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 08:53:12 -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/t12599420701lp2a33jqq4fqde.htm/, Retrieved Fri, 04 Dec 2009 16:54:34 +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/t12599420701lp2a33jqq4fqde.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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6,3 6,2 6,1 6,3 6,5 6,6 6,5 6,2 6,2 5,9 6,1 6,1 6,1 6,1 6,1 6,4 6,7 6,9 7 7 6,8 6,4 5,9 5,5 5,5 5,6 5,8 5,9 6,1 6,1 6 6 5,9 5,5 5,6 5,4 5,2 5,2 5,2 5,5 5,8 5,8 5,5 5,3 5,1 5,2 5,8 5,8 5,5 5 4,9 5,3 6,1 6,5 6,8 6,6 6,4 6,4

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 0.246754817664138 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 6.1 6.03467521072268 0.0653247892773168 14 6.1 6.07200226160909 0.0279977383909085 15 6.1 6.06829092762826 0.0317090723717426 16 6.4 6.3793628103621 0.0206371896379007 17 6.7 6.71297328595974 -0.0129732859597445 18 6.9 6.96258780495377 -0.0625878049537683 19 7 6.72565654791084 0.274343452089161 20 7 6.71522590731944 0.284774092680556 21 6.8 7.0296064004095 -0.229606400409505 22 6.4 6.49033861010353 -0.0903386101035304 23 5.9 6.62815115470167 -0.72815115470167 24 5.5 5.90474480817848 -0.40474480817848 25 5.5 5.49512339291347 0.00487660708652804 26 5.6 5.47712037184531 0.122879628154688 27 5.8 5.57284229303788 0.22715770696212 28 5.9 6.06684389281614 -0.166843892816141 29 6.1 6.19054758525098 -0.0905475852509827 30 6.1 6.34147574865495 -0.241475748654952 31 6 5.94889139243784 0.051108607562159 32 6 5.75946568092377 0.240534319076234 33 5.9 6.02893705636667 -0.128937056366667 34 5.5 5.63445987453335 -0.134459874533352 35 5.6 5.69950618174338 -0.0995061817433776 36 5.4 5.60574281217049 -0.205742812170494 37 5.2 5.39565284360838 -0.195652843608376 38 5.2 5.17967942696342 0.0203205730365772 39 5.2 5.17648338536558 0.0235166146344223 40 5.5 5.44180605772423 0.0581939422757731 41 5.8 5.77260702468397 0.0273929753160287 42 5.8 6.03091972050554 -0.230919720505543 43 5.5 5.65760445913547 -0.157604459135467 44 5.3 5.28158556772593 0.0184144322740734 45 5.1 5.32846851553668 -0.228468515536680 46 5.2 4.87367877624875 0.326321223751254 47 5.8 5.38995785742395 0.410042142576055 48 5.8 5.80507747617582 -0.005077476175817 49 5.5 5.79353504082876 -0.293535040828759 50 5 5.47712037184531 -0.477120371845312 51 4.9 4.97830393152943 -0.078303931529427 52 5.3 5.12928714017827 0.170712859821730 53 6.1 5.56363674440047 0.536363255599533 54 6.5 6.34147574865495 0.158524251345049 55 6.8 6.33727397017434 0.462726029825659 56 6.6 6.52407386204031 0.0759261379596925 57 6.4 6.62933866279237 -0.229338662792369 58 6.4 6.10994806096123 0.290051939038771 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 59 6.62815115470167 6.14889022584609 7.10741208355725 60 6.6304736370162 5.95382276644273 7.30712450758968 61 6.61961172940279 5.79334783370942 7.44587562509615 62 6.58718160749335 5.63807390964351 7.53628930534319 63 6.55103785208851 5.4954946166103 7.60658108756672 64 6.84922234805195 5.64972625073375 8.04871844537015 65 7.1823438858699 5.84053919001367 8.52414858172612 66 7.46190380961345 5.99252866692951 8.9312789522974 67 7.27124067295492 5.76619521594528 8.77628612996456 68 6.97446695431055 5.45826589530098 8.49066801332013 69 7.00405626432803 5.41294495984037 8.59516756881568 70 6.6843912962127 -11.7859559638598 25.1547385562852

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/04/t12599420701lp2a33jqq4fqde/1jkzm1259941990.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599420701lp2a33jqq4fqde/1jkzm1259941990.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599420701lp2a33jqq4fqde/291of1259941990.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599420701lp2a33jqq4fqde/291of1259941990.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599420701lp2a33jqq4fqde/3603g1259941990.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t12599420701lp2a33jqq4fqde/3603g1259941990.ps (open in new window)

Parameters (Session):

par1 = multiplicative ; par2 = 12 ;

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')

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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