# Ex 2.18 Finding the root of the charcteristic polynomials
z <- seq(-3,3,length=200)
a1 <- (-1+sqrt(1+12))/2
a2 <- a1+1
A <- function(z) 1+a1*z+a2*z^2 - z^3 # The characteristic polynomial
plot(z,A(z),type="l")
uniroot(A,c(2,10)) # Finding the real root
z1 <- uniroot(A,c(2,10))$root
z <- seq(-2,2,length=200)
y <- A(z)/(z-z1) # The quadratic part
lm(y~z+I(z^2)) # Fitting the quadratic part
b <- lm(y~z+I(z^2))$coef
(b[2]^2 + abs(b[2]^2-4*b[1]*b[3]))/(4*b[1]^2)

z <- seq(-3,3,length=200)
a1 <- (-1-sqrt(1+12))/2
a2 <- a1+1
A <- function(z) 1+a1*z+a2*z^2 - z^3 # The characteristic polynomial
plot(z,A(z),type="l")
uniroot(A,c(-1,3)) # Finding the real root


# Ex 2.23 Plot the autocorrelation functions of MA(p)-processes for different
# values of p.
ma.autocorr <- function(k,b)
{ 

   q <- length(b)
   b <- c(1,b)
   gamma <- rep(0,k+1) 
   for(s in 0:(min(k,q)))
   {
      gamma[s+1] <- sum(b[1:(q-s+1)]*b[(s+1):(q+1)])
   }
   rho <- gamma/gamma[1]
   return(rho)
}
k<-10
plot(c(0,k),c(-1,1),type="n",xlab="lag",ylab="rho")
lines(0:k,ma.autocorr(k,0.5),type="b")
lines(0:k,ma.autocorr(k,c(-0.5,0.3)),type="b",col="red")
lines(0:k,ma.autocorr(k,c(0,0.7,-0.2)),type="b",col="blue")
lines(0:k,ma.autocorr(k,0.8^(1:7)),type="b",col="green")