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 12:29:39 -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/t1259955026c64xu2glnn6agsz.htm/, Retrieved Fri, 04 Dec 2009 20:30:30 +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/2009/Dec/04/t1259955026c64xu2glnn6agsz.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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7.3 7.6 7.5 7.6 7.9 7.9 8.1 8.2 8 7.5 6.8 6.5 6.6 7.6 8 8.1 7.7 7.5 7.6 7.8 7.8 7.8 7.5 7.5 7.1 7.5 7.5 7.6 7.7 7.7 7.9 8.1 8.2 8.2 8.2 7.9 7.3 6.9 6.6 6.7 6.9 7 7.1 7.2 7.1 6.9 7 6.8 6.4 6.7 6.6 6.4 6.3 6.2 6.5 6.8 6.8 6.4 6.1 5.8 6.1 7.2 7.3 6.9 6.1 5.8 6.2 7.1 7.7 7.9 7.7 7.4 7.5

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 1 beta 0 gamma 1 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 6.6 6.60287968271151 -0.00287968271150874 14 7.6 7.63453373333942 -0.0345337333394165 15 8 8.02308015304527 -0.0230801530452709 16 8.1 8.09173264343856 0.0082673565614435 17 7.7 7.65374855683443 0.0462514431655654 18 7.5 7.42652453687695 0.0734754631230476 19 7.6 8.13654148519345 -0.536541485193451 20 7.8 7.71990832193341 0.0800916780665908 21 7.8 7.58505231302218 0.214947686977819 22 7.8 7.26887694179002 0.531123058209976 23 7.5 7.05725778766112 0.442742212338883 24 7.5 7.18983318049289 0.310166819507109 25 7.1 7.65028232418406 -0.550282324184056 26 7.5 8.213173299187 -0.713173299186991 27 7.5 7.91746455154151 -0.417464551541512 28 7.6 7.58576499145864 0.0142350085413643 29 7.7 7.18107612474252 0.518923875257484 30 7.7 7.42652453687695 0.273475463123048 31 7.9 8.3536149252818 -0.453614925281800 32 8.1 8.02479047681556 0.0752095231844354 33 8.2 7.87692627653793 0.323073723462064 34 8.2 7.64181536949755 0.558184630502446 35 8.2 7.41932730833437 0.78067269166563 36 7.9 7.86116156867636 0.0388384313236427 37 7.3 8.05845907272605 -0.75845907272605 38 6.9 8.44462912552602 -1.54462912552602 39 6.6 7.28377094251895 -0.683770942518955 40 6.7 6.67502321789478 0.0249767821052238 41 6.9 6.33026574697706 0.569734253022936 42 7 6.65458219350604 0.345417806493963 43 7.1 7.59385788497258 -0.493857884972579 44 7.2 7.21177139712982 -0.0117713971298175 45 7.1 7.00130438599067 0.098695614009328 46 6.9 6.61623469330184 0.283765306698156 47 7 6.2426013661463 0.757398633853702 48 6.8 6.71031290321899 0.089687096781014 49 6.4 6.93597301423556 -0.535973014235562 50 6.7 7.40307790700039 -0.703077907000387 51 6.6 7.07253973951144 -0.472539739511437 52 6.4 6.67502321789478 -0.275023217894776 53 6.3 6.04666228772191 0.253337712278086 54 6.2 6.07562543597785 0.124374564022149 55 6.5 6.72556412461918 -0.225564124619183 56 6.8 6.60200708736551 0.197992912634492 57 6.8 6.612139101303 0.187860898697001 58 6.4 6.3365308725212 0.0634691274788057 59 6.1 5.79001446530473 0.309985534695268 60 5.8 5.84717640412596 -0.0471764041259561 61 6.1 5.91553114288058 0.184468857119423 62 7.2 7.05589416749184 0.144105832508158 63 7.3 7.60061774703023 -0.300617747030233 64 6.9 7.38337793066667 -0.483377930666666 65 6.1 6.51933471981383 -0.419334719813833 66 5.8 5.88263985013512 -0.0826398501351218 67 6.2 6.29141724444248 -0.0914172444424848 68 7.1 6.29712493248335 0.802875067516647 69 7.7 6.90401306481875 0.795986935181247 70 7.9 7.17564233486314 0.724357665136859 71 7.7 7.14777516782943 0.552224832170569 72 7.4 7.38164129140245 0.0183587085975470 73 7.5 7.54823813704856 -0.0482381370485570 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 74 8.67608495186505 7.78736342894481 9.5648064747853 75 9.15959374764899 7.86698028629065 10.4522072090073 76 9.26516649493567 7.68377542517103 10.8465575647003 77 8.75523271874095 7.01605730324904 10.4944081342329 78 8.44474805901014 6.5457718943849 10.3437242236354 79 9.16194004112406 6.91741263587784 11.4064674463703 80 9.30726714038216 6.85919941098014 11.7553348697842 81 9.05149242749017 6.50932908998186 11.5936557649985 82 8.4357009872802 5.90438508574616 10.9670168888142 83 7.63267766705125 5.17507846799298 10.0902768661095 84 7.31707644387783 4.79815219424181 9.83600069351385 85 7.46361946825984 -216.657454182507 231.584693119026

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/04/t1259955026c64xu2glnn6agsz/1790a1259954975.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259955026c64xu2glnn6agsz/1790a1259954975.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259955026c64xu2glnn6agsz/2b9n61259954975.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259955026c64xu2glnn6agsz/2b9n61259954975.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259955026c64xu2glnn6agsz/34wl01259954975.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259955026c64xu2glnn6agsz/34wl01259954975.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')

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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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