ws 9 Exponentional 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 08:41:39 -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/04/t1259941359er0owouh9lj8nb2.htm/, Retrieved Fri, 04 Dec 2009 16:42:43 +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/04/t1259941359er0owouh9lj8nb2.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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9,3 9,3 8,7 8,2 8,3 8,5 8,6 8,5 8,2 8,1 7,9 8,6 8,7 8,7 8,5 8,4 8,5 8,7 8,7 8,6 8,5 8,3 8 8,2 8,1 8,1 8 7,9 7,9 8 8 7,9 8 7,7 7,2 7,5 7,3 7 7 7 7,2 7,3 7,1 6,8 6,4 6,1 6,5 7,7 7,9 7,5 6,9 6,6 6,9 7,7 8 8 7,7 7,3 7,4 8,1 8,3

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 0.916379251616883 gamma 0.00180441050470130 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 8.7 8.68099051787152 0.0190094821284745 14 8.7 8.7154941714524 -0.0154941714523904 15 8.5 8.49290955300777 0.00709044699222972 16 8.4 8.39529785596176 0.0047021440382391 17 8.5 8.5079238029641 -0.00792380296410578 18 8.7 8.7262674174135 -0.0262674174135054 19 8.7 8.59795356137622 0.102046438623784 20 8.6 8.73231590394013 -0.132315903940132 21 8.5 8.29187468279727 0.208125317202725 22 8.3 8.54770867588768 -0.247708675887676 23 8 8.00543315104749 -0.00543315104748565 24 8.2 8.60627807175544 -0.406278071755439 25 8.1 7.82075295426395 0.279247045736052 26 8.1 7.88579241130282 0.214207588697183 27 8 7.88854939448731 0.111450605512690 28 7.9 7.977636400544 -0.0776364005440007 29 7.9 8.00250399280299 -0.102503992802985 30 8 8.02218989935545 -0.0221898993554515 31 8 7.82089820754594 0.179101792454064 32 7.9 8.0226269351786 -0.122626935178609 33 8 7.60920432506505 0.390795674934951 34 7.7 8.21451830560136 -0.514518305601364 35 7.2 7.3398493281955 -0.139849328195498 36 7.5 7.51467812018654 -0.0146781201865442 37 7.3 7.2514634582626 0.0485365417373931 38 7 7.01303738473855 -0.0130373847385457 39 7 6.54118278383354 0.458817216166462 40 7 7.03475850042076 -0.03475850042076 41 7.2 7.17621141072429 0.0237885892757124 42 7.3 7.5084592509359 -0.208459250935899 43 7.1 7.16101698409946 -0.0610169840994583 44 6.8 6.94161668171015 -0.141616681710153 45 6.4 6.34662004678788 0.053379953212116 46 6.1 6.13380440161523 -0.0338044016152281 47 6.5 5.72432220097245 0.775677799027552 48 7.7 7.60251798562167 0.0974820143783273 49 7.9 8.41426891467136 -0.514268914671362 50 7.5 8.05704408252355 -0.557044082523548 51 6.9 6.97615247416296 -0.0761524741629582 52 6.6 6.40106699065521 0.198933009344787 53 6.9 6.44773968975057 0.452260310249427 54 7.7 7.27099811009575 0.429001889904248 55 8 8.20412774118276 -0.204127741182761 56 8 8.34961669172671 -0.349616691726709 57 7.7 7.82129410731772 -0.121294107317724 58 7.3 7.5684626041225 -0.268462604122502 59 7.4 6.82957093029714 0.570429069702856 60 8.1 8.30774672381223 -0.207746723812228 61 8.3 8.26043686603904 0.0395631339609626 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 62 8.40633507206294 7.87985281273271 8.93281733139316 63 8.30859843729656 7.19277501688213 9.42442185771099 64 8.30079764425668 6.4782385666954 10.1233567218179 65 8.49850130618337 5.81544076444694 11.1815618479198 66 8.82526110627407 5.11383304556625 12.5366891669819 67 8.84518589433897 4.13424361576123 13.5561281729167 68 8.9062678635162 3.10947880022750 14.7030569268049 69 8.73461287818849 1.96800380169636 15.5012219546806 70 8.73539125662888 0.842315961569622 16.6284665516881 71 8.60729026555166 -0.319642553957442 17.5342230850608 72 9.4649336707083 -1.65416294248618 20.5840302839028 73 9.67545508936314 -3.94744934482535 23.2983595235516

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/04/t1259941359er0owouh9lj8nb2/1b08r1259941297.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259941359er0owouh9lj8nb2/1b08r1259941297.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259941359er0owouh9lj8nb2/2scml1259941297.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259941359er0owouh9lj8nb2/2scml1259941297.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259941359er0owouh9lj8nb2/32kn01259941297.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259941359er0owouh9lj8nb2/32kn01259941297.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')

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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