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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: Thu, 09 Dec 2010 18:18:02 +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/09/t1291918567o6uxvxe3arrgoqk.htm/, Retrieved Thu, 09 Dec 2010 19:16:08 +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/09/t1291918567o6uxvxe3arrgoqk.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 «

10 10 10 10 10 9,94 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 10,06 9,94 9,94 9,94 9,94 9,94 9,94 10,06 10,06 9,94 10,06 10,06 10,06 10,18 10,28 10,28 10,18 10,28 10,28 10,28 10,18 10,28 10,28 10,18 10,18 10,18 10,28 10,28 10,18 10,18 10,18 10,18 10,18 10,18 10,28 10,28 10,28 10,18 10,18 10,18 10,28 10,18 10,18 10,28 10,18 10,18 10,18 10,28 10,28 10,28 10,28 10,28 10,28 10,18 10,18 10,18 10,18 10,18 10,18 10,18 10,18 10,18 10,28 10,28 10,28 10,28 10,28 10,28 10,28 10,28 10,18 10,28 10,28 10,28 10,28 10,18 10,28 10,28 10,28 10,18 10,18 10,28 10,28 10,28 10,28 10,28 10,28 10,28 10,18 10,28 10,28 10,28 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,34 10,34 10,34 10,42 10,42 10,42 10,42 10,34 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,42 10,34 10,34 10,34 10,3 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	'RServer@AstonUniversity' @ vre.aston.ac.uk

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.6515049537628
beta	FALSE
gamma	FALSE

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
2	10	10	0
3	10	10	0
4	10	10	0
5	10	10	0
6	9.94	10	-0.0600000000000005
7	10.06	9.96090970277423	0.0990902972257697
8	10.06	10.0254675222866	0.034532477713352
9	10.06	10.0479656025826	0.0120343974174002
10	10.06	10.0558060721156	0.00419392788441364
11	10.06	10.058538436908	0.00146156309199341
12	10.06	10.0594906525027	0.000509347497322565
13	10.06	10.0598224949204	0.000177505079630436
14	10.06	10.0599381403591	6.18596409331928e-05
15	10.06	10.0599784422216	2.15577784263843e-05
16	10.06	10.059992487221	7.51277898913827e-06
17	10.06	10.0599973818337	2.61816626156985e-06
18	10.06	10.059999087582	9.12417972287471e-07
19	10.06	10.0599996820269	3.1797314292703e-07
20	10.06	10.0599998891879	1.10812065301502e-07
21	9.94	10.0599999613825	-0.119999961382545
22	9.94	9.98181939209047	-0.0418193920904724
23	9.94	9.95457385098018	-0.0145738509801809
24	9.94	9.94507891487119	-0.00507891487119139
25	9.94	9.94176997667287	-0.00176997667287004
26	9.94	9.94061682810245	-0.00061682810245145
27	10.06	9.94021496153808	0.119785038461917
28	10.06	10.0182555074827	0.0417444925173101
29	9.94	10.04545225115	-0.105452251150032
30	10.06	9.97674958714035	0.0832504128596536
31	10.06	10.0309876435212	0.0290123564787912
32	10.06	10.0498893374875	0.0101106625125258
33	10.18	10.0564764842002	0.123523515799791
34	10.28	10.13695266665	0.14304733335003
35	10.28	10.2301487129501	0.0498512870499273
36	10.18	10.2626270734146	-0.0826270734145513
37	10.28	10.20879512577	0.0712048742299505
38	10.28	10.2551854540629	0.0248145459370814
39	10.28	10.2713522536663	0.00864774633369869
40	10.18	10.2769863032416	-0.0969863032415894
41	10.28	10.2137992462326	0.066200753767447
42	10.28	10.2569293652549	0.0230706347451246
43	10.18	10.2719599980778	-0.0919599980777761
44	10.18	10.2120476037821	-0.0320476037820878
45	10.18	10.1911684311618	-0.0111684311618294
46	10.28	10.1838921429341	0.0961078570658618
47	10.28	10.2465068879081	0.0334931120919251
48	10.18	10.2683278163529	-0.088327816352896
49	10.18	10.2107818064439	-0.0307818064439331
50	10.18	10.1907273070599	-0.010727307059943
51	10.18	10.1837384133699	-0.00373841336985592
52	10.18	10.1813028185402	-0.00130281854018222
53	10.18	10.1804540258074	-0.000454025807400171
54	10.28	10.1801582257447	0.0998417742552569
55	10.28	10.2452056362645	0.0347943637354895
56	10.28	10.2678743366012	0.012125663398793
57	10.18	10.2757742663732	-0.09577426637318
58	10.18	10.2133768573881	-0.0333768573880544
59	10.18	10.1916316694587	-0.0116316694587031
60	10.28	10.1840535791858	0.0959464208141725
61	10.18	10.2465631476421	-0.0665631476420714
62	10.18	10.2031969272152	-0.023196927215217
63	10.28	10.1880840142224	0.0919159857775718
64	10.18	10.2479677342865	-0.0679677342865066
65	10.18	10.2036864187028	-0.0236864187028143
66	10.18	10.188254599581	-0.00825459958103103
67	10.28	10.1828766870627	0.0971233129373381
68	10.28	10.2461530065672	0.0338469934328085
69	10.28	10.2682044904586	0.011795509541356
70	10.28	10.275889323357	0.00411067664300546
71	10.28	10.2785674495532	0.00143255044677026
72	10.28	10.2795007632658	0.000499236734183839
73	10.18	10.2798260184712	-0.0998260184712372
74	10.18	10.2147888729228	-0.0347888729228085
75	10.18	10.1921237498778	-0.0121237498777749
76	10.18	10.1842250667742	-0.00422506677422341
77	10.18	10.1814724148408	-0.00147241484083871
78	10.18	10.180513129278	-0.000513129278038704
79	10.18	10.1801788230115	-0.00017882301147587
80	10.18	10.1800623189337	-6.23189336526053e-05
81	10.18	10.1800217178397	-2.17178396653139e-05
82	10.28	10.1800075685595	0.0999924314404605
83	10.28	10.2451531329818	0.0348468670182136
84	10.28	10.2678560394673	0.0121439605327343
85	10.28	10.2757678899126	0.00423211008735791
86	10.28	10.2785251305994	0.00147486940057462
87	10.28	10.2794860153201	0.000513984679946233
88	10.28	10.2798208788852	0.000179121114802783
89	10.28	10.2799375771788	6.2422821185848e-05
90	10.18	10.279978245956	-0.099978245956045
91	10.28	10.2148419234472	0.0651580765528337
92	10.28	10.257292733099	0.0227072669010067
93	10.28	10.2720866299714	0.0079133700285876
94	10.28	10.277242229746	0.00275777025400537
95	10.18	10.2790389307278	-0.0990389307278186
96	10.28	10.2145145767433	0.0654854232567263
97	10.28	10.2571786543943	0.0228213456057151
98	10.28	10.2720468741079	0.00795312589205821
99	10.18	10.2772283750245	-0.0972283750245158
100	10.18	10.2138836070497	-0.0338836070497361
101	10.28	10.1918082692055	0.0881917307945184
102	10.28	10.249265618699	0.0307343813009737
103	10.28	10.2692892203674	0.0107107796325554
104	10.28	10.2762673463567	0.00373265364328468
105	10.28	10.278699188696	0.00130081130400406
106	10.28	10.2795466737045	0.00045332629553485
107	10.28	10.2798420180317	0.000157981968323284
108	10.18	10.2799449440666	-0.0999449440666442
109	10.28	10.2148303179037	0.0651696820963199
110	10.28	10.2572886886246	0.0227113113754207
111	10.28	10.2720852204921	0.00791477950788533
112	10.42	10.2772417385494	0.142758261450558
113	10.42	10.370249453075	0.0497505469249546
114	10.42	10.4026621808491	0.0173378191509386
115	10.42	10.4139578559133	0.00604214408665804
116	10.42	10.4178943427171	0.00210565728285239
117	10.42	10.4192661888679	0.000733811132146656
118	10.42	10.4197442704556	0.000255729544427652
119	10.42	10.4199108795206	8.91204794086775e-05
120	10.42	10.4199689419544	3.10580455913367e-05
121	10.42	10.419989176425	1.08235750335695e-05
122	10.34	10.4199962280377	-0.079996228037718
123	10.34	10.3678782891888	-0.0278782891888056
124	10.34	10.3497154456799	-0.00971544567986626
125	10.42	10.3433857846914	0.0766142153085809
126	10.42	10.3933003254936	0.026699674506391
127	10.42	10.4106952956984	0.00930470430162345
128	10.42	10.4167573566442	0.00324264335581859
129	10.34	10.4188699548538	-0.0788699548537828
130	10.42	10.3674857885635	0.052514211436506
131	10.42	10.4016990574573	0.018300942542675
132	10.42	10.4136222121824	0.00637778781759302
133	10.42	10.4177773725396	0.00222262746038382
134	10.42	10.4192254253404	0.000774574659574867
135	10.42	10.4197300645682	0.000269935431802537
136	10.42	10.4199059288392	9.40711607864131e-05
137	10.42	10.4199672166665	3.27833335269645e-05
138	10.42	10.4199885751707	1.14248293332508e-05
139	10.42	10.4199960185036	3.98149642677481e-06
140	10.42	10.4199986124682	1.38753178191564e-06
141	10.42	10.419999516452	4.83547951901642e-07
142	10.42	10.4199998314859	1.68514066700709e-07
143	10.42	10.4199999412737	5.87263180307218e-08
144	10.42	10.4199999795342	2.04658316960149e-08
145	10.42	10.4199999928678	7.13224146409175e-09
146	10.42	10.4199999975144	2.48555132031925e-09
147	10.34	10.4199999991338	-0.079999999133797
148	10.34	10.3678796033971	-0.0278796033971087
149	10.34	10.3497159036749	-0.0097159036749499
150	10.34	10.3433859443004	-0.00338594430043848

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
151	10.3411799848155	10.244646564867	10.437713404764
152	10.3411799848155	10.2259666826561	10.456393286975
153	10.3411799848155	10.2099187612115	10.4724412084196
154	10.3411799848155	10.1956296064252	10.4867303632058
155	10.3411799848155	10.1826230030169	10.4997369666142

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/09/t1291918567o6uxvxe3arrgoqk/16z631291918677.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/09/t1291918567o6uxvxe3arrgoqk/16z631291918677.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/09/t1291918567o6uxvxe3arrgoqk/26z631291918677.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/09/t1291918567o6uxvxe3arrgoqk/26z631291918677.ps (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/09/t1291918567o6uxvxe3arrgoqk/3g8nn1291918677.png (open in new window)

http://www.freestatistics.org/blog/date/2010/Dec/09/t1291918567o6uxvxe3arrgoqk/3g8nn1291918677.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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