5.4 General Kalman Filter

The more general formulation of the state space model described in the previous section as an observation equation $$y_t = A_t x_t + V_t$$ and a state equation $$x_t = \Theta x_{t-1} + W_t$$ where $y_t$ is a $p\times 1$ vector, $x_t$ is a $k\times 1$ vector, $A_t$ is a $p\times k$ matrix and $\Theta$ is $k\times k$ matrix. We can think of $V_t\sim\mathcal{N}(0, S)$ and $W_t\sim\mathcal{N}(0, R)$.

Given an initial state $x_0^0$ and $P_0^0$, the prediction equations are (analogous to above) $$\begin{eqnarray*} x_1^0 & = & \Theta x_0^0\\ P_1^0 & = & \Theta P_0^0\Theta^\prime + R \end{eqnarray*}$$ and the updating equations are, given a new observation $y_1$, $$\begin{eqnarray*} x_1^1 & = & x_1^0 + K_1(y_1 - A_1 x_1^0)\\ P_1^1 & = & (I - K_1A_1) P_1^0 \end{eqnarray*}$$ where $$K_1 = P_1^0A_1^\prime(A_1P_1^0A_1^\prime + S)^{-1}.$$ In general, given the current state $x_{t-1}^{t-1}$ and $P_{t-1}^{t-1}$ and a new observation $y_t$, we have $$\begin{eqnarray*} x_t^{t-1} & = & \Theta x_{t-1}^{t-1}\\ P_t^{t-1} & = & \Theta P_{t-1}^{t-1}\Theta^\prime + R \end{eqnarray*}$$ and the updating equations are, given a new observation $y_t$, $$\begin{eqnarray*} x_t^t & = & x_t^{t-1} + K_t(y_t - A_t x_t^{t-1})\\ P_t^t & = & (I - K_tA_t) P_t^{t-1} \end{eqnarray*}$$ where $$K_t = P_t^{t-1}A_t^\prime(A_tP_t^{t-1}A_t^\prime + S)^{-1}.$$