ts1129
1 formula
\[ w_{t-1}= \left [ \begin{array}{cc} \alpha \\ 1 \end{array} \right ]~,~\hat{x}_{t-1} = \left [ \begin{array}{cc} l_{t-1} \\ P_1 \end{array} \right ] ~,~H=\left [ \begin{array}{cc} 1 & z_t^* \end{array} \right ]\\ Y=\varepsilon_{t-1}~,~Z_{t-1}=HX+Y \]
1.1 formula (1)
\[ E(x_{t-1}w_{t-1}^T|Z_{t-1})\\ =E[\left [ \begin{array}{cc} x_1 \\ x_2 \end{array} \right ] \left [ \begin{array}{cc} \alpha Y & 0 \end{array} \right ]|Z_{t-1}]\\ =\alpha Z_{t-1} \hat{x}_{t-1} - \alpha E[(HX)X|Z_{t-1}]......(1) \]
1.2 formula (2)
\[ \hat{x}_{t-1}E(w_{t-1}^T|Z_{t-1})\\ =\left [ \begin{array}{cc} \alpha ~l_{t-1}(Z_{t-1}-l_{t-1}-P_1 Z_t^*) & 0\\ \alpha ~P_1(Z_{t-1}-l_{t-1}-P_1 Z_t^*) & 0 \end{array} \right ]\\ =\alpha \hat{x}_{t-1} Z_{t-1} - \alpha \hat{x}_{t-1}l_{t-1} - \alpha \hat{x}_{t-1} P_1......(2) \]
1.3 formula (3)
\[ E[XX^T|Z_{t-1}]\\ = E(~\left [ \begin{array}{cc} x_1^2 & x_1 x_2 \\ x_1 x_2 & x_2^2 \end{array} \right ]|Z_{t-1}~)\\ =\hat{x}_{t-1}\hat{x}_{t-1}^T +P_{t-1}......(3) \]
1.4 formula (4)
\[ E[(HX)X|Z_{t-1}]\\ =E[\left [ \begin{array}{cc} 1 & Z_t^* \end{array} \right ] \left [ \begin{array}{cc} x_1 \\ x_2 \end{array} \right ] \left [ \begin{array}{cc} x_1 \\ x_2 \end{array} \right ]|Z_{t-1}]\\ =E[ \left [ \begin{array}{cc} x_1^2 & x_1 x_2 \\ x_1 x_2 & x_2^2 \end{array} \right ]|Z_{t-1}]\\ =E[XX^T|Z_{t-1}] \left [ \begin{array}{cc} 1 \\ z_t^* \end{array} \right ]\\ =(3) \left [ \begin{array}{cc} 1 \\ z_t^* \end{array} \right ]\\ =(\hat{x}_{t-1}\hat{x}_{t-1}^T +P_{t-1}) \left [ \begin{array}{cc} 1 \\ z_t^* \end{array} \right ]......(4) \]
2 Derivative of formula
\[ E[(x_{t-1}-\hat{x}_{t-1})w_{t-1}^T|Z_{t-1}]\\ =E(x_{t-1}w_{t-1}^T|Z_{t-1})-\hat{x}_{t-1}E(w_{t-1}^T|Z_{t-1})\\ =(1)-(2) \]
\[ =\alpha Z_{t-1} \hat{x}_{t-1} - \alpha E[(HX)X|Z_{t-1}] - \alpha \hat{x}_{t-1} Z_{t-1} + \alpha \hat{x}_{t-1}l_{t-1} + \alpha \hat{x}_{t-1} P_1\\ = \alpha \hat{x}_{t-1}l_{t-1} + \alpha \hat{x}_{t-1} P_1 - \alpha E[(HX)X|Z_{t-1}]\\ =\alpha \hat{x}_{t-1}l_{t-1} + \alpha \hat{x}_{t-1} P_1 - \alpha ~(4) \]
\[ =\alpha \hat{x}_{t-1}l_{t-1} + \alpha \hat{x}_{t-1} P_1 - \alpha (\hat{x}_{t-1}\hat{x}_{t-1}^T +P_{t-1}) \left [ \begin{array}{cc} 1 \\ z_t^* \end{array} \right ]\\ =\alpha P_1 (1-Z_t^*) \hat{x}_{t-1} - \alpha P_{t-1} H^T \]
\[ P_t^-=AP_{t-1} A^T + 2 A E[(x_{t-1}-\hat{x}_{t-1})w_{t-1}^T|Z_{t-1}] + Q\\ =AP_{t-1} A^T +2 \alpha P_1 (1-Z_t^*) A \hat{x}_{t-1} - 2 \alpha AP_{t-1} H^T + Q \]