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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 16:04:11 +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/t1293206515saocbugbdf498vu.htm/, Retrieved Fri, 24 Dec 2010 17:01:55 +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/t1293206515saocbugbdf498vu.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 «

14.458 13.594 17.814 20.235 21.811 21.439 21.393 19.831 20.468 21.080 21.600 17.390 17.848 19.592 21.092 20.899 25.890 24.965 22.225 20.977 22.897 22.785 22.769 19.637 20.203 20.450 23.083 21.738 26.766 25.280 22.574 22.729 21.378 22.902 24.989 21.116 15.169 15.846 20.927 18.273 22.538 15.596 14.034 11.366 14.861 15.149 13.577 13.026 13.190 13.196 15.826 14.733 16.307 15.703 14.589 12.043 15.057 14.053 12.698 10.888 10.045 11.549 13.767 12.434 13.116 14.211 12.266 12.602 15.714 13.742 12.745 10.491 10.057 10.900 11.771 11.992 11.933 14.504 11.727 11.477 13.578 11.555 11.846 11.397 10.066 10.269 14.279 13.870 13.695 14.420 11.424 9.704 12.464 14.301 13.464 9.893 11.572 12.380 16.692 16.052 16.459 14.761 13.654 13.480 18.068 16.560 14.530 10.650 11.651 13.735 13.360 17.818 20.613 16.231 13.862 12.004 17.734 15.034 12.609 12.320 10.833 11.350 13.648 14.890 16.325 18.045 15.616 11.926 16 etc...

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	2 seconds
R Server	'Sir Ronald Aylmer Fisher' @ 193.190.124.24

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.447201661696798
beta	0
gamma	0.474929455987073

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	17.848	16.0586436237374	1.78935637626262
14	19.592	18.5204301019031	1.07156989809688
15	21.092	20.3506796076230	0.74132039237702
16	20.899	20.3169493189437	0.582050681056295
17	25.89	25.4484933507038	0.441506649296159
18	24.965	24.5786025245859	0.386397475414068
19	22.225	23.4831084509999	-1.25810845099988
20	20.977	20.9673135944513	0.00968640554871314
21	22.897	21.2221453711085	1.67485462889146
22	22.785	22.4188931442494	0.36610685575063
23	22.769	22.9049917384996	-0.135991738499648
24	19.637	18.3173010070656	1.31969899293444
25	20.203	19.6536672624031	0.549332737596895
26	20.45	21.3724652931722	-0.922465293172213
27	23.083	22.2242751719428	0.858724828057163
28	21.738	22.2012336378836	-0.463233637883551
29	26.766	26.8284263250821	-0.0624263250820825
30	25.28	25.7187074611296	-0.438707461129585
31	22.574	23.8224761508690	-1.24847615086896
32	22.729	21.6438360079938	1.08516399200619
33	21.378	22.8167967702696	-1.43879677026962
34	22.902	22.2775155432228	0.624484456777235
35	24.989	22.7473399575732	2.24166004242681
36	21.116	19.6051164097721	1.51088359022795
37	15.169	20.8247286131131	-5.65572861311307
38	15.846	19.3822067268321	-3.53620672683214
39	20.927	19.5327811289706	1.39421887102939
40	18.273	19.4021461621031	-1.12914616210312
41	22.538	23.8367696996340	-1.29876969963403
42	15.596	22.0753670941416	-6.47936709414157
43	14.034	17.2651459130930	-3.23114591309303
44	11.366	14.8125269406535	-3.44652694065349
45	14.861	13.2962667886879	1.56473321131210
46	15.149	14.6418634554185	0.507136544581474
47	13.577	15.4837833129012	-1.90678331290119
48	13.026	10.2945107969368	2.73148920306324
49	13.19	10.1784559565142	3.01154404348576
50	13.196	13.1684125344169	0.0275874655831316
51	15.826	16.2071566133214	-0.381156613321352
52	14.733	14.6200859837136	0.112914016286412
53	16.307	19.5656278970912	-3.2586278970912
54	15.703	15.5676582999848	0.135341700015152
55	14.589	14.5683345675782	0.0206654324217919
56	12.043	13.5133848512927	-1.47038485129271
57	15.057	14.1965160385504	0.860483961449637
58	14.053	14.9495096151991	-0.896509615199067
59	12.698	14.5299655414359	-1.83196554143592
60	10.888	10.5918847165021	0.296115283497945
61	10.045	9.4602522701558	0.584747729844194
62	11.549	10.5815329052276	0.967467094772395
63	13.767	13.9332809581206	-0.166280958120614
64	12.434	12.5720164874451	-0.138016487445114
65	13.116	16.5201765268627	-3.40417652686269
66	14.211	13.3481708452732	0.862829154726803
67	12.266	12.6440735796332	-0.378073579633224
68	12.602	11.0193464538816	1.58265354611840
69	15.714	13.6797482827775	2.03425171722251
70	13.742	14.4963712805018	-0.754371280501832
71	12.745	13.8947969070686	-1.14979690706864
72	10.491	10.8204898243408	-0.329489824340847
73	10.057	9.48486335845752	0.572136641542476
74	10.9	10.7009835178861	0.199016482113857
75	11.771	13.411424723346	-1.640424723346
76	11.992	11.3983412714638	0.593658728536237
77	11.933	14.8162093273194	-2.8832093273194
78	14.504	12.9974416281841	1.50655837181589
79	11.727	12.2554342685767	-0.528434268576699
80	11.477	11.0782353122247	0.398764687775271
81	13.578	13.3277627569255	0.250237243074533
82	11.555	12.6144458385062	-1.05944583850623
83	11.846	11.7726264871401	0.0733735128598703
84	11.397	9.46068684585417	1.93631315414583
85	10.066	9.37504444240165	0.690955557598347
86	10.269	10.5463415719972	-0.277341571997182
87	14.279	12.5608273762253	1.71817262377473
88	13.87	12.6362509866776	1.23374901332236
89	13.695	15.4275507978102	-1.73255079781023
90	14.42	15.2758500566134	-0.855850056613422
91	11.424	12.9431023187442	-1.51910231874418
92	9.704	11.5663019766636	-1.86230197666362
93	12.464	12.7656822246914	-0.301682224691426
94	14.301	11.4617015265481	2.83929847345190
95	13.464	12.6608177450433	0.803182254956747
96	9.893	11.1643463581869	-1.27134635818687
97	11.572	9.31727684065678	2.25472315934321
98	12.38	10.9336765045623	1.44632349543769
99	16.692	14.2428909153154	2.44910908468457
100	16.052	14.5180098472874	1.53399015271255
101	16.459	16.6648050074222	-0.205805007422200
102	14.761	17.4260367209542	-2.6650367209542
103	13.654	14.1100873095826	-0.456087309582628
104	13.48	13.1185653343451	0.361434665654912
105	18.068	15.7221297477755	2.34587025222454
106	16.56	16.4267726616529	0.133227338347059
107	14.53	15.8811669839622	-1.35116698396216
108	10.65	12.8766198513743	-2.22661985137434
109	11.651	11.5280854863124	0.122914513687613
110	13.735	11.9788993606364	1.75610063936356
111	13.36	15.689918168187	-2.32991816818701
112	17.818	13.5875926510712	4.23040734892884
113	20.613	16.4834638681830	4.12953613181703
114	16.231	18.5378201226600	-2.30682012266002
115	13.862	15.9620038206483	-2.10000382064834
116	12.004	14.4499520443661	-2.44595204436612
117	17.734	16.3190425997773	1.41495740022269
118	15.034	16.0264719950507	-0.992471995050687
119	12.609	14.5877385017339	-1.97873850173393
120	12.32	11.0726987603369	1.24730123966314
121	10.833	11.8945549349842	-1.06155493498421
122	11.35	12.2444491387021	-0.894449138702052
123	13.648	13.6973934286187	-0.0493934286186999
124	14.89	14.3372716302824	0.552728369717578
125	16.325	15.5619959457684	0.763004054231564
126	18.045	14.4210311120424	3.62396888795764
127	15.616	14.5517711069576	1.06422889304242
128	11.926	14.3639441375175	-2.43794413751746
129	16.855	17.2502598343074	-0.395259834307392
130	15.083	15.516110045447	-0.433110045447014
131	12.52	14.0685895256775	-1.54858952567754
132	12.355	11.5928782854899	0.762121714510064

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
133	11.5915937758937	8.10705777593714	15.0761297758502
134	12.4600891020489	8.64298813913186	16.2771900649660
135	14.534893640132	10.4119664776403	18.6578208026237
136	15.3549418509889	10.9473578173802	19.7625258845975
137	16.3876909442985	11.7127507693411	21.062631119256
138	15.6566277495631	10.7288154291169	20.5844400700092
139	13.4946881495897	8.32636125477636	18.6630150444030
140	11.9114749935637	6.51333906365603	17.3096109234714
141	16.4243301340399	10.8057769266857	22.0428833413940
142	14.8570039973481	9.02636005606865	20.6876479386276
143	13.3103127880576	7.27502676367656	19.3455988124386
144	12.1337879807172	5.90057484284054	18.3670011185939

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293206515saocbugbdf498vu/1jlsi1293206647.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293206515saocbugbdf498vu/1jlsi1293206647.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293206515saocbugbdf498vu/2jlsi1293206647.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293206515saocbugbdf498vu/2jlsi1293206647.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293206515saocbugbdf498vu/3tcrl1293206647.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293206515saocbugbdf498vu/3tcrl1293206647.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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