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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, 10 Dec 2010 20:04:18 +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/10/t12920115132z31axz8zkev0vw.htm/, Retrieved Fri, 10 Dec 2010 21:05:14 +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/10/t12920115132z31axz8zkev0vw.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 «

42.33600 42.14710 40.25640 39.18980 39.13170 38.15070 38.27070 39.13350 40.12190 41.28450 42.57490 43.90190 43.18350 43.61880 44.76240 45.19720 44.38810 43.55520 43.56780 44.21350 45.14510 45.80790 42.32820 37.89990 34.79640 35.21440 36.37270 36.25020 36.82610 36.77230 36.90420 37.04940 36.82590 36.13570 36.03000 35.79270 35.91740 35.40080 35.17230 34.92110 35.02920 34.77390 34.89990 34.90540 34.56800 34.40600 34.45780 34.73160 34.26020 33.88490 34.05490 34.27550 34.13930 34.15870 34.53860 33.79870 33.49730 33.68020 34.32840 34.15380 33.91840 34.32620 34.77500 35.01190 34.55130 34.69510 35.47300 35.97940 36.47890 36.39100 36.67040 37.41620 37.11850 36.30010 35.70020 35.58590 35.67770 35.24080 34.80160 34.43890 34.98810 36.06800 36.35660 36.12540 34.87100 35.21990 34.33900 33.82340 34.51990 35.53000 35.79660 33.84840 33.98710 34.11610 33.82350 32.45400 31.87740 31.11500 31.04360 30.88680 31.30010 30.05070 28.6 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	'RServer@AstonUniversity' @ vre.aston.ac.uk

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.99993997895847
beta	FALSE
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
2	42.1471	42.336	-0.188899999999997
3	40.2564	42.1471113379747	-1.89071133797475
4	39.1898	40.2565134824637	-1.06671348246374
5	39.1317	39.1898640252542	-0.0581640252542286
6	38.1507	39.1317034910654	-0.98100349106538
7	38.2707	38.1507588808513	0.11994111914872
8	39.1335	38.2706928010091	0.862807198990893
9	40.1219	39.1334482134133	0.988451786586722
10	41.2845	40.1218406720943	1.16265932790574
11	42.5749	41.2844302159762	1.29046978402381
12	43.9019	42.5748225446595	1.3270774553405
13	43.1835	43.9018203474289	-0.718320347428936
14	43.6188	43.1835431143354	0.435256885664593
15	44.7624	43.6187738754284	1.14362612457161
16	45.1972	44.7623313583689	0.43486864163112
17	44.3881	45.1971738987312	-0.8090738987312
18	43.5552	44.3881485614581	-0.832948561458082
19	43.5678	43.5552499944402	0.0125500055598025
20	44.2135	43.5677992467356	0.645700753264407
21	45.1451	44.2134612443683	0.931638755631724
22	45.8079	45.1450440820716	0.66285591792844
23	42.3282	45.8078602146974	-3.47966021469742
24	37.8999	42.3284088528303	-4.42850885283026
25	34.7964	37.9001658037138	-3.10376580371377
26	35.2144	34.7965862912562	0.4178137087438
27	36.3727	35.214374922386	1.15832507761397
28	36.2502	36.3726304761224	-0.122430476122418
29	36.8261	36.2502073484047	0.575892651595304
30	36.7723	36.8260654343232	-0.053765434323239
31	36.9042	36.7723032270574	0.131896772942632
32	37.0494	36.9041920834183	0.145207916581683
33	36.8259	37.0493912844696	-0.22349128446961
34	36.1357	36.8259134141797	-0.690213414179667
35	36.03	36.135741427328	-0.105741427327999
36	35.7927	36.0300063467106	-0.237306346710596
37	35.9174	35.7927142433741	0.124685756625908
38	35.4008	35.917392516231	-0.516592516231029
39	35.1723	35.4008310064209	-0.228531006420866
40	34.9211	35.172313716669	-0.251213716669021
41	35.0292	34.9211150781089	0.108084921891077
42	34.7739	35.0291935126304	-0.255293512630416
43	34.8999	34.7739153229825	0.125984677017478
44	34.9054	34.8998924382685	0.00550756173152678
45	34.568	34.9053996694304	-0.337399669430411
46	34.406	34.5680202510796	-0.162020251079568
47	34.4578	34.4060097246242	0.0517902753757795
48	34.7316	34.4577968914937	0.273803108506272
49	34.2602	34.7315835660523	-0.471383566052253
50	33.8849	34.2602282929326	-0.375328292932593
51	34.0549	33.8849225275951	0.169977472404945
52	34.2755	34.0548897977751	0.220610202224925
53	34.1393	34.2754867587459	-0.136186758745893
54	34.1587	34.1393081740711	0.0193918259288992
55	34.5386	34.1586988360824	0.379901163917587
56	33.7987	34.5385771979365	-0.739877197936472
57	33.4973	33.7987444082	-0.30144440820002
58	33.6802	33.4973180930073	0.182881906992655
59	34.3284	33.6801890232375	0.648210976762535
60	34.1538	34.328361093702	-0.174561093702046
61	33.9184	34.1538104773387	-0.235410477338654
62	34.3262	33.918414129582	0.407785870417968
63	34.775	34.3261755242673	0.448824475732664
64	35.0119	34.7749730610875	0.236926938912497
65	34.5513	35.0118857793984	-0.460585779398357
66	34.6951	34.5513276448382	0.143772355161808
67	35.473	34.6950913706335	0.7779086293665
68	35.9794	35.4729533091138	0.50644669088615
69	36.4789	35.9793696025421	0.499530397457868
70	36.391	36.4788700176653	-0.0878700176652742
71	36.6704	36.39100527405	0.27939472595002
72	37.4162	36.6703832304376	0.745816769562452
73	37.1185	37.4161552353007	-0.297655235300709
74	36.3001	37.1185178655772	-0.818417865577239
75	35.7002	36.3001491222927	-0.599949122292699
76	35.5859	35.7002360095712	-0.114336009571183
77	35.6777	35.5859068625664	0.0917931374336192
78	35.2408	35.6776944904803	-0.436894490480285
79	34.8016	35.2408262228624	-0.439226222862359
80	34.4389	34.8016263628154	-0.362726362815366
81	34.9881	34.4389217712141	0.549178228785919
82	36.068	34.9880670377507	1.07993296224927
83	36.3566	36.0679351812988	0.288664818701179
84	36.1254	36.3565826740369	-0.23118267403693
85	34.871	36.1254138758249	-1.25441387582487
86	35.2199	34.8710752912273	0.348824708772668
87	34.339	35.2198790631777	-0.880879063177673
88	33.8234	34.3390528712788	-0.515652871278832
89	34.5199	33.8234309500224	0.696469049977601
90	35.53	34.5198581972022	1.01014180279778
91	35.7966	35.5299393702369	0.266660629763095
92	33.8484	35.7965839947513	-1.94818399475127
93	33.9871	33.8485169320325	0.138583067967545
94	34.1161	33.9870916820999	0.12900831790008
95	33.8235	34.1160922567864	-0.292592256786392
96	32.454	33.823517561692	-1.369517561692
97	31.8774	32.4540821998704	-0.576682199870447
98	31.115	31.8774346130663	-0.762434613066272
99	31.0436	31.1150457621196	-0.0714457621195699
100	30.8868	31.0436042882491	-0.156804288249056
101	31.3001	30.8868094115567	0.413290588443303
102	30.0507	31.3000751938684	-1.24937519386843
103	28.6799	30.0507749888004	-1.3708749888004
104	27.6438	28.6799822813446	-1.03618228134464
105	27.2394	27.6438621927397	-0.404462192739739
106	26.8549	27.2394242762421	-0.384524276242068
107	27.0158	26.8549230795476	0.160876920452445
108	26.9188	27.0157903439997	-0.096990343999675
109	26.4966	26.9188058214615	-0.422205821461464
110	26.785	26.4966253412331	0.288374658766855
111	26.8398	26.7849826914526	0.0548173085473707
112	26.4478	26.839796709808	-0.391996709808048
113	25.1728	26.4478235280508	-1.2750235280508
114	24.8784	25.1728765282401	-0.294476528240128
115	25.4015	24.8784176747879	0.523082325212069
116	25.7716	25.401468604054	0.370131395945965
117	26.1181	25.7715777843281	0.346522215671886
118	26.3969	26.1180792013757	0.278820798624299
119	26.6571	26.3968832648853	0.260216735114735
120	25.1839	26.6570843815205	-1.47318438152053
121	23.8394	25.1839884220609	-1.34458842206094
122	23.8619	23.8394807035975	0.0224192964024752
123	24.2581	23.8618986543705	0.396201345629521
124	25.1098	24.2580762195826	0.85172378041742
125	26.1617	25.1097488786516	1.0519511213484
126	26.8087	26.1616368607981	0.647063139201943
127	25.6577	26.8086611625965	-1.15096116259645
128	27.0982	25.6577690818877	1.44043091811226
129	27.4665	27.098113543836	0.368386456163957
130	28.289	27.4664778890612	0.822522110938785
131	28.6933	28.2889506313662	0.404349368633778
132	27.2401	28.6932757305298	-1.45317573052975
133	27.2798	27.2401872211209	0.0396127788791283
134	27.6404	27.2797976223998	0.360602377600244
135	26.831	27.6403783562697	-0.809378356269718
136	26.2467	26.8310485797319	-0.584348579731934
137	25.2212	26.2467350732104	-1.02553507321037
138	25.3653	25.2212615536832	0.144038446316781
139	26.2592	25.3652913546624	0.893908645337568
140	27.2279	26.2591463466721	0.968753653327926
141	26.3315	27.2278418543967	-0.896341854396745
142	25.8981	26.3315537993717	-0.433453799371666
143	26.6898	25.8981260163485	0.79167398365151

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
144	26.6897524829029	24.9975380500968	28.3819669157091
145	26.6897524829029	24.2966717002631	29.0828332655428
146	26.6897524829029	23.7588683879782	29.6206365778277
147	26.6897524829029	23.3054759688566	30.0740289969493
148	26.6897524829029	22.906027668759	30.4734772970469

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/10/t12920115132z31axz8zkev0vw/1kqst1292011454.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/10/t12920115132z31axz8zkev0vw/1kqst1292011454.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/10/t12920115132z31axz8zkev0vw/2kqst1292011454.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/10/t12920115132z31axz8zkev0vw/2kqst1292011454.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/10/t12920115132z31axz8zkev0vw/3disf1292011454.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/10/t12920115132z31axz8zkev0vw/3disf1292011454.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Single ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Single ; par3 = additive ;

R code (references can be found in the software module):

par1 <- 5

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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