WS 8: 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, 04 Dec 2009 16:46:53 -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/05/t1259970471n5v5bqgx9oyn6op.htm/, Retrieved Sat, 05 Dec 2009 00:47:55 +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/05/t1259970471n5v5bqgx9oyn6op.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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114 116 153 162 161 149 139 135 130 127 122 117 112 113 149 157 157 147 137 132 125 123 117 114 111 112 144 150 149 134 123 116 117 111 105 102 95 93 124 130 124 115 106 105 105 101 95 93 84 87 116 120 117 109 105 107 109 109 108 107

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 gamma 0.432015235872734 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 112 113.387857793709 -1.38785779370852 14 113 112.927073783159 0.0729262168412532 15 149 149.043505980836 -0.0435059808360734 16 157 157.095249460890 -0.095249460890301 17 157 157.095519289391 -0.0955192893914898 18 147 147.043274160734 -0.043274160733688 19 137 135.316709557186 1.68329044281415 20 132 132.968509499453 -0.968509499452836 21 125 127.102166225199 -2.1021662251992 22 123 122.177831576290 0.822168423710437 23 117 118.220013167857 -1.22001316785700 24 114 112.157090642521 1.84290935747875 25 111 109.017543423488 1.98245657651211 26 112 111.916570237376 0.0834297626242346 27 144 147.721620291083 -3.72162029108301 28 150 151.811910439922 -1.81191043992195 29 149 150.075689261063 -1.07568926106282 30 134 139.533919002382 -5.53391900238236 31 123 123.323179263295 -0.323179263294890 32 116 119.350423394794 -3.35042339479357 33 117 111.661430062584 5.33856993741581 34 111 114.340650102666 -3.34065010266555 35 105 106.660214202392 -1.66021420239208 36 102 100.627396667060 1.37260333293960 37 95 97.5160407698656 -2.51604076986555 38 93 95.7485135048482 -2.74851350484822 39 124 122.605792185775 1.39420781422534 40 130 130.678554356048 -0.678554356048267 41 124 130.019032037267 -6.01903203726678 42 115 116.067184132535 -1.06718413253455 43 106 105.794173449147 0.205826550853416 44 105 102.814175981993 2.18582401800698 45 105 101.045923950786 3.95407604921364 46 101 102.584877892230 -1.58487789222956 47 95 97.027048397838 -2.02704839783799 48 93 91.0193183541763 1.98068164582367 49 84 88.8899137796488 -4.88991377964878 50 87 84.6329745012355 2.36702549876446 51 116 114.674478047256 1.32552195274377 52 120 122.225211922499 -2.22521192249881 53 117 119.990703425369 -2.99070342536878 54 109 109.496498368977 -0.496498368977143 55 105 100.258697928889 4.74130207111078 56 107 101.841455545946 5.15854445405408 57 109 102.976015971113 6.02398402888674 58 109 106.503468629042 2.49653137095846 59 108 104.733581041481 3.26641895851873 60 107 103.509820160926 3.49017983907439 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 61 102.308333542208 96.819429944689 107.797237139727 62 103.133610463014 95.3302552717974 110.936965654231 63 136.001266842365 124.316650915753 147.685882768978 64 143.359906637913 129.848042135511 156.871771140314 65 143.416785436193 128.797127305835 158.036443566552 66 134.293126366769 119.51308930069 149.073163432849 67 123.593611567894 108.894567638771 138.292655497017 68 119.927841497958 104.612405020897 135.243277975019 69 115.451977828743 99.6855754390117 131.218380218474 70 112.824133767499 96.4323023235536 129.215965211444 71 108.417432505592 91.7001157424877 125.134749268697 72 103.910892581333 81.0390277227304 126.782757439935

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/05/t1259970471n5v5bqgx9oyn6op/13le51259970411.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/05/t1259970471n5v5bqgx9oyn6op/13le51259970411.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/05/t1259970471n5v5bqgx9oyn6op/2kbqg1259970411.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/05/t1259970471n5v5bqgx9oyn6op/2kbqg1259970411.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/05/t1259970471n5v5bqgx9oyn6op/3cg0q1259970411.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/05/t1259970471n5v5bqgx9oyn6op/3cg0q1259970411.ps (open in new window)

Parameters (Session):

par1 = FALSE ; par2 = -0.5 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 2 ; par9 = 1 ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ; par4 = 1 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 2 ; par9 = 1 ;

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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