Exponential smoothing multiplicative

*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, 28 Dec 2010 11:57:54 +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/28/t1293537374pzq8okgz5ms92gc.htm/, Retrieved Tue, 28 Dec 2010 12:56:15 +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/28/t1293537374pzq8okgz5ms92gc.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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621 587 655 517 646 657 382 345 625 654 606 510 614 647 580 614 636 388 356 639 753 611 639 630 586 695 552 619 681 421 307 754 690 644 643 608 651 691 627 634 731 475 337 803 722 590 724 627 696 825 677 656 785 412 352 839 729 696 641 695 638 762 635 721 854 418 367 824 687 601 676 740 691 683 594 729 731 386 331 706 715 657 653 642 643 718 654 632 731 392 344 792 852 649 629 685 617 715 715 629 916 531 357 917 828 708 858 775 785 1006 789 734 906 532 387 991 841 892 782 813 793 978 775 797 946 594 438 1022 868 795

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 'RServer@AstonUniversity' @ vre.aston.ac.uk Estimated Parameters of Exponential Smoothing Parameter Value alpha 0.100728105613193 beta 0 gamma 0.706402297696408 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 614 651.525172725983 -37.5251727259829 14 647 670.684799400148 -23.6847994001478 15 580 582.366193489376 -2.36619348937563 16 614 614.082108335153 -0.0821083351529523 17 636 638.174882180815 -2.17488218081451 18 388 385.959140339202 2.04085966079765 19 356 394.926976308915 -38.9269763089152 20 639 352.672465590974 286.327534409026 21 753 693.557227890671 59.4427721093285 22 611 732.547969407503 -121.547969407503 23 639 664.801621910674 -25.8016219106745 24 630 570.204342613149 59.795657386851 25 586 684.778644320277 -98.7786443202771 26 695 708.953919341885 -13.9539193418846 27 552 629.015895313806 -77.0158953138058 28 619 656.890366325891 -37.8903663258913 29 681 677.18892234409 3.81107765590946 30 421 412.132031059866 8.86796894013384 31 307 394.297058880402 -87.2970588804023 32 754 548.078347243632 205.921652756368 33 690 738.007640476895 -48.0076404768953 34 644 650.364283790538 -6.36428379053825 35 643 655.016731898568 -12.0167318985677 36 608 615.297809380185 -7.29780938018473 37 651 621.45786180542 29.5421381945806 38 691 714.167953646476 -23.1679536464757 39 627 590.361326062452 36.6386739375477 40 634 656.629737764733 -22.6297377647328 41 731 706.927738944956 24.0722610550442 42 475 435.746172981773 39.2538270182273 43 337 354.502513101708 -17.5025131017079 44 803 720.352028684078 82.647971315922 45 722 736.604638027393 -14.6046380273928 46 590 676.10183672607 -86.1018367260698 47 724 669.123010458944 54.8769895410561 48 627 637.505502594523 -10.505502594523 49 696 667.903375102016 28.0966248979839 50 825 729.271384717009 95.7286152829913 51 677 650.188277027583 26.8117229724173 52 656 679.113810957953 -23.1138109579528 53 785 763.609137573266 21.3908624267337 54 412 486.525784079443 -74.5257840794429 55 352 353.932010390557 -1.93201039055725 56 839 799.58096909401 39.4190309059899 57 729 748.091207851041 -19.0912078510413 58 696 638.025230518301 57.9747694816986 59 641 739.369895717159 -98.3698957171586 60 695 648.518206853014 46.4817931469859 61 638 711.124747901819 -73.1247479018185 62 762 806.64707788869 -44.64707788869 63 635 669.168037572516 -34.1680375725161 64 721 660.229436337253 60.7705636627469 65 854 782.134785031731 71.8652149682692 66 418 444.066913202401 -26.0669132024013 67 367 360.664586686199 6.33541331380138 68 824 845.017958361628 -21.0179583616283 69 687 748.612709808728 -61.6127098087283 70 601 682.190300654261 -81.1903006542611 71 676 669.410901471898 6.58909852810211 72 740 680.98018402519 59.0198159748098 73 691 668.260798330375 22.7392016696249 74 683 793.770722055046 -110.770722055046 75 594 654.717447856971 -60.7174478569707 76 729 703.435798355649 25.564201644351 77 731 828.604696928258 -97.6046969282577 78 386 419.108486405942 -33.1084864059417 79 331 356.880824472789 -25.8808244727888 80 706 806.5251716464 -100.525171646401 81 715 680.869110641995 34.1308893580053 82 657 613.040899260461 43.9591007395389 83 653 668.228632670744 -15.2286326707435 84 642 710.107472507583 -68.1074725075835 85 643 662.749894114399 -19.7498941143987 86 718 696.896524087989 21.1034759120112 87 654 604.375117566558 49.624882433442 88 632 718.842688245944 -86.8426882459437 89 731 752.99602607945 -21.9960260794497 90 392 394.765325432344 -2.76532543234441 91 344 340.100045600303 3.8999543996967 92 792 747.68238166773 44.3176183322699 93 852 721.199203322578 130.800796677422 94 649 666.471440466248 -17.4714404662477 95 629 678.199883454367 -49.199883454367 96 685 682.839299327259 2.16070067274075 97 617 672.847062803514 -55.8470628035141 98 715 731.041817232163 -16.0418172321632 99 715 650.772655258232 64.2273447417683 100 629 679.867208951556 -50.8672089515558 101 916 761.290149999896 154.709850000104 102 531 414.330995039189 116.669004960811 103 357 371.603349387028 -14.6033493870279 104 917 836.962376447361 80.0376235526387 105 828 869.245091697302 -41.2450916973019 106 708 693.441293776717 14.5587062232828 107 858 687.486984844789 170.513015155211 108 775 750.861332399602 24.1386676003976 109 785 701.03140276128 83.9685972387201 110 1006 809.343633894797 196.656366105203 111 789 796.57032178313 -7.57032178312954 112 734 737.773987383306 -3.77398738330567 113 906 984.09426384761 -78.0942638476107 114 532 542.458664190362 -10.4586641903621 115 387 392.075888372369 -5.07588837236949 116 991 962.677697694665 28.322302305335 117 841 908.330735465524 -67.3307354655235 118 892 755.047910374334 136.952089625666 119 782 865.880879179644 -83.8808791796441 120 813 807.95388326262 5.04611673738043 121 793 792.723487392503 0.276512607497125 122 978 967.742541741162 10.2574582588376 123 775 803.9309183551 -28.9309183550996 124 797 744.67136301732 52.3286369826798 125 946 953.043956280599 -7.04395628059876 126 594 550.643450395871 43.3565496041286 127 438 403.543542647334 34.4564573526662 128 1022 1027.68207330509 -5.68207330508676 129 868 903.510592234765 -35.5105922347655 130 795 880.689443126828 -85.6894431268275 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 131 827.429548585132 730.229994148165 924.6291030221 132 834.599505198186 736.481453704931 932.71755669144 133 815.2589480673 716.331868304118 914.186027830481 134 1001.63937840422 900.527979785676 1102.75077702276 135 806.666389269466 706.34963912771 906.983139411222 136 801.510713192643 700.379589898783 902.641836486502 137 970.7218365002 866.57756163545 1074.86611136495 138 591.883665574239 491.727316897714 692.040014250764 139 432.044100698021 332.789271702236 531.298929693807 140 1031.55768421552 918.657027025342 1144.4583414057 141 887.78499969121 778.181198893442 997.38880048898 142 835.49543354432 750.709349608741 920.2815174799

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/28/t1293537374pzq8okgz5ms92gc/1hqln1293537469.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/28/t1293537374pzq8okgz5ms92gc/1hqln1293537469.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/28/t1293537374pzq8okgz5ms92gc/2sz381293537469.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/28/t1293537374pzq8okgz5ms92gc/2sz381293537469.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/28/t1293537374pzq8okgz5ms92gc/3sz381293537469.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/28/t1293537374pzq8okgz5ms92gc/3sz381293537469.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=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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