| | 119 | \form#118:$ y_t $ |
| | 120 | \form#119:$ dt = [y_t psi_t] $ |
| | 121 | \form#120:$ [d_1, d_2, \ldots d_t] $ |
| | 122 | \form#121:\begin{eqnarray} x_t &= &A x_{t-1} + B u_{t} + v_t,\\ y_t &= &C x_{t} + D u_{t} + w_t, \end{eqnarray} |
| | 123 | \form#122:$ x_t $ |
| | 124 | \form#123:$ A, B, C, D$ |
| | 125 | \form#124:$v_t, w_t$ |
| | 126 | \form#125:$Q, R\$, respectively. Both prior and posterior densities on the state are Gaussian, i.e. of the class enorm. There is a range of classes that implements this functionality, namely: - KalmanFull which implements the estimation algorithm on full matrices, - KalmanCh which implements the estimation algorithm using choleski decompositions and QR algorithm. \section ekf Extended Kalman Filtering Extended Kalman filtering arise by linearization of non-linear state space model: \f{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \f} where $ |
| | 127 | \form#126:$Q, R$ |
| | 128 | \form#127:\begin{eqnarray} x_t &= &g( x_{t-1}, u_{t}) + v_t,\\ y_t &= &h( x_{t} , u_{t}) + w_t, \end{eqnarray} |
| | 129 | \form#128:$ g(), h() $ |
| | 130 | \form#129:\[ V_t = \sum_{i=0}^{n} \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} \] |
| | 131 | \form#130:\[ \nu_t = \sum_{i=0}^{n} 1 \] |
| | 132 | \form#131:$ \theta_t , r_t $ |
| | 133 | \form#132:\[ V_t = V_{t-1} + \phi \left[\begin{array}{c}y_{t}\\ \psi_{t}\end{array}\right] \begin{array}{c} [y_{t}',\,\psi_{t}']\\ \\\end{array} +(1-\phi) V_0 \] |
| | 134 | \form#133:\[ \nu_t = \nu_{t-1} + \phi + (1-\phi) \nu_0 \] |
| | 135 | \form#134:$ \phi $ |
| | 136 | \form#135:$ \phi \in [0,1]$ |
| | 137 | \form#136:\[ \mathrm{win_length} = \frac{1}{1-\phi}\] |
| | 138 | \form#137:$ \phi=0.9 $ |
| | 139 | \form#138:$ V_0 , \nu_0 $ |
| | 140 | \form#139:$ V_t , \nu_t $ |
| | 141 | \form#140:$ \phi<1 $ |
