Workshop 9: Exponentional 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 10:47:44 -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/t1259776143r37t4zkwpvcd990.htm/, Retrieved Wed, 02 Dec 2009 18:49:07 +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/t1259776143r37t4zkwpvcd990.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)

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Dataseries X:

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8.1 7.7 7.5 7.6 7.8 7.8 7.8 7.5 7.5 7.1 7.5 7.5 7.6 7.7 7.7 7.9 8.1 8.2 8.2 8.2 7.9 7.3 6.9 6.6 6.7 6.9 7.0 7.1 7.2 7.1 6.9 7.0 6.8 6.4 6.7 6.6 6.4 6.3 6.2 6.5 6.8 6.8 6.4 6.1 5.8 6.1 7.2 7.3 6.9 6.1 5.8 6.2 7.1 7.7 7.9 7.7 7.4 7.5 8.0 8.1

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 0.899913987353234 gamma 0 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 7.6 7.48025965567942 0.119740344320576 14 7.7 7.78467255465862 -0.0846725546586216 15 7.7 7.70842463477782 -0.0084246347778194 16 7.9 7.92194022754802 -0.0219402275480167 17 8.1 8.14557182661695 -0.0455718266169516 18 8.2 8.25328008681465 -0.053280086814647 19 8.2 7.87789875075827 0.322101249241725 20 8.2 8.12887706082737 0.0711229391726338 21 7.9 8.47725542963737 -0.577255429637374 22 7.3 7.23800628747794 0.0619937125220567 23 6.9 7.51292475603305 -0.61292475603305 24 6.6 6.15182789529414 0.448172104705860 25 6.7 6.33759038458112 0.362409615418879 26 6.9 6.752693353889 0.147306646110998 27 7 6.99586839052863 0.00413160947137303 28 7.1 7.30222370767404 -0.202223707674045 29 7.2 7.25567270945402 -0.0556727094540204 30 7.1 7.25623961196135 -0.156239611961346 31 6.9 6.64746693363021 0.252533066369794 32 7 6.65528848226268 0.34471151773732 33 6.8 7.30460093243933 -0.504600932439332 34 6.4 6.2858232815591 0.114176718440899 35 6.7 6.70563966404874 -0.00563966404873995 36 6.6 6.63678201396038 -0.0367820139603783 37 6.4 6.58758138338824 -0.18758138338824 38 6.3 6.20383984264064 0.0961601573593631 39 6.2 6.10839136401051 0.0916086359894877 40 6.5 6.26696731931987 0.233032680680133 41 6.8 6.81855803213871 -0.0185580321387135 42 6.8 7.06518701134419 -0.265187011344190 43 6.4 6.47238707388988 -0.0723870738898826 44 6.1 6.00243536171957 0.0975646382804323 45 5.8 6.01284942528561 -0.212849425285614 46 6.1 5.21884205296817 0.881157947031832 47 7.2 6.98708299739394 0.212917002606056 48 7.3 7.9258423645877 -0.625842364587703 49 6.9 7.55703420929963 -0.657034209299631 50 6.1 6.54700458599799 -0.447004585997989 51 5.8 5.26417707774255 0.535822922257449 52 6.2 5.6066809847681 0.593319015231902 53 7.1 6.58532982399115 0.514670176008847 54 7.7 7.93762349963424 -0.237623499634238 55 7.9 7.91563004175484 -0.0156300417548421 56 7.7 8.0495831772975 -0.349583177297499 57 7.4 7.80935188173757 -0.409351881737571 58 7.5 6.74958608058079 0.750413919419208 59 8 8.35852704878512 -0.358527048785119 60 8.1 8.10831121120325 -0.00831121120325129 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 61 8.30490283499975 7.61950937424279 8.9902962957567 62 8.49775805306428 7.01795520568898 9.97756090043958 63 8.58035388480222 6.16434922001359 10.9963585495909 64 8.9106804482209 5.33418431463342 12.4871765818084 65 9.29518723987377 4.36113087220876 14.2292436075388 66 9.62827260347205 3.19743986858971 16.0591053383544 67 9.45704141410634 1.78298227721081 17.1311005510019 68 9.2510623302481 0.36225198917524 18.1398726713210 69 9.37349463238458 -1.08251807225750 19.8295073370266 70 8.97444624167821 -2.4686557961377 20.4175482794941 71 9.58078539597907 -4.20706164343723 23.3686324353954 72 9.6755968305694 -6.64847059456097 25.9996642556998

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259776143r37t4zkwpvcd990/1go641259776062.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259776143r37t4zkwpvcd990/1go641259776062.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259776143r37t4zkwpvcd990/2u2pr1259776062.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259776143r37t4zkwpvcd990/2u2pr1259776062.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259776143r37t4zkwpvcd990/321oy1259776062.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259776143r37t4zkwpvcd990/321oy1259776062.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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