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Last week (in the MAT8181 course) in order to identify the orders of an ARMA process, we've seen the eacf method, and I mentioned the scan method, introduced in Tsay and Tiao (1985). The code below - to produce the output of the scan procedure - has been adapted from an old code by Steve Chen (where I included a visualization of the p-values, with the following colors)The procedure was described in the course, last Thursday,arma.scan=function(z,ar.max=15,ma.max=15,alpha=0.01){  ym=function(z,t,m){return(z[t:(t-m)])}  n=length(z)  z=z - mean(z)  cmax=ma.max + 1  rmax=ar.max + 1  corref=matrix(0,nrow=rmax,ncol=cmax)  cmj.table=matrix(0,nrow=rmax,ncol=cmax)  pv=matrix(0,nrow=rmax,ncol=cmax)  mark=matrix(rep("X",(rmax)*(cmax)),nrow=rmax,ncol=cmax)  Rnames=paste("AR",0:(ar.max),sep="-")  Cnames=paste("MA",0:(ma.max),sep="-")  rownames(corref)=Rnames  colnames(corref)=Cnames  rownames(cmj.table)=Rnames  colnames(cmj.table)=Cnames  rownames(pv)=Rnames  colnames(pv)=Cnames  rownames(mark)=Rnames  colnames(mark)=Cnames  for (m in 0:ar.max)  {   m1=m+1   for (j in 0:ma.max)   {   j1=j+1    if (m == 0 &#38;&#38; j != 0)     {      racf=acf(z,plot=FALSE)$acf[1:(j+1)]          lamb=racf[j+1]^2          corref[m1,j]=round(lamb,4)      dmj=1 + 2*sum(racf[1:j]^2)      cmj=-1*(n-m-j)*log(1.0 - lamb/dmj)      pvalue =pchisq(cmj,1,lower.tail=FALSE)      pv[m1,j]=round(pvalue,4)      cmj.table[m1,j]=round(cmj,4)      mark[m1,j]=ifelse(pvalue &#62; alpha,"O","X")        }     else if (m != 0 &#38;&#38; j == 0)     {      racf=pacf(z,plot=FALSE)$acf[1:(m+1)]      lamb=racf[m+1]^2      corref[m1,j1]=round(lamb,4)      dmj = 1      cmj=-1*(n-m-j)*log(1.0 - lamb/dmj)          pvalue =pchisq(cmj,1,lower.tail=FALSE)      pv[m1,j1]=round(pvalue,4)      cmj.table[m1,j1]=round(cmj,4)          mark[m1,j1]=ifelse(pvalue &#62; alpha,"O","X")    }     else    {              mat1=matrix(0,nrow=m1,ncol=m1)      mat2=matrix(0,nrow=m1,ncol=m1)       mat3=matrix(0,nrow=m1,ncol=m1)      mat4=matrix(0,nrow=m1,ncol=m1)           for (t in (j+m+2):n)      {         tj1=t-j-1         ym1=ym(z,tj1,m)         ym2=ym(z,t,m)             mat1=mat1 + as.matrix(ym1)%*%ym1             mat2=mat2 + as.matrix(ym1)%*%ym2             mat3=mat3 + as.matrix(ym2)%*%ym2             mat4=mat4 + as.matrix(ym2)%*%ym1                      }        b1=solve(mat1)%*%mat2      b2=solve(mat3)%*%mat4      A=b2%*%b1      eig &#60;-eigen(A)      eig.val &#60;-eig$values      eig.val=Re(eig.val)      eig.len=length(eig.val)      eig.vector=eig$vectors      lamb=min(eig.val)      eig.vector0=eig.vector[,which.min(eig.val)]      eig.vector0 = eig.vector0/eig.vector0[1]      resid=(1:n)*0       for (t in (j+m+1):n)      {        z0=z[seq(t,t-m,-1)]              resid[t]=sum(z0 * eig.vector0)      }       jm1=j + m + 1      rx=Re(resid[jm1:n])      racf=acf(rx,plot=FALSE)$acf[1:j]      dmj=1 + 2*sum(racf^2)      cmj=-1*(n-m-j)*log(1.0 - lamb/dmj)           pvalue =pchisq(cmj,df=1,lower.tail=FALSE)      corref[m1,j1]=round(lamb,4)           pv[m1,j1]=round(pvalue,4)      cmj.table[m1,j1]=round(cmj,4)          mark[m1,j1]=ifelse(pvalue &#62; alpha,"O","X")    }   }   }   cat("\n\nSCAN: Smallest CANonical Correlation Method for ARIMA(p,d,q)\n\n")  cat("Estimates of Squared Canonical Correlation \n\n")  print(corref)  cat("\n\nC(m,j)\n\n")  print(cmj.table)  cat("\n\nChi-Square(1) Test p-value\n\n")  print(pv)  cat("\nSCAN Matrix \n\n")  print(mark)plot(0:1,0:1,col="white",xlim=c(0,nrow(pv)-1),ylim=c(0,ncol(pv)-1),axes=FALSE,xlab="AR",ylab="MA")axis(1); axis(2)library(RColorBrewer)CL=brewer.pal(6, "RdBu")[c(1,2,3,5)]cpv=matrix(as.numeric(cut(as.vector(pv),c(-1,.01,.05,.1,2))),nrow(pv),ncol(pv))for(i in 1:nrow(pv)){for(j in 1:ncol(pv)){ polygon(c(i-1,i-1,i,i)-.5,c(j-1,j,j,j-1)-.5, col=CL[cpv[i,j]])}}}Consider the following simulated time series,&#62; s=arima.sim(n=200,model=list(ar=c(0,0,0,.4,0,0,0,.5),ma=c(0,0,1))) &#62; plot(s,type="l")The output is here&#62; arma.scan(s,6,6)SCAN: Smallest CANonical Correlation Method for ARIMA(p,d,q)Estimates of Squared Canonical Correlation        MA-0   MA-1   MA-2   MA-3   MA-4   MA-5   MA-6AR-0 0.0614 0.0104 0.1862 0.3516 0.0971 0.0128 0.0000AR-1 0.0302 0.0294 0.1501 0.0943 0.0855 0.0127 0.0385AR-2 0.3070 0.2781 0.2140 0.0006 0.1589 0.1884 0.2243AR-3 0.1627 0.0037 0.1927 0.2311 0.1379 0.0207 0.0376AR-4 0.2087 0.3947 0.3653 0.3075 0.1502 0.1364 0.1013AR-5 0.1677 0.1219 0.0110 0.0263 0.0332 0.0350 0.0044AR-6 0.0114 0.0485 0.0561 0.0427 0.0009 0.0089 0.0308C(m,j)        MA-0    MA-1    MA-2    MA-3   MA-4   MA-5    MA-6AR-0  4.1161  0.6585 12.0315 20.6512 4.5388 0.5620  0.0000AR-1  6.1127  1.9499  9.9356  4.9145 4.7219 0.4642  1.9015AR-2 72.6011 19.1679 14.3512  0.0337 7.9668 9.6479 11.4573AR-3 34.9724  0.2386 10.1620 13.4082 6.7875 0.8725  1.4071AR-4 45.8691 27.5070 19.1422 20.2835 7.3339 5.5374  3.5874AR-5 35.7981  8.0498  0.6280  1.3543 1.8470 1.7930  0.2338AR-6  2.2147  3.1466  3.5990  1.9904 0.0511 0.4816  1.6440Chi-Square(1) Test p-value       MA-0   MA-1   MA-2   MA-3   MA-4   MA-5   MA-6AR-0 0.0425 0.4171 0.0005 0.0000 0.0331 0.4534 0.0000AR-1 0.0134 0.1626 0.0016 0.0266 0.0298 0.4957 0.1679AR-2 0.0000 0.0000 0.0002 0.8543 0.0048 0.0019 0.0007AR-3 0.0000 0.6252 0.0014 0.0003 0.0092 0.3503 0.2355AR-4 0.0000 0.0000 0.0000 0.0000 0.0068 0.0186 0.0582AR-5 0.0000 0.0046 0.4281 0.2445 0.1741 0.1806 0.6287AR-6 0.1367 0.0761 0.0578 0.1583 0.8212 0.4877 0.1998SCAN Matrix      MA-0 MA-1 MA-2 MA-3 MA-4 MA-5 MA-6AR-0 "O"  "O"  "X"  "X"  "O"  "O"  "X" AR-1 "O"  "O"  "X"  "O"  "O"  "O"  "O" AR-2 "X"  "X"  "X"  "O"  "X"  "X"  "X" AR-3 "X"  "O"  "X"  "X"  "X"  "O"  "O" AR-4 "X"  "X"  "X"  "X"  "X"  "O"  "O" AR-5 "X"  "X"  "O"  "O"  "O"  "O"  "O" AR-6 "O"  "O"  "O"  "O"  "O"  "O"  "O"with the following graphOf course, it is possible to ask for larger values,&#62; arma.scan(s,12,12)The graph is now