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Last update Jul 10, 2004

STATE SPACE MODEL NOTATION - QUICK GUIDE

We write the state space model for exponential family observations {Yt} and Gaussian latent process {qt} for t=1,2,,n as
Observation equation:
p(yt | ht)
=
exp{ytTht - b(ht) + c(yt)},
Link function:
g{b(ht)}
=
lt,
Signal:
lt
=
FtTqt,
System equation:
qt
=
Gtqt-1 + wt,
 wt
~
Np(0,Wt),
Initial prior:
q0
~
Np(m0,C0),
for known values of the prior mean m0 and the prior variance C0.
In the definition we have introduced the system matrices which are needed for specifying an object of class state space model
Ft:
The p × d design matrix usually consisting of known functions of covariates at time t. It links the state vector to the observation vector by means of a dynamic linear regression, whose coefficient vector is determined by the first order vector auto-regression in the system equation.

Gt:
The p × p evolution transfer matrix at time t. It is the design matrix for the latent process, and usually consists of known functions of covariates and possible previous observations. Often the evolution transfer matrices are block-diagonal; each block representing a certain aspect of the model e.g. trend or seasonal effects.

Wt:
The evolution variance at time time. It can depend on covariates and an unknown parameter vector. When applying the iterated extended Kalman filter and smoother Wt is allowed to be singular. However, when simulating from a state space model it must be a proper variance matrix.