A linear Gaussian state space model can be written in the following form (Durbin and Koopman, 2012):
The measurement or observation equation with for is given by
The state equation (transition dynamics) with , for is given by
- The system matrices , , , , , and are predetermined matrices. Usually they are time-invariant and functions of unknown parameters.
- Initialisation oftentimes plays an important role in inference. For non-stationary components in , say , conventionally diffuse initialisation is used. In such a case,
and is diagonal with being a large number. For stationary components in , say , unconditional distribution is used to initialise. It follows,
and is such that
where is the dimension of stationary components in .
The Kalman filter computes and via a forward recursion. Prediction errors are produced as a by product. It follows