Exponential Smoothing

*Unverified author*

R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)

Title produced by software: Exponential Smoothing

Date of computation: Sun, 11 Jul 2010 16:01:25 +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/Jul/11/t1278864170rpzuyid9690txa8.htm/, Retrieved Sun, 11 Jul 2010 18:02:54 +0200

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/Jul/11/t1278864170rpzuyid9690txa8.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:

Febiri Lordina

Dataseries X:

» Textbox « » Textfile « » CSV «

213 212 211 209 229 228 213 203 204 204 205 207 205 208 201 201 220 218 203 185 179 182 182 185 183 192 177 172 188 182 162 150 141 135 139 148 142 156 143 134 146 142 117 106 104 99 105 106 96 104 96 85 91 98 73 70 62 60 70 82 72 73 68 53 61 73 46 50 52 45 58 73 58 49 44 35 46 61 29 33 37 31 44 57 42 34 27 22 30 47 12 13 18 11 26 41 21 24 30 34 48 64 35 44 55 53 73 94 73 78 87 87 91 104 73 84 103 111 131 155

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 0.575651073368057 beta 0.0869583411263661 gamma 0.402742687472037 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 205 209.192478598406 -4.19247859840641 14 208 209.632082374054 -1.63208237405388 15 201 202.032536769601 -1.03253676960068 16 201 201.916846630005 -0.916846630004585 17 220 220.811708811239 -0.81170881123856 18 218 218.685886150255 -0.685886150254959 19 203 198.712804324687 4.28719567531294 20 185 190.858097501976 -5.85809750197632 21 179 187.294217200712 -8.29421720071221 22 182 181.112639390593 0.88736060940741 23 182 181.100159314246 0.899840685754185 24 185 182.073315385259 2.92668461474102 25 183 180.230383720094 2.76961627990596 26 192 184.175768148198 7.82423185180241 27 177 182.689366602584 -5.68936660258356 28 172 179.543655784908 -7.54365578490805 29 188 191.372081448318 -3.37208144831806 30 182 187.116246159165 -5.11624615916494 31 162 167.284067282252 -5.28406728225201 32 150 152.944275949712 -2.94427594971205 33 141 149.328674507793 -8.32867450779273 34 135 143.138082233170 -8.1380822331703 35 139 135.997513541148 3.00248645885151 36 148 136.348949267036 11.6510507329643 37 142 138.832755683937 3.16724431606301 38 156 141.655536448269 14.3444635517309 39 143 142.369424298934 0.630575701065709 40 134 141.842565480813 -7.84256548081308 41 146 149.737292151698 -3.73729215169757 42 142 144.542665587422 -2.54266558742177 43 117 129.013747026312 -12.0137470263116 44 106 112.682713645966 -6.6827136459658 45 104 105.194591570029 -1.19459157002912 46 99 102.169049253627 -3.16904925362729 47 105 98.7446736891192 6.25532631088082 48 106 101.366485883294 4.63351411670573 49 96 98.7245163533632 -2.72451635336317 50 104 97.4367811841935 6.56321881580647 51 96 92.889276171447 3.11072382855301 52 85 91.4800289150223 -6.48002891502232 53 91 94.2741655201472 -3.27416552014725 54 98 88.5812307281074 9.41876927189264 55 73 82.3381903051804 -9.33819030518038 56 70 70.1580679263133 -0.158067926313279 57 62 67.2101684585941 -5.21016845859411 58 60 61.102021148952 -1.10202114895200 59 70 58.9448470669737 11.0551529330263 60 82 63.4252368036296 18.5747631963704 61 72 69.4333869408486 2.56661305915139 62 73 72.5449129311942 0.455087068805838 63 68 66.4581167395394 1.54188326046064 64 53 63.8455425495228 -10.8455425495228 65 61 61.7379184546695 -0.737918454669504 66 73 59.6693204883252 13.3306795116748 67 46 56.7766086813169 -10.7766086813169 68 50 46.7232795210231 3.27672047897689 69 52 45.9089701530545 6.09102984694554 70 45 48.2520618502894 -3.25206185028943 71 58 47.2614130239069 10.7385869760931 72 73 53.4153583517955 19.5846416482045 73 58 59.525784897736 -1.52578489773599 74 49 60.572620874895 -11.572620874895 75 44 49.4320353622459 -5.43203536224589 76 35 41.9761871672008 -6.97618716720078 77 46 41.5540333916603 4.4459666083397 78 61 44.4422974261529 16.5577025738471 79 29 42.7666858290369 -13.7666858290369 80 33 33.7033191132543 -0.703319113254338 81 37 31.4106339444663 5.58936605553374 82 31 32.475571950384 -1.47557195038399 83 44 33.5162528869359 10.4837471130641 84 57 40.1635473865073 16.8364526134927 85 42 43.5059419744658 -1.50594197446578 86 34 42.6721193844814 -8.67211938448144 87 27 35.1708476984624 -8.17084769846244 88 22 27.0296209699263 -5.02962096992629 89 30 27.3511959517069 2.64880404829310 90 47 29.6164918695055 17.3835081304945 91 12 27.7540167311256 -15.7540167311256 92 13 18.6342160669002 -5.63421606690021 93 18 13.7716270638491 4.22837293615093 94 11 13.5886758044485 -2.58867580444850 95 26 11.9705218550925 14.0294781449075 96 41 19.6406741559448 21.3593258440552 97 21 25.9952352375190 -4.99523523751904 98 24 21.9305444788722 2.06945552112780 99 30 21.4364281173417 8.56357188265828 100 34 24.6424380278853 9.3575619721147 101 48 38.7865367504351 9.21346324956492 102 64 52.083562991078 11.916437008922 103 35 34.7471509256928 0.252849074307171 104 44 42.8344527948753 1.16554720512467 105 55 51.4261390729795 3.57386092702050 106 53 48.2597099385473 4.74029006145269 107 73 72.1526172665678 0.847382733432212 108 94 79.3749216267448 14.6250783732552 109 73 65.2423194591332 7.75768054086677 110 78 74.8683241257462 3.13167587425376 111 87 78.948350674488 8.05164932551202 112 87 81.6328093937419 5.36719060625806 113 91 110.822142408706 -19.8221424087059 114 104 118.864830003997 -14.8648300039972 115 73 62.9584714041431 10.0415285958569 116 84 85.1059561337918 -1.10595613379181 117 103 100.885862847317 2.11413715268318 118 111 92.696727952884 18.3032720471161 119 131 144.705635126675 -13.7056351266745 120 155 153.549940015822 1.4500599841777 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 121 113.160167846310 100.882829002242 125.437506690379 122 119.191249417548 103.011611990825 135.370886844272 123 122.860746197940 103.174515090349 142.546977305532 124 118.070439283392 96.0733979277513 140.067480639033 125 145.862307720869 116.524862847368 175.199752594370 126 175.438910078245 137.800668843947 213.077151312543 127 104.957803906518 78.905076171792 131.010531641243 128 125.755668064754 92.3688327937673 159.142503335740 129 150.286273132161 108.078689547376 192.493856716946 130 139.298549589954 97.5094619666208 181.087637213287 131 184.209649024615 126.568799455319 241.850498593910 132 209.099935895541 141.799453764276 276.400418026806

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Jul/11/t1278864170rpzuyid9690txa8/1r9541278864082.png (open in new window) http://www.freestatistics.org/blog/date/2010/Jul/11/t1278864170rpzuyid9690txa8/1r9541278864082.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Jul/11/t1278864170rpzuyid9690txa8/2104p1278864082.png (open in new window) http://www.freestatistics.org/blog/date/2010/Jul/11/t1278864170rpzuyid9690txa8/2104p1278864082.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Jul/11/t1278864170rpzuyid9690txa8/3104p1278864082.png (open in new window) http://www.freestatistics.org/blog/date/2010/Jul/11/t1278864170rpzuyid9690txa8/3104p1278864082.ps (open in new window)

Parameters (Session):

par1 = multiplicative ; par2 = 12 ;

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=F, beta=F) if (par2 == 'Double') fit <- HoltWinters(x, gamma=F) 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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