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, 03 Dec 2009 09:44:51 -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/03/t1259858780sqqnkuffm6gymzp.htm/, Retrieved Thu, 03 Dec 2009 17:46:24 +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/03/t1259858780sqqnkuffm6gymzp.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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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 8.2 8.1 8.1 8 7.9 7.9 8 8 7.9 8 7.7 7.2 7.5 7.3 7 7 7 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 8 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 3 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Estimated Parameters of Exponential Smoothing Parameter Value alpha 1 beta 0 gamma 1 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 9.3 9.4006281885072 -0.100628188507207 14 8.7 8.67269616555066 0.0273038344493415 15 8.2 8.1461784881448 0.0538215118552028 16 8.3 8.25402075291598 0.0459792470840252 17 8.5 8.4986807867695 0.00131921323049866 18 8.6 8.64507808414717 -0.045078084147173 19 8.5 8.4308931797946 0.0691068202054073 20 8.2 7.64507210354858 0.554927896451415 21 8.1 7.8776586027846 0.222341397215402 22 7.9 8.35547852570824 -0.455478525708243 23 8.6 9.4390561812499 -0.839056181249903 24 8.7 9.12829638060467 -0.428296380604673 25 8.7 8.71244853084964 -0.0124485308496372 26 8.5 8.11391533770584 0.386084662294159 27 8.4 7.95915923286362 0.440840767136384 28 8.5 8.45507252995357 0.0449274700464297 29 8.7 8.70319974235725 -0.00319974235724807 30 8.7 8.84822534501423 -0.148225345014229 31 8.6 8.52879795713377 0.0712020428662257 32 8.5 7.73489701427511 0.765102985724888 33 8.3 8.16551616955018 0.134483830449824 34 8 8.56154363891267 -0.561543638912672 35 8.2 9.5583892456576 -1.35838924565759 36 8.1 8.70430345736025 -0.604303457360253 37 8.1 8.11244853084964 -0.0124485308496372 38 8 7.55513450986103 0.444865490138973 39 7.9 7.49161109466066 0.408388905339338 40 7.9 7.95244308735959 -0.0524430873595847 41 8 8.089642875594 -0.0896428755940075 42 8 8.13720993197953 -0.137209931979534 43 7.9 7.8434645157595 0.0565354842405066 44 8 7.10612263918942 0.893877360810585 45 7.7 7.68575355827421 0.0142464417257875 46 7.2 7.94334829929938 -0.743348299299385 47 7.5 8.60372473039606 -1.10372473039606 48 7.3 7.96231584168252 -0.662315841682517 49 7 7.31244853084964 -0.312448530849637 50 7 6.53070299214553 0.469297007854468 51 7 6.55651481825475 0.443485181745245 52 7.2 7.04771009069041 0.152289909309589 53 7.3 7.3738265310369 -0.073826531036894 54 7.1 7.42619451894484 -0.326194518944841 55 6.8 6.96232151970685 -0.162321519706847 56 6.4 6.1180486211976 0.281951378802395 57 6.1 6.15051320219113 -0.0505132021911283 58 6.5 6.29482739366395 0.205172606336047 59 7.7 7.76839327954222 -0.0683932795422235 60 7.9 8.17431230330473 -0.274312303304727 61 7.5 7.91244853084964 -0.412448530849637 62 6.9 6.99635368201621 -0.0963536820162112 63 6.6 6.46300519061416 0.136994809385835 64 6.9 6.64560653661522 0.254393463384778 65 7.7 7.06704809765527 0.632951902344727 66 8 7.83248904067895 0.167510959321048 67 8 7.8434645157595 0.156535484240506 68 7.7 7.19594754991594 0.504052450084058 69 7.3 7.39789599150863 -0.0978959915086346 70 7.4 7.53121807289053 -0.131218072890525 71 8.1 8.84239085921145 -0.742390859211447 72 8.3 8.59830522654915 -0.298305226549147 73 8.2 8.31244853084964 -0.112448530849639 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 74 7.64826464783516 6.81624987474706 8.48027942092327 75 7.162704676571 6.02360229465643 8.30180705848557 76 7.21127041247505 5.79582487073715 8.62671595421294 77 7.38535159597909 5.71583246215451 9.05487072980367 78 7.51288923361376 5.62398575283356 9.40179271439396 79 7.36655980453386 5.3386894931374 9.39443011593032 80 6.62696045983253 4.62440460583673 8.62951631382834 81 6.36828748792243 4.2743863940608 8.46218858178405 82 6.57125085151428 4.25897435248902 8.88352735053955 83 7.8534191040708 4.9713985476697 10.7354396604719 84 8.33693383934602 5.17076795754196 11.5030997211501 85 8.34938237019565 -76.1943207884345 92.8930855288258

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/03/t1259858780sqqnkuffm6gymzp/1x0ua1259858688.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/03/t1259858780sqqnkuffm6gymzp/1x0ua1259858688.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/03/t1259858780sqqnkuffm6gymzp/28c551259858688.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/03/t1259858780sqqnkuffm6gymzp/28c551259858688.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/03/t1259858780sqqnkuffm6gymzp/3vpco1259858688.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/03/t1259858780sqqnkuffm6gymzp/3vpco1259858688.ps (open in new window)

Parameters (Session):

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

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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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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