# The poisson autogression model 
# models integer valued time series using a GLM type approach. 
# This model is studied in Fokianos, Rahbek and Tjostheim (2009)

# parameters need to be positive in this specification

poisson.autoregressive = function(d1,a,b,n,n0){

         m = (n+n0)
    
      	y <- rep(1,m)
	lambda <- rep(1,m)
         
        for(i in c(2:m)){

            lambda[i] = (d1+a*lambda[i-1]+b*y[i-1])

            y[i] = rpois(1,lambda = lambda[i])

        }

         y = y[-c(1:n0)]
          
         return(y)

     }
            


test1 = poisson.autoregressive(d1=1,a=0.6, b=0.3, n0=100, n=200)

# Make acf plots to check the correlation

par(mfrow=c(2,1))

plot(c(1:200),test1,col="blue")

acf(test1)

# without the a component 

test2 = poisson.autoregressive(d1=1,a=0, b=0.9, n0=100, n=200)

# Make acf plots to check the correlation

#par(mfrow=c(2,1))

plot(c(1:200),test2,col="blue")

acf(test2)

#################################

# Next we give a model that allows for negative correlation and negative parameters.
# This model is studied in Fokianos and Tjostheim (2011)


poisson.autoregressive.log = function(d1,a,b,n,n0){

         m = (n+n0)
    
      	y <- rep(1,m)
	lambda <- rep(1,m)
        eta <- rep(1,m)
    
        for(i in c(2:m)){

            eta[i] = (d1+a*eta[i-1]+b*log(1+y[i-1]))
            lambda[i] = exp(eta[i])

            y[i] = rpois(1,lambda = lambda[i])

        }

         y = y[-c(1:n0)]
          
         return(y)

     }
         

test3 = poisson.autoregressive.log(d1=0.5,a=-0.5, b=-0.25, n0=100, n=200)

# Make acf plots to check the correlation

par(mfrow=c(2,1))

plot(c(1:200),test3,col="blue")

acf(test3)