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paper- exponential smoothing

*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: Tue, 21 Dec 2010 15:32:41 +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/21/t1292945580ogr85u9h7cdatm8.htm/, Retrieved Tue, 21 Dec 2010 16:33:04 +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/21/t1292945580ogr85u9h7cdatm8.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 «

631 923 654 294 671 833 586 840 600 969 625 568 558 110 630 577 628 654 603 184 656 255 600 730 670 326 678 423 641 502 625 311 628 177 589 767 582 471 636 248 599 885 621 694 637 406 595 994 696 308 674 201 648 861 649 605 672 392 598 396 613 177 638 104 615 632 634 465 638 686 604 243 706 669 677 185 644 328 644 825 605 707 600 136 612 166 599 659 634 210 618 234 613 576 627 200 668 973 651 479 619 661 644 260 579 936 601 752 595 376 588 902 634 341 594 305 606 200 610 926 633 685 639 696 659 451 593 248 606 677 599 434 569 578 629 873 613 438 604 172 658 328 612 633 707 372 739 770 777 535 685 030 730 234 714 154 630 872 719 492 677 023 679 272 718 317 645 672

Output produced by software:

Summary of computational transaction
Raw Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	4 seconds
R Server	'George Udny Yule' @ 72.249.76.132

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.372612088044863
beta	0
gamma	1

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	670326	668237.219818376	2088.78018162376
14	678423	676144.043939775	2278.95606022526
15	641502	641317.479892395	184.520107605262
16	625311	625905.5453581	-594.545358099393
17	628177	628846.98828057	-669.988280570717
18	589767	591452.903591505	-1685.90359150525
19	582471	562903.359910489	19567.6400895109
20	636248	640338.851852142	-4090.85185214237
21	599885	637432.82037809	-37547.8203780904
22	621694	597915.7346686	23778.2653313997
23	637406	657393.031473608	-19987.0314736081
24	595994	595061.516318865	932.48368113453
25	696308	667074.9658235	29233.0341765003
26	674201	685215.38115173	-11014.3811517294
27	648861	644121.53516968	4739.46483031951
28	649605	629918.05184364	19686.9481563602
29	672392	640369.292435601	32022.7075643989
30	598396	614519.528423495	-16123.5284234949
31	613177	593924.567599103	19252.4324008974
32	638104	656399.557486453	-18295.5574864526
33	615632	627210.183362094	-11578.1833620943
34	634465	635844.94318856	-1379.94318856043
35	638686	658490.169206887	-19804.1692068871
36	604243	609351.44167522	-5108.44167521992
37	706669	696869.392651568	9799.60734843183
38	677185	682517.936367155	-5332.93636715494
39	644328	653424.837925338	-9096.83792533842
40	644825	643443.6512916	1381.34870839992
41	605707	654813.310587734	-49106.3105877343
42	600136	568527.527255988	31608.4727440118
43	612166	587912.537248203	24253.4627517974
44	599659	628693.81652344	-29034.8165234402
45	634210	599717.263990958	34492.7360090421
46	618234	631916.85789054	-13682.8578905402
47	613576	638418.732481697	-24842.7324816969
48	627200	596622.497178211	30577.5028217891
49	668973	706790.592195716	-37817.5921957165
50	651479	665202.416758017	-13723.4167580174
51	619661	630621.497558669	-10960.4975586688
52	644260	626519.776450768	17740.2235492324
53	579936	612309.603114105	-32373.6031141051
54	601752	582898.108231167	18853.8917688327
55	595376	592916.162812658	2459.83718734165
56	588902	592144.451494083	-3242.45149408269
57	634341	612634.864485779	21706.1355142212
58	594305	609845.231212133	-15540.2312121327
59	606200	608653.455634225	-2453.45563422504
60	610926	609969.721233806	956.278766193544
61	633685	666190.334254506	-32505.3342545058
62	639696	641697.964758653	-2001.96475865284
63	659451	613218.022371085	46232.9776289148
64	593248	648433.766962864	-55185.7669628644
65	606677	575609.678958357	31067.3210416434
66	599434	601976.550341887	-2542.55034188682
67	569578	593736.600279411	-24158.6002794114
68	629873	579468.990406653	50404.009593347
69	613438	635601.165189725	-22163.1651897246
70	604172	593097.379931351	11074.6200686492
71	658328	610033.104466228	48294.8955337722
72	612633	632398.045305151	-19765.0453051509
73	707372	659904.230972865	47467.7690271351
74	739770	684348.25177381	55421.7482261892
75	777535	707527.098772627	70007.9012273734
76	685030	687972.772886987	-2942.77288698684
77	730234	688729.200773638	41504.7992263619
78	714154	697898.775669105	16255.2243308946
79	630872	683101.45524303	-52229.4552430304
80	719492	705153.985607069	14338.0143929311
81	677023	702319.766347463	-25296.7663474628
82	679272	679501.348109869	-229.348109868704
83	718317	715576.628365014	2740.37163498602
84	645672	678267.418763399	-32595.4187633987

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
85	743173.90718524	691554.613566771	794793.200803709
86	754921.093855583	699834.82349248	810007.364218685
87	766600.303599613	708252.699609698	824947.907589527
88	675191.816349674	613755.763414846	736627.869284503
89	704930.626446057	640554.122121435	769307.130770679
90	682793.733366484	615605.341378692	749982.125354277
91	618973.059742036	549085.82394928	688860.295534792
92	702250.542260668	629764.878605022	774736.205916314
93	669207.423190179	594213.309124831	744201.537255528
94	671541.881068287	594120.547746691	748963.214389882
95	709565.785471356	629791.049106133	789340.521836578
96	649066.232517482	567005.558190568	731126.906844396

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/21/t1292945580ogr85u9h7cdatm8/1sng51292945556.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t1292945580ogr85u9h7cdatm8/1sng51292945556.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t1292945580ogr85u9h7cdatm8/2lef81292945556.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t1292945580ogr85u9h7cdatm8/2lef81292945556.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t1292945580ogr85u9h7cdatm8/3lef81292945556.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/21/t1292945580ogr85u9h7cdatm8/3lef81292945556.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')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

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