#'@rdname get_model_AR2
#'@title get_model_AR2
#'@description Auto-regressive model of order 2
#'@export
get_model_AR2 <- function(nu0, sigma02){
model = list()
model$observation_type = 'continuous'
# dimension of parameter
model$dimtheta = 3
model$dimY = 1
# uniform distribution on the triangle that guarantees causality of the AR(2) process
model$runiftriangle = function(){
phi1 = runif(1,-2,2)
phi2 = runif(1,-1,1)
accept = (-(1-phi2) < phi1)&&(phi1 < 1-phi2)
while (!accept){
phi1 = runif(1,-2,2)
phi2 = runif(1,-1,1)
accept = (-(1-phi2) < phi1)&&(phi1 < 1-phi2)
}
return (rbind(phi1,phi2))
}
model$rprior = function(Ntheta){
phi = sapply(1:Ntheta,function(i)model$runiftriangle())
sigma2 = rinvchisq(Ntheta,nu0,sigma02)
return (rbind(phi,sigma2))
}
# prior distribution density on parameters
model$dprior = function(theta, log = TRUE){
if ((-1<theta[2])&&(theta[2]<1)&&(-(1-theta[2])<theta[1])&&(theta[1]<(1-theta[2]))){
logd = log(1/4) + dinvchisq(theta[3],nu0,sigma02,log = TRUE)
} else {
logd = -Inf
}
if (log==TRUE) {return (logd)}
else {return (exp(logd))}
}
# one-step predicitve density of the observation at time t given all the past from 1 to (t-1)
model$dpredictive = function(observations,t,theta,log = TRUE){
if ((t==1)||(t==2)) {
mu = 0
sigma2 = theta[3]/(1-theta[2]^2-(theta[1]^2)*(1+theta[2])/(1-theta[2]))
} else {
mu = theta[1]*observations[,t-1]+theta[2]*observations[,t-2]
sigma2 = theta[3]
}
return (dnorm(observations[,t],mean = mu,sd = sqrt(sigma2), log))
}
# OPTIONAL: derivatives of the predicitve density with respect to the observation at time t
# inputs: observations (dimY by T matrix, with T >= t), time index t (int), theta (single vector),
# byproduct (OPTIONAL: auxiliary object needed to compute likelihood, e.g. Kalman filter)
# outputs: list with the following fields
# jacobian >> the transpose of the gradient (1 by dimY)
# hessiandiag >> the Hessian diagonal coefficients (1 by dimY)
# NB: if missing, this field is automatically filled with numerical derivatives
# via set_default_model in util_default.R)
model$derivativelogdpredictive = function(observations,t,theta,byproduct) {
if ((t==1)||(t==2)) {
mu = 0
sigma2 = theta[3]/(1-theta[2]^2-(theta[1]^2)*(1+theta[2])/(1-theta[2]))
} else {
mu = theta[1]*observations[,t-1]+theta[2]*observations[,t-2]
sigma2 = theta[3]
}
deriv1 <- -(observations[,t]-mu)/sigma2
deriv2 <- -1/sigma2
return (list(jacobian = matrix(deriv1, 1, 1), hessiandiag = matrix(deriv2, 1, 1)))
}
# OPTIONAL: simulate observations
model$robs = function(nobservations,theta){
Y = matrix(NA, nrow = 1, ncol = nobservations)
Y[,1] = rnorm(1, mean = 0, sd = sqrt(theta[3]/(1-theta[2]^2-(theta[1]^2)*(1+theta[2])/(1-theta[2]))))
Y[,2] = rnorm(1, mean = 0, sd = sqrt(theta[3]/(1-theta[2]^2-(theta[1]^2)*(1+theta[2])/(1-theta[2]))))
if (nobservations > 2){
for (t in 3:nobservations){
Y[,t] = theta[1]*Y[,t-1] + theta[2]*Y[,t-2] + rnorm(1, mean = 0, sd = sqrt(theta[3]))
}
}
return (Y)
}
# OPTIONAL: simulate Ny draws of y_t given theta and the past y_1 to y_(t-1)
# (with the convention y_0 = NULL)
# outputs: matrix of Ny draws of Yt given theta and past (dimY by Ny matrix)
model$rpredictive = function(Ny,t,theta,y_past){
if ((t==1)||(t == 2)) {
Yt = matrix(rnorm(1, mean = 0, sd = sqrt(theta[3]/(1-theta[2]^2-(theta[1]^2)*(1+theta[2])/(1-theta[2])))), ncol = Ny)
} else if (t >= 3) {
Yt = theta[1]*repeat_column(Ny,y_past[,t-1,drop=FALSE]) + theta[2]*repeat_column(Ny,y_past[,t-2,drop=FALSE]) + matrix(rnorm(Ny*model$dimY, mean = 0, sd = sqrt(theta[3])),ncol=Ny)
}
return (Yt)
}
return(model)
}
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.