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*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: Fri, 24 Dec 2010 12:52:36 +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/24/t1293195708q77qklkg9p4qd1f.htm/, Retrieved Fri, 24 Dec 2010 14:01:52 +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/24/t1293195708q77qklkg9p4qd1f.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 «

6.4 7.7 9.2 8.6 7.4 8.6 6.2 6 6.6 5.1 4.7 5 3.6 1.9 -0.1 -5.7 -5.6 -6.4 -7.7 -8 -11.9 -15.4 -15.5 -13.4 -10.9 -10.8 -7.3 -6.5 -5.1 -5.3 -6.8 -8.4 -8.4 -9.7 -8.8 -9.6 -11.5 -11 -14.9 -16.2 -14.4 -17.3 -15.7 -12.6 -9.4 -8.1 -5.4 -4.6 -4.9 -4 -3.1 -1.3 0 -0.4 3 0.4 1.2 0.6 -1.3 -3.2 -1.8 -3.6 -4.2 -6.9 -8 -7.5 -8.2 -7.6 -3.7 -1.7 -0.7 0.2 0.6 2.2 3.3 5.3 5.5 6.3 7.7 6.5 5.5 6.9 5.7 6.9 6.1 4.8 3.7 5.8 6.8 8.5 7.2 5 4.7 2.3 2.4 0.1 1.9 1.7 2 -1.9 0.5 -1.3 -3.3 -2.8 -8 -13.9 -21.9 -28.8 -27.6 -31.4 -31.8 -29.4 -27.6 -23.6 -22.8 -18.2 -17.8 -14.2 -8.8 -7.9 -7 -7 -3.6 -2.4 -4.9 -7.7 -6.5 -5.1 -3.4 -2.8 0.8

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	3 seconds
R Server	'George Udny Yule' @ 72.249.76.132

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.949476818409096
beta	0.434986722767703
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
3	9.2	9	0.199999999999999
4	8.6	10.5724973255986	-1.97249732559855
5	7.4	9.26759805763575	-1.86759805763575
6	8.6	7.29096189470873	1.30903810529127
7	6.2	8.87111370760506	-2.67111370760506
8	6	5.56900747665307	0.430992523346931
9	6.6	5.39028334023191	1.20971665976809
10	5.1	6.4505645665793	-1.35056456657931
11	4.7	4.52212170542846	0.177878294571544
12	5	4.11836538977327	0.881634610226726
13	3.6	4.7469331241338	-1.14693312413380
14	1.9	2.97572818894263	-1.07572818894263
15	-0.1	0.82784439459281	-0.92784439459281
16	-5.7	-1.56283600193754	-4.13716399806246
17	-5.6	-8.70938027988442	3.10938027988442
18	-6.4	-7.69129419502735	1.29129419502735
19	-7.7	-7.86612153200947	0.166121532009465
20	-8	-9.04066440693171	1.04066440693171
21	-11.9	-8.95504448686306	-2.94495551313694
22	-15.4	-13.8699738036022	-1.53002619639780
23	-15.5	-18.0733763624424	2.5733763624424
24	-13.4	-17.3178636336238	3.91786363362379
25	-10.9	-13.6676752948746	2.76767529487457
26	-10.8	-9.9664870740466	-0.833512925953393
27	-7.3	-10.0287926024562	2.72879260245624
28	-6.5	-5.58175349841656	-0.918246501583441
29	-5.1	-4.97673829226389	-0.123261707736109
30	-5.3	-3.66781174780649	-1.63218825219351
31	-6.8	-4.46568573723669	-2.33431426276331
32	-8.4	-6.89430678647731	-1.50569321352269
33	-8.4	-9.15803742562658	0.758037425626584
34	-9.7	-8.95933140696588	-0.740668593034119
35	-8.8	-10.4895154052073	1.68951540520733
36	-9.6	-9.01450959685643	-0.58549040314357
37	-11.5	-9.94138234519615	-1.55861765480385
38	-11	-12.4359412411738	1.43594124117376
39	-14.9	-11.4941780654694	-3.40582193453063
40	-16.2	-16.5561946539795	0.356194653979514
41	-14.4	-17.899151815014	3.499151815014
42	-17.3	-14.8127599855701	-2.48724001442994
43	-15.7	-18.4375629488454	2.73756294884538
44	-12.6	-15.9708962655097	3.37089626550967
45	-9.4	-11.5106810549306	2.11068105493059
46	-8.1	-7.37527899241413	-0.724721007585874
47	-5.4	-6.23134234447072	0.831342344470715
48	-4.6	-3.26660707238492	-1.33339292761508
49	-4.9	-2.90794211827162	-1.99205788172838
50	-4	-3.99740371559970	-0.00259628440030468
51	-3.1	-3.19898993607543	0.098989936075426
52	-1.3	-2.26323858049044	0.963238580490443
53	0	-0.109076188979906	0.109076188979906
54	-0.4	1.27912834867521	-1.67912834867521
55	3	0.275977651699932	2.72402234830007
56	0.4	4.57856442083763	-4.17856442083763
57	1.2	0.601516969838464	0.598483030161536
58	0.6	1.40734469632935	-0.807344696329346
59	-1.3	0.544930306540998	-1.84493030654100
60	-3.2	-2.06462188185782	-1.13537811814218
61	-1.8	-4.4693930162999	2.66939301629990
62	-3.6	-2.15913665787112	-1.44086334212888
63	-4.2	-4.34656412405781	0.146564124057806
64	-6.9	-4.96623358922543	-1.93376641077457
65	-8	-8.35979316893831	0.35979316893831
66	-7.5	-9.42707298788098	1.92707298788098
67	-8.2	-8.21035690299861	0.0103569029986090
68	-7.6	-8.80924080564856	1.20924080564856
69	-3.7	-7.77038391988683	4.07038391988683
70	-1.7	-2.33382948532013	0.633829485320131
71	-0.7	0.101573973465097	-0.801573973465097
72	0.2	0.843037208921463	-0.643037208921463
73	0.6	1.46944675196572	-0.869446751965718
74	2.2	1.52179564495035	0.678204355049652
75	3.3	3.32370843856912	-0.0237084385691246
76	5.3	4.44937948840314	0.850620511596856
77	5.5	6.75652022360631	-1.25652022360631
78	6.3	6.54402449931889	-0.244024499318891
79	7.7	7.19208548198325	0.507914518016746
80	6.5	8.763868808836	-2.26386880883600
81	5.5	6.76890809552944	-1.26890809552944
82	6.9	5.19456802382144	1.70543197617856
83	5.7	7.14865503600485	-1.44865503600485
84	6.9	5.50970080630211	1.39029919369789
85	6.1	7.14047501148008	-1.04047501148008
86	4.8	6.03355907168681	-1.23355907168681
87	3.7	4.23384229541535	-0.533842295415354
88	5.8	2.87800827288838	2.92199172711162
89	6.8	6.01021978983968	0.789780210160322
90	8.5	7.4441328737121	1.05586712628790
91	7.2	9.56677279690345	-2.36677279690345
92	5	7.4621950730862	-2.46219507308620
93	4.7	4.2501053917804	0.4498946082196
94	2.3	3.98878824247629	-1.68878824247629
95	2.4	0.999355194055063	1.40064480594494
96	0.1	1.52174725173442	-1.42174725173442
97	1.9	-1.22285208352967	3.12285208352967
98	1.7	1.63730884337346	0.0626911566265382
99	2	1.61780997224168	0.382190027758315
100	-1.9	2.05951610334665	-3.95951610334665
101	0.5	-3.2564460812639	3.7564460812639
102	-1.3	0.305168040868629	-1.60516804086863
103	-3.3	-1.88689630212747	-1.41310369787253
104	-2.8	-4.48022569298612	1.68022569298612
105	-8	-3.44256084200316	-4.55743915799684
106	-13.9	-10.2096812468672	-3.69031875313278
107	-21.9	-17.6776287740778	-4.22237122592221
108	-28.8	-27.3946285263889	-1.40537147361111
109	-27.6	-35.0173845211269	7.41738452112689
110	-31.4	-31.1996856557017	-0.200314344298256
111	-31.8	-34.6975470618145	2.89754706181455
112	-29.4	-34.0573455159193	4.65734551591927
113	-27.6	-29.8227267480842	2.22272674808417
114	-23.6	-26.9817141109894	3.38171411098943
115	-22.8	-21.6435887389409	-1.15641126105912
116	-18.2	-21.0919174014170	2.89191740141703
117	-17.8	-15.50206159028	-2.29793840972
118	-14.2	-15.7889246677368	1.58892466773679
119	-8.8	-11.7290598823533	2.92905988235334
120	-7.9	-5.18703731288784	-2.71296268711216
121	-7	-5.12246458488257	-1.87753541511743
122	-7	-5.04011357293581	-1.95988642706419
123	-3.6	-5.84540525785245	2.24540525785245
124	-2.4	-1.73049557530838	-0.669504424691619
125	-4.9	-0.659736959053438	-4.24026304094656
126	-7.7	-4.73060110419315	-2.96939889580685
127	-6.5	-8.82120007671058	2.32120007671058
128	-5.1	-6.92981956763588	1.82981956763588
129	-3.4	-4.74926002974388	1.34926002974388
130	-2.8	-2.46772300498234	-0.332276995017664
131	0.8	-1.92000006296976	2.72000006296976

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
132	2.64917589703852	-1.58252381401007	6.88087560808711
133	4.63577485118573	-2.51613545164312	11.7876851540146
134	6.62237380533295	-3.75069305575822	16.9954406664241
135	8.60897275948016	-5.29672796132532	22.5146734802856
136	10.5955717136274	-7.14035101378613	28.3314944410409
137	12.5821706677746	-9.26486547423865	34.4292068097878
138	14.5687696219218	-11.6546503847594	40.792189628603
139	16.555368576069	-14.2958292236786	47.4065663758166
140	18.5419675302162	-17.1762082267385	54.260143287171
141	20.5285664843634	-20.2850577450619	61.3421907137888
142	22.5151654385107	-23.6128854676838	68.6432163447051
143	24.5017643926579	-27.1512383962716	76.1547671815873

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293195708q77qklkg9p4qd1f/1z0t91293195153.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293195708q77qklkg9p4qd1f/1z0t91293195153.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293195708q77qklkg9p4qd1f/2z0t91293195153.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293195708q77qklkg9p4qd1f/2z0t91293195153.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293195708q77qklkg9p4qd1f/3rrsc1293195153.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293195708q77qklkg9p4qd1f/3rrsc1293195153.ps (open in new window)

Parameters (Session):

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

Parameters (R input):

par1 = 12 ; par2 = Double ; 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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