# ::Free Statistics and Forecasting Software::

v1.2.1

### :: ARIMA Backward Selection - Free Statistics Software (Calculator) ::

All rights reserved. The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this software when results are used in publications.

This free online software (calculator) computes the ARIMA Backward Selection approach as designed by Romain Francois. This (adapted) implementation includes seasonality and the Box-Cox transform.

Enter (or paste) your data delimited by hard returns.

 Send output to: Browser Blue - Charts White Browser Black/White CSV Data[reset data] 112 118 132 129 121 135 148 148 136 119 104 118 115 126 141 135 125 149 170 170 158 133 114 140 145 150 178 163 172 178 199 199 184 162 146 166 171 180 193 181 183 218 230 242 209 191 172 194 196 196 236 235 229 243 264 272 237 211 180 201 204 188 235 227 234 264 302 293 259 229 203 229 242 233 267 269 270 315 364 347 312 274 237 278 284 277 317 313 318 374 413 405 355 306 271 306 315 301 356 348 355 422 465 467 404 347 305 336 340 318 362 348 363 435 491 505 404 359 310 337 360 342 406 396 420 472 548 559 463 407 362 405 417 391 419 461 472 535 622 606 508 461 390 432 Include mean? FALSETRUE Box-Cox lambda transformation parameter (lambda) 1-2.0-1.9-1.8-1.7-1.6-1.5-1.4-1.3-1.2-1.1-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.0 Degree of non-seasonal differencing (d) 012 Degree of seasonal differencing (D) 01 Seasonal Period (s) 1234612 Maximum AR(p) order 0123 Maximum MA(q) order 01 Maximum SAR(P) order 012 Maximum SMA(Q) order 01 Chart options Width: Height:

 Source code of R 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,'Menu of Residual Diagnostics',2,TRUE) a <- table.row.end(a) a <- table.row.start(a) a <- table.element(a,'Description',1,TRUE) a <- table.element(a,'Link',1,TRUE) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Histogram',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_histogram.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Central Tendency',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_centraltendency.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'QQ Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_fitdistrnorm.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Kernel Density Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_density.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Skewness/Kurtosis Test',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_skewness_kurtosis.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Skewness-Kurtosis Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_skewness_kurtosis_plot.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Harrell-Davis Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_harrell_davis.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Bootstrap Plot -- Central Tendency',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_bootstrapplot1.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Blocked Bootstrap Plot -- Central Tendency',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_bootstrapplot.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'(Partial) Autocorrelation Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_autocorrelation.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Spectral Analysis',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_spectrum.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Tukey lambda PPCC Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_tukeylambda.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <-table.element(a,'Box-Cox Normality Plot',1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_boxcoxnorm.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a <- table.row.start(a) a <- table.element(a,"Summary Statistics",1,header=TRUE) a <- table.element(a,hyperlink( paste("https://supernova.wessa.net/rwasp_summary1.wasp?convertgetintopost=1&data=",paste(as.character(resid),sep="",collapse=" "),sep="") ,"Compute","Click here to examine the Residuals."),1) a <- table.row.end(a) a<-table.end(a) table.save(a,file="mytable7.tab")
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 Cite this software as: Wessa P., (2017), ARIMA Backward Selection (v1.0.6) in Free Statistics Software (v1.2.1), Office for Research Development and Education, URL https://www.wessa.net/rwasp_arimabackwardselection.wasp/ The R code is based on : Romain Francois, website: http://addictedtor.free.fr/graphiques/graphcode.php?graph=29
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