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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: Wed, 02 Dec 2009 14:26:47 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61.htm/, Retrieved Wed, 02 Dec 2009 22:27:46 +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/2009/Dec/02/t1259789261b0qm9choeh99l61.htm/},
    year = {2009},
}
@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 = {2009},
    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 «

0.0314796223103059 -3.00870920563557 -2.07677512619799 -1.25010391965540 0.817975239137125 0.0252076485413113 0.554937772830776 0.230027371950115 2.35672227418686 1.41350455171120 2.73311719024401 1.31551925971717 -2.70076272244080 -0.721411049152714 -0.149388576811997 -0.118199629770334 -0.676562489695275 1.79699928690761 1.79845572032988 0.245100010770855 1.80710848932636 -1.75934771184948 -0.0186697168761931 0.189651523600062 -1.84149562719087 -1.07019530156943 -0.507291477584104 0.866365633831705 -1.76077926699189 -0.580719393339347 -0.435702079860853 -0.994868534845203 1.63136048315789 -1.1949403709466 -1.00525975426991 1.32302234837564 -0.628357549594746 0.632048410440518 -2.16903155809288 2.53779364144266 -0.632933703679292 -1.41749196342200 -0.455343045381255 0.812255211942954 0.627897309219833 0.650904313655623 -1.29800419154382 0.74391671726854 -1.50461634127457 -1.42734677658523 0.263353807408564 -0.430830854870631 0.379576092518008 1.70309353400 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	'Gwilym Jenkins' @ 72.249.127.135

Estimated Parameters of Exponential Smoothing
Parameter	Value
alpha	0.00511311737670761
beta	0.260114412291294
gamma	0.525051146760498

Interpolation Forecasts of Exponential Smoothing
t	Observed	Fitted	Residuals
13	-2.7007627224408	-3.07843489276669	0.377672170325891
14	-0.721411049152714	-1.14414607616718	0.422735027014468
15	-0.149388576811997	-0.541679990965137	0.39229141415314
16	-0.118199629770334	-0.346851142738269	0.228651512967935
17	-0.676562489695275	-0.650349492584578	-0.0262129971106967
18	1.79699928690761	1.99144775679617	-0.194448469888564
19	1.79845572032988	0.628194406684996	1.17026131364488
20	0.245100010770855	0.335905416592872	-0.090805405822017
21	1.80710848932636	2.29450177275622	-0.487393283429857
22	-1.75934771184948	1.22865088406288	-2.98799859591236
23	-0.0186697168761931	2.551450621411	-2.57012033828719
24	0.189651523600062	1.10909622888707	-0.919444705287008
25	-1.84149562719087	-2.84152733828757	1.00003171109670
26	-1.07019530156943	-0.885920041691832	-0.184275259877598
27	-0.507291477584104	-0.308670759806620	-0.198620717777484
28	0.866365633831705	-0.209339592390441	1.07570522622215
29	-1.76077926699189	-0.64750923678777	-1.11327003020412
30	-0.580719393339347	1.89353370868427	-2.47425310202362
31	-0.435702079860853	1.22115451235218	-1.65685659221303
32	-0.994868534845203	0.241563466848861	-1.23643200169406
33	1.63136048315789	1.97150653754671	-0.340146054388816
34	-1.1949403709466	-0.415259403324323	-0.779680967622277
35	-1.00525975426991	1.12461887306691	-2.12987862733682
36	1.32302234837564	0.534860207874224	0.788162140501416
37	-0.628357549594746	-2.41400381177643	1.78564626218169
38	0.632048410440518	-1.08161881916941	1.71366722960993
39	-2.16903155809288	-0.508231423848648	-1.66080013424423
40	2.53779364144266	0.241271235302312	2.29652240614035
41	-0.632933703679292	-1.34049728118701	0.707563577507715
42	-1.417491963422	0.494951825516196	-1.91244378893820
43	-0.455343045381255	0.249200970004793	-0.704544015386047
44	0.812255211942954	-0.507863305722332	1.32011851766529
45	0.627897309219833	1.70478342316529	-1.07688611394546
46	0.650904313655623	-0.914885767482161	1.56579008113778
47	-1.29800419154382	-0.0647312780017505	-1.23327291354207
48	0.74391671726854	0.87915529562683	-0.135238578358290
49	-1.50461634127457	-1.54983196119136	0.0452156199167872
50	-1.42734677658523	-0.262714263898025	-1.16463251268720
51	0.263353807408564	-1.46934961426520	1.73270342167377
52	-0.430830854870631	1.36659353103896	-1.79742438590959
53	0.379576092518008	-1.06965522176257	1.44923131428058
54	1.70309353400146	-0.601560086531992	2.30465362053345
55	-3.12314448117342	-0.191715463240113	-2.93142901793331
56	-1.32526207118689	0.0975546571912894	-1.42281672837818
57	-0.60032490743804	1.04051827031719	-1.64084317775523
58	1.23607137604666	-0.20588229765998	1.44195367370664
59	0.738007075905376	-0.822955745751313	1.56096282165669
60	0.899100896289585	0.70805274367383	0.191048152615755

Extrapolation Forecasts of Exponential Smoothing
t	Forecast	95% Lower Bound	95% Upper Bound
61	-1.62531067371396	-4.41179533567949	1.16117398825156
62	-0.970776169290266	-3.75731866930500	1.81576633072447
63	-0.656804300749603	-3.44343097913932	2.12982237764011
64	0.325135427083718	-2.46160669614828	3.11187755031571
65	-0.404711546120209	-3.19160530503655	2.38218221279613
66	0.502157150503387	-2.28492935695182	3.28924365795859
67	-1.83866182775345	-4.62598711557029	0.948663460063385
68	-0.746181142481291	-3.53379615763881	2.04143387267623
69	0.0922334312146882	-2.69572716875365	2.88019403118303
70	0.468820442929303	-2.31954650420794	3.25718739006654
71	-0.0911293248799895	-2.87996827980696	2.69770963004698
72	0.71655434253189	-2.07282717133810	3.50593585640188

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61/1ij1j1259789205.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61/1ij1j1259789205.ps (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61/2hzqm1259789205.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61/2hzqm1259789205.ps (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61/3c39o1259789205.png (open in new window)

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259789261b0qm9choeh99l61/3c39o1259789205.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=0, beta=0)

if (par2 == 'Double') fit <- HoltWinters(x, gamma=0)

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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