WS 9 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, 11 Dec 2009 02:19:15 -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/11/t1260523195t6yjvt4mduhhksg.htm/, Retrieved Fri, 11 Dec 2009 10:20:00 +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/11/t1260523195t6yjvt4mduhhksg.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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108.8 128.4 121.1 119.5 128.7 108.7 105.5 119.8 111.3 110.6 120.1 97.5 107.7 127.3 117.2 119.8 116.2 111 112.4 130.6 109.1 118.8 123.9 101.6 112.8 128 129.6 125.8 119.5 115.7 113.6 129.7 112 116.8 127 112.1 114.2 121.1 131.6 125 120.4 117.7 117.5 120.6 127.5 112.3 124.5 115.2 104.7 130.9 129.2 113.5 125.6 107.6 107 121.6 110.7 106.3 118.6 104.6

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 0.0293236060805783 beta 1 gamma 0.305240513880531 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 107.7 109.000462287793 -1.30046228779317 14 127.3 128.040935508667 -0.74093550866688 15 117.2 117.545829304800 -0.345829304800148 16 119.8 119.924079425371 -0.124079425370951 17 116.2 115.851102524165 0.348897475835187 18 111 110.407302739617 0.592697260382536 19 112.4 106.085403826194 6.31459617380602 20 130.6 121.049305423344 9.55069457665634 21 109.1 113.445066947439 -4.34506694743868 22 118.8 113.142499562598 5.65750043740182 23 123.9 124.196068395081 -0.296068395080965 24 101.6 101.682542715042 -0.0825427150418108 25 112.8 112.075376208134 0.724623791866364 26 128 132.53585326085 -4.53585326085005 27 129.6 122.096265728768 7.50373427123212 28 125.8 125.538927739499 0.261072260500995 29 119.5 122.087070511082 -2.58707051108176 30 115.7 116.890654805018 -1.19065480501759 31 113.6 114.535241138508 -0.935241138508147 32 129.7 131.449881827212 -1.74988182721250 33 112 118.633291926168 -6.63329192616793 34 116.8 121.258301258138 -4.45830125813841 35 127 130.274150488050 -3.27415048804951 36 112.1 106.190005289299 5.90999471070063 37 114.2 117.193078542629 -2.99307854262878 38 121.1 136.319538941663 -15.2195389416634 39 131.6 128.148154075951 3.45184592404902 40 125 128.495473334345 -3.49547333434472 41 120.4 123.065198909847 -2.66519890984704 42 117.7 117.327457479859 0.372542520140584 43 117.5 114.262520923978 3.23747907602250 44 120.6 130.289944674282 -9.68994467428217 45 127.5 114.98394300478 12.51605699522 46 112.3 118.42720039909 -6.12720039909006 47 124.5 127.185239111114 -2.68523911111416 48 115.2 105.832093407342 9.36790659265831 49 104.7 113.879099312683 -9.17909931268328 50 130.9 128.327786557001 2.57221344299941 51 129.2 126.246913408497 2.95308659150284 52 113.5 124.536763862736 -11.0367638627363 53 125.6 119.016299567763 6.5837004322372 54 107.6 114.593978691425 -6.99397869142524 55 107 112.033842372140 -5.03384237214023 56 121.6 123.150062922684 -1.55006292268445 57 110.7 114.684985896835 -3.98498589683486 58 106.3 111.507173290213 -5.20717329021346 59 118.6 120.130039457263 -1.5300394572631 60 104.6 102.627675731307 1.97232426869348 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 61 103.979136153675 100.487834879532 107.470437427819 62 120.342186971249 116.774592556106 123.909781386392 63 117.674048896664 113.965704533188 121.382393260140 64 111.362319327877 107.457820384689 115.266818271066 65 110.861494135351 106.639913774821 115.083074495881 66 102.300489185400 97.8030932889347 106.797885081866 67 100.217040956392 95.291027793013 105.143054119772 68 111.005284488298 105.156228910919 116.854340065677 69 102.419021713785 96.1855485756125 108.652494851957 70 99.1216579607233 92.2968068678402 105.946509053606 71 107.951438520377 99.7743151695281 116.128561871225 72 93.1042989867176 85.0506999635212 101.157898009914

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/11/t1260523195t6yjvt4mduhhksg/1f03f1260523153.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260523195t6yjvt4mduhhksg/1f03f1260523153.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260523195t6yjvt4mduhhksg/2s2zf1260523153.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260523195t6yjvt4mduhhksg/2s2zf1260523153.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260523195t6yjvt4mduhhksg/38qes1260523153.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/11/t1260523195t6yjvt4mduhhksg/38qes1260523153.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')

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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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