Paper - ES

*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: Tue, 14 Dec 2010 10:21:18 +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/Dec/14/t1292321955bf89juhfjju8r97.htm/, Retrieved Tue, 14 Dec 2010 11:19:19 +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/2010/Dec/14/t1292321955bf89juhfjju8r97.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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235.1 280.7 264.6 240.7 201.4 240.8 241.1 223.8 206.1 174.7 203.3 220.5 299.5 347.4 338.3 327.7 351.6 396.6 438.8 395.6 363.5 378.8 357 369 464.8 479.1 431.3 366.5 326.3 355.1 331.6 261.3 249 205.5 235.6 240.9 264.9 253.8 232.3 193.8 177 213.2 207.2 180.6 188.6 175.4 199 179.6 225.8 234 200.2 183.6 178.2 203.2 208.5 191.8 172.8 148 159.4 154.5 213.2 196.4 182.8 176.4 153.6 173.2 171 151.2 161.9 157.2 201.7 236.4 356.1 398.3 403.7 384.6 365.8 368.1 367.9 347 343.3 292.9 311.5 300.9 366.9 356.9 329.7 316.2 269 289.3 266.2 253.6 233.8 228.4 253.6 260.1 306.6 309.2 309.5 271 279.9 317.9 298.4 246.7 227.3 209.1

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 'George Udny Yule' @ 72.249.76.132 Estimated Parameters of Exponential Smoothing Parameter Value alpha 0.931014256544288 beta 0.206391957488703 gamma 1 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 299.5 239.394497863248 60.105502136752 14 347.4 351.546379551372 -4.14637955137209 15 338.3 347.761267230222 -9.46126723022172 16 327.7 334.364066404668 -6.66406640466812 17 351.6 356.944740044102 -5.34474004410174 18 396.6 403.143379598236 -6.54337959823573 19 438.8 386.106233025412 52.6937669745876 20 395.6 440.036691546718 -44.4366915467175 21 363.5 394.211126862163 -30.7111268621633 22 378.8 340.717163831117 38.0828361688831 23 357 415.401624919934 -58.4016249199345 24 369 374.768913175063 -5.76891317506329 25 464.8 445.996743604339 18.8032563956606 26 479.1 503.369179174461 -24.2691791744608 27 431.3 464.722125847542 -33.4221258475416 28 366.5 408.845143482560 -42.3451434825595 29 326.3 371.076136331253 -44.7761363312532 30 355.1 345.6828958946 9.4171041053998 31 331.6 315.860575405554 15.7394245944456 32 261.3 289.853347712182 -28.5533477121820 33 249 223.982266884870 25.0177331151295 34 205.5 202.046980971670 3.4530190283304 35 235.6 206.108794757452 29.4912052425480 36 240.9 236.099672258271 4.80032774172892 37 264.9 306.056870999756 -41.1568709997564 38 253.8 280.306744485138 -26.506744485138 39 232.3 214.187661657891 18.1123383421087 40 193.8 190.819583638426 2.98041636157421 41 177 188.936240573508 -11.9362405735080 42 213.2 198.02091035018 15.1790896498201 43 207.2 175.271356359930 31.9286436400704 44 180.6 165.663899251459 14.9361007485408 45 188.6 156.7173676347 31.8826323652999 46 175.4 153.744471659366 21.6555283406338 47 199 194.105755786364 4.89424421363597 48 179.6 212.323203513490 -32.7232035134896 49 225.8 249.794787152436 -23.9947871524365 50 234 249.950934783951 -15.9509347839508 51 200.2 207.683360947197 -7.48336094719704 52 183.6 165.468941877368 18.131058122632 53 178.2 185.600786619152 -7.40078661915223 54 203.2 210.588865862496 -7.38886586249606 55 208.5 173.457450296597 35.0425497034026 56 191.8 171.648936288595 20.1510637114047 57 172.8 175.800849222989 -3.00084922298859 58 148 140.016585341255 7.98341465874466 59 159.4 164.236672863252 -4.83667286325192 60 154.5 166.673623033259 -12.1736230332594 61 213.2 223.702169031232 -10.5021690312323 62 196.4 239.39057988577 -42.9905798857699 63 182.8 169.752613061597 13.0473869384029 64 176.4 149.584465280711 26.8155347192886 65 153.6 178.873926475988 -25.2739264759875 66 173.2 186.621865566447 -13.4218655664472 67 171 145.040725366441 25.9592746335586 68 151.2 130.242787571883 20.9572124281166 69 161.9 130.197524143635 31.7024758563652 70 157.2 130.798125576262 26.4018744237381 71 201.7 178.13865380437 23.5613461956299 72 236.4 218.822203796322 17.577796203678 73 356.1 321.695685829852 34.4043141701481 74 398.3 403.611023511471 -5.31102351147098 75 403.7 406.818941593635 -3.1189415936347 76 384.6 403.342954976271 -18.7429549762708 77 365.8 408.662580727233 -42.8625807272334 78 368.1 419.512325649181 -51.4123256491812 79 367.9 356.637719841075 11.2622801589251 80 347 336.346972645868 10.6530273541316 81 343.3 334.005020078399 9.29497992160105 82 292.9 315.627954770634 -22.7279547706338 83 311.5 309.841166936280 1.65883306372035 84 300.9 318.320944264907 -17.420944264907 85 366.9 371.646305877894 -4.74630587789386 86 356.9 388.724539548477 -31.8245395484767 87 329.7 336.657017800339 -6.95701780033909 88 316.2 297.050191389958 19.1498086100416 89 269 311.786148591763 -42.7861485917631 90 289.3 297.933467787587 -8.633467787587 91 266.2 263.246590625195 2.95340937480455 92 253.6 217.617902786234 35.9820972137661 93 233.8 226.070834134544 7.72916586545617 94 228.4 191.032815136388 37.3671848636118 95 253.6 241.431279955095 12.1687200449046 96 260.1 258.952674786756 1.14732521324379 97 306.6 334.580689426555 -27.9806894265545 98 309.2 327.835752397094 -18.6357523970942 99 309.5 291.973343988169 17.5266560118306 100 271 283.877458081835 -12.8774580818352 101 279.9 265.284004888885 14.6159951111154 102 317.9 319.020759041934 -1.12075904193443 103 298.4 305.362418845651 -6.96241884565143 104 246.7 264.109866220531 -17.4098662205309 105 227.3 221.975001427004 5.32499857299592 106 209.1 187.351231640874 21.7487683591257 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 107 219.077216722963 170.752129011988 267.402304433938 108 219.777597650389 147.104849866391 292.450345434387 109 287.376112509797 190.827733352295 383.924491667299 110 307.750955103168 186.80240432511 428.699505881226 111 295.739011578135 149.53822630925 441.93979684702 112 269.865917216459 97.4350712451952 442.296763187724 113 268.270479181712 68.5852591143745 467.955699249049 114 307.617663811998 79.6428952150977 535.592432408898 115 295.118875171702 37.8257063089802 552.412044034423 116 261.484664602242 -26.1409255037481 549.110254708233 117 242.329343696241 -76.6237331483823 561.282420540865 118 208.060039970237 -143.194830840926 559.3149107814

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/14/t1292321955bf89juhfjju8r97/1p3xj1292322075.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/14/t1292321955bf89juhfjju8r97/1p3xj1292322075.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/14/t1292321955bf89juhfjju8r97/2idx41292322075.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/14/t1292321955bf89juhfjju8r97/2idx41292322075.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/14/t1292321955bf89juhfjju8r97/3idx41292322075.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/14/t1292321955bf89juhfjju8r97/3idx41292322075.ps (open in new window)

Parameters (Session):

par1 = 12 ; par2 = Triple ; par3 = additive ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; 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')

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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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