exponential smooting US Airlines

*Unverified author*

R Software Module: /rwasp_exponentialsmoothing.wasp (opens new window with default values)

Title produced by software: Exponential Smoothing

Date of computation: Thu, 19 May 2011 08:27:12 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq.htm/, Retrieved Thu, 19 May 2011 10:23:43 +0200

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

KDGP2W102

Dataseries X:

» Textbox « » Textfile « » CSV «

6827 6178 7084 8162 8462 9644 10466 10748 9963 8194 6848 7027 7269 6775 7819 8371 9069 10248 11030 10882 10333 9109 7685 7602 8350 7829 8829 9948 10638 11253 11424 11391 10665 9396 7775 7933 8186 7444 8484 9948 10252 12282 11637 11577 12417 9637 8094 9280 8334 7899 9994 10078 10801 12950 12222 12246 13281 10366 8730 9614 8639 8772 10894 10455 11179 10588 10794 12770 13812 10857 9290 10925 9491 8919 11607 8852 12537 14759 13667 13731 15110 12185 10645 12161 10840 10436 13589 13402 13103 14933 14147 14057 16234 12389 11595 12772

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 0.120618354224454 beta 0 gamma 0.444031255824324 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 7269 7028.04555893714 240.954441062862 14 6775 6589.60133762837 185.398662371631 15 7819 7651.76756439805 167.232435601953 16 8371 8204.9718412555 166.028158744501 17 9069 8881.74618834833 187.253811651666 18 10248 10045.4986564522 202.501343547778 19 11030 10965.4438240899 64.5561759100929 20 10882 11266.6213746771 -384.621374677072 21 10333 10383.7901131434 -50.7901131433737 22 9109 8537.60367918035 571.396320819646 23 7685 7199.13792787709 485.862072122906 24 7602 7439.3413446404 162.658655359601 25 8350 7811.81173514451 538.188264855487 26 7829 7334.10759647993 494.892403520066 27 8829 8531.70250674818 297.297493251821 28 9948 9152.55924664116 795.440753358836 29 10638 9984.98032679143 653.019673208571 30 11253 11343.2792317032 -90.27923170318 31 11424 12265.7510932665 -841.751093266523 32 11391 12288.1771235432 -897.177123543206 33 10665 11397.9005365729 -732.900536572924 34 9396 9559.71015530468 -163.710155304678 35 7775 7978.7111309595 -203.711130959495 36 7933 8010.66540394782 -77.6654039478235 37 8186 8528.28931383278 -342.289313832784 38 7444 7896.39580195039 -452.395801950387 39 8484 8936.3393618554 -452.339361855402 40 9948 9674.77925484802 273.220745151981 41 10252 10396.4340258043 -144.434025804298 42 12282 11373.7551472545 908.24485274554 43 11637 12130.112944568 -493.112944567951 44 11577 12170.7395779461 -593.739577946071 45 12417 11361.4221282407 1055.5778717593 46 9637 9894.75711420503 -257.757114205033 47 8094 8220.57368637548 -126.573686375483 48 9280 8313.93932571252 966.060674287484 49 8334 8875.68319024482 -541.683190244825 50 7899 8138.88375700173 -239.883757001729 51 9994 9264.54938107381 729.450618926187 52 10078 10505.2105270501 -427.210527050149 53 10801 11010.4108378166 -209.410837816557 54 12950 12474.1971982292 475.802801770838 55 12222 12632.9488724434 -410.948872443438 56 12246 12643.2223261401 -397.222326140136 57 13281 12487.834995547 793.165004452952 58 10366 10353.0782793023 12.92172069765 59 8730 8664.05338292564 65.9466170743617 60 9614 9233.23078732424 380.769212675757 61 8639 9124.33509691525 -485.335096915247 62 8772 8479.89391555331 292.106084446692 63 10894 10138.247229165 755.752770834988 64 10455 10969.9547417632 -514.954741763173 65 11179 11585.3784954611 -406.378495461136 66 10588 13390.9502407925 -2802.95024079251 67 10794 12799.4909429901 -2005.4909429901 68 12770 12622.3394092947 147.660590705334 69 13812 12999.9830083534 812.016991646606 70 10857 10521.9073662575 335.092633742499 71 9290 8859.06189099036 430.938109009641 72 10925 9610.27776690527 1314.72223309473 73 9491 9255.43619720476 235.563802795235 74 8919 8986.46215097725 -67.4621509772496 75 11607 10851.0505378656 755.949462134431 76 8852 11193.4179729864 -2341.41797298642 77 12537 11643.1123415451 893.887658454852 78 14759 12668.9864990244 2090.01350097557 79 13667 12963.5412770187 703.458722981279 80 13731 14046.1055865017 -315.105586501653 81 15110 14682.9948203013 427.005179698723 82 12185 11695.7616546418 489.238345358208 83 10645 9918.86907658675 726.130923413251 84 12161 11144.1293323947 1016.8706676053 85 10840 10238.667075591 601.332924409033 86 10436 9850.52478860235 585.475211397654 87 13589 12345.5555433791 1243.44445662094 88 13402 11399.3041205281 2002.69587947186 89 13103 13953.0351718078 -850.035171807769 90 14933 15410.2747374316 -477.274737431553 91 14147 14787.2270003629 -640.227000362882 92 14057 15366.6920823981 -1309.69208239809 93 16234 16259.4499723878 -25.4499723878071 94 12389 12962.7672731057 -573.767273105737 95 11595 11006.1988737279 588.801126272117 96 12772 12416.299652881 355.700347119044 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 97 11183.3518537012 10399.4703321603 11967.2333752422 98 10684.3981813208 9880.96733643282 11487.8290262088 99 13498.4781653489 12651.1667452774 14345.7895854204 100 12648.2946440496 11791.2269120756 13505.3623760236 101 13864.9636527499 12971.2709261263 14758.6563793735 102 15611.950569055 14668.758547805 16555.1425903049 103 14961.5698703663 14014.6058189549 15908.5339217777 104 15363.3274416976 14390.692677899 16335.9622054962 105 16983.3549338981 15954.4967268903 18012.2131409059 106 13313.3217716288 12361.2244137956 14265.4191294619 107 11803.5877342728 10871.244680323 12735.9307882226 108 13104.0961915316 12508.9289219767 13699.2634610865

Charts produced by software:

http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq/1uaat1305793628.png (open in new window) http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq/1uaat1305793628.ps (open in new window) http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq/2zyid1305793628.png (open in new window) http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq/2zyid1305793628.ps (open in new window) http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq/3aj1h1305793628.png (open in new window) http://www.freestatistics.org/blog/date/2011/May/19/t1305793422gcgqtxxo271q6tq/3aj1h1305793628.ps (open in new window)

Parameters (Session):

par1 = 48 ; par2 = 1 ; par3 = 1 ; par4 = 0 ; par5 = 12 ; par6 = White Noise ; par7 = 0.95 ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ; par4 = 0 ; par5 = 12 ; par6 = White Noise ; par7 = 0.95 ;

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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