WS9 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: Fri, 04 Dec 2009 07:20:29 -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/04/t12599364788plxyn3eusumub8.htm/, Retrieved Fri, 04 Dec 2009 15:21:22 +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/04/t12599364788plxyn3eusumub8.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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7.1 6.9 6.8 7.5 7.6 7.8 8.0 8.1 8.2 8.3 8.2 8.0 7.9 7.6 7.6 8.3 8.4 8.4 8.4 8.4 8.6 8.9 8.8 8.3 7.5 7.2 7.4 8.8 9.3 9.3 8.7 8.2 8.3 8.5 8.6 8.5 8.2 8.1 7.9 8.6 8.7 8.7 8.5 8.4 8.5 8.7 8.7 8.6 8.5 8.3 8.0 8.2 8.1 8.1 8.0 7.9 7.9 8.0 8.0 7.9 8.0 7.7 7.2 7.5 7.3 7.0 7.0 7.0 7.2 7.3 7.1 6.8 6.4 6.1 6.5 7.7 7.9 7.5 6.9 6.6 6.9 7.7 8.0 8.0 7.7 7.3 7.4 8.1 8.3 8.2

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 1 beta 1 gamma 1 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 7.9 7.54485690255072 0.355143097449276 14 7.6 7.9583753600631 -0.358375360063103 15 7.6 7.6 -8.88178419700125e-16 16 8.3 8.28734756097561 0.0126524390243912 17 8.4 8.39151515151516 0.00848484848484432 18 8.4 8.41266968325791 -0.0126696832579114 19 8.4 8.60279870828849 -0.202798708288487 20 8.4 8.27211959423385 0.127880405766152 21 8.6 8.39909643753924 0.200903562460763 22 8.9 8.79618983094593 0.103810169054075 23 8.8 8.98232931726907 -0.182329317269073 24 8.3 8.60118740657386 -0.301187406573861 25 7.5 7.93479492873633 -0.434794928736326 26 7.2 6.5623107267289 0.637689273271094 27 7.4 7.18491873650457 0.215081263495430 28 8.8 8.28579375267437 0.51420624732563 29 9.3 9.61602621010358 -0.316026210103576 30 9.3 9.70384342801068 -0.403843428010676 31 8.7 9.5245271413194 -0.824527141319397 32 8.2 7.95314850125848 0.246851498741519 33 8.3 7.69476148906675 0.60523851093325 34 8.5 8.39557782914431 0.104422170855692 35 8.6 8.49355421686746 0.106445783132536 36 8.5 8.59333011416109 -0.0933301141610894 37 8.2 8.51806716099568 -0.318067160995684 38 8.1 7.7159694152301 0.384030584769903 39 7.9 8.30988091697806 -0.409880916978059 40 8.6 8.3979454161318 0.202054583868204 41 8.7 8.66716915508926 0.0328308449107393 42 8.7 8.70954587581093 -0.00954587581093058 43 8.5 8.91004151929879 -0.410041519298787 44 8.4 8.1657958965754 0.234204103424608 45 8.5 8.29822514516636 0.201774854833642 46 8.7 8.59545197482787 0.104548025172129 47 8.7 8.6887751004016 0.0112248995983961 48 8.6 8.59725876036748 0.00274123963252393 49 8.5 8.61939915186189 -0.119399151861886 50 8.3 8.19694139324851 0.103058606751489 51 8 8.42190687992139 -0.421906879921387 52 8.2 8.39872232028242 -0.198722320282416 53 8.1 7.75313925558694 0.346860744413058 54 8.1 7.91552090715804 0.184479092841961 55 8 8.29555589727818 -0.295555897278181 56 7.9 7.77580848142781 0.124191518572191 57 7.9 7.79827892864616 0.101721071353839 58 8 7.8950287545009 0.104971245499104 59 8 7.9071686746988 0.0928313253012085 60 7.9 7.90552529688963 -0.00552529688963332 61 8 7.91007521579845 0.0899247842015463 62 7.7 7.89757326654945 -0.197573266549445 63 7.2 7.70377031587386 -0.503770315873858 64 7.5 7.30983566003424 0.190164339965764 65 7.3 7.21924482619473 0.080755173805267 66 7 7.02371476857656 -0.0237147685765633 67 7 6.86452406581578 0.135475934184220 68 7 6.89343299519487 0.106567004805127 69 7.2 6.9992470312827 0.200752968717303 70 7.3 7.39707081116096 -0.0970708111609593 71 7.1 7.22389558232931 -0.123895582329315 72 6.8 6.82637286098008 -0.0263728609800795 73 6.4 6.60511919100673 -0.205119191006725 74 6.1 5.8660319602411 0.233968039758905 75 6.5 6.04344526189214 0.45655473810786 76 7.7 7.52093161638853 0.179068383611465 77 7.9 8.38669837243712 -0.486698372437122 78 7.5 8.01801232077631 -0.518012320776311 79 6.9 7.27510379824697 -0.375103798246966 80 6.6 6.18055144535123 0.419448554648769 81 6.9 6.2966761809479 0.603323819052103 82 7.7 7.19806037591188 0.501939624088116 83 8 8.30267068273092 -0.302670682730922 84 8 8.19019655690384 -0.190196556903835 85 7.7 8.1065592692964 -0.406559269296396 86 7.3 7.22825684978167 0.0717431502183272 87 7.4 7.19317438357404 0.206825616425958 88 8.1 8.17752660997005 -0.0775266099700502 89 8.3 8.20774581579908 0.0922541842009164 90 8.2 8.41384724418034 -0.213847244180336 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 91 8.2964785483623 7.7194606008618 8.87349649586279 92 8.26896880482038 6.97418270942029 9.56375490022047 93 8.2403299031477 6.0665713873039 10.4140884189915 94 8.20627626050421 5.01569259002946 11.3968599309790 95 7.97669642857143 3.74659520416572 12.2067976529771 96 7.66464765844291 2.3929648393623 12.9363304775235 97 7.4698192239859 1.03827428476558 13.9013641632062 98 7.10277513363111 -0.354662565361127 14.5602128326234 99 7.01999124471036 -1.78744497918500 15.8274274686057 100 7.56525479094078 -3.59065004059843 18.72115962248 101 7.54727272727274 -5.35323383772776 20.4477792922733 102 7.45067873303169 -9.89242901410205 24.7937864801654

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/04/t12599364788plxyn3eusumub8/1hh2m1259936427.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599364788plxyn3eusumub8/1hh2m1259936427.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599364788plxyn3eusumub8/2hrpw1259936427.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599364788plxyn3eusumub8/2hrpw1259936427.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599364788plxyn3eusumub8/3jtok1259936427.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599364788plxyn3eusumub8/3jtok1259936427.ps (open in new window)

Parameters (Session):

par1 = 0.01 ; par2 = 0.99 ; par3 = 0.005 ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ;

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')

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Software written by Ed van Stee & Patrick Wessa

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