Given the observation, $(y_1, \dots, y_T; x_1,\dots, x_T)$ estimate the unknown parameters $(F, A, H, Q, R)$.

Idea: Sequentially updating a (linear) projection of the dynamical system (Kalman Filter)


\[\begin{aligned} &\xi_{t+1} = F\xi_t + V_{t+1} \\ &y_t = A^T x_t + H^T\xi_t + W_t \end{aligned}\]

At each time step, recursively compute the optimal linear predicted value of $\xi_{t+1}$ based on the data observed up to time point $t$, i.e.

given $D_t = (y_t,\dots, y_1; x_t,\dots, x_1)$ predict

\[\hat{\xi}_{t+1|t} = \mathbb{E}(\xi_{t+1}|D_t)\]

The MSE matrix

\[P_{t+1|t} = \mathbb{E}(\hat\xi_{t+1|t} - \xi_{t+1}) (\hat\xi_{t+1|t} - \xi_{t+1})^T\]

Initialization: $\xi_{1|0} = \mathbb{E}(\xi_1 | D_0) $ $= \mathbb{E}[\xi_1]$ where $D_0$ means there is no data. The expected MSE is

\[P_{1|0} = \mathbb{E}(\mathbb{E}\xi_1-\xi_1) (\mathbb{E}\xi_1-\xi)^T = \text{Cov}(\xi_1)\]
Recursion: Given $\hat\xi_{t t-1}$ and $ P_{t t-1}$ and data $D_t = D_{t-1} \cup {y_t,x_t}$ compute $\xi_{t t-1}$ and $P_{t+1 t}$.
Note $\mathbb{E} (\xi_t x_t, D_{t-1}) = \mathbb{E} (\xi_t D_{t-1}) = \xi_{t t-1}$.

(i) Predict the value of $y_t$:

\[\begin{aligned} \hat{y}_{t|t-1 } &=\hat{\mathbb{E}}(y_t|x_t, D_{t-1})\\ &= \hat{\mathbb{E}} (A^T x_t + H^T \xi_t + W_t | x_t, D_{t-1}) \\ &=A^T x_t + H^T \hat\xi_{t|t-1} + \mathbb{E} (w_t | x_t, D_{t-1} ) \\ &= A^T x_t + H^T \hat\xi_{t|t-1} \end{aligned}\]

MSE matrix:

\[\begin{aligned} &\mathbb{E}(y_t - y_{t|t-1})(y_t - \hat y_{t|t-1}) ^T \\ &=H^T \mathbb{E}(\xi_t - \hat\xi_{t|t-1})(\xi_t-\hat\xi_{t|t-1})^T H + \mathbb{E}w_tw_t^T \\ &= H^T P_{t+1|t} H + R \end{aligned}\]

(ii) update the inference about $\xi_t$ with new observation $y_t$. compute

\[\hat\xi_{t+1|t} = \mathbb{E} (\xi_t | y_t ,x_t , D_{t-1})\]

optimal prediction:

\[\hat\xi_{t+1|t} = \hat\xi_{t|t-1} + [\mathbb{E}(\xi_t - \hat\xi_{t|t-1}) (y_t -\hat y_{t|t-1})^T] \times [\mathbb{E}[y_t-\hat y_{t|t-1}][y_t-\hat y_{t|t-1}]]^{-1} (y_t-\hat y_{t|t-1})\]

The term

\[\mathbb{E}(\xi-\hat\xi_{t|t-1}) (y_t - \hat y_{t|t-1})^T = P_{t|t-1} H\]

This then becomes that

\[\hat\xi_{t | t} = \hat\xi_{t|t-1} + P_{t|t-1} H(H^T P_{t+1|t} H + R)^{-1} [y_t - A^T x_t - H^T\hat\xi_{t|t-1}]\]


\[\begin{aligned} &P_{t|t} = \mathbb{E}(\hat\xi_{t|t} - \xi) (\hat\xi_{t|t} - \xi_t)^T \\ &= P_{t|t-1} - P_{t|t-1} H (H^T P_{t|t-1} H +R)^{-1} H^TP_{t|t-1} \end{aligned}\]
(iii) predict $\xi_{t+1 t}$,
\[\hat\xi_{t+1} = \mathbb{E} (\xi_{t+1}|D_t) = \mathbb{E}(F\xi_t+V_{t+1} |D_t) = F\hat\xi_{t|t}\]

combining we could have

\[\hat\xi_{t+1|t} = F\hat\xi_{t|t-1} + FP_{t|t-1} H(H^T P_{t+1|t} H + R)^{-1}[y_t - A^T x_t - H^T \hat\xi_{t|t-1}]\]

the second term is called the Kalman gain. and

\[P_{t+1|t} = \mathbb{E}(\xi_{t+1}-\hat\xi_{t+1|t}) (\xi_{t+1}-\hat\xi_{t+1|t})^T = FP_{t|t} F^T +Q\]


\[P_{t+1|t} = F[P_{t|t-1} - P_{t|t-1} H (H^T P_{t|t-1} H + R)^{-1}H^T P_{t|t-1} ] F^T +Q\]

the two updating equations are the Kalman Filter.


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