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 \[ K_t = P_t^{t-1}A_t^\prime(A_tP_t^{t-1}A_t^\prime + S)^{-1}. \]