Paper Triple 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: Tue, 07 Dec 2010 17:44:26 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf.htm/, Retrieved Tue, 07 Dec 2010 18:42:30 +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/2010/Dec/07/t12917437465j7t1qir1iq2mgf.htm/}, year = {2010}, } @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 = {2010}, 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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17.848 19.592 21.092 20.899 25.890 24.965 22.225 20.977 22.897 22.785 22.769 19.637 20.203 20.450 23.083 21.738 26.766 25.280 22.574 22.729 21.378 22.902 24.989 21.116 15.169 15.846 20.927 18.273 22.538 15.596 14.034 11.366 14.861 15.149 13.577 13.026 13.190 13.196 15.826 14.733 16.307 15.703 14.589 12.043 15.057 14.053 12.698 10.888 10.045 11.549 13.767 12.434 13.116 14.211 12.266 12.602 15.714 13.742 12.745 10.491 10.057 10.900 11.771 11.992 11.933 14.504 11.727 11.477 13.578 11.555 11.846 11.397 10.066 10.269 14.279 13.870 13.695 14.420 11.424 9.704 12.464 14.301 13.464 9.893 11.572 12.380 16.692 16.052 16.459 14.761 13.654 13.480 18.068 16.560 14.530 10.650 11.651 13.735 13.360 17.818 20.613 16.231 13.862 12.004 17.734 15.034 12.609 12.320 10.833 11.350 13.648 14.890 16.325 18.045 15.616 11.926 16.855 15.083 12.520 12.355

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 0.473971450972799 beta 0.0065615366876307 gamma 0.328170676425391 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 20.203 19.7520502136752 0.450949786324788 14 20.45 20.1878033285003 0.262196671499719 15 23.083 22.9987416156329 0.0842583843670681 16 21.738 21.8157292766096 -0.0777292766096025 17 26.766 26.7729060076284 -0.00690600762840177 18 25.28 25.1928794686559 0.0871205313440981 19 22.574 23.0523981013263 -0.478398101326281 20 22.729 21.505930904748 1.22306909525199 21 21.378 23.9528809725080 -2.57488097250796 22 22.902 22.5604949715285 0.341505028471538 23 24.989 22.6939130825353 2.29508691746474 24 21.116 20.6662442462503 0.449755753749734 25 15.169 21.5631443600254 -6.39414436002542 26 15.846 18.7070404533627 -2.86104045336273 27 20.927 19.9823294163029 0.944670583697079 28 18.273 19.1572331801393 -0.884233180139319 29 22.538 23.7199366443331 -1.18193664433310 30 15.596 21.5711168793076 -5.97511687930761 31 14.034 16.4127369567014 -2.37873695670139 32 11.366 14.2064255749357 -2.84042557493574 33 14.861 14.0062712656517 0.854728734348273 34 15.149 14.6880411272139 0.46095887278606 35 13.577 15.1608610980324 -1.58386109803237 36 13.026 10.9096068221584 2.11639317784157 37 13.19 11.3536651987690 1.83633480123103 38 13.196 12.9727423565462 0.223257643453822 39 15.826 16.340720977083 -0.514720977083002 40 14.733 14.4775090737517 0.255490926248314 41 16.307 19.5018747509537 -3.19487475095367 42 15.703 15.5381437967695 0.164856203230496 43 14.589 13.8964608467448 0.69253915325517 44 12.043 13.0613905059975 -1.01839050599748 45 15.057 14.3636230692343 0.693376930765746 46 14.053 14.9013499437283 -0.84834994372828 47 12.698 14.3969400845724 -1.69894008457240 48 10.888 10.7258846941654 0.162115305834634 49 10.045 10.1852264162700 -0.140226416269973 50 11.549 10.5727629558904 0.976237044109611 51 13.767 14.1563327990870 -0.389332799086965 52 12.434 12.4719963747322 -0.0379963747322165 53 13.116 16.7472035154773 -3.63120351547726 54 14.211 13.1408618118764 1.07013818812356 55 12.266 12.0063803550454 0.259619644954562 56 12.602 10.6564505296472 1.94554947035279 57 15.714 13.6539062511671 2.06009374883288 58 13.742 14.5724271360866 -0.830427136086591 59 12.745 13.9288861337279 -1.1838861337279 60 10.491 10.8240310030629 -0.333031003062949 61 10.057 9.99576593988131 0.0612340601186876 62 10.9 10.6714182488754 0.228581751124628 63 11.771 13.6624585669115 -1.89145856691147 64 11.992 11.3197081958025 0.672291804197467 65 11.933 15.3063961280774 -3.37339612807743 66 14.504 12.6297382988172 1.87426170118276 67 11.727 11.7348830753153 -0.00788307531533405 68 11.477 10.5467820680298 0.930217931970176 69 13.578 13.0771956355327 0.500804364467331 70 11.555 12.7472486974918 -1.19224869749179 71 11.846 11.8596459251731 -0.0136459251731296 72 11.397 9.44841929888116 1.94858070111884 73 10.066 9.7688170299487 0.297182970051301 74 10.269 10.5851080689890 -0.316108068989035 75 14.279 12.9502274095353 1.32877259046466 76 13.87 12.5845853906473 1.28541460935272 77 13.695 16.1736250841754 -2.47862508417542 78 14.42 14.8398787055740 -0.419878705573966 79 11.424 12.5385497457272 -1.11454974572725 80 9.704 10.9902129166719 -1.28621291667189 81 12.464 12.3914309301873 0.0725690698126975 82 14.301 11.5603717979225 2.7406282020775 83 13.464 12.7466564260035 0.71734357399654 84 9.893 11.0292620950649 -1.13626209506491 85 11.572 9.60149309408338 1.97050690591662 86 12.38 11.1092619909836 1.27073800901636 87 16.692 14.5196279475516 2.17237205244843 88 16.052 14.5581426298284 1.49385737017159 89 16.459 17.6086504737641 -1.14965047376408 90 14.761 17.2767763721974 -2.51577637219739 91 13.654 13.8721952174728 -0.218195217472841 92 13.48 12.7319213087774 0.748078691222647 93 18.068 15.3510749175141 2.71692508248591 94 16.56 16.2613451640843 0.298654835915684 95 14.53 15.9607364858548 -1.43073648585482 96 10.65 12.9183565913382 -2.26835659133818 97 11.651 11.4999247022647 0.151075297735261 98 13.735 12.0284823762784 1.70651762372159 99 13.36 15.8063437290606 -2.44634372906059 100 17.818 13.5295266092454 4.28847339075461 101 20.613 17.4478903258188 3.16510967418117 102 16.231 18.9383066349928 -2.70730663499282 103 13.862 15.8520225188341 -1.99002251883408 104 12.004 14.0456998503392 -2.04169985033923 105 17.734 15.6807199395748 2.05328006042518 106 15.034 15.8551839135026 -0.821183913502637 107 12.609 14.7179826843134 -2.10898268431336 108 12.32 11.2001470869607 1.11985291303927 109 10.833 11.8064362583755 -0.97343625837554 110 11.35 12.0681687829837 -0.71816878298368 111 13.648 13.9700103316430 -0.322010331642952 112 14.89 13.8593944091262 1.03060559087377 113 16.325 16.0262823841779 0.298717615822106 114 18.045 15.1220414626911 2.92295853730894 115 15.616 14.8233442591850 0.792655740815043 116 11.926 14.3308440601420 -2.40484406014198 117 16.855 16.5033528068680 0.351647193131967 118 15.083 15.3724927828561 -0.289492782856074 119 12.52 14.2640544045649 -1.74405440456491 120 12.355 11.4767694426125 0.878230557387475 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 121 11.6066269208744 8.14454377072808 15.0687100710206 122 12.3762828431977 8.54038424560877 16.2121814407867 123 14.6916133920903 10.5109555553541 18.8722712288265 124 14.9728306421709 10.4697653210477 19.4758959632940 125 16.5274033384787 11.7197592248363 21.3350474521211 126 15.9361710864376 10.8385573587785 21.0337848140966 127 13.8768136227748 8.50145312115497 19.2521741243946 128 12.4466635549264 6.80395337289073 18.0893737369621 129 16.2323455480606 10.3312498360947 22.1334412600265 130 14.8205436174045 8.6688780226641 20.9722092121449 131 13.5955270380566 7.20017036162475 19.9908837144884 132 12.0902829189264 5.45733826766284 18.7232275701900

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf/155fr1291743862.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf/155fr1291743862.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf/2yxxu1291743862.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf/2yxxu1291743862.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf/3yxxu1291743862.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/07/t12917437465j7t1qir1iq2mgf/3yxxu1291743862.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Triple ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = additive ;

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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