Workshop 9 (ARIMA)

*The author of this computation has been verified*

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

Title produced by software: ARIMA Backward Selection

Date of computation: Wed, 15 Dec 2010 16:00:56 +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/15/t1292428765qm4izwomrs64ckk.htm/, Retrieved Wed, 15 Dec 2010 16:59:32 +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/15/t1292428765qm4izwomrs64ckk.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:

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Dataseries X:

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37 30 47 35 30 43 82 40 47 19 52 136 80 42 54 66 81 63 137 72 107 58 36 52 79 77 54 84 48 96 83 66 61 53 30 74 69 59 42 65 70 100 63 105 82 81 75 102 121 98 76 77 63 37 35 23 40 29 37 51 20 28 13 22 25 13 16 13 16 17 9 17 25 14 8 7 10 7 10 3

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 21 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 ARIMA Parameter Estimation and Backward Selection Iteration ar1 ar2 ar3 ma1 sar1 sar2 sma1 Estimates ( 1 ) -0.1052 -0.0694 -0.2205 -0.4445 -0.7085 0.2229 0.8863 (p-val) (0.6994 ) (0.685 ) (0.088 ) (0.1043 ) (0.3893 ) (0.1714 ) (0.3594 ) Estimates ( 2 ) 0 -0.0244 -0.1931 -0.536 -0.7484 0.22 0.9319 (p-val) (NA ) (0.8455 ) (0.0916 ) (0 ) (0.1371 ) (0.1385 ) (0.1678 ) Estimates ( 3 ) 0 0 -0.1901 -0.5448 -0.6944 0.2218 0.8714 (p-val) (NA ) (NA ) (0.0952 ) (0 ) (0.54 ) (0.2206 ) (0.5012 ) Estimates ( 4 ) 0 0 -0.1864 -0.5524 0 0.1182 0.1598 (p-val) (NA ) (NA ) (0.1011 ) (0 ) (NA ) (0.3899 ) (0.1966 ) Estimates ( 5 ) 0 0 -0.1853 -0.5554 0 0 0.1462 (p-val) (NA ) (NA ) (0.1019 ) (0 ) (NA ) (NA ) (0.2046 ) Estimates ( 6 ) 0 0 -0.1824 -0.5766 0 0 0 (p-val) (NA ) (NA ) (0.1087 ) (0 ) (NA ) (NA ) (NA ) Estimates ( 7 ) 0 0 0 -0.6263 0 0 0 (p-val) (NA ) (NA ) (NA ) (0 ) (NA ) (NA ) (NA ) Estimates ( 8 ) NA NA NA NA NA NA NA (p-val) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) Estimates ( 9 ) NA NA NA NA NA NA NA (p-val) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) Estimates ( 10 ) NA NA NA NA NA NA NA (p-val) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) Estimates ( 11 ) NA NA NA NA NA NA NA (p-val) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) Estimates ( 12 ) NA NA NA NA NA NA NA (p-val) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) Estimates ( 13 ) NA NA NA NA NA NA NA (p-val) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) (NA ) Estimated ARIMA Residuals Value 0.0369999745003413 -5.96231865345569 13.1997505098284 -4.38134104589191 -8.55664024884683 11.1858500863415 43.1600828165327 -18.0555098072061 -1.03390376614919 -21.4802077626364 12.9526883697128 92.7450087063382 -7.62959923195752 -36.3793071824434 6.34670394256463 5.44381424702743 11.2068476674050 -9.34880084836778 70.7983973267327 -21.4398345681963 19.3537016545635 -24.3407859715554 -47.8930389462538 -5.2312039931975 15.0447379468876 2.6617408591111 -18.5463802158022 24.2312685099367 -22.3926123684638 30.8921837761603 10.2858183735463 -17.6363037071706 -6.41302975388917 -14.0694124837839 -34.2139388466662 23.3594254468519 7.01011296852435 -10.1536114378739 -14.8280599607298 13.5377144258887 10.9818654022284 33.2311335254277 -13.6424737521364 35.0456012905473 2.68076272967238 -6.20395765707651 -1.91546067166381 21.6997226801847 31.330072611802 -6.02900023963369 -20.5509575909647 -7.3840144026248 -22.4535557582985 -42.9605320191498 -26.5894843267677 -29.8859887511151 -4.97592444425357 -14.2340692365998 -2.39675628753506 15.7192136404274 -23.9426635204241 -4.34641981168204 -14.9522778718257 -5.27697234969494 1.41659439714148 -13.9195439801644 -3.38446142179613 -4.4042725759719 -1.72869716449047 0.550474508657317 -8.22986146893105 3.80177197508502 10.3746038349681 -6.47719721085457 -8.27547947873524 -4.31240532432229 -1.49329926015154 -4.95561884743177 -0.0399352265722914 -6.47575127816825

Charts produced by software:

http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/1bsm71292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/1bsm71292428834.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/2l1ls1292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/2l1ls1292428834.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/3l1ls1292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/3l1ls1292428834.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/4l1ls1292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/4l1ls1292428834.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/5l1ls1292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/5l1ls1292428834.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/6wb2v1292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/6wb2v1292428834.ps (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/7wb2v1292428834.png (open in new window) http://www.freestatistics.org/blog/date/2010/Dec/15/t1292428765qm4izwomrs64ckk/7wb2v1292428834.ps (open in new window)

Parameters (Session):

par1 = FALSE ; par2 = 1 ; par3 = 1 ; par4 = 0 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 2 ; par9 = 1 ;

Parameters (R input):

par1 = FALSE ; par2 = 1 ; par3 = 1 ; par4 = 0 ; par5 = 12 ; par6 = 3 ; par7 = 1 ; par8 = 2 ; par9 = 1 ;

R code (references can be found in the software module):

library(lattice) if (par1 == 'TRUE') par1 <- TRUE if (par1 == 'FALSE') par1 <- FALSE par2 <- as.numeric(par2) #Box-Cox lambda transformation parameter par3 <- as.numeric(par3) #degree of non-seasonal differencing par4 <- as.numeric(par4) #degree of seasonal differencing par5 <- as.numeric(par5) #seasonal period par6 <- as.numeric(par6) #degree (p) of the non-seasonal AR(p) polynomial par7 <- as.numeric(par7) #degree (q) of the non-seasonal MA(q) polynomial par8 <- as.numeric(par8) #degree (P) of the seasonal AR(P) polynomial par9 <- as.numeric(par9) #degree (Q) of the seasonal MA(Q) polynomial armaGR <- function(arima.out, names, n){ try1 <- arima.out$coef try2 <- sqrt(diag(arima.out$var.coef)) try.data.frame <- data.frame(matrix(NA,ncol=4,nrow=length(names))) dimnames(try.data.frame) <- list(names,c('coef','std','tstat','pv')) try.data.frame[,1] <- try1 for(i in 1:length(try2)) try.data.frame[which(rownames(try.data.frame)==names(try2)[i]),2] <- try2[i] try.data.frame[,3] <- try.data.frame[,1] / try.data.frame[,2] try.data.frame[,4] <- round((1-pt(abs(try.data.frame[,3]),df=n-(length(try2)+1)))*2,5) vector <- rep(NA,length(names)) vector[is.na(try.data.frame[,4])] <- 0 maxi <- which.max(try.data.frame[,4]) continue <- max(try.data.frame[,4],na.rm=TRUE) > .05 vector[maxi] <- 0 list(summary=try.data.frame,next.vector=vector,continue=continue) } arimaSelect <- function(series, order=c(13,0,0), seasonal=list(order=c(2,0,0),period=12), include.mean=F){ nrc <- order[1]+order[3]+seasonal$order[1]+seasonal$order[3] coeff <- matrix(NA, nrow=nrc*2, ncol=nrc) pval <- matrix(NA, nrow=nrc*2, ncol=nrc) mylist <- rep(list(NULL), nrc) names <- NULL if(order[1] > 0) names <- paste('ar',1:order[1],sep='') if(order[3] > 0) names <- c( names , paste('ma',1:order[3],sep='') ) if(seasonal$order[1] > 0) names <- c(names, paste('sar',1:seasonal$order[1],sep='')) if(seasonal$order[3] > 0) names <- c(names, paste('sma',1:seasonal$order[3],sep='')) arima.out <- arima(series, order=order, seasonal=seasonal, include.mean=include.mean, method='ML') mylist[[1]] <- arima.out last.arma <- armaGR(arima.out, names, length(series)) mystop <- FALSE i <- 1 coeff[i,] <- last.arma[[1]][,1] pval [i,] <- last.arma[[1]][,4] i <- 2 aic <- arima.out$aic while(!mystop){ mylist[[i]] <- arima.out arima.out <- arima(series, order=order, seasonal=seasonal, include.mean=include.mean, method='ML', fixed=last.arma$next.vector) aic <- c(aic, arima.out$aic) last.arma <- armaGR(arima.out, names, length(series)) mystop <- !last.arma$continue coeff[i,] <- last.arma[[1]][,1] pval [i,] <- last.arma[[1]][,4] i <- i+1 } list(coeff, pval, mylist, aic=aic) } arimaSelectplot <- function(arimaSelect.out,noms,choix){ noms <- names(arimaSelect.out[[3]][[1]]$coef) coeff <- arimaSelect.out[[1]] k <- min(which(is.na(coeff[,1])))-1 coeff <- coeff[1:k,] pval <- arimaSelect.out[[2]][1:k,] aic <- arimaSelect.out$aic[1:k] coeff[coeff==0] <- NA n <- ncol(coeff) if(missing(choix)) choix <- k layout(matrix(c(1,1,1,2, 3,3,3,2, 3,3,3,4, 5,6,7,7),nr=4), widths=c(10,35,45,15), heights=c(30,30,15,15)) couleurs <- rainbow(75)[1:50]#(50) ticks <- pretty(coeff) par(mar=c(1,1,3,1)) plot(aic,k:1-.5,type='o',pch=21,bg='blue',cex=2,axes=F,lty=2,xpd=NA) points(aic[choix],k-choix+.5,pch=21,cex=4,bg=2,xpd=NA) title('aic',line=2) par(mar=c(3,0,0,0)) plot(0,axes=F,xlab='',ylab='',xlim=range(ticks),ylim=c(.1,1)) rect(xleft = min(ticks) + (0:49)/50*(max(ticks)-min(ticks)), xright = min(ticks) + (1:50)/50*(max(ticks)-min(ticks)), ytop = rep(1,50), ybottom= rep(0,50),col=couleurs,border=NA) axis(1,ticks) rect(xleft=min(ticks),xright=max(ticks),ytop=1,ybottom=0) text(mean(coeff,na.rm=T),.5,'coefficients',cex=2,font=2) par(mar=c(1,1,3,1)) image(1:n,1:k,t(coeff[k:1,]),axes=F,col=couleurs,zlim=range(ticks)) for(i in 1:n) for(j in 1:k) if(!is.na(coeff[j,i])) { if(pval[j,i]<.01) symb = 'green' else if( (pval[j,i]<.05) & (pval[j,i]>=.01)) symb = 'orange' else if( (pval[j,i]<.1) & (pval[j,i]>=.05)) symb = 'red' else symb = 'black' polygon(c(i+.5 ,i+.2 ,i+.5 ,i+.5), c(k-j+0.5,k-j+0.5,k-j+0.8,k-j+0.5), col=symb) if(j==choix) { rect(xleft=i-.5, xright=i+.5, ybottom=k-j+1.5, ytop=k-j+.5, lwd=4) text(i, k-j+1, round(coeff[j,i],2), cex=1.2, font=2) } else{ rect(xleft=i-.5,xright=i+.5,ybottom=k-j+1.5,ytop=k-j+.5) text(i,k-j+1,round(coeff[j,i],2),cex=1.2,font=1) } } axis(3,1:n,noms) par(mar=c(0.5,0,0,0.5)) plot(0,axes=F,xlab='',ylab='',type='n',xlim=c(0,8),ylim=c(-.2,.8)) cols <- c('green','orange','red','black') niv <- c('0','0.01','0.05','0.1') for(i in 0:3){ polygon(c(1+2*i ,1+2*i ,1+2*i-.5 ,1+2*i), c(.4 ,.7 , .4 , .4), col=cols[i+1]) text(2*i,0.5,niv[i+1],cex=1.5) } text(8,.5,1,cex=1.5) text(4,0,'p-value',cex=2) box() residus <- arimaSelect.out[[3]][[choix]]$res par(mar=c(1,2,4,1)) acf(residus,main='') title('acf',line=.5) par(mar=c(1,2,4,1)) pacf(residus,main='') title('pacf',line=.5) par(mar=c(2,2,4,1)) qqnorm(residus,main='') title('qq-norm',line=.5) qqline(residus) residus } if (par2 == 0) x <- log(x) if (par2 != 0) x <- x^par2 (selection <- arimaSelect(x, order=c(par6,par3,par7), seasonal=list(order=c(par8,par4,par9), period=par5))) bitmap(file='test1.png') resid <- arimaSelectplot(selection) dev.off() resid bitmap(file='test2.png') acf(resid,length(resid)/2, main='Residual Autocorrelation Function') dev.off() bitmap(file='test3.png') pacf(resid,length(resid)/2, main='Residual Partial Autocorrelation Function') dev.off() bitmap(file='test4.png') cpgram(resid, main='Residual Cumulative Periodogram') dev.off() bitmap(file='test5.png') hist(resid, main='Residual Histogram', xlab='values of Residuals') dev.off() bitmap(file='test6.png') densityplot(~resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test7.png') qqnorm(resid, main='Residual Normal Q-Q Plot') qqline(resid) dev.off() ncols <- length(selection[[1]][1,]) nrows <- length(selection[[2]][,1])-1 load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a,'ARIMA Parameter Estimation and Backward Selection', ncols+1,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Iteration', header=TRUE) for (i in 1:ncols) { a<-table.element(a,names(selection[[3]][[1]]$coef)[i],header=TRUE) } a<-table.row.end(a) for (j in 1:nrows) { a<-table.row.start(a) mydum <- 'Estimates (' mydum <- paste(mydum,j) mydum <- paste(mydum,')') a<-table.element(a,mydum, header=TRUE) for (i in 1:ncols) { a<-table.element(a,round(selection[[1]][j,i],4)) } a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'(p-val)', header=TRUE) for (i in 1:ncols) { mydum <- '(' mydum <- paste(mydum,round(selection[[2]][j,i],4),sep='') mydum <- paste(mydum,')') a<-table.element(a,mydum) } 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,'Estimated ARIMA Residuals', 1,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Value', 1,TRUE) a<-table.row.end(a) for (i in (par4*par5+par3):length(resid)) { a<-table.row.start(a) a<-table.element(a,resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable1.tab')

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