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I went through the answer Kalman filter in practice and it seems we must know all the first and second order properties of random variables to apply the Kalman filter.
But when I only have a set of data $y_1,\dots,y_n$ and a given model $G$ and $F$ with assumed noise processes, could someone please explain to me how to apply Kalman filter?
Thanks.
Signal Model
The first thing you really need when you start to use a Kalman filter is what I would call the "signal model". That means, you need to take a guess (and sometimes that's all it is, though usually you can have an educated guess) at how the signal you're measuring, $y$, was generated.
The usual signal model comprises $F$, $G$, $H$, $Q$ and $R$:
$$ x_{k+1} = F x_k + G u_k + w_k\\ y_k = H x_k + v_k $$
where $Q$ is the covariance of $w_k$ and $R$ is the covariance of $v_k$.
What are you trying to achieve?
Implicit in the signal model is that you've chose the "state variables" (the components of $x_k$) that you are working with. Usually, what you are trying to achieve is to measure something and, from that, infer the value of one or more of the state variables.
What's missing?
Based on what you've said, the only thing that seems to be missing is how you get from the states $x_k$ to your measurements $y$ (i.e. $H$).
