WS 9 Estimation of Box-Jenkins ARIMA models

*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: Fri, 04 Dec 2009 08:08:40 -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/04/t1259939380q926p21kykfbmts.htm/, Retrieved Fri, 04 Dec 2009 16:09:45 +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/04/t1259939380q926p21kykfbmts.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:

WS 9 Estimation of Box-Jenkins ARIMA models

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

14,5 14,3 15,3 14,4 13,7 14,2 13,5 11,9 14,6 15,6 14,1 14,9 14,2 14,6 17,2 15,4 14,3 17,5 14,5 14,4 16,6 16,7 16,6 16,9 15,7 16,4 18,4 16,9 16,5 18,3 15,1 15,7 18,1 16,8 18,9 19 18,1 17,8 21,5 17,1 18,7 19 16,4 16,9 18,6 19,3 19,4 17,6 18,6 18,1 20,4 18,1 19,6 19,9 19,2 17,8 19,2 22 21,1 19,5 22,2 20,9 22,2 23,5 21,5 24,3 22,8 20,3 23,7 23,3 19,6 18 17,3 16,8 18,2 16,5 16 18,4

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 0.597126235287043 beta 0 gamma 0.901410325583061 Interpolation Forecasts of Exponential Smoothing t Observed Fitted Residuals 13 14.2 13.4812493346188 0.718750665381153 14 14.6 14.2962834091534 0.303716590846596 15 17.2 17.0001134024304 0.199886597569616 16 15.4 15.3360780794504 0.0639219205496122 17 14.3 14.2640249819681 0.0359750180319054 18 17.5 17.4257003412120 0.0742996587880462 19 14.5 14.9514570741622 -0.451457074162191 20 14.4 13.0560618632243 1.34393813677572 21 16.6 17.033454253127 -0.433454253127014 22 16.7 17.9212732352252 -1.22127323522519 23 16.6 15.6003547386059 0.999645261394079 24 16.9 17.0655402785585 -0.165540278558492 25 15.7 16.4074650629215 -0.707465062921482 26 16.4 16.2315260448530 0.168473955147029 27 18.4 19.0948308769003 -0.694830876900333 28 16.9 16.6765728270487 0.223427172951286 29 16.5 15.5752117924012 0.924788207598825 30 18.3 19.6653161652278 -1.36531616522776 31 15.1 15.9178336435388 -0.81783364353884 32 15.7 14.3570648429723 1.34293515702767 33 18.1 17.8212344386997 0.278765561300325 34 16.8 18.8869887133395 -2.08698871333949 35 18.9 16.7925139442958 2.10748605570420 36 19 18.5233180659821 0.476681934017947 37 18.1 17.9446208974422 0.155379102557834 38 17.8 18.6647243046389 -0.864724304638944 39 21.5 20.8371566701604 0.662843329839617 40 17.1 19.2870598574822 -2.18705985748221 41 18.7 16.9172196603500 1.78278033965004 42 19 20.9004375180540 -1.90043751805396 43 16.4 16.8022581308116 -0.40225813081161 44 16.9 16.2073823658934 0.69261763410659 45 18.6 19.0288496196914 -0.428849619691373 46 19.3 18.7536861663317 0.546313833668307 47 19.4 19.7639881593884 -0.363988159388381 48 17.6 19.4139860794003 -1.81398607940025 49 18.6 17.3885821703684 1.21141782963165 50 18.1 18.3600001031061 -0.260000103106115 51 20.4 21.5078361206708 -1.10783612067084 52 18.1 17.9004122077839 0.199587792216132 53 19.6 18.3690053332061 1.23099466679392 54 19.9 20.681164758908 -0.781164758907984 55 19.2 17.6412309618206 1.55876903817939 56 17.8 18.5972258934997 -0.797225893499718 57 19.2 20.2573983934826 -1.05739839348260 58 22 19.9680947725864 2.03190522741363 59 21.1 21.5543092064775 -0.454309206477525 60 19.5 20.5047320951071 -1.00473209510708 61 22.2 20.0442219844667 2.15577801553334 62 20.9 20.9816621659555 -0.0816621659554997 63 22.2 24.3564489374252 -2.15644893742518 64 23.5 20.2628275972412 3.23717240275884 65 21.5 23.0318681203166 -1.53186812031663 66 24.3 23.0485978670448 1.25140213295523 67 22.8 21.6791065830458 1.12089341695419 68 20.3 21.3494133167639 -1.04941331676385 69 23.7 23.0593191734165 0.640680826583509 70 23.3 25.1425232705637 -1.84252327056371 71 19.6 23.4451898225841 -3.84518982258406 72 18 20.1582924440604 -2.15829244406035 73 17.3 20.0717158947412 -2.77171589474115 74 16.8 17.4705444768559 -0.670544476855923 75 18.2 19.2418463363559 -1.0418463363559 76 16.5 17.8493142704545 -1.34931427045453 77 16 16.4096486766431 -0.409648676643094 78 18.4 17.6466439408779 0.753356059122133 Extrapolation Forecasts of Exponential Smoothing t Forecast 95% Lower Bound 95% Upper Bound 79 16.5056129124019 14.0406509716108 18.970574853193 80 15.2338185101497 12.3972713069801 18.0703657133193 81 17.4746598922364 14.0583317215505 20.8909880629224 82 18.0783333296844 14.2772779723994 21.8793886869694 83 16.9662120332049 13.0451830875728 20.887240978837 84 16.6132143608070 12.4810608311121 20.7453678905019 85 17.4405438234093 12.9056757733348 21.9754118734839 86 17.2512182130528 12.5284549324076 21.9739814936979 87 19.3228837879826 13.9342068149707 24.7115607609946 88 18.3559261478736 13.0098184152905 23.7020338804567 89 18.0144542672288 12.5679813320519 23.4609272024057 90 20.1300023440808 -0.095618151551303 40.3556228397129

Charts produced by software:

http://www.freestatistics.org/blog/date/2009/Dec/04/t1259939380q926p21kykfbmts/121d01259939318.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259939380q926p21kykfbmts/121d01259939318.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259939380q926p21kykfbmts/2ktiq1259939318.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259939380q926p21kykfbmts/2ktiq1259939318.ps (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259939380q926p21kykfbmts/3sqzb1259939318.png (open in new window) http://www.freestatistics.org/blog/date/2009/Dec/04/t1259939380q926p21kykfbmts/3sqzb1259939318.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=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

Disclaimer

Information provided on this web site is provided "AS IS" without warranty of any kind, either express or implied, including, without limitation, warranties of merchantability, fitness for a particular purpose, and noninfringement. We use reasonable efforts to include accurate and timely information and periodically update the information, and software without notice. However, we make no warranties or representations as to the accuracy or completeness of such information (or software), and we assume no liability or responsibility for errors or omissions in the content of this web site, or any software bugs in online applications. Your use of this web site is AT YOUR OWN RISK. Under no circumstances and under no legal theory shall we be liable to you or any other person for any direct, indirect, special, incidental, exemplary, or consequential damages arising from your access to, or use of, this web site.

Privacy Policy

We may request personal information to be submitted to our servers in order to be able to:

• personalize online software applications according to your needs
• enforce strict security rules with respect to the data that you upload (e.g. statistical data)
• manage user sessions of online applications
• alert you about important changes or upgrades in resources or applications

We NEVER allow other companies to directly offer registered users information about their products and services. Banner references and hyperlinks of third parties NEVER contain any personal data of the visitor.

We do NOT sell, nor transmit by any means, personal information, nor statistical data series uploaded by you to third parties.

We carefully protect your data from loss, misuse, alteration, and destruction. However, at any time, and under any circumstance you are solely responsible for managing your passwords, and keeping them secret.

We store a unique ANONYMOUS USER ID in the form of a small 'Cookie' on your computer. This allows us to track your progress when using this website which is necessary to create state-dependent features. The cookie is used for NO OTHER PURPOSE. At any time you may opt to disallow cookies from this website - this will not affect other features of this website.

We examine cookies that are used by third-parties (banner and online ads) very closely: abuse from third-parties automatically results in termination of the advertising contract without refund. We have very good reason to believe that the cookies that are produced by third parties (banner ads) do NOT cause any privacy or security risk.

FreeStatistics.org is safe. There is no need to download any software to use the applications and services contained in this website. Hence, your system's security is not compromised by their use, and your personal data - other than data you submit in the account application form, and the user-agent information that is transmitted by your browser - is never transmitted to our servers.

As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

FreeStatistics.org is powered by