Some notes on Kalman Filtering

State Space form

Measurement Equation

\displaystyle \boxed{\mathbf{\underbrace{y_{t}}_{N \times 1}=\underbrace{Z_{t}}_{N \times m}\underbrace{a_{t}}_{m \times 1}+d_{t}+\varepsilon_{t}}}

\displaystyle Var(\varepsilon_{t})= \mathbf{H_{t}}

Transition Equation

\displaystyle \boxed{\mathbf{\underbrace{a_{t}}_{m \times 1} =\underbrace{T_{t}}_{m \times m} a_{t-1}+c_{t}+\underbrace{R_{t}}_{m \times g} \underbrace{\eta_{t}}_{g \times 1}}}

\displaystyle Var(\eta_{t})=\mathbf{Q}_{t}

\displaystyle E(a_{0})= \mathbf{a_{0} \; \; \; \; Var(a_{0})=P_{0}} \; \; \; \; E(\varepsilon_{t}a_{0}^{\top}) \; \; \; E(\eta_{t}a_{0}^{\top})

Future form

\displaystyle \mathbf{a_{t+1}=T_{t}a_{t}+c_{t}+R_{t}\eta_{t}}

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