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: Mon, 16 Aug 2010 16:54:09 +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/Aug/16/t1281977686e8xjhxghrjs0o31.htm/, Retrieved Mon, 16 Aug 2010 18:54:47 +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/Aug/16/t1281977686e8xjhxghrjs0o31.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:

Dataseries X:

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257 256 255 253 251 250 251 253 254 254 255 257 260 251 255 252 245 239 240 247 251 255 259 260 259 266 261 243 232 225 229 238 240 241 239 246 242 251 246 219 203 192 197 203 208 207 208 212 208 215 200 170 162 150 148 152 155 150 146 147 142 146 130 102 98 90 86 104 108 94 88 92 93 95 81 50 49 34 27 45 47 42 32

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 2 seconds R Server 'RServer@AstonUniversity' @ vre.aston.ac.uk Estimated Parameters of Exponential Smoothing Parameter Value alpha 1 beta 0.0285328075927933 gamma FALSE Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 3 255 255 0 4 253 254 -1 5 251 251.971467192407 -0.971467192407204 6 250 249.943748505924 0.0562514940764629 7 251 248.945353518981 2.05464648101918 8 253 250.003978351695 2.99602164830503 9 254 252.08946326093 1.9105367390701 10 254 253.143976238105 0.856023761895244 11 255 253.168400999398 1.83159900060224 12 257 254.220661661269 2.77933833873089 13 260 256.299963987323 3.70003601267661 14 251 259.40553640296 -8.40553640295951 15 255 250.16570285006 4.83429714994037 16 252 254.303638920485 -2.30363892048527 17 245 251.237909634404 -6.23790963440379 18 239 244.059924559024 -5.0599245590241 19 240 237.915550705147 2.08444929485256 20 247 238.975025895814 8.02497410418562 21 251 246.204000937866 4.79599906213375 22 255 250.340844256321 4.65915574367867 23 259 254.473783050701 4.52621694929942 24 260 258.602928728038 1.39707127196181 25 259 259.642791093834 -0.64279109383449 26 266 258.624450459232 7.37554954076825 27 261 265.83489559517 -4.8348955951696 28 243 260.696942449421 -17.6969424494214 29 232 242.191998995531 -10.1919989955313 30 225 230.901192649206 -5.90119264920588 31 229 223.732815054778 5.26718494522191 32 238 227.883102629376 10.1168973706243 33 240 237.171766115488 2.8282338845122 34 241 239.252463568742 1.747536431258 35 239 240.302325689496 -1.30232568949648 36 246 238.265166681175 7.73483331882505 37 242 245.485863192023 -3.48586319202329 38 251 241.38640172827 9.6135982717295 39 246 250.660704678032 -4.66070467803218 40 219 245.527721688207 -26.527721688207 41 203 217.770811309402 -14.7708113094023 42 192 201.349358592322 -9.34935859232164 43 197 190.082595142491 6.9174048575091 44 203 195.279968124332 7.72003187566835 45 208 201.50024230845 6.49975769154966 46 207 206.685698644063 0.314301355936919 47 208 205.694666544178 2.3053334558218 48 212 206.76044418011 5.2395558198896 49 208 210.909943418191 -2.90994341819101 50 215 206.826914562534 8.17308543746617 51 200 214.060115636761 -14.0601156367605 52 170 198.658941062564 -28.6589410625644 53 162 167.841221011413 -5.84122101141307 54 150 159.674554576187 -9.67455457618743 55 148 147.398512371919 0.601487628080918 56 152 145.415674502681 6.58432549731944 57 155 149.603543795224 5.39645620477609 58 150 152.757519841798 -2.75751984179772 59 146 147.678840058718 -1.67884005871841 60 147 143.630938038344 3.36906196165609 61 142 144.727066835064 -2.72706683506405 62 146 139.649255961766 6.35074403823353 63 130 143.83046051948 -13.8304605194805 64 102 127.435838650558 -25.4358386505584 65 98 98.7100827603807 -0.710082760380686 66 90 94.6898221056038 -4.68982210560378 67 86 86.5560083138202 -0.556008313820172 68 104 82.540143835582 21.4598561644181 69 108 101.15245378249 6.8475462175097 70 94 105.347833501197 -11.3478335011973 71 88 91.0240479513126 -3.02404795131255 72 92 84.9377633729664 7.06223662703363 73 93 89.1392688118203 3.8607311881797 74 95 90.2494263119801 4.75057368801987 75 81 92.3849735169758 -11.3849735169758 76 50 78.0601282581669 -28.0601282581669 77 49 46.2594940175475 2.74050598245251 78 34 45.3376883474517 -11.3376883474517 79 27 30.0141922672868 -3.01419226728681 80 45 22.9281888992766 22.0718111007234 81 47 41.557959638638 5.44204036136195 82 42 43.713236329181 -1.713236329181 83 32 38.6643528866395 -6.6643528866395 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 84 28.4742001879945 10.2948197337542 46.6535806422349 85 24.9484003759891 -1.13048790817408 51.0272886601522 86 21.4226005639836 -10.97174676512 53.8169478930872 87 17.8968007519781 -20.0358232738825 55.8294247778388 88 14.3710009399727 -28.6303619631845 57.3723638431299 89 10.8452011279672 -36.9107576056697 58.6011598616041 90 7.31940131596176 -44.9679320657645 59.606734697688 91 3.7936015039563 -52.8602266270973 60.4474296350099 92 0.267801691950837 -60.6275408337909 61.1631442176926 93 -3.25799812005462 -68.2984689671373 61.782472727028 94 -6.78379793206009 -75.8942506835535 62.3266548194333 95 -10.3095977440655 -83.4311154622521 62.811919974121

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Aug/16/t1281977686e8xjhxghrjs0o31/16a6d1281977645.png (open in new window) http://www.freestatistics.org/blog/date/2010/Aug/16/t1281977686e8xjhxghrjs0o31/16a6d1281977645.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Aug/16/t1281977686e8xjhxghrjs0o31/2h1ng1281977645.png (open in new window) http://www.freestatistics.org/blog/date/2010/Aug/16/t1281977686e8xjhxghrjs0o31/2h1ng1281977645.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Aug/16/t1281977686e8xjhxghrjs0o31/3h1ng1281977645.png (open in new window) http://www.freestatistics.org/blog/date/2010/Aug/16/t1281977686e8xjhxghrjs0o31/3h1ng1281977645.ps (open in new window)

Parameters (Session):

par1 = 0.1 ; par2 = 0.9 ; par3 = 0.1 ;

Parameters (R input):

par1 = 12 ; par2 = Double ; par3 = additive ;

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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