I I am trying to understand how the start up values (initialisation) are calculated in the Kalman filter. As an example, I simulated the MA(2) model below.
y
Extracted matrices needed for the Kalman Filter.
sigma2 <- fit$sigma2
T <- fit$model$T
R <- c(1,fit$coef)
Z <- t(fit$model$Z)
I <- diag(9)
And calculated start up values for predicted state vector a_pred1 and variance P_pred1 as below,
A <- solve(I-kronecker(T, T))
vecc <- A%%vec(R%%t(R))
P_pred1 <- as.matrix(cbind(vecc[1:3],vec[4:6],vec[7:9]))
a_pred1 <- c(0,0,0)
But I do not get the same start up values as produced by KalmanRun.
F <- as.numeric(Z%%P_pred1%%t(Z))
v <- as.numeric(y[i]-Z%%a_pred1)
a_filt1 <- a_pred1 + P_pred1%%t(Z)*(F^-1)*v
kfrun <- KalmanRun(y, fit$model)
cbind(kfrun$states[1,],a_filt1)
[,1] [,2]
[1,] 1.1964116 1.1964116
[2,] 0.7583928 0.6427514
[3,] 0.5353670 0.2493952
y <- arima.sim(n = 120, model = list(order = c(0,0,2), ma = c(0.6,0.4)))
fit <- arima(y, order = c(0,0,2), include.mean = FALSE)
Extracted matrices needed for the Kalman Filter.
sigma2 <- fit$sigma2
T <- fit$model$T
R <- c(1,fit$coef)
Z <- t(fit$model$Z)
I <- diag(9)
And calculated start up values for predicted state vector a_pred1 and variance P_pred1 as below,
A <- solve(I-kronecker(T, T))
vecc <- A%*%vec(R%*%t(R))
P_pred1 <- as.matrix(cbind(vecc[1:3],vec[4:6],vec[7:9]))
a_pred1 <- c(0,0,0)
But I do not get the same start up values as produced by KalmanRun.
F <- as.numeric(Z%*%P_pred1%*%t(Z))
v <- as.numeric(y[i]-Z%*%a_pred1)
a_filt1 <- a_pred1 + P_pred1%*%t(Z)*(F^-1)*v
kfrun <- KalmanRun(y, fit$model)
cbind(kfrun$states[1,],a_filt1)
[,1] [,2]
[1,] 1.1964116 1.1964116
[2,] 0.7583928 0.6427514
[3,] 0.5353670 0.2493952
Can any one explain what I am doing wrong in my calculations?
