#AR2 Series  

e<- rnorm(200) 
seq<- 1:200; Y<- 1:200; Y[1]<- 0; Y[2]<- 0

postscript(file="AR2plot.ps",height=16,width=10,horizontal=F)
par(mfrow=c(2,1))
sink("AR2sim.txt")

#AR2-Funktionen

AR2Y<- function(alph1,alph2,e,Y){
ne<- length(e)
for (i in 3:ne)
{Y[i]<- alph1*Y[i-1] + alph2*Y[i-2] + e[i]}
invisible(Y)                    #return ohne Ausdruck
}

AR2p<- function(alph1,alph2,Y){
plot(1:200,Y,type="l",lty=1,xlim=c(0,200),ylim=c(-5,5),xlab="t",
   ylab="AR(2) Series")
abline(h=c(-4,-2,0,2,4),lty=2)
title(main="AR(2) Processes",cex=0.85)
text(100,-4.5,"a2 =",font=13); text(120,-4.5,alph2)
text(100,-5,"a1 =",font=13); text(120,-5,alph1)
}

AR2m<- function(alph1,alph2,Y){
ny<- length(Y)
my<- mean(Y) ; sy<-sqrt(var(Y))
sigmay<- sqrt((1-alph2)/((1+alph2)*(1-alph1-alph2)*(1+alph1-alph2)))
om<- acos(-alph1*(1-alph2)/(4*alph2)); Tp<- 2*pi/om
list("alph_1"=alph1,"alph_2"=alph2,"length_y"=ny,"mean_y"=my,"standev_y"=sy,
   "sigma_y"=sigmay,"omega_max"=om,"Tperiod_max"=Tp)
}

alph1<- 1.33; alph2<- -0.66
Y<- AR2Y(alph1,alph2,e,Y)
AR2p(alph1,alph2,Y)
unlist(AR2m(alph1,alph2,Y))

alph1<- -1.33; alph2<- -0.66
Y<- AR2Y(alph1,alph2,e,Y) 
AR2p(alph1,alph2,Y)
unlist(AR2m(alph1,alph2,Y))

dev.off()

rm(AR2Y,AR2m)
rm(alph1,alph2)
rm(e,Y,seq)