WS09 - 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, 02 Dec 2009 13:49:23 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su.htm/, Retrieved Wed, 02 Dec 2009 21:51:29 +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/2009/Dec/02/t1259787084q1n5qsey8g852su.htm/}, year = {2009}, } @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 = {2009}, 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:

» Textbox « » Textfile « » CSV «

423.4 404.1 500 472.6 496.1 562 434.8 538.2 577.6 518.1 625.2 561.2 523.3 536.1 607.3 637.3 606.9 652.9 617.2 670.4 729.9 677.2 710 844.3 748.2 653.9 742.6 854.2 808.4 1819 1936.5 1966.1 2083.1 1620.1 1527.6 1795 1685.1 1851.8 2164.4 1981.8 1726.5 2144.6 1758.2 1672.9 1837.3 1596.1 1446 1898.4 1964.1 1755.9 2255.3 1881.2 2117.9 1656.5 1544.1 2098.9 2133.3 1963.5 1801.2 2365.4 1936.5 1667.6 1983.5 2058.6 2448.3 1858.1 1625.4 2130.6 2515.7 2230.2 2086.9 2235 2100.2 2288.6 2490 2573.7 2543.8 2004.7 2390 2338.4 2724.5 2292.5 2386 2477.9 2337 2605.1 2560.8 2839.3 2407.2 2085.2 2735.6 2798.7 3053.2 2405 2471.9 2727.3 2790.7 2385.4 3206.6 2705.6 3518.4 1954.9 2584.3 2535.8 2685.9 2866 2236.6 2934.9 2668.6 2371.2 3165.9 2887.2 3112.2 2671.2 2432.6 2812.3 3095.7 2862.9 2607.3 2862.5

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 1 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Estimated Parameters of Exponential Smoothing Parameter Value alpha 0.346213598844670 beta 0 gamma 0.383242980338082 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 523.3 468.445487276567 54.8545127234325 14 536.1 498.66352637662 37.4364736233803 15 607.3 579.707630854539 27.5923691454614 16 637.3 617.181151035876 20.1188489641239 17 606.9 595.825725879453 11.074274120547 18 652.9 641.33105790874 11.5689420912598 19 617.2 544.60488343514 72.5951165648603 20 670.4 703.977363450823 -33.5773634508229 21 729.9 741.541064358148 -11.6410643581482 22 677.2 658.65927351803 18.5407264819694 23 710 798.861606007522 -88.8616060075216 24 844.3 690.582334067552 153.717665932448 25 748.2 709.718189935114 38.4818100648864 26 653.9 727.902110578053 -74.0021105780528 27 742.6 786.505003080898 -43.9050030808976 28 854.2 801.412484657752 52.7875153422481 29 808.4 776.430913035321 31.9690869646787 30 1819 838.502991683396 980.497008316604 31 1936.5 1013.07365058068 923.426349419323 32 1966.1 1561.72553117971 404.37446882029 33 2083.1 1819.37286406244 263.727135937559 34 1620.1 1707.58414162003 -87.4841416200336 35 1527.6 1923.92162632109 -396.321626321093 36 1795 1723.31752789012 71.6824721098778 37 1685.1 1593.67868858838 91.4213114116171 38 1851.8 1559.50749420858 292.292505791415 39 2164.4 1866.573332006 297.826667993998 40 1981.8 2090.32148568062 -108.521485680624 41 1726.5 1916.46122352916 -189.961223529157 42 2144.6 2338.72818934595 -194.128189345952 43 1758.2 1890.06218712522 -131.862187125217 44 1672.9 1948.98889262301 -276.08889262301 45 1837.3 1934.50892901993 -97.2089290199256 46 1596.1 1620.97248104066 -24.8724810406634 47 1446 1770.08600238853 -324.086002388528 48 1898.4 1710.77199199425 187.628008005747 49 1964.1 1624.89663314946 339.203366850536 50 1755.9 1720.62343820042 35.2765617995792 51 2255.3 1936.06528564962 319.234714350378 52 1881.2 2064.92035206239 -183.720352062393 53 2117.9 1844.95589521561 272.944104784394 54 1656.5 2460.67229645203 -804.17229645203 55 1544.1 1822.49542625217 -278.395426252171 56 2098.9 1788.4901141119 310.409885888100 57 2133.3 2028.5216907021 104.778309297903 58 1963.5 1775.65577873815 187.844221261852 59 1801.2 1927.59225463942 -126.392254639415 60 2365.4 2095.74516966627 269.654830333734 61 1936.5 2044.36729662966 -107.867296629662 62 1667.6 1898.40249141378 -230.802491413779 63 1983.5 2099.91542005586 -116.415420055862 64 2058.6 1954.26619078847 104.333809211530 65 2448.3 1944.10187951795 504.198120482055 66 1858.1 2356.05138517422 -497.951385174217 67 1625.4 1926.84740134125 -301.447401341247 68 2130.6 2047.35555316413 83.2444468358653 69 2515.7 2160.27464522433 355.425354775673 70 2230.2 1987.49318770981 242.706812290192 71 2086.9 2078.60273538847 8.29726461152904 72 2235 2426.31980271082 -191.319802710819 73 2100.2 2108.74017584760 -8.54017584759731 74 2288.6 1954.13452021044 334.465479789559 75 2490 2431.65227243603 58.3477275639693 76 2573.7 2388.39421469764 185.305785302360 77 2543.8 2502.83815920172 40.9618407982825 78 2004.7 2489.16653498205 -484.466534982046 79 2390 2081.16901205823 308.830987941765 80 2338.4 2586.96443620870 -248.564436208704 81 2724.5 2674.25593695877 50.2440630412284 82 2292.5 2319.63032211609 -27.1303221160888 83 2386 2253.35109159517 132.648908404832 84 2477.9 2621.84724829203 -143.947248292027 85 2337 2342.08424960309 -5.08424960309412 86 2605.1 2260.78152706651 344.318472933492 87 2560.8 2701.82903771605 -141.029037716049 88 2839.3 2616.73847733390 222.561522666096 89 2407.2 2708.00758084818 -300.807580848181 90 2085.2 2430.14911313109 -344.949113131093 91 2735.6 2262.61109751585 472.988902484153 92 2798.7 2700.83031488706 97.8696851129407 93 3053.2 3011.35850078951 41.8414992104899 94 2405 2586.50809709010 -181.508097090097 95 2471.9 2503.40548652555 -31.5054865255488 96 2727.3 2761.54246769244 -34.2424676924356 97 2790.7 2537.14929618599 253.55070381401 98 2385.4 2627.68142370859 -242.281423708586 99 3206.6 2749.27354456528 457.32645543472 100 2705.6 2965.37509735931 -259.775097359306 101 3518.4 2749.47352120628 768.926478793724 102 1954.9 2787.89219909764 -832.992199097638 103 2584.3 2665.32078027908 -81.0207802790833 104 2535.8 2823.53661753304 -287.736617533041 105 2685.9 2983.28465081925 -297.384650819251 106 2866 2409.97324870202 456.026751297976 107 2236.6 2585.36521520188 -348.765215201880 108 2934.9 2730.88054591979 204.01945408021 109 2668.6 2655.56376111757 13.0362388824269 110 2371.2 2538.00990550250 -166.809905502497 111 3165.9 2854.37120129727 311.528798702729 112 2887.2 2840.98259500262 46.2174049973773 113 3112.2 2973.4596939424 138.740306057599 114 2671.2 2406.86670493956 264.333295060444 115 2432.6 2883.44867870295 -450.848678702945 116 2812.3 2864.25108391293 -51.9510839129271 117 3095.7 3118.23997266485 -22.5399726648461 118 2862.9 2789.23550889622 73.6644911037783 119 2607.3 2616.46986007017 -9.16986007017249 120 2862.5 3055.59449714028 -193.094497140276 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 121 2785.13059086808 2448.17398860440 3122.08719313175 122 2608.84471389662 2225.71231194584 2991.9771158474 123 3133.37653500486 2668.85830315633 3597.89476685338 124 2939.46984980711 2453.54099035311 3425.39870926110 125 3081.70095831986 2544.27379281056 3619.12812382917 126 2491.15640509781 1989.24487286823 2993.06793732739 127 2685.16432359776 2122.10561980854 3248.22302738699 128 2928.00879444682 2296.64827910602 3559.36930978763 129 3216.24627907143 2508.81157814971 3923.68097999316 130 2907.89127131893 2230.93372920066 3584.84881343719 131 2682.80604129979 2021.83293745790 3343.77914514167 132 3088.46687493822 2398.25635645366 3778.67739342277

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su/1gpsp1259786961.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su/1gpsp1259786961.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su/22e9v1259786961.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su/22e9v1259786961.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su/33ki01259786961.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/02/t1259787084q1n5qsey8g852su/33ki01259786961.ps (open in new window)

Parameters (Session):

par1 = FALSE ; par2 = 1 ; par3 = 1 ; par4 = 1 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 2 ; par9 = 1 ;

Parameters (R input):

par1 = 12 ; par2 = Triple ; par3 = multiplicative ; par4 = 1 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 2 ; par9 = 1 ;

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=0, beta=0) if (par2 == 'Double') fit <- HoltWinters(x, gamma=0) 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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