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: Mon, 07 Dec 2009 15:52:45 -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/07/t1260226433sr14819w76bkq38.htm/, Retrieved Mon, 07 Dec 2009 23:53:57 +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/07/t1260226433sr14819w76bkq38.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:

Exponential Smoothing

Dataseries X:

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220206 220115 218444 214912 210705 209673 237041 242081 241878 242621 238545 240337 244752 244576 241572 240541 236089 236997 264579 270349 269645 267037 258113 262813 267413 267366 264777 258863 254844 254868 277267 285351 286602 283042 276687 277915 277128 277103 275037 270150 267140 264993 287259 291186 292300 288186 281477 282656 280190 280408 276836 275216 274352 271311 289802 290726 292300 278506 269826 265861 269034 264176 255198 253353 246057 235372 258556 260993 254663 250643 243422 247105 248541 245039 237080 237085 225554 226839 247934 248333 246969 245098 246263 255765 264319 268347 273046 273963 267430 271993 292710 295881 293299 288576

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 0.94869489322508 beta 0.175963834108314 gamma 1 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 244752 231608.737070407 13143.2629295928 14 244576 245903.325396201 -1327.32539620149 15 241572 243364.053924208 -1792.05392420804 16 240541 242194.826493501 -1653.82649350128 17 236089 237747.699578116 -1658.69957811607 18 236997 238452.877674112 -1455.87767411213 19 264579 264064.067305385 514.932694614516 20 270349 270969.512147259 -620.512147259142 21 269645 270902.763389931 -1257.76338993129 22 267037 271021.465996317 -3984.46599631675 23 258113 262498.400155499 -4385.400155499 24 262813 259257.607244243 3555.39275575674 25 267413 267640.15118384 -227.151183839946 26 267366 265802.200089692 1563.79991030786 27 264777 263606.737937451 1170.26206254878 28 258863 263586.260963481 -4723.26096348133 29 254844 253916.724584628 927.275415371842 30 254868 255630.863548618 -762.863548617752 31 277267 282406.233486272 -5139.23348627193 32 285351 281661.788677228 3689.2113227715 33 286602 283862.58845968 2739.41154032008 34 283042 286552.969876638 -3510.96987663762 35 276687 277165.589420007 -478.589420007367 36 277915 277811.580782719 103.419217281102 37 277128 282081.067667089 -4953.06766708888 38 277103 274147.275501131 2955.72449886915 39 275037 271729.164078105 3307.83592189493 40 270150 272344.444975788 -2194.44497578795 41 267140 264574.142356461 2565.85764353914 42 264993 267479.568305000 -2486.56830500049 43 287259 292846.336443541 -5587.33644354122 44 291186 291608.149839614 -422.14983961446 45 292300 288468.751850337 3831.24814966315 46 288186 290689.352216807 -2503.35221680679 47 281477 281330.075128912 146.924871087947 48 282656 281752.933756179 903.066243821231 49 280190 285843.824962593 -5653.82496259286 50 280408 276783.244551763 3624.75544823677 51 276836 274247.327824989 2588.67217501142 52 275216 273063.196717606 2152.80328239402 53 274352 269461.337633015 4890.66236698517 54 271311 274586.638501698 -3275.63850169827 55 289802 299879.632619517 -10077.6326195168 56 290726 294123.130899924 -3397.13089992350 57 292300 287324.517490892 4975.48250910785 58 278506 289449.341261054 -10943.3412610535 59 269826 270233.965110454 -407.965110453952 60 265861 267864.81854471 -2003.81854471005 61 269034 265899.390021311 3134.60997868871 62 264176 264427.39238183 -251.392381830257 63 255198 256578.07084208 -1380.07084207979 64 253353 249372.686347478 3980.31365252161 65 246057 245949.790026124 107.20997387619 66 235372 243298.843766005 -7926.8437660053 67 258556 256173.131506351 2382.86849364921 68 260993 259995.073624063 997.926375937037 69 254663 256667.881829140 -2004.88182913963 70 250643 249270.767757247 1372.23224275277 71 243422 242496.607111015 925.392888984788 72 247105 241117.827937300 5987.1720627005 73 248541 247958.652836620 582.347163380327 74 245039 244818.258880736 220.741119264363 75 237080 238556.846896415 -1476.84689641459 76 237085 232533.710664992 4551.28933500778 77 225554 230671.340467827 -5117.34046782652 78 226839 222743.733283671 4095.26671632929 79 247934 248819.419647614 -885.419647614093 80 248333 250936.961365441 -2603.96136544063 81 246969 245160.266243138 1808.73375686168 82 245098 243276.927837336 1821.07216266438 83 246263 238711.154690835 7551.84530916536 84 255765 246658.431531802 9106.56846819786 85 264319 259583.240083798 4735.75991620217 86 268347 264185.23828359 4161.76171641017 87 273046 265592.365082251 7453.63491774903 88 273963 273840.104030974 122.895969025616 89 267430 271387.864162814 -3957.86416281387 90 271993 270032.189286958 1960.81071304181 91 292710 303696.52220238 -10986.52220238 92 295881 300566.414224755 -4685.41422475548 93 293299 296066.680917328 -2767.68091732817 94 288576 291927.205471524 -3351.20547152415 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 95 283461.88276079 275236.469642982 291687.295878599 96 284760.263821161 272429.52429436 297091.003347962 97 287916.972938466 271595.978121189 304237.967755742 98 285852.314594405 265795.920499146 305908.708689664 99 280557.074910050 256977.311817304 304136.838002797 100 277528.933204777 250236.452854669 304821.413554886 101 271010.202892415 240329.884257648 301690.521527182 102 270727.860029076 235942.025856382 305513.69420177 103 298107.574925040 255284.997019572 340930.152830509 104 303992.394017023 255614.335110779 352370.452923267 105 302952.803588062 249934.329966301 355971.277209823 106 300745.422996905 243823.839403356 357667.006590455

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/07/t1260226433sr14819w76bkq38/1d4361260226363.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/07/t1260226433sr14819w76bkq38/1d4361260226363.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/07/t1260226433sr14819w76bkq38/2z10f1260226363.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/07/t1260226433sr14819w76bkq38/2z10f1260226363.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/07/t1260226433sr14819w76bkq38/3cdwr1260226363.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/07/t1260226433sr14819w76bkq38/3cdwr1260226363.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')

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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