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 09:10:30 -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/t125977027793qc5dff6stv62i.htm/, Retrieved Wed, 02 Dec 2009 17:11:21 +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/t125977027793qc5dff6stv62i.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:

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108.5 112.3 116.6 115.5 120.1 132.9 128.1 129.3 132.5 131 124.9 120.8 122 122.1 127.4 135.2 137.3 135 136 138.4 134.7 138.4 133.9 133.6 141.2 151.8 155.4 156.6 161.6 160.7 156 159.5 168.7 169.9 169.9 185.9 190.8 195.8 211.9 227.1 251.3 256.7 251.9 251.2 270.3 267.2 243 229.9 187.2 178.2 175.2 192.4 187 184 194.1 212.7 217.5 200.5 205.9 196.5 206.3

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 1 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Estimated Parameters of Exponential Smoothing Parameter Value alpha 1 beta 0 gamma 0 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 122 116.516397726313 5.48360227368678 14 122.1 122.182945499263 -0.0829454992628911 15 127.4 127.718432479231 -0.318432479230864 16 135.2 135.610161653526 -0.410161653525876 17 137.3 137.418292153351 -0.118292153351263 18 135 134.885920174669 0.114079825330947 19 136 138.310626170459 -2.31062617045879 20 138.4 137.030300688663 1.36969931133706 21 134.7 141.695776982594 -6.99577698259381 22 138.4 132.661548225744 5.73845177425594 23 133.9 131.147073582721 2.75292641727867 24 133.6 129.443017082836 4.15698291716396 25 141.2 135.244335778506 5.955664221494 26 151.8 141.293851029031 10.5061489709687 27 155.4 158.595650753974 -3.19565075397423 28 156.6 165.233514265211 -8.63351426521129 29 161.6 159.037664152692 2.56233584730774 30 160.7 158.614813407126 2.08518659287409 31 156 164.483294211696 -8.48329421169589 32 159.5 157.059908883138 2.44009111686202 33 168.7 163.169619692391 5.53038030760914 34 169.9 165.938596346448 3.961403653552 35 169.9 160.819270729613 9.08072927038745 36 185.9 164.043735085615 21.8562649143846 37 190.8 187.893437119827 2.90656288017317 38 195.8 190.663690314266 5.13630968573352 39 211.9 204.339677827668 7.56032217233187 40 227.1 225.009207928077 2.09079207192346 41 251.3 230.260361627157 21.0396383728433 42 256.7 246.2066538578 10.4933461422000 43 251.9 262.248902459119 -10.3489024591186 44 251.2 253.101880175646 -1.90188017564591 45 270.3 256.494329478571 13.8056705214294 46 267.2 265.378246024787 1.82175397521314 47 243 252.473390805565 -9.47339080556546 48 229.9 234.302415252370 -4.40241525237033 49 187.2 232.18713614503 -44.9871361450299 50 178.2 187.080395527435 -8.8803955274349 51 175.2 186.042066998191 -10.8420669981906 52 192.4 186.181456469189 6.21854353081119 53 187 195.204650955328 -8.20465095532813 54 184 183.417854069859 0.582145930141422 55 194.1 188.211822046747 5.88817795325252 56 212.7 195.216312493613 17.4836875063870 57 217.5 217.312199415671 0.187800584329267 58 200.5 213.700947766753 -13.2009477667525 59 205.9 189.643690815164 16.2563091848360 60 196.5 198.644453088395 -2.14445308839490 61 206.3 198.564191884989 7.73580811501066 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 62 206.091765090902 185.623894483243 226.559635698562 63 215.039377394812 185.514198788967 244.564556000658 64 228.330596626208 190.985166291131 265.676026961285 65 231.503573133845 188.580906818577 274.426239449112 66 226.875489269687 180.231011058486 273.519967480888 67 231.875866706714 180.151528484447 283.600204928982 68 233.048102960723 177.350094443693 288.746111477753 69 238.020823227108 177.753870261532 298.287776192684 70 233.785430766897 171.349699910565 296.22116162323 71 220.997718238484 158.725035072443 283.270401404525 72 213.155338956595 149.900135738223 276.410542174967 73 215.330704860129 147.258284662347 283.403125057911

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/02/t125977027793qc5dff6stv62i/1joov1259770228.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t125977027793qc5dff6stv62i/1joov1259770228.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t125977027793qc5dff6stv62i/2pufs1259770228.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t125977027793qc5dff6stv62i/2pufs1259770228.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t125977027793qc5dff6stv62i/398bj1259770228.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t125977027793qc5dff6stv62i/398bj1259770228.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = periodic ; par3 = 0 ; par5 = 1 ; par7 = 1 ; par8 = FALSE ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ; par5 = 1 ; par7 = 1 ; par8 = FALSE ;

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')

Copyright

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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