# 2/27
quasi.period=114/11.5
arima(log(lynx),order=c(2,0,0))
quasi.period

library(TSA)
data(lynx)
plot(log(lynx))
?polyroot
# 1-phi1*x-phi2*x^2
phi1=1.38;phi2=-0.74
# ? is it quasi-periodic
phi1^2+4*phi2

# is is negative so quasi-periodic
theta=acos(phi1/(2*sqrt(-phi2)))
theta

quasi.period=2*pi/theta

R=sqrt(-phi2)
R
quasi.period

z=c(1,-phi1,-phi2)
roots=polyroot(z)
roots
Mod(roots) # magnitude 
# stationary AR(2)

1/roots
Mod(1/roots)  # R
# all magnitude of the roots >1, so stationarity

Arg(1/roots)  # -theta and theta

ar(log(lynx)) # tells the AR order, based AIC

roots=polyroot(c(1,-arima(log(lynx),order=c(11,0,0))$coef)[-12])
roots
Mod(roots)
# Bad, because the magnitude are less than 1


roots=polyroot(c(1,-arima(log(lynx),order=c(3,0,0))$coef[-4]))
roots
Mod(1/roots)
Arg(1/roots)

# Once all roots have magnitude > 1, then the process is stationary.
# Then, look at the reciprocol of the roots. If some of them are complex
# we have quasiperiodicity.
# The pair of (reciprocols of) roots having the largest magnitude is the
# dominating pair.  The Arg of this pair of (reciprocol) of roots is the theta.
# Quais-period=2*pi/theta.

plot(y=c(1,ARMAtoMA(ar=c(phi1,phi2),lag.max=100)), x=0:100,type='h')

abline(h=0)
  


Mod(roots)
abs(phi2)  #?<1
phi2+phi1   #?<1
phi2-phi1   #?<1