NbbTools > Class library guide
While the general linear gaussian state-space model can be written in many different ways, we use a very simple presentation that encompasses most of the usual model. Apart from the disturbance term(s) in the measurement equation - we simply add them to the state vector - , it mainly follows the approach of Durbin and Koopman ([1], 2001).
with
is
the m observations at period t, 
is
the r x 1 state vector. The 
are
assumed to be serially independent at all time
points. They will
often be modelised as
where
is
a vector of e x 1 residuals (0 < e ≤ r),
is
a e x e covariance matrix, 
is
a k x e matrix (weights of the disturbances) and
is
a r x k matrix composed of columns of
,
that identifies the items of the state vector modified by the residuals.
The initial conditions of the filter are defined as follows:
where
is
arbitrary large [1].
is
the variance of the stationary elements of the initial state vector and
models
the diffuse part. Both matrices are r x r [2].
We also suppose that the rank of 
is
d.
[1]
The 
factor
is absorbed in
[2] We do not require that diffuse/non diffuse elements reside in separate items of the initial state vector, but we must be able to split them in independent parts.
Bibliography
[1] Durbin J. and Koopman S.J. (2001), "Time Series Analysis by State Space Methods". Oxford University Press.