Linear and Gaussian state space model

A linear Gaussian state space model can be written in the following form (Durbin and Koopman, 2012):

The measurement or observation equation with y_{t}\in\mathbb{R}^{N} for t=1,...,T is given by
y_{t}=d_{t}+Z_{t}\alpha_{t}+\epsilon_{t},\ \ \ \ \epsilon_{t}\sim N(0, H_{t}).
The state equation (transition dynamics) with \alpha_{t}\in\mathbb{R}^{n}, for t=1,...,T-1 is given by
\alpha_{t}=c_{t}+T_{t}\alpha_{t}+R_{t}\eta_{t},\ \ \ \ \eta_{t}\sim N(0, Q_{t}),
with initialisaiton
\alpha_{1}\sim N(a_{1}, P_{1}).


  1. The system matrices c_{t}, d_{t}, Z_{t}, T_{t}, R_{t}, H_{t} and Q_{t} are predetermined matrices. Usually they are time-invariant and functions of unknown parameters.
  2. Initialisation oftentimes plays an important role in inference. For non-stationary components in \alpha_{t}, say \alpha_{tj}, conventionally diffuse initialisation is used. In such a case,
    a_{1j}=0 and P_{1} is diagonal with P_{1,jj} being a large number. For stationary components in \alpha_{t}, say \alpha_{t}^{*}, unconditional distribution is used to initialise. It follows,
    and P_{1}^{*} is such that
    vec(P_{1}^{*})=(I_{{n^{*}}^{2}}-T_{1}^{*}\otimes T_{1}^{*})^{-1}vec\big((R_{1}Q_{1}R_{1}')^{*}\big),
    where n^{*}\in\{n^{*}\in \mathbb{N}:n^{*}\leq n\} is the dimension of stationary components in \alpha_{t}.

The Kalman filter computes a_{t}=E(\alpha_{t}|y_{1:t-1}) and P_{t}=\text{Var}(\alpha_{t}|y_{1:t-1}) via a forward recursion. Prediction errors are produced as a by product. It follows