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: Wed, 29 Dec 2010 09:43:45 +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/29/t1293615798ewh7702p1aybc9d.htm/, Retrieved Wed, 29 Dec 2010 10:43:18 +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/29/t1293615798ewh7702p1aybc9d.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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1775 2197 2920 4240 5415 6136 6719 6234 7152 3646 2165 2803 1615 2350 3350 3536 5834 6767 5993 7276 5641 3477 2247 2466 1567 2237 2598 3729 5715 5776 5852 6878 5488 3583 2054 2282 1552 2261 2446 3519 5161 5085 5711 6057 5224 3363 1899 2115 1491 2061 2419 3430 4778 4862 6176 5664 5529 3418 1941 2402 1579 2146 2462 3695 4831 5134 6250 5760 6249 2917 1741 2359 1511 2059 2635 2867 4403 5720 4502 5749 5627 2846 1762 2429 1169 2154 2249 2687 4359 5382 4459 6398 4596 3024 1887 2070 1351 2218 2461 3028 4784 4975 4607 6249 4809 3157 1910 2228 1594 2467 2222 3607 4685 4962 5770 5480 5000 3228 1993 2288 1588 2105 2191 3591 4668 4885 5822 5599 5340 3082 2010 2301

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 'Sir Ronald Aylmer Fisher' @ 193.190.124.24 Estimated Parameters of Exponential Smoothing Parameter Value alpha 0.0535977728080357 beta 0.000530028422734916 gamma 1 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 1615 1538.19070512821 76.8092948717931 14 2350 2260.4341115385 89.5658884614995 15 3350 3281.07212070563 68.9278792943678 16 3536 3537.06427000638 -1.06427000638359 17 5834 5834.92996575642 -0.929965756420643 18 6767 6774.80283349635 -7.80283349634556 19 5993 6672.55710916704 -679.557109167037 20 7276 6147.70421336274 1128.29578663726 21 5641 7098.19692601847 -1457.19692601847 22 3477 3521.77992444771 -44.7799244477064 23 2247 2046.52238964722 200.477610352776 24 2466 2647.79080771315 -181.790807713152 25 1567 1522.96614664932 44.0338533506792 26 2237 2255.50181624142 -18.5018162414208 27 2598 3250.78879452445 -652.788794524454 28 3729 3401.81032399214 327.189676007861 29 5715 5717.35864362219 -2.35864362219036 30 5776 6650.61223710473 -874.612237104733 31 5852 5866.09488908419 -14.0948890841892 32 6878 7087.82137027955 -209.821370279547 33 5488 5519.59598532189 -31.5959853218883 34 3583 3356.2611774559 226.738822544102 35 2054 2127.63499569831 -73.6349956983131 36 2282 2352.39039501135 -70.3903950113454 37 1552 1447.21916351783 104.780836482169 38 2261 2123.79021822474 137.209781775262 39 2446 2527.10018503946 -81.1001850394614 40 3519 3636.20080115686 -117.200801156859 41 5161 5616.01693622107 -455.016936221075 42 5085 5699.46486948967 -614.464869489669 43 5711 5743.25232569529 -32.2523256952891 44 6057 6778.73506519813 -721.735065198128 45 5224 5351.69603916262 -127.696039162623 46 3363 3427.64728149998 -64.6472814999765 47 1899 1899.06888669762 -0.068886697616108 48 2115 2130.77993654896 -15.7799365489645 49 1491 1394.26167264435 96.7383273556547 50 2061 2101.03578936866 -40.0357893686601 51 2419 2288.17501080885 130.824989191147 52 3430 3374.41292250402 55.5870774959753 53 4778 5043.72935077699 -265.729350776988 54 4862 4986.37536557658 -124.375365576579 55 6176 5607.40626614289 568.593733857108 56 5664 6022.55057533152 -358.550575331516 57 5529 5177.17316318898 351.826836811018 58 3418 3338.50474765172 79.4952523482793 59 1941 1878.78280243959 62.217197560411 60 2402 2098.97863869733 303.021361302675 61 1579 1486.05937008565 92.940629914353 62 2146 2063.21092230959 82.78907769041 63 2462 2418.66410600368 43.3358939963173 64 3695 3429.03278599895 265.967214001049 65 4831 4805.56183037692 25.4381696230757 66 5134 4897.6310664831 236.368933516904 67 6250 6193.87436954402 56.1256304559829 68 5760 5704.13534467181 55.8646553281851 69 6249 5553.31945782004 695.680542179962 70 2917 3475.4024122172 -558.4024122172 71 1741 1965.17725698159 -224.177256981593 72 2359 2397.95112283166 -38.9511228316551 73 1511 1567.90284137867 -56.9028413786732 74 2059 2127.43223157852 -68.4322315785166 75 2635 2437.45397897118 197.546021028818 76 2867 3666.80340626359 -799.803406263588 77 4403 4758.55866983873 -355.558669838734 78 5720 5029.80821921765 690.191780782354 79 4502 6179.78119562558 -1677.78119562558 80 5749 5596.80082469422 152.199175305781 81 5627 6056.61335608884 -429.613356088836 82 2846 2731.42611714609 114.573882853915 83 1762 1573.51149640558 188.488503594418 84 2429 2203.64254960891 225.357450391094 85 1169 1370.71937694711 -201.719376947109 86 2154 1911.51967316080 242.480326839205 87 2249 2489.88107473489 -240.881074734888 88 2687 2751.77863475017 -64.7786347501728 89 4359 4303.32524571644 55.6747542835556 90 5382 5586.28967745597 -204.289677455972 91 4459 4447.21326151398 11.7867384860228 92 6398 5686.68318588384 711.316814116162 93 4596 5625.84610379992 -1029.84610379992 94 3024 2783.50229160411 240.497708395894 95 1887 1702.28799668004 184.712003319959 96 2070 2367.10751149119 -297.107511491186 97 1351 1101.97809781242 249.021902187579 98 2218 2087.32469437971 130.675305620286 99 2461 2202.23209534194 258.767904658057 100 3028 2657.58046995612 370.419530043876 101 4784 4346.46945290073 437.530547099273 102 4975 5403.89979933126 -428.899799331258 103 4607 4457.30381344760 149.696186552395 104 6249 6366.22994747587 -117.229947475873 105 4809 4613.14835207163 195.85164792837 106 3157 3038.7944538962 118.205546103800 107 1910 1898.26541302465 11.7345869753472 108 2228 2097.84930476264 130.150695237362 109 1594 1372.52085362760 221.479146372397 110 2467 2244.42973561878 222.570264381223 111 2222 2485.53423166698 -263.534231666984 112 3607 3018.58549318573 588.414506814272 113 4685 4782.7085014647 -97.7085014647 114 4962 4991.48037623507 -29.4803762350712 115 5770 4613.90901720555 1156.09098279445 116 5480 6324.21687882173 -844.216878821731 117 5000 4828.51156540994 171.488434590063 118 3228 3179.40676063280 48.593239367196 119 1993 1934.41967581065 58.5803241893482 120 2288 2248.62236763414 39.3776323658581 121 1588 1604.89825765683 -16.8982576568255 122 2105 2465.09263190931 -360.092631909306 123 2191 2214.93011791436 -23.9301179143608 124 3591 3567.12941779773 23.8705822022721 125 4668 4651.64935610467 16.3506438953264 126 4885 4931.11260714286 -46.1126071428598 127 5822 5674.68351019995 147.316489800049 128 5599 5437.80517075069 161.194829249307 129 5340 4957.25969883867 382.740301161328 130 3082 3203.18148086508 -121.181480865077 131 2010 1958.55406918362 51.4459308163841 132 2301 2254.20812113126 46.7918788687393 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 133 1557.62919925444 787.213613065321 2328.04478544356 134 2093.93727103644 1322.41471214799 2865.45982992488 135 2181.23801040174 1408.60889418985 2953.86712661364 136 3579.97741851713 2806.24215795643 4353.71267907784 137 4656.11920045374 3881.27820613201 5430.96019477547 138 4875.60840903364 4099.66208916608 5651.55472890121 139 5804.7315588039 5027.68031924723 6581.78279836057 140 5573.10667547203 4794.95091973856 6351.2624312055 141 5293.60286899197 4514.34299826352 6072.86273972043 142 3042.09727468024 2261.73368782193 3822.46086153855 143 1967.34267814184 1185.87577171580 2748.80958456788 144 2255.83606692871 1473.26623520769 3038.40589864974

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/29/t1293615798ewh7702p1aybc9d/1ybge1293615821.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/29/t1293615798ewh7702p1aybc9d/1ybge1293615821.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/29/t1293615798ewh7702p1aybc9d/29lgz1293615821.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/29/t1293615798ewh7702p1aybc9d/29lgz1293615821.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/29/t1293615798ewh7702p1aybc9d/3jux21293615821.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/29/t1293615798ewh7702p1aybc9d/3jux21293615821.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')

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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