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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 15:31:52 +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/t1293204584gh0lmosjcr8ruz8.htm/, Retrieved Fri, 24 Dec 2010 16:29:47 +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/t1293204584gh0lmosjcr8ruz8.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	3 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.480421268200843
beta	0.0199237820561068
gamma	0.495517190826123

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	17.848	16.2568319978632	1.59116800213676
14	19.592	18.8962650104645	0.695734989535488
15	21.092	20.8016307350822	0.290369264917782
16	20.899	20.7987378365668	0.100262163433207
17	25.89	25.9419731335047	-0.0519731335046565
18	24.965	25.0729905458009	-0.107990545800888
19	22.225	23.9751040036675	-1.75010400366747
20	20.977	21.3866845709481	-0.409684570948105
21	22.897	21.6419763863665	1.25502361363350
22	22.785	22.9062922648777	-0.121292264877731
23	22.769	23.3828607372404	-0.613860737240394
24	19.637	18.7676630819247	0.869336918075337
25	20.203	20.0863003403317	0.116699659668310
26	20.45	21.7894375673915	-1.33943756739151
27	23.083	22.5958258132405	0.487174186759457
28	21.738	22.6235491927387	-0.88554919273869
29	26.766	27.2295612400831	-0.463561240083109
30	25.28	26.1200570971972	-0.840057097197231
31	22.574	24.2123205430961	-1.63832054309614
32	22.729	21.988408056635	0.740591943365004
33	21.378	23.2016226397241	-1.82362263972410
34	22.902	22.5797859899142	0.322214010085844
35	24.989	23.0940938538796	1.89490614612038
36	21.116	20.0415252770837	1.07447472291629
37	15.169	21.2424042607255	-6.0734042607255
38	15.846	19.5149993446043	-3.66899934460429
39	20.927	19.5684110683126	1.3585889316874
40	18.273	19.5656146982232	-1.29261469822315
41	22.538	23.9850682666054	-1.44706826660541
42	15.596	22.197079140875	-6.601079140875
43	14.034	17.1519060148900	-3.11790601488996
44	11.366	14.6512841657522	-3.28528416575221
45	14.861	13.0533038409512	1.80769615904876
46	15.149	14.5463605626839	0.602639437316109
47	13.577	15.4208436360836	-1.84384363608365
48	13.026	10.1456323651496	2.88036763485039
49	13.19	10.1758527170071	3.01414728299293
50	13.196	13.3223700100955	-0.126370010095545
51	15.826	16.2950800490145	-0.469080049014472
52	14.733	14.6370991184400	0.0959008815600324
53	16.307	19.6025959512777	-3.29559595127774
54	15.703	15.5006249145450	0.202375085455026
55	14.589	14.5869184264698	0.00208157353019445
56	12.043	13.5381397873835	-1.49513978738354
57	15.057	14.1245855421042	0.932414457895812
58	14.053	14.8916690997237	-0.83866909972367
59	12.698	14.4348338803330	-1.73683388033304
60	10.888	10.4193419854109	0.468658014589092
61	10.045	9.29429751914311	0.750702480856885
62	11.549	10.4921131811662	1.05688681883380
63	13.767	13.9036402629653	-0.136640262965344
64	12.434	12.5126021152704	-0.0786021152703604
65	13.116	16.4811897961192	-3.36518979611921
66	14.211	13.2058077877365	1.00519221226353
67	12.266	12.5933436186570	-0.327343618657027
68	12.602	10.9647935177839	1.63720648221611
69	15.714	13.6750321530902	2.03896784690981
70	13.742	14.5222842222188	-0.780284222218846
71	12.745	13.8673557549460	-1.12235575494603
72	10.491	10.7258793510906	-0.234879351090576
73	10.057	9.33970242977608	0.717297570223916
74	10.9	10.6042268312998	0.295773168700189
75	11.771	13.3394546704025	-1.56845467040255
76	11.992	11.2584215443654	0.733578455634605
77	11.933	14.7617431838948	-2.82874318389483
78	14.504	12.8651263389304	1.63887366106957
79	11.727	12.2159321633832	-0.488932163383241
80	11.477	11.0159091248546	0.461090875145414
81	13.578	13.2536610004937	0.324338999506342
82	11.555	12.5240182546296	-0.969018254629624
83	11.846	11.6612361489509	0.184763851049114
84	11.397	9.35961715722292	2.03738284277708
85	10.066	9.31538014075278	0.750619859247223
86	10.269	10.4928556446657	-0.223855644665703
87	14.279	12.4989726223900	1.78002737761003
88	13.87	12.6518487445950	1.21815125540495
89	13.695	15.5079970441796	-1.81299704417958
90	14.42	15.2965058733267	-0.876505873326696
91	11.424	12.9138739684875	-1.48987396848749
92	9.704	11.4908188866763	-1.78681888667628
93	12.464	12.6051509499796	-0.141150949979597
94	14.301	11.3061653531070	2.99483464689295
95	13.464	12.6699731209887	0.794026879011284
96	9.893	11.1690821105613	-1.27608211056126
97	11.572	9.20102938021473	2.37097061978527
98	12.38	10.9209104549542	1.4590895450458
99	16.692	14.2824222906082	2.40957770939178
100	16.052	14.6300650664331	1.42193493356693
101	16.459	16.8426439494949	-0.383643949494942
102	14.761	17.6115636959491	-2.85056369594911
103	13.654	14.1563493489163	-0.502349348916315
104	13.48	13.1744377867609	0.305562213239071
105	18.068	15.7808832743803	2.28711672561972
106	16.56	16.5423185433153	0.0176814566846772
107	14.53	15.9671594359957	-1.43715943599574
108	10.65	12.8979731468994	-2.24797314689945
109	11.651	11.4292561808993	0.221743819100698
110	13.735	11.8885399112734	1.84646008872657
111	13.36	15.6912834972136	-2.33128349721361
112	17.818	13.4720762388857	4.34592376111428
113	20.613	16.6175639347857	3.99543606521426
114	16.231	18.8900960167736	-2.65909601677363
115	13.862	16.1682140241811	-2.3062140241811
116	12.004	14.5472018155865	-2.54320181558646
117	17.734	16.2874533012097	1.44654669879025
118	15.034	16.0449672498692	-1.01096724986921
119	12.609	14.5754094216707	-1.96640942167075
120	12.32	11.0124916618311	1.30750833816887
121	10.833	11.8910735500971	-1.05807355009709
122	11.35	12.1448715172094	-0.794871517209405
123	13.648	13.5688441851703	0.0791558148296971
124	14.89	14.2156342095986	0.674365790401389
125	16.325	15.4607031867267	0.864296813273342
126	18.045	14.439434042353	3.605565957647
127	15.616	14.8017878941969	0.814212105803133
128	11.926	14.6324538911486	-2.70645389114858
129	16.855	17.3334884643763	-0.478488464376301
130	15.083	15.5270479253007	-0.444047925300689
131	12.52	14.0828736784164	-1.56287367841638
132	12.355	11.5596009462472	0.795399053752753

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
133	11.5810840335186	8.04435956370176	15.1178085033355
134	12.4190673231092	8.48058960369755	16.3575450425208
135	14.4656490169454	10.1490216861198	18.7822763477709
136	15.2426047130626	10.5654574728026	19.9197519533226
137	16.2210897326469	11.1970273167615	21.2451521485323
138	15.4905848508294	10.1303559285690	20.8508137730898
139	13.3677969913248	7.68004249952992	19.0555514831196
140	11.8587847058867	5.85054150431082	17.8670279074626
141	16.4174932378915	10.0945467917341	22.7404396840489
142	14.8381993395343	8.20533876773776	21.4710599113309
143	13.3119559448242	6.3731634553543	20.2507484342941
144	12.1542957289867	4.91289052468055	19.3957009332929

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293204584gh0lmosjcr8ruz8/1e4q91293204709.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293204584gh0lmosjcr8ruz8/1e4q91293204709.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293204584gh0lmosjcr8ruz8/27w7c1293204709.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293204584gh0lmosjcr8ruz8/27w7c1293204709.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293204584gh0lmosjcr8ruz8/37w7c1293204709.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/24/t1293204584gh0lmosjcr8ruz8/37w7c1293204709.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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