')
 




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