State Space Models
Description
Construct a
Usage
SS(F.=NULL, G=NULL, H=NULL, K=NULL, Q=NULL, R=NULL, z0=NULL, P0=NULL,
description=NULL, names=NULL, input.names=NULL, output.names=NULL)
is.SS(obj)
is.innov.SS(obj)
is.non.innov.SS(obj)
Details
State space models have a
further sub-class: innov or non-innov, indicating an innovations form
or a non-innovations form.
The state space (SS) model is defined by:
z(t) = Fz(t-1) + Gu(t) + Qe(t)
y(t) = Hz(t) + Rw(t)
or the innovations model:
z(t) = Fz(t-1) + Gu(t) + Kw(t-1)
y(t) = Hz(t) + w(t)
F(nxn) is the state transition matrix F.
H(pxn)is the output matrix H.
Q(nxn) is the input matrix of the system noise and the noise is
assumed to be white. Some authors (eg. Harvey) modify this
as rt*qt*rt' where rt is the matrix for the system noise and qt is
the noise cov, but that is redundant.
R(pxp) is the input matrix of the output (measurement) noise, which
is assumed white. (probably need R if p>n )
G(nxp)is the control (input) matrix.
K(nxp)is the Kalman gain.
yis the p dimensional output data.
uis the m dimensional exogenous (input) data.
zis the n dimensional (estimated) state at time t, E[z(t)|y(t-1), u(t)]
denoted E[z(t)|t-1]. An initial value for z can be specified as z0 and
an initial one step ahead state tracking error (for non-innovations
models) as P0.
z0An initial value for z can be specified as z0.
P0An initial one step ahead state tracking error (for non-innovations
models) can be specified as P0.
K, Q, RFor sub-class innov the Kalman gain K is specified but not Q and R.
For sub-class non-innov Q and R are specified but not the Kalman gain K.
Value
An SS TSmodel
See Also
TSmodel
ARMA
Examples
f <- array(c(.5,.3,.2,.4),c(2,2))
h <- array(c(1,0,0,1),c(2,2))
k <- array(c(.5,.3,.2,.4),c(2,2))
ss <- SS(F=f,G=NULL,H=h,K=k)
is.SS(ss)